Apm 2010 Book of Abstracts

XXXVIII Summer School – Conference “Advanced Problems in Mechanics”

July 1 – 5, 2010, St. Petersburg (Repino), Russia

APM 2010

BOOK OF ABSTRACTS

IPME RAS 2010

http://apm-conf.spb.ru GENERAL INFORMATION The International Summer School – Conference “Advanced Problems in Mechanics 2010” is the thirty eight in a series of annual summer schools held by Russian Academy of Sciences. The Conference is organized by the Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences (IPME RAS) under the patronage of the Russian Academy of Sciences (RAS). The main purpose of the meeting is to gather specialists from different branches of mechanics to provide a platform for cross-fertilisation of ideas.

The list of problems under investigation is not limited to questions of mechanical engineering, but includes practically all advanced problems in mechanics, which is reﬂected in the name of the conference. The main attention is given to problems on the boundary between mechanics and other research areas, which stimulates the investigation in such domains as micro- and nanomechanics, material science, physics of solid states, molecular physics, astro- physics and many others. The conference “Advanced Problems in Mechanics” helps us to maintain the existing contacts and to establish new ones between foreign and Russian scientists.

Young scientists’ school-conference “Modern Ways in Mechanics” (MWM), which is held within the annual international conference “Advanced Problems in Mechanics” (APM), is meant for broadening scientific outlook of young researchers in the field of mechanics and also for organizing their scientific dialogue. It is supposed that students, Ph.D. students and young Ph.D.’s under 30 (date of birth is later than 12/31/1979) from different all over the world, specializing in the sphere of theoretical and applied mechanics become the main participants of the conference. In order to attract the largest possible number of various scientific areas and schools, organizing committee suggests a partial compensation for the costs connected with participation in conference, as well as extensive cultural program. One of the major purposes of conference is transfer of scientific experience from well- known scientists to their young colleagues.

HISTORY OF THE SCHOOL The first Summer School was organized by Ya.G. Panovko and his colleagues in 1971. In the early years the main focus of the Schoolwas on nonlinearoscillations of mechanicalsystems with a finite numberof degreesof freedom. The School specialized in this way because at that time in Russia (USSR) there were held regular National Meetings on Theoretical and Applied Mechanics, and also there were many conferences on mechanics with a more particular specialization. After 1985 many conferences and schools on mechanics in Russia were terminated due to financial problems. In 1994 the Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences restarted the Summer School. The traditional name of “Summer School” has been kept, but the topics covered by the School have been much widened, and the School has been transformed into an international conference. The topics of the conference cover now all fields of mechanics and associated into interdisciplinary problems.

SCIENTIFIC COMMITTEE

• H. Altenbach (Martin-Luter University Halle Wittenberg, Germany) • V.V.Beletsky (Keldysh Institute of Applied Mathematics RAS, Moscow, Russia) • A. K.Belyaev (IPME RAS, St. Petersburg, Russia) • E. K. Bezoyan (Yerevan State University, Armenia) • I. I. Blekhman (IPME RAS, Mekhanobr-tekhnika Corp., St.Petersburg, Russia) • M. Cartmell (University of Glasgow, UK) • A. Castellanos (University of Seville, Spain) • E. P. Chen (Sandia National Laboratories, Livermore, USA) • V.A. Eremeyev (Southern Scientiﬁc Center of RAS, Rostov State University, Rostov-on-Don, Russia) • A. B. Freidin (IPME RAS, St. Petersburg, Russia) • S. N. Gavrilov (IPME RAS, St. Petersburg, Russia)

ISBN 978-5-89319-135-6

2 • J. Goddard (University of California at San Diego, USA) • I. G. Goryacheva (Institute for Problems in Mechanics RAS, Moscow, Russia) • E. F. Grekova (IPME RAS, St. Petersburg, Russia) • D. Harris (University of Manchester, UK)

• D. A. Indeitsev (IPME RAS, St. Petersburg, Russia) — Co-Chairman • H. Irschik (Johannes Kepler University of Linz, Austria) • B. L. Karihaloo (Institute of Theoretical, Applied and Computational Mechanics, UK) • A. M. Krivtsov (St. Petersburg State Polytechnical University, IPME RAS, Russia) — Co-Chairman • O. S. Loboda (St. Petersburg State Polytechnical University, IPME RAS, Russia) — Scientiﬁc Secretary • E. V.Lomakin (Moscow State University, Russia) • A. Metrikine (TU Delft, The Netherlands) • N. F.Morozov (St. Petersburg State University, IPME RAS, Russia) • V.A. Palmov (St. Petersburg State Polytechnical University, Russia) • E. Pavlovskaia (University of Aberdeen, UK) • S. H. Sargsyan (National Academy of Sciences, Armenia) • E. N. Vilchevskaya (St. Petersburg State Polytechnical University, IPME RAS, Russia) • V.A. Vladimirov (University of Hull, UK) • M. Wiercigroch (University of Aberdeen, UK) • M. V.Zakrzhevsky (Institute of Mechanics Riga Technical University, Latvia)

LOCAL ORGANIZING COMMITTEE

• I. N. Alexandrov, St.Petersburg State University of Aerospace Instrumentation • M. A. Antimonov, IPME RAS, St. Petersburg • I. E. Berinskiy, St.Petersburg State Polytechnical University, IPME RAS, St. Petersburg • N.G.Dvas, St.Petersburg State Polytechnical University • A. M. Kudarova, St.Petersburg State Polytechnical University • V.A. Kuz’kin, St.Petersburg State Polytechnical University • A. A. Le-Zakharov, St.Petersburg State Polytechnical University • O. S. Loboda, St.Petersburg State Polytechnical University, IPME RAS, St. Petersburg • E. A. Podol’skaya, St.Petersburg State Polytechnical University • E. V.Serogo, IPME RAS, St. Petersburg

• S. V.Vasiliev, IPME RAS, St. Petersburg

The conference is organized with help of our service agency “Monomax” (ﬁnd the information further).

3 SCIENTIFIC PROGRAM Presentations devoted to fundamental aspects of mechanics, or spreading the ﬁeld of applications of mechanics, are invited. We are particularly keen to receive contributions that show new effects and phenomena or develop new mathematical models. The topics of the conference cover all ﬁelds of mechanics, including, but not restricted, to • mechanics of media with microstructure • nonlinear dynamics, chaos and vibration

• nanomechanics • dynamics of rigid bodies and multibody dynamics • molecular and particle dynamics • solids and structures • phase transitions • ﬂuid and gas • computational mechanics • mechanical and civil engineering applications • wave motion • aerospace mechanics

MINISYMPOSIA AND CHAIRS

[MS1] “Nano Structures and Materials” Prof. A. M. Krivtsov (Saint-Petersburg, Russia)

[MS2] “Discrete and continuum models in dynamics of condensed matter” Dr. E. L. Aero, Dr. A. V.Porubov (Saint-Petersburg, Russia)

Four different forms of presentations are offered, namely, plenary lectures (40 minutes), presentations at minisymposia (30 minutes), short communications (20 minutes), and posters.

The working language for oral presentations is English. Regrettably we can not provide simultaneous translation, and due to the international nature of the conference all the oral presentations must be in English.

The working languages for poster sessions are English and Russian.

Attention: each participant may only give ONE oral presentation. The number of posters for each participant is not limited.

The Summer School – Conference has two main purposes: to gather specialists from different branches of mechanics to provide a platform for cross-fertilization of ideas, and to give the young scientists a possibility to learn from their colleagues and to present their work. Thus the Scientiﬁc Committee encouraged the participation of young researchers, and did its best to gather at the conference leading scientists belonging to various scientiﬁc schools of the world.

We believe that the signiﬁcance of Mechanics as of fundamental and applied science should much increase in the eyes of the world scientiﬁc community, and we hope that APM conference makes its contribution into this process.

We are happy to express our sincere gratitude for the help in organization to Russian Foundation for Basic Re- search, Russian Academy of Sciences (RAS), St. Petersburg Scientiﬁc Center of RAS, Scientiﬁc committee of RAS for problems of mechanical engineering and technological processes, Federal Agency for Science and Inno- vations. This support has helped substantially to organize the conference and to increase the participation of young researchers.

4 License N 0009465; Certiﬁcate N POCC RU Y062.Y01614 Member of St. Petersburg Chamber of Commerce and Industry (N 88.872) Member of St. Petersburg Chamber of Trade and Industry Member of the Russian Association of Travel Agencies (RATA) Member of Exhibition Association of SPbCCI Member of ICCA (Inter. Congress & Convention Association, ref. CA/5788)

Monomax Ltd. professional congress organizer in Saint Petersburg and Northwest Russia

Monomax Ltd is a fully licensed professional travel and business cooperation company with 19 years of practical experience in destination management. Our activity past 19 years helped us to gain the reputation of a reliable partner in Russia and abroad. Company’s staff of highly trained professionals always offers its full assistance. Monomax Ltd provided services for about 150 international and more than 1000 local events held in St. Petersburg in 1991- 2010. Monomax Ltd offers from A to Z services to the major international conferences, congresses, symposia, festivals and exhibitions as well as to incentive and educational tourism groups. We plan all events imaginatively and organize them efficiently and economically to the last detail. Services we offer include: • consulting on congress planning & management • participants registration and administration, database developing and on-line registration • financial management of the event budget, developing of bid, preliminary budget, accumulation of registration fees, fundraising • highly skilled congress secretariat, informative work on event, original soft ware for congress management • congress venue selection, all kinds of presentation equipment, including simultaneous translation equipment • interpreter services, simultaneous translation, guides-interpreters, individual and group guidance, written translation • exhibitions and poster sessions facilities, exposition project, fitting, technical assistance • handling and printing all kinds of congress materials, including programs, announcements, abstracts, proceedings, brochures, etc. • participant packets, congress bags • hotel accommodation & visa support • catering & meal services: coffee breaks, lunches, dinners, fourchettes, banquets, receptions • transportation services, transfers, shuttle buses • concert and special programs, theater events • social and cultural programs, excursions, post-congress tours • special programs for VIP, corporative clients • presentations

Address for correspondence: PO box 168, 195112 Saint Petersburg, Russia tel. +7 (812) 335 2055, fax +7 (812) 335 2039 E-mail: [email protected] Web site: http://www.monomax.ru

5 JULY 1, THURSDAY

Room A Room B Room C Morning Session CO-CHAIRPERSONS D.A. INDEITSEV,A.M. KRIVTSOV

9:30 – 9:40 OPENING CEREMONY 9:40 – 10:20 E.M. Galimov. Cavitational synthesis of carbon nanostructures 10:20 – 11:00 W.G. Hoover, C.G. Hoover. Well-posed two-temperature constitutive equations for stable dense-ﬂuid shockwaves using molecular dynamics and generalizations of Navier-Stokes-Fourier continuum mechanics 11:00 – 11:40 M. Wiercigroch. Complex dynamics of pendula systems Coffee break 12:00 – 12:40 I.G. Goryacheva. Modeling of wear and contact fatigue 12:40 – 13:20 H. Altenbach. On some advanced models of inelastic behavior 13:20 – 14:00 F. Spahn. Saturns Rings - our natural “Dy- namical Labs” in Space

Evening Session

Evening Session Evening Session Evening Session CHAIRPERSON E.PAVLOVSKAIA Nonlinear Dynamics CHAIRPERSON S.LURIE CHAIRPERSON G.FILIPPENKO Nanomechanics Mechanical and Civil Engineering Applications 15:30 – 15:50 A.V. Sylyvonyuk, N.P. Yaroshevich. Exper- imental research of a run-up process of a vibrating machine 15:30 – 15:50 S.V. Bobylev, N.F. Morozov, I.A. Ovid’ko. 15:30 – 15:50 N.V. Banichuk, S.Yu. Ivanova, E.V. Ma- with an unbalance excitation Accomodation of grain boundary sliding by grain boundary keev. Contact optimization problems with uncertainties 15:50 – 16:10 G. Radons. On simple dynamical systems migration in deformed nanomaterials 15:50 – 16:10 V. Monin, J.T. de Assis, S. Marrero Igle- with hysteresis operators 15:50 – 16:10 V.A. Eremeyev, H. Altenbach, N.F. Moro- sias. Application of X-ray tensometry for study of superficial 16:10 – 16:30 S.A. Rumyantsev, D.Y. Tarasov. Numerical zov. On the theories of thin-walled structures with surface stresses with strong gradients simulation of non-linear dynamics of vibration transport ma- stresses 16:10 – 16:30 M. Shevtsova, A. Nasedkin. Finite element chines in cases of four and three independently rotating vibra- 16:10 – 16:30 M. Yu. Gutkin, I.A. Ovidko. Micromechan- modeling of piezocomposite materials with different connection exciters ics of nanotube glide in deformed ceramic nanocomposites tivity: models, results and applications 16:30 – 16:50 S.V. Semendyaev, A.A.Kireenkov. The ex- 16:30 – 16:50 L.V. Mirantsev, A.K. Abramyan. Influence 16:30 – 16:50 P. Wahi, S. Kumar, I. Sharma. Nonlinear perimental determination of the coefficients of the coupled of wetability of bounding walls on fluid flow in nanoscopic response of a flexible member supported non-ideally at one two-dimensional model of the combined dry friction tubes end 16:50 – 17:10 A.A. Kireenkov. Coulomb law in the gener- 16:50 – 17:10 A.G. Sheinerman, N.F. Morozov, 16:50 – 17:10 G. Filippenko. The vibrations of the built-up alized differential form in problem about torsion oscillations I.A. Ovid’ko, E.C. Aifantis. Effects of grain size and constructions partially submerged into the liquid of the cylindrical rod on the infinite belt rotational deformation on fracture toughness of nanocrys- 17:10 – 17:30 A.S. Kuleshov. Mathematical model of the talline materials boomerang 17:30 – 18:00 Welcome party JULY 2, FRIDAY

Room A Room B Room C Morning Session CHAIRPERSON M.WIERCIGROCH 9:30 – 10:10 N.N. Smirnov. Traffic flows modelling and simulations Morning Session 10:10 – 10:50 P.E. Tovstik, N.F. Morozov. The volume and surface stability of materials Morning Session Minisymposium “Discrete and continuum models in 10:50 – 11:30 A.M. Krivtsov. Modelling of solids with mi- dynamics of condensed matter, mechanics of mediums with CHAIRPERSON YU.A. KOSEVICH crostructure at different scale using particles microstructure, nanomechanics” Coffee break Nonlinear Dynamics CHAIRPERSON A.M. KRIVTSOV CO-CHAIRPERSONS E. AERO,A.PORUBOV Minisymposium Nano Structures and Materials 11:50 – 12:10 L.I. Manevitch . Limiting phase trajectories 11:50 – 12:10 I.E. Berinskiy, A.M. Krivtsov. Multybody and “wave particle” duality in Classical Physics 11:50 – 12:20 E.L. Aero, A.N. Bulygin . Theory of nonlin- potentials to determine elastic properties of diamond and 12:10 – 12:30 K.E. Starkov. Towards computing bounds ear waves in solids undergoing strong rearrangements of their graphene for compact invariant sets of some Hamiltonian systems with crystalline structure 12:10 – 12:30 O.S. Loboda, A.M. Krivtsov. Moment in- 2 degrees of freedom 12:20 – 12:50 I.S. Pavlov, A.I. Potapov. Description of teraction in diatomic crystals. The crystals with the diamond 12:30 – 12:50 D.S. Shepelev, V.V. Smirnov, L.I. Mane- nanostructured media by means of quantum and classical ap- and sphalerite structure vitch. Transition to energy localization in asymmetric periodic proaches 12:30 – 12:50 V.A. Kuzkin, A.M. Krivtsov. Computer Fermi-Pasta-Ulam oscillatory chain 12:50 – 13:20 A.T. Il’ichev. Stability of soliton-aneurysms simulation of solids using particles with rotational degrees of 12:50 – 13:10 P.L. Denisenko, S.O. Gladkov. Some ana- in a thin-walled hyperelastic tube freedom lytical cases LotkaVolterra’s equation 13:20 – 13:50 A.K. Abramyan, L.V. Mirantsev, Y. Kuch- 12:50 – 13:10 A.A. Le-Zakharov, A.M. Krivtsov. Using 13:10 – 13:30 Yu.A. Kosevich, L.I. Manevitch. Efficient min. Fluid behavior in nano-channels multiprocessor computers for modeling of heat conductivity irreversible transfer of vibration energy in linear and nonlinear in crystals coupled parametric systems 13:10 – 13:30 E. Pavlovskaia, O.K. Ajibose, M. Wierci- 13:30 – 13:50 M. Cherkay, S. Vakulenko. Relay control in groch. Dynamics of impact oscillator with different models noisy vibration systems of contact 13:50 – 14:10 I.I. Blekhman, V.S. Sorokin, V.B. Vasilkov. 13:30 – 13:50 J. Ing, E. Pavlovskaia, M. Wiercigroch. Motion of an elastic body in fluid under vibration Particle dynamics to model brittle fracture in rocks Evening Session 13:50 – 14:10 I.E. Asonov, I.E. Berinskiy, A.M. Krivtsov, CHAIRPERSON S.N. GAVRILOV S.A. Le-Zakharov. Particle dynamics model of the percussive Evening Session Phase Transitions drilling process Evening Session CHAIRPERSON S.PALIS Computational Mechanics CHAIRPERSON L.I. MANEVITCH 15:30 – 15:50 M.Yu. Filimonov, A.G. Kravets, Mechanics of Media with Microstructure N.A. Vaganova . Simulation of thermal interaction in a 15:30 – 15:50 B. Tarasov. Shear rupture mechanism of 15:30 – 15:50 A. Carini. Bounds for the overall properties zone of extractive well in permafrost super-brittleness of hard rocks of inelastic composites 15:50 – 16:10 E.V. Shishkina, S.N. Gavrilov. On new 15:50 – 16:10 S. Melin, P. Hansson. On the unpredictibil- 15:50 – 16:10 F. Genna, S. Cresseri. Numerical analysis of phase nucleation due to shock waves collision ity of short fatigue crack growth. A study using a dislocation thrust on structures for active defense against avalanches 16:10 – 16:30 M. Antimonov, A. Freidin. On transforma- based model. 16:10 – 16:30 A. Panteghini, F. Genna. Some constitutive tion surfaces construction for phase transitions in deformable 16:10 – 16:30 M.A.S. Quintanilla, M.J. Espin, J.M. Val- modelling issues in wire drawing analysis solids verde, A. Castellanos. Jamming transition and tensile Coffee break Coffee break strength of magnetic stabilized beds 16:50 – 17:10 V.A. Bogoyavlenskaya, I.N. Shardakov. 16:50 – 17:10 S. Sazhenkov. Kinetic formulation for the Coffee break Estimation of surface deformation sensitivity to changes in the Darcy-Stefan problem on phase transitions in a saturated 16:50 – 17:10 S. Lurie, D. Volkov-Bogorodskii, characteristic parameters of the volcanic system porous ground N. Tuchkova. Generalized self-consistency eshelby so- 17:10 – 17:30 A.V. Zaitsev, V.S. Koksharov. Probabilistic 17:10 – 17:30 G. Kitavtsev, B. Wagner. Critical slippage lution of gradient thermomechanics of structured materials methods for the analysis of random stress and strain fields in for droplet migration in nanometric liquid films (filled composites, ceramics) 2D and 3D matrix-inclusion composites 17:30 – 17:50 G.V. Kharlamov, A.A. Onischuk, 17:10 – 17:30 V. Polyanskiy, A. Belyaev, D. Indeitsev, 17:30 – 17:50 Ya.N. Parshakova, T.P. Lyubimova. 3-D S.V. Vosel, P.A. Purtov. Molecular dynamics of nanodrops Y. Yakovlev. Two-continuum models of the dissolved hydro- numerical modeling of a directional solidification of binary 17:50 – 18:10 N. Kizilova. Thermodynamics of biological gen influence on strength of materials alloys under the action of rotational vibrations growth: experimental data, general regularities and mathemat- 17:30 – 17:50 A.V. Fedorov, A.V. Shulgin. Towards the 17:50 – 18:10 V.D. Sulimov, P.M. Shkapov. Hybrid algo- ical models micro and nano particle ignition and burning theory ritms in computational model updating procedures for hy- 18:10 – 18:30 E. Vilchevskaya. Modeling mechanochem- 17:50 – 18:10 M. Perelmuter. Bridged cracks growth crite- dromechanical systems istry of the diffusion controlled chemical reaction front prop- rion 18:10 – 18:30 L. Sharipova, M. Jabareen, M.B. Rubin. agation in elastic solids 18:10 – 18:30 Y. Yakovlev, A. Polyanskiy, V. Polyanskiy, A three dimensional brick element for nonlinear orthotropic E. Chernyaeva. SMART Hydrogen and its determination in elastic materials based on the theory of a Cosserat point micro and nanostructures in materials JULY 3, SATURDAY

Room A Room B Room C Morning Session Morning Session CHAIRPERSON S. ANTONYUK Morning Session 9:30 – 10:10 A. Castellanos, M.J. Espin, M.A.S. Quin- CHAIRPERSON M. ITSKOV tanilla, J.M. Valverde. The role of electrical forces in the Computational Mechanics Minisymposium “Discrete and continuum models in fluidization of fine powders dynamics of condensed matter, mechanics of mediums with 10:10 – 10:50 D. Kuznetsov. Frontiers of Computational 11:50 – 12:10 S. Palis, F. Palis. Mechanical system simu- microstructure, nanomechanics” Science lation via automatic differentiation 10:50 – 11:30 I.I. Kudish. Regularization of numerical so- 12:10 – 12:30 V.A. Skripnyak, E.G. Skripnyak, S.V. Ra- CO-CHAIRPERSONS E. AERO,A.PORUBOV lutions of isothermal EHL problems. Stable solutions zorenov. Mechanical behavior of ultra fine-grained alloys at high strain rates 11:50 – 12:15 E. Grekova. Soils in terms of nonlinear elas- Coffee break 12:30 – 12:50 V. Koshelev. Numerical evaluation of tic reduced Cosserat medium: small perturbations near ax- CHAIRPERSON A.V. METRIKINE Wave Motion asymptote exponents at singular points in anisotropic isymmetric prestressed state composites 12:15 – 12:40 S.N. Gavrilov, E.F. Grekova, E.V. Shishk- 11:50 – 12:10 D.P. Kouzov. On the influence of gravity 12:50 – 13:10 N.N. Fedorova, I.A. Fedorchenko, ina, A. Savitski. Thermodynamics of hypoplasticity force on elastic wave process in an ideal gas M.A. Goldfeld, Yu.V. Zakharova. Supesonic flows in 12:40 – 13:05 Yu.I. Meshcheryakov, A.K. Divakov, 12:10 – 12:30 L.A. Molotkov. Propagation of low fre- channel of variable cross-section with mass supply N.I. Zhigacheva, I.P. Makarevich, S.Yu. Mushnikova, quency waves in porous Biot layer surrounded by two elastic 13:10 – 13:30 A. Linkov, A. Dobroskok. Two-stage BEM G.Yu. Kalinin. Mechanisms of micro-macro energy ex- half-space for efficient solving transient thermo-(poro-) elastic problems change and macroscopic response of metals on impact 12:30 – 12:50 J.S. Hoving, A.V. Metrikine. On the cou- 13:30 – 13:50 N. Dvas, V. Kuzkin. On the design of ex- 13:05 – 13:30 E.A. Ivanova. On one model of internal pling of nonlinear discrete element models with continua in tended inertial navigation system damping and thermal conductivity the time domain 12:50 – 13:10 A.M. Kudarova, A.V. Metrikine. On the use of gradient continua for minimizing the wave reflection Evening Session at a discrete-continuum interface Evening Session 13:10 – 13:30 A. Kiselev. Structured elastic waves Minisymposium “Discrete and continuum models in 13:30 – 13:50 M. N. Smirnova, K. A. Kondrat’ev, CHAIRPERSON N.N.SMIRNOV dynamics of condensed matter, mechanics of mediums with A.V. Zvyaguin. Wave formation on free surface on interac- Fluid and Gas microstructure, nanomechanics” tion with a body moving under free surface 15:30 – 15:50 E. Semenko. The force on aerofoil in steady CO-CHAIRPERSONS E. AERO,A.PORUBOV Evening Session irrotational flow with immobile point vortex CHAIRPERSON A.V. ZAITSEV 15:50 – 16:10 A.O. Ivantsov, D.V. Lyubimov, T.P. Lyubi- 15:15 – 15:40 S. Vakulenko. Robustness of layered struc- Molecular and Particle Dynamics mova, K.B. Tsiberkin. Numerical modeling of gas release tures in biology process from methane hydrates 15:30 – 15:50 S. Antonyuk, M. Dosta, L. Fries, S. Hein- 15:40 – 16:05 A.V. Porubov. Application of nonlinear 16:10 – 16:30 A.I. Lutsik, A.I. Mizev. Investigation of sur- rich. DEM study of the breakage process in a fluidised bed waves for the problem of growth of long bones face activity dynamics of multicomponent surfactant solutions apparatus 16:05 – 16:30 A. Pozdnyakov, B. Ginzburg. Molecular 15:50 – 16:10 K. P. Zolnikov, A. V. Abdrashitov, Coffee break mechanisms of polymer composites wear as revealed by mass- S.G. Psakhie. Simulation of dusty plasma shape and spectrometry structure in background state at different configuration of 16:50 – 17:10 M.O. Denisova, K.G. Kostarev. Arising of confinement solutocapillary convection in water-alcohol systems Coffee break 17:10 – 17:30 C.A. Bushueva, K.G. Kostarev, A.V. Lebe- 16:10 – 16:30 I.P. Kikot et al. New coarse-grain model for 16:50 – 17:15 A.N. Bulygin, E.L. Aero, Yu.V Pavlov. dev. Dynamics of the ferrofluid layer with a stable rupture of analysis of localized nonlinear excitations in DNA Functionally invariant solutions of Sine-Gordon equation with the surface under the action of external magnetic field variable amplitude Coffee break 17:30 – 17:50 A.E. Mamontov. On existence and unique- 17:15 – 17:40 V. Odintsova. The influence of vapor com- ness of solutions in several boundary-value problems for the 16:50 – 17:10 S.N. Korobeynikov et al . Computer simu- pressibility on the stability of phase transition interface in Euler equations lation of nanotubes buckling and post-buckling deformation geothermal systems 17:50 – 18:10 S.A. Chivilikhin. Evolution of free surface with accounting for van der Waals forces 17:40 – 18:05 A.G. Grechnev, A.S. Kovalev, of high viscous liquid due to capillary forces 17:10 – 17:30 O.E. Khvostova, E.L. Averbukh. Mod- M.L. Pankratova. Exchange bias problem in discrete 18:10 – 18:30 A.A. Alabuzhev. The effect of contact angle elling of landslide motion with lagrangian particles systems hysteresis on the cylindrical drops dynamics 17:30 – 17:50 M.G. Zeitlin, A.N. Fedorova. Quantum me- 18:05 – 18:30 D.V. Laptev, M.M. Bogdan. Exact descrip- 18:30 – 18:50 E.V. Prozorova. Influence of the dispersion chanics at different scales. Resolution of a (quantum) point tion of interaction and reflection of discrete breathers in an- in nonequilibrium models for continuous mechanics. 17:50 – 18:10 A.N. Fedorova, M.G. Zeitlin. Quantum me- harmonic lattice chanics at different scales. Localization, multiscales and com- 18:30 – 18:55 V.G. Mozhaev. A new approach to the prob- plex quantum patterns lem of Rayleigh wave propagation in discrete lattices 18:10 – 18:30 M. Karyakin, O. Pustovalova. On the stability of compressible elastic bodies with microstructure at tension 21:00 Banquet JULY 4, SUNDAY

Room A Room B Morning Session CHAIRPERSON E. GREKOVA

10:10 – 10:50 A.G. Knyazeva, S.N. Sorokova. Self-similar regimes of solid phase conversions in media of various rheology 10:50 – 11:30 M. Itskov. Modeling of active, passive and softening behavior of soft biological tissues based on polyconvex strain energies Coffee break Morning Session CHAIRPERSON S.V. PETINOV CHAIRPERSON A.G. KNYAZEVA Mechanical and Civil Engineering Applications Solids and Structures 11:50 – 12:10 K.L. Muratikov, A.L. Glazov. Comparison of thermoelastic 11:50 – 12:10 S.A. Lychev, A.V. Manzhirov. Finite deformations of accreted stress analysis and photoacoustic microscopy approaches to the problem of resid- solids ual stress detection 12:10 – 12:30 A. Zhigalin. Three-dimensional dynamic problem of coupled 12:10 – 12:30 E. Bezoyan. Calculation of the spatial structures with a glance thermoelasticity for sphere of rheological and heterogeneous behaviors of the materials 12:30 – 12:50 S.B. Bogdanova, S.O. Gladkov . On the magnetic susceptibility 12:30 – 12:50 L.A. Nazarova, I.N. Eltsov, L.A. Nazarov. Origination and of fractal ferrite dielectric evolution of failure zones in the vicinity of inhomogeneities in the rock mass 12:50 – 13:10 A.S. Smirnov, A.V. Konovalov. Simulation of rheology of plas- 12:50 – 13:10 L.A. Nazarov, A.I. Artemova, L.A. Nazarova. Statistical ap- tically deformable 304 steel in a condition dynamic recrystallization proach to constructing equivalent models of rocks based on the inverse problem 13:10 – 13:30 E. Rudoy. Invariant integrals in linear elasticity theory for bod- solution ies with rigid inclusions and cracks 13:10 – 13:30 P. Dyatlova, A. Majorov, V. Lutov. On the problem of interac- 13:30 – 13:50 V. Hakobyan. Contact problem for elastic wedge strengthened tion between the soft shell and the solid covered by it by rigid inclusion 13:30 – 13:50 N. Nikitina. Application of the acoustoelastic effect to evalua- 13:50 – 14:10 E.I. Ryzhak. Ever-topical questions regarding the ”ever- tion of stresses in technological pipelines Hookean” material 13:50 – 14:10 I. Neygebauer. On the 2D equations of mathematical physics

Evening Session CHAIRPERSON I.E. BERINSKIY Opening of the fourth young scientists’ school - conference “Modern Ways in Mechanics (MWM) 2010”

15:00 – 15:40 A. Metrikine. The synchronization phenomenon in modern structural dynamics: vortex-, pedestrian- and ice- induced vibrations of structures Break

CHAIRPERSON P. DYATLOVA Mechanical and Civil Engineering Applications 15:50 ENGINEERINGCOMPETITIONSFORPARTICIPANTSOF 15:50 – 16:10 A.F. Arrieta, S.A. Neild, D.J. Wagg, P. Hagedorn. Dynamics “MWM 2010” of snap-through of bi-stable composite plates 16:10 – 16:30 S.V. Petinov, I.M. Afanasyeva. Fatigue assessment of structures in high-cycle segment: technique and problems Coffee break 16:50 – 17:10 R.A. Arutyunyan, K.S. Yakimova. Fatigue and creep fracture of aging elastic viscous materials subjected to cyclic compression 17:10 – 17:30 E.V. Shilko, S.V. Astafurov, A.S. Grigoriev, S.G. Psakhie. Study of features of deformation response of interfacial zones in block-structured medium to low-amplitude dynamic mechanical actions 17:30 – 17:50 V. Kolykhalin, S. Neganov. Quantities available from two- microfone measurement of acoustical intensity JULY 5, MONDAY

Room A Room B Morning Session CHAIRPERSON I.E. BERINSKIY THE FOURTH YOUNG SCIENTISTS’ SCHOOL - CONFERENCE “MODERN WAYS IN MECHANICS (MWM) 2010”

10:00 – 10:40 G. Rega. Dynamical integrity and control of nonlinear mechanical and structural systems 10:40 – 11:20 C.G. Hoover, W.G. Hoover. Smooth particle applied mechanics: the method, with three example problems 11:20 – 11:40 M. Yumashev, N. Smirnov. The Mechanical Olympiad in Moscow State University Coffee break

CO-CHAIRPERSONS V. KUZKIN,M.ANTIMONOV Posters presentations Evening Session 12:00 – 12:10 Opening ceremony 12:10 – 14:10 Posters presentations (PS1) 15:30 – 17:00 MWM 2010 posters session (PS1) 17:00 – 18:30 APM 2010 posters session (PS2)

18:45 Closing ceremony POSTER SESSION

1. T. Abid, A. Boubertakh, K. Taleb, C. Serrar, S. Hamamda. Effect of room temperature ageing on the precipitation hardening. 2. O. Adiguzel. Thermal memory and mechanical shape reversibility in copper-based shape memory alloys. 3. A. A. Alabuzhev. Non-linear dynamics of gas-vapour bubble under vibrations and temperature gradient. 4. A. A. Alabuzhev, K. Yu. Lebedeva. Forced oscillations and parametrical instability of cylindrical drop. 5. S. Allag, M. Boukezzata. Control of the electric parameters evolution of a semi conductor structure of a polycrystalline silicon using the Monté-carlo method. 6. S. Antonyuk, V.Salikov, K. Brandt, M. Wolff, S. Heinrich, G. Schneider. Novel hierarchical ceramic- polymer composites with extremely small amounts of polymer. 7. Y. Aouabdia, A. Boubertakh, K. Taleb, C. Serrar, S. Hamamda. Precipitation kinetics of the hardening phase in AlMgSi alloys. 8. T. S. Argunova, M.Yu. Gutkin, L. S. Kostina, I. V.Grekhov, E. I. Belyakova, J. H. Je. Crack prevention in wafer bonded Ge/Si by grooved patterns at the interface. 9. T. S. Argunova, M. Yu. Gutkin, J.-H. Je, E. N. Mokhov, S. S. Nagalyuk, Y. Hwu. Evolution mechanisms of pores and micropipes during SiC growth studied by sinchrotron radiation phase contrast imaging. 10. G. P.Bashin, V.V.Shadrin, L. A. Komar. Influence of temperature and filler concentration on the dynamic modules of polyethylene-based nanocomposites. 11. A. Boubertakh, K. Taleb, C. Serrar, S. Hamamda. Investigation of some rapidly solidified. 12. N. Y. Bykov, D. V.Leshchev. Kinetic model of water cluster formation and growth in rarefied gas flows. 13. V.N. Demidov. Calculation of the micro parameters of dislocation kinetics of plastic deformation for aluminum alloys using the results of computer modeling. 14. A. I. Dmitriev, W. Oesterle, H. Kloss. The influence of copper on tribological characteristics of automotive pad-disc interface. Results of nano-scale modeling. 15. M. E. Fedina, I. A. Kuzmina. Analysis and accurate determination of eigenvalues in a nonlinear eigenvalue problem for mode II crack problems under plane strain conditions in a power-law material. 16. G. Filippenko. The vibrations of the built-up constructions partially submerged into the liquid. 17. O. K. Garishin. Polymer/silicate nanocomposites. Nonlinear elastoplastic model with damage accumulation under deformation. 18. L. A. Golotina, L. L. Kozhevnikova. Investigation of the dependence of potential energy density on structural parameters in modeling the mechanical behavior of coarse-grained elastomeric composites. 19. L. A. Golotina, I. N. Shardakov. Investigation of the influence of relaxation transition in the amorphous component of amorphous-crystalline polymer on deformation processes . 20. Yu. V.Grinyaev, N. V.Chertova. Wave process in media with defects. 21. M. Yu.Gutkin, S. V.Panpurin. Spontaneous formation of quantum dots in pentagonal nanowires of inhomogeneous composition. 22. K. Hamouda. Technological basis for treatment details of surface plastic deformation in granular flexible workspaces. 23. Yu. Khokhryakova. Modeling of effective characteristics of multicomponent randomly reinforced composite materials weakened by cracks noninteracting. 24. L. A. Komar, S. N. Lebedev. Formation of polymer nanofiber in the gap between filler particles in deformed elastomeric nanocomposites.

11 25. S. Yu. Korostelev, A.Yu.Smolin, N.V.Roman. MD simulation of high speed rotating particle collisions. 26. V.A. Kovalev. Transient oscillation of the elliptic thermoviscoelastic plates. 27. I. A. Kuzmina M. E. Fedina. For mode II crack problems under plane strain conditions in a power-law material. 28. T. B. Lavrova. The extent of softening, that can be achieved under stable deformation with the Tresca elastic-plastic material. 29. O. N. Lyubimova, K. N. Pestov, E. A. Gridasova. Numerical solution of problem fusion penetration metal, when glass welds with metal by diffusion welding. 30. V.V.Moshev, S. E. Evlampieva. Role of relaxation in formation of cyclic resistance of elastomeric nanocomposites. 31. I. A. Ovid'ko, N. V.Skiba, A. K. Mukherjee. Nucleation of nanograins near cracks as a new toughening mechanism in deformed nanocrystalline materials. 32. V.E. Petrov. Forced quasi 2D turbulence with chemical reactions: analytical theory. 33. M. Podolsky. Direct tensor calculus in the problem of mechanical energy dissipation in ﬂuids. 34. M. A. Potapenko. Improvement of the dynamic behaviours of the plant with selfsynchronizing vibroexciters by means of additional degrees of freedom. 35. S. Sargsyan. Mathematical models of thermoelasticity of micropolar thin elastic shells, plates and bars. 36. E. G.Skripnyak, V.A. Skripnyak, V.V.Skripnyak. Mechanical behavior of nanoceramics under shock waves loading. 37. A. Yu. Smolin, A. I. Dmitriev, S. A. Dobrynin, E. A. Kolubaev. Time-frequencies analysis of acoustic vibration in friction: comparison of experiment and simulation. 38. I. Yu. Smolin, A. V.Kuznetsova. Numerical analysis of rate-dependant roof fall in coal mines. 39. I. N. Soldatov, N. V.Klyueva. Wave propagation in a viscous layer inside a rotating elastic cylindrical shell. 40. Yu. P.Stefanov. Numerical investigation of inelastic deformation and fracture of rocks. 41. T. V.Suvorova, U. V.Stoliarov, S. A. Usoshin, H. A. Usoshina. Wave propagation in the composite porous and liguid layer generated by moving and oscillating load . 42. E. Svijazheninov. Ignition distributor. 43. P.E. Tovstik, T. P.Tovstik. Two-dimensional models of plates made of an anisotropic material. 44. A. Vorobev. Numerical modelling of incompressible miscible ﬂuids. 45. P.Wahi. Analytical study of Hopf bifurcations in frictional chatter models . 46. Ya. S. Zinkevich, D. D. Leshchenko, A. L. Rachinskaya. Optimal breaking of rotations of a rigid body.

12 YOUNG SCIENTISTS SESSION “MODERN WAYS IN MECHANICS (MWM) - 2010”

1. A. S. Alashova, A. N. Alekhin, S. A. Rumyantsev. Estimating parameters of the non-linear model of ground. 2. O. Alexeyeva, S. Rumyantsev, G. Vasilyeva. The research of dynamics of two-mass vibrational transport machines with two unbalanced vibration exciters mounted on the lower mass. 3. O. V.Antonova, A. I. Borovkov, A. A. Michailov, I. B. Voynov. CFD-analysis of hydrogenerator thrust bearing capacity . 4. I. E. Asonov, I. E. Berinskiy, A. M. Krivtsov, S. A. Le-Zakharov. Particle dynamics model of the percussive drilling process. 5. S. V.Astafurov, E. V.Shilko, V.V.Sergeev, S. G. Psakhie. Investigation of influence of nonequiaxial compression on general features of mechanical response of shearing fault zones. 6. S. V.Astafurov, E. V.Shilko, A. Yu. Smolin, S. G. Psakhie. Development of movable cellular automaton method formalism for computer-aided simulation of elastic-plastic response of monocrystalline metal materials. 7. M. Babenkov. Dispersion relations in the coupled problem of thermoelasticity. 8. K. L. Belov, V.A. Eremeyev. On the flows in a free-suspended visoelastic micropolar fluid film. 9. I. N. Borodin, V.S. Krasnikov, A. E. Mayer. Microcrystal material dynamics at irradiation by powerful beams of charged particles. 10. C. A. Bushueva, M. O. Denisova, K. G. Kostarev. Terrestrial modeling of surfactant diffusion through the drop-surrounding fluid interface. 11. U. Cakmak, Z. Major. Mechanics of peel-test: soft materials. 12. Yu. Chumakov. The stress-strain state of porous burner under various regimes of gas filtration burning. 13. M. O. Denisova, R. V.Birikh, A. I. Mizev. Development and stability of a convective mass transfer induced by local heterogeneity of surface-active substance distribution near a free liquid surface. 14. N. Dvas, V.Kuzkin. On the design of extended inertial navigation system. 15. P.Dyatlova, V.Lutov, V.Korovkin. The strain-stressed state of the band winded up on the ridged body. 16. A. A. Dyda, E. V.Lyubimov. Symbolic synthesis of control laws for underwater vehicle and robot manipulator. 17. N. K. Evstigneev, A. G. Knyazeva. Simulation of intermetallic compound synthesis during solid-phase extrusion through tapered die. 18. A. A. Fukalov, A. V.Kutergin, A. V.Zaitsev. Exact analytical solutions to problems on equilibrium state of thick-walled heavy anisotropic axial- and central-symmetric bodies and their applications. 19. E. A. Gerasimenko, A. V.Zavertan. The use of near-front expansions for shock loading simulation. 20. E. S. Gorkunov, S. M. Zadvorkin, A. N. Mushnikov, E. I. Yakushenko. Influence of elastic deformation on set of magnetic characteristics for structural steels. 21. I. G. Goryacheva, A. V.Sakharov. Multiscale modelling friction of rough surfaces. 22. A. G. Grechnev, A. S. Kovalev, M. L. Pankratova. Exchange bias problem in discrete systems. 23. D. A. Indeitsev, V.A. Polyanskiy, Y. A. Yakovlev. The hydrogen material interaction during cyclic mechanical loading. 24. K. Ivanov. Vibrational translation due to transversal strain in a Chelomei pendulum-like system. 25. K. Kazakov. On a new solution of the contact problem for foundations with coatings. 26. O. E. Khvostova, E. L. Averbukh, A. A. Kurkin. Cape Canaille cliff falling: hypothetic tsunami consequences estimation.

13 27. G. Kitavtsev, B. Wagner. Critical slippage for droplet migration in nanometric liquid films. 28. L. S. Klimenko, D. V.Lyubimov, T. P.Lyubimova. Convection in steady thermal stratified medium. 29. Ig. S. Konovalenko, A. I. Dmitriev, A. Yu. Smolin, S. G. Psakhie. On strength properties estimation of porous ceramics coating. 30. Iv. S. Konovalenko, D. S. Kryzhevich, K. P.Zolnikov, S. G. Psakhie. Investigation of structure-phase changing in titanium crystallite under different rate mechanical loading. 31. D. A. Konovalov, E. N. Perunov, S. V.Smirnov. Identification of strain hardening curve by portable hardness tester . 32. I. K. Korolev, S. V.Petinov, A. B. Freidin. FEM simulation of damage accumulation and fatigue crack growth in elastic materials. 33. M. Kovaleva, L. Manevitch. Localized and periodic solutions of the Oregonator model of BZ Reaction. 34. A. U. Koverga, E. P.Kubyshkin. Nonlinear vibrations of the distributed rotor from a material with nonlinear-hereditary properties. 35. E. N. Kozhevnikov, Ya. V.Kucherenko. Spatially modulated structure in a nematic liquid crystal layer. 36. N. V.Kozlov, S. V.Subbotin. Vibrational dynamics of a spherical body in a rotating cylinder with liquid. 37. V.G. Kozlov, A. A. Vjatkin, R. R. Sabirov. Heat transfer and overcritical convection in rotating horizontal annulus. 38. C. Kroener, H. Altenbach, K. Naumenko, A. Kutschke. Generalized fractional elements in rheological modeling of polymer creep. 39. D. O. Kucheyeva, A. B. Freidin. The influence of external strains and material parameters on the kinetics of moving interfaces. 40. A. M. Kudarova, A, M. Krivtsov. Description of equivalent elastic continuum for graphene lattice. 41. A. Kutschke, K. Naumenko, H. Altenbach, C. Kroener. The application of Neuber’s rule for lifetime prediction of notched structures subjected to complex loading. 42. T. P.Lyubimova, Ya. N. Parshakova. Numerical investigation of a dilution and transport of heavy pollutants in water bodies taking into account density stratification effects. 43. P.V.Markov, A. L. Svistkov. Variational formulation for solving problems of complex incompressible media mechanics. 44. A. Melentyev, E. Tarunin. Modulation of gravity in problem of free convection. 45. M. A. Mikolaychuk, A. G. Knyazeva. Influence mechanisms of strains and stresses to diffusion in loaded plate. 46. I. A. Morozov, A. L. Svistkov, O. K. Garishin, O. S. Gileva, E. S. Erofeeva. Investigation of the influence of bleaching on human teeth enamel microstructure. 47. A. Nasedkina. Modeling of failure processes in problems of geomechanics of coal beds. 48. A. Yu. Nikonov, A. I. Dmitriev. Study of the influence of the structure of grain boundaries on their behavior under shear loading. 49. V.N. Pilishkin, V.V.Sosnovtsev. Complex dynamic systems synthesis, based onmethod of Lyapunov’s edge conditions. 50. E. Polyakova, V.Lutov, V.Chaikin. On the problem of soft toroidal shells designing. 51. I. Yu. Popov, O. A. Rodygina, S. A. Chivilikhin, V.V.Gusarov. Solutions and flows in nanotube. 52. S. G. Psakhie, N. V.Roman, A. Yu. Smolin. Influence of porosity percolation on strength and elastic properties of ceramic materials. 3D simulation using movable cellular automata. 53. S. Polyanskiy. The Fourier series expansion in the thermal stress problem on rotating work rolls during hot rolling.

14 54. S. G. Psakhie, K. P.Zolnikov, D. S. Kryzhevich. Plastic deformation nucleation on initial stage of copper crystalline nanoindentation investigation. 55. V.D. Schipitsyn, A. A. Ivanova. Vibrational lift force acting the cylinder in viscous liquid. 56. M. A. Shepelov. Application of genetic algorithms to the methods of molecular dynamics. 57. D. Sheydakov. Influence of solid coating on stability of highly porous elastic rod subject to combined loads. 58. A. Shihov, S. Rumyantsev, D. Tarasov. Starting dynamics of vibrational transport machines with three unbalanced vibration exciters. 59. A. S. Smirnov, A. V.Konovalov. Method of identification of model of strain resistance by results of experiments at compression samples. 60. S. V.Smirnov, I. A. Golubkova. Investigation of failure of multilayered composite materials received by explosion welding and rolling. 61. E. O. Smirnova, S. V.Smirnov. Determination of stress-strain curve by indentation and scratch test. 62. A. Yu. Smolin, V.V.Sergeev, S. Yu. Korostelev, S. A. Dobrinin. Modeling tribospectroscopy using movable cellular automata. 63. V.S. Sorokin, I. I. Blekhman, V.B. Vasilkov. Motion of an elastic body in fluid under vibration. 64. A. L. Svistkov, A. G. Pelevin A.G., A.A.Adamov, B.Lauke, K.W.Stockelhuber,¨ G. Heinrich. Structural- phenomenological model of the mechanical behavior of elastomeric nanocomposites based on styrene butadiene rubber. 65. A. V.Zaitsev, N. G. Zlobin, O. Yu. Isaev, D. V.Smirnov, A. M. Khanov . Mathematical model for thermoelastic deformation of nanostructured flexible graphite o-ring seal packs. 66. S. Zavsek, A. V.Dimaki, E. V.Shilko, A. I. Dmitriev, J. Pezdic, S. G. Psakhie. Symbiotic cellular automata method and its application to simulation of the mechanical response of lignite. 67. E. Zdanchuk, V.Lalin. The theory of continuous medium with free rotation without coupled stresses. 68. E. Zdanchuk, V.Lalin. Asymmetric continuous medium without coupled stresses. Wave propagation in linear isotropic space. 69. F.A. Zvezdin, V.G. Kozlov. Pendulum thermovibrational convection in vertical annulus.

15 SIMULATION OF DUSTY PLASMA SHAPE AND FLUID BEHAVIOR IN NANO-CHANNELS STRUCTURE IN BACKGROUND STATE AT DIFFERENT CONFIGURATION OF CONFINEMENT A.K.ABRAMYAN L. V. MIRANTSEV Y. KUCHMIN [email protected] A.V.ABDRASHITOV K.P.ZOLNIKOV S.G.PSAKHIE We consider behavior of a simple fluid in nano-channels of no [email protected] more then 50 nm widths. On the basis of 3D MD simulations and with the help of signal The influence of anisotropy degree of a confinement field on a analysis methods (wavelets, Fourie spectrum etc.) the correlation dusty plasma structure was investigated on the base of the molec- between forces caused by external excitation and a structure of ular dynamics method. Dust particles represented melamine fluid has been found and analyzed. It has been showed that for formaldehyde balls with diameters of 7,1 µm. The simulated Couette flow the friction force and forces inside fluid strongly dusty plasma system was confined by the electrical force in the depends on the crystalline lattice constant of the bounding walls. horizontal plane and by combination of electrical and gravity The fluid structure and forces dependences on the velocity of forces in the vertical direction. The Yukawa potential was used the moving bounding wall have been investigated, as well as the to describe the interparticle interaction. A screening parameter dependence on the channel width. The reological models of the of the interparticle interactions was determined by adjusting the fluid behavior for different excitation velocities are suggested. In- interparticle distances to the experimental data. stead of classical explanation of so call stick-slip behavior a new The simulation showed that the dusty plasma shape and struc- interpretation of the fluid behavior is suggested. The stick-slip ture sufficiently depended not only on anisotropy value of the corresponds to the process of appearence and vanishing of spe- confinement field but also on the dust particle number of the sim- cific fluid structure.The behavior of the fluid in nano-tubes with ulated system. The dusty plasma system represented a Coulomb amorphous and crystalline walls is considered for Poiseuille flow. ball with a typical shell structure in the background state for the It has been showed that depending on hydrophilling or hy- spherically symmetric confinement field. The simulated system drophobing nature of the bounding walls we can get essential tended to the transition from a bulk to plane shape at the decrease decreasing or increasing of the flow rate. of the horizontal confinement field component to compare with vertical one. At the same time the simulated system tended to the transition from the plane to bulk shape and to form the shell Andrey K. Abramyan, Industrialny 11-2-44, St.Petersburg, structure in the background state at increase of the dust particle 195426, Russia number.

Konstantin P. Zolnikov, Akademicheskii, 2/4, Tomsk, 634021, THERMAL MEMORY AND MECHANICAL SHAPE Russia REVERSIBILITY IN COPPER-BASED SHAPE MEMORY ALLOYS

EFFECT OF ROOM TEMPERATURE AGEING ON THE O.ADIGUZEL PRECIPITATION HARDENING [email protected]

T. ABID A.BOUBERTAKH K.TALEB A series of alloy systems exhibit a peculiar property called C.SERRAR S.HAMAMDA shape memory effect or thermal memory effect. This property [email protected] involves the repeated recovery of macroscopic shape of material at different temperatures by means of thermal manner. These ma- The effect of pre-ageing and maturing at room temperature on terials are also called smart materials due to their capacity of re- hardening response of an Al–Mg–Si alloy is investigated using sponding to changes in the environment. Shape memory alloys differential scanning calorimetry (DSC), hardness measurements are widely used as a shape memory component in devices. Shape (Hv) and scanning electron microscopy (SEM). Two experimen- memory alloys cycle between the original and deformed shapes tal conditions are examined. First, natural ageing for different through heating and cooling in a temperature interval, and these times (3 weeks and 1 month) followed by artificial ageing at alloys can be used as a thermal sensor or actuator in devices due 180oC for different times. Second pre-ageing at temperatures in to this property. the range 75–100oC followed by artificial ageing at 180oC after Copper-based alloys exhibit this property in metastable beta natural ageing for the same periods. The present results indi- phase field. The bcc- structures of ternary or binary copper alloys cate that the effect of the pre-ageing just after the solution treat- are subjected to a shearing motion upon cooling from high tem- ment and quenching is used in order to correct the undesirable peratures, following two ordered reactions. The material has little effect of ageing at room temperature. However, during the arti- resistance to this motion, and undergoes a diffusionless transition ficial ageing, the alloy hardening becomes faster. Aged samples called martensitic transition. which have already undergone pre-ageing and maturing reveal Shape memory materials can be permanently deformed when better hardening response. stress is applied and then are able to recover their original shape just by thermal means, on heating over the reverse transition tem- Khelifa Taleb, Facutédes sciences UMC, Constantine, 25000, perature after removing the strain. Also, they cycle between the Algeria deformed and original shapes on cooling and heating. Due to this

16 property, these alloys exhibit thermoelasticity, and this phenom- means that the complicated lattice energy should be a periodic ena can be called the thermal memory, as well as shape memory function of the relative rigid displacement of sub-lattices invari- effect. ant to such a translation. The variation equations of macro- and The shape memory effect is based on the martensitic transition micro-displacements turned out to be a nonlinear generalization in crystallographic manner, and lattices of materials exhibit two of the linear equations of acoustic and optical modes obtained by different configurations of the crystals: austenite and martensite. Karman, Born and Huang. Some exact solutions are obtained for These two phases have different structures, the former is highly the one-dimensional case, and their specific features are revealed. symmetric and the latter is internally twinned and complex. Also, several types of microscopic deformation involving changes can Eron Aero, 61, Bol’shoy pr., V.O., Saint Petersburg, 199178, Rus- occur in the stacking sequence of close-packed planes of material sia with martensite formation. Martensitic transformations in beta-phase copper alloys occur by two or more lattice invariant shears on a 110 - plane of par- FUNCTIONALLY INVARIANT SOLUTIONS OF ent phase called basal plane of martensite as a step of transition, SINE-GORDON EQUATION WITH VARIABLE and these planes turn into unusual layered structures which con- AMPLITUDE sist of an array of close-packed planes with complicated stacking E.L.AERO A.N.BULYGIN YU.V.PAVLOV sequences called as 3R, 9R or 18R martensites depending on the stacking sequences on the close-packed planes of the ordered lat- bulygin [email protected] tice. Copper based beta phase alloys are widely used as a shape A new approach to solution of sine-Gordon (SG) equation with memory component in devices. variable amplitude The product martensitic phase occurs following two ordering ∂2U ∂2U ∂2U 1 ∂2U transitions and has a close packed-structure. The martensitic + + = n(x,y,z,t) sin U (1) structures are closely related to the austenitic structures and in- ∂x2 ∂y2 ∂z2 − v2 ∂t2 herit the order in the parent phase due to the diffusionless charac- is represented. The SG equation appears in many branches of ter of the transformation. modern natural sciences. It describes deformation of a nonlinear On the other hand, the basal plane of the martensite is sub- crystal lattice, the orientation structure of liquid crystal (LC), sur- jected to hexagonal distortions when martensite forms and this is face metric, etc. In present time there are many effective methods strongly affected by the sizes of atoms. In case the atoms occupy- for SG equation solution. However the main methods have been ing the lattice sites have the same size, the basal plane of marten- developed for case n(x,y,z,t) = const. It is strong limited the site becomes regular hexagonal. Otherwise deviations occur from domains of the SG equation application. So in mechanics of non- the hexagonal arrangement. The knowledge of the associated dis- linear crystal lattice the case n = const describes deformation of tortion allows us to obtain information on the ordering degree in ideal lattice by a homogeneous stress field. The deformation of the martensite. real lattice with structure defects (dislocations, disclinations) by In the present contribution, x-ray diffraction and transmission nonhomogeneous stress field is described by SG equation with electron microscope (TEM) studies have been carried out on two n = const. In mechanics of LC the case n = const models copper based ternary alloys, which have the following alloy com- of6 the axes orientation by a homogeneous electromagnetic field. positions in weight, respectively; Cu-26.1 For nonhomogeneous fields n = const. In differential geometry 6 SG equation with n = const describes the metrics of Chebyshev Osman Adiguzel, Firat University Department of Physics, Elazig, nets on a surface with constant curvature [1]. If the curvature is 23169, Turkey changing then n = const. It is clear from these examples, that the domains of SG6 equation applications are extended if one find the solutions of SG equation with a variable amplitude. THEORY OF NONLINEAR WAVES IN SOLIDS A new approach to SG equation with n = const is found on UNDERGOING STRONG REARRANGEMENTS OF the next results: 6 THEIR CRYSTALLINE STRUCTURE 1. If function ϕ(x,y,z,t) obeys simultaneously to equations

E.L.AERO A.N.BULYGIN ∂2ϕ ∂2ϕ ∂2ϕ 1 ∂2ϕ + + = 0 , ∂x2 ∂y2 ∂z2 − v2 ∂t2 [email protected] (2) ∂ϕ 2 ∂ϕ 2 ∂ϕ 2 1 ∂ϕ 2 + + = n(x,y,z,t) , The nonlinear theory of propagation of nonlinear localized ∂x ∂y ∂z − v2 ∂t waves (like kinks and solitons), connected with the movement of defects in crystals, is developed. It is assumed that crystals then U = 4 arc tg eϕ(x,y,z,t) is a solution of (1). possess a complicated lattice consisting of two sub-lattices. The 2. If function ϕ(x,y,z,t) obeys simultaneously to equations arbitrary large displacements of sub-lattices are considered. An ∂2ϕ ∂2ϕ ∂2ϕ 1 ∂2ϕ additional element of translational symmetry is employed in the + + = n(x,y,z,t) , theory. The element is typical for complicated lattices, however, ∂x2 ∂y2 ∂z2 − v2 ∂t2 (3) it has not been introduced in physics of solid state before. It is ev- ∂ϕ 2 ∂ϕ 2 ∂ϕ 2 1 ∂ϕ 2 + + = 0 , ident that the relative displacement of sub-lattices for one period ∂x ∂y ∂z − v2 ∂t (or for an integer of periods) to a superposition of the sub-lattice with itself reproduces the structure of a complicated lattice. This then U = 2 arc tg eϕ(x,y,z,t) is a solution of (1).

17 A solutions of Eqs. (2), (3) can be found by method of con- Ekaterina E. Pavlovskaia, School of Engineering, King’s College, struction of functionally-invariant solutions of differential equa- Aberdeen, AB24 3UE, United Kingdom tions [2–5]. New solutions of SG equation with n = const are found and the approach to constructing of SG equation solution with variable amplitude are represented. THE EFFECT OF CONTACT ANGLE HYSTERESIS ON THE CYLINDRICAL DROP’S DYNAMICS

REFERENCES A.A.ALABUZHEV [email protected] [1] P.L. Chebyshev, Uspekhi Matem. Nauk 1 (1946) 38. [2] H. Bateman, The Mathematical Analises of Electrical and The free and forced oscillations of a cylindrical drop are con- Optical Wave-Motion: On the Basis of Maxwell’s Equations, sidered in the present work. The drop is suspended in the differ- Dover, New York, 1915. ent fluid and confined by two parallel rigid plates, subjected to [3] E. L. Aero, A. N. Bulygin, Yu.V.Pavlov, TMF 158 (2009) vibrations. The vibration axis is perpendicular to the symmetry 370 [Theor. Math. Phys. 158 (2009) 313]. axis. The amplitude of vibrations is small in comparison with the [4] E. L. Aero, A. N. Bulygin, Yu.V.Pavlov, Nelineinii Mir 7 drop radius. The equilibrium contact angle is right. The specific (2009) 513. boundary conditions, assumed by Hocking (1987), is applied to [5] E. L. Aero, A. N. Bulygin, Yu. V.Pavlov, Proc. Int. Conf. take into account: the contact line starts to slide only when the “Days on Diffraction 2009”, pp. 7–12. deviation of the contact angle exceeds a certain critical value. As a result, the stick-slip dynamics can be observed. This work was supported by RFBR (grants No 08-01-91959- Anatoly Bulygin, Bol’shoy pr., 61, Saint Petersburg, 199178, NNIO a). Russia Aleksey A. Alabuzhev, 1, Koroleva, Perm, 614013, Russia DYNAMICS OF IMPACT OSCILLATOR WITH DIFFERENT MODELS OF CONTACT FORCED OSCILLATIONS AND PARAMETRICAL O.K.AJIBOSE M.WIERCIGROCH E.PAVLOVSKAIA INSTABILITY OF CYLINDRICAL DROP [email protected] A.A.ALABUZHEV K.YU.LEBEDEVA [email protected] [email protected] [email protected]

Accurate description of the contact force between two impact- The forced oscillations of a cylindrical drop are considered in ing solid bodies is important in various engineering applications. the present work. The drop is suspended in the different fluid and This paper examines the effects of contact force models on the confined by two parallel rigid plates, subjected to vibrations. The global and local dynamics of a drifting oscillator. First three con- vibrations axis is parallel to the cylinder axis. The both vibration tact force models are considered in this study, namely the Kelvin- amplitude and the thickness of viscous boundary layers (between Voigt (KV), the Hertz stiffness (HS) and the non-linear contact fluid and rigid plate) are small in comparison with drop radius stiffness and damping (NSD) models. The non-linear analysis of and thickness of the layer. The equilibrium contact line is right. the system shows that the local and global dynamic behaviour The parametrical instability of forced oscillation is investi- depends on the choice of contact force model. The short-term gated for fixed or free contact line. The viscous boundary layers (local) dynamic responses for the system are almost identical for are taken into account for case of pinned contact line. all the three models. However, results for the long-term (global) This work was supported by RFBR (grant No 08-01-91959- dynamics depend on the choice of contact force model; while the NNIO a) results for HS model are topologically similar to those of the KV model, the corresponding results for NSD are different from the A. Alabuzhev, 1, Koroleva str., Perm, 614013, Russia other two models. In the second part of the paper the dynamic response of the semi-infinite elasto-plastic media subjected to re- peatable impacts by a rigid body with conical or spherical con- NON-LINEAR DYNAMICS OF GAS-VAPOUR BUBBLE tacting surfaces is considered in order to formulate the relevant UNDER VIBRATIONS AND TEMPERATURE force penetration relationship during the loading and unloading GRADIENT phase of the contact. These relationships are then used to de- A.A.ALABUZHEV velop a physical and mathematical model of a new drifting oscillator where the time histories of the progression through the [email protected] media include both the loading and unloading phases of the contact. The nonlinear dynamic analysis of such system shows that The non-linear oscillations of a gas-vapour bubble in an in- the maximum progressive motion of the oscillator occurs when compressible liquid under the linear-polarized vibrations are con- the system exhibit periodic motion. In addition the dynamic re- sidered in the present work. Both the presence of inert gas inside sponse for both contact geometries show a topological similarity the bubble and the phase transition at the interface liquid-vapour for a range of the static load when similar parameter values were become essential in this case. We take into account the kinetics used. of phase transition and apply the perfect gas law to describe the

18 thermodynamics of both the gas and vapour. The ﬂuid viscos- CONTROL OF THE ELECTRIC PARAMETERS ity and the heat conductivity of both liquid and gas/vapour are EVOLUTION OF A SEMI CONDUCTOR STRUCTURE considered. There is linear temperature gradient in liquid. OF A POLYCRYSTALLINE SILICON USING THE This work supported by project ADTP RSP 2.1.1/4463. MONTE-CARLO´ METHOD

S.ALLAG M.BOUKEZZATA Aleksey A. Alabuzhev, 1, Koroleva str., Perm, 614013, Russia [email protected]

ESTIMATING PARAMETERS OF THE NON-LINEAR Among the various applications of the Monte Carlo method, MODEL OF GROUND it is possible to determine the particle trajectories in given target, taking, only into account the basic equations governing the physi- A.S.ALASHOVA A.N.ALEKHIN cal phenomena (ion-structure interactions) in the semi-conductor. S.A.RUMYANTSEV This aim is obtained with a computer program called TRIM (Transport of Ions in Matter), written in Fortran 77 and in order ugt [email protected] [email protected] to control many parameters of ions implantation, and to simulate the doping profiles of different ions in silicon and polysilicon In this report we present the stable method of estimating pa- targets Indeed, using the computer results and only based on the rameters of the non-linear model of ground. This method corre- known ions stories, we are be able to construct the simulated pro- sponds with government standard (GOST) 12248-96 (Grounds. files with a very good precision. When we compare them with the Methods of laboratorial estimating of strength and deformability experimental data, it appears clearly that this process is consid- specifications) and is based on complex using of the laboratorial ered as very interesting in the parameters control system of high and environmental tests results as well as the results of the nu- speed oscillators circuits. This study may be very attractive to merical simulation. be used for resolving the fundamental problem linked in the nar- Suggested method lets to constrict the class of the possible row junctions of MOS transistors used in UHF/VHF oscillators. parameters values and thus lead to stable and unique solution of When we can control the narrow junction parameters, themselves the inverse problem. On the base of these results we propose to linked to the ion implantation process, we can realize easily these extend the dependence of the shear modulus G (by GOST 12248- high frequency equipments. 96) by adding to it the coefficient considering the stressed state type by Lode-Nadai. Saadoun Allag, Faculté Science de l’ingénieur, Constantine, The report is illustrated with a big amount of graphic materials. 25000, Algeria

ON SOME ADVANCED MODELS OF INELASTIC Anna S. Alashova, 66, Kolmogorova st., Ekaterinburg, 620034, BEHAVIOR Russia H.ALTENBACH [email protected] THE RESEARCH OF DYNAMICS OF TWO-MASS VIBRATIONAL TRANSPORT MACHINES WITH TWO Any material behavior can be classified as elastic, viscoelastic, UNBALANCED VIBRATION EXCITERS MOUNTED ON plastic and viscoplastic (Haupt, 2002). The first and the third one THE LOWER MASS. are models of the spontanuous response of a material on external O.ALEXEYEVA S.RUMYANTSEV G.VASILYEVA loads, the second and the fourth one express the time-dependent response. The formulation of the specific material laws is up to [email protected] now an actual research topic since a universal material law does [email protected] not exist and cannot be established in the future. In this case one has three possibilities to present the constitu- In this report we present the results of the researching of tive equations for advanced materials: starting dynamics of two-mass vibrational transport machines • the inductive approach, (TVTM) with two unbalanced vibration exciters (VE) mounted • the deductive approach, and on the lower mass. The main attention was paid to transient dynamic processes accompanying the start-up of the machine and • the method of rheological models the impact effects. Let us note that all three possebilities have advantages, but also The research was carried out using the mathematical model of disadvantages. The inductive approach is more common for engi- TVTM dynamics. In the report we present the TVTM general- neers. The reason is that this approach is straight-forward related ized coordinates curves at the start-up and after the impact with to the material testing. Some models of the inealastic behavior various values of the TVTM geometrical and physical parame- will be discussed as examples. Last but not least a brief discus- ters. sion concerning the application possibilities of the deductive approach and the rheological models is given.

Olga Alexeyeva, 66, Kolmogorova st., Ekaterinburg, 620034, Holm Altenbach, Kurt-Mothes-Str. 1, Halle (Saale), 06120, Ger- Russia many

19 341/33-1. ON THE THEORIES OF THIN-WALLED STRUCTURES WITH SURFACE STRESSES REFERENCES H.ALTENBACH V.A.EREMEYEV N. F. MOROZOV [email protected] [1] Duan, H. L., Wang, J. & Karihaloo, B. L. Theory of elastic- [email protected] ity at the nanoscale. Advances in Applied Mechanics, Else- [email protected] vier. 2008. 42. 1–68. [2] Lu, P., He, L.H., Lee, H.P. & Lu, C. Thin plate theory The development of nanotechnologies extends the field of ap- including surface effects. Int. J. Solids Structures, 2006. plication of the classical or non-classical theories of mechanics 43(16). 4631–4647. towards the new materials. The discussions on the application [3] Huang, D. W. Size-dependent response of ultra-thin films of the continuum mechanics and the mechanics of structures in with surface effects. Int. J. Solids Structures, 2008. 45(2). the nanoscale are very extensive, see [1] among others. In gen- 568–579. eral, modern nanomaterials have physical properties which are different from the bulk material. The classical elasticity can be [4] Eremeyev, V.A., Altenbach, H. & Morozov, N. F. The influ- extended to the nanoscale by implementation of the theory of ence of surface tension on the effective stiffness of nanosize elasticity taking into account the surface stresses, cf. [1]. In plates. Doklady Physics, 2009. 54(2). 98–100. particular, the surface stresses are responsible for the size-effect, [5] Altenbach, H., Eremeyev, V.A. & Morozov, N. F. Linear that means the material properties of a specimen depend on its theory of shells taking into account surface stresses. Dok- size. For example, Young’s modulus of a cylindrical specimen lady Physics, 2009, 54(12). 531–535. increases significantly, when the cylinder diameter becomes very [6] Altenbach, H. & Eremeyev V.A. On the shell and plate the- small. Let us note that even for such nanostructures as nanoshells, ories with surface stresses. Shell Structures. Theory and Ap- nanofilms, and nanoplates the continuum approach gives a satis- plications. Vol. 2. W. Pietraszkiewicz, I. Kreja (Eds). Boca fying coincidence with atomistic simulations, if one takes into ac- Raton, CRC Press, 2010. 47–50. count the appropriate constitutive equations. The surface stresses are the generalization of the scalar surface tension which is a [7] Libai, A. & Simmonds, J. G. The Nonlinear Theory of well-known phenomenon in the theory of capillarity. The in- Elastic Shells, (2nd ed.). Cambridge, Cambridge University vestigations of the surface phenomena were initiated by Laplace, Press. 1998. Young, and Gibbs. [8] Chró´scielewski, J., Makowski, J. & Pietraszkiewicz, W. We discuss here the methods of derivation of the governing Statyka i dynamika powłok wielopłatowych: Nieliniowa plate and shell equations taking into account the surface stresses. teoria i metoda elementów skończonych. Warszawa, The theory of elasticity with surface stresses was applied recently Wydawnictwo IPPT PAN, 2004. to the modifications of the two-dimensional theories of nano- [9] Eremeyev, V.A. & Zubov, L. M., Mechanics of Elastic sized plates and shells, see, for example, [2–6] and the presented Shells. Nauka, Moscow, 2008 (in Russ.). there references. The most popular in nanomechanics are the Kirchhoff–Love, the Mindlin–Reissner-Naghdi, the von Kàrmàn, and more general theories of plate and shells. We show that Victor A. Eremeyev, Milchakova str., 8a, Rostov on Don, 344090, both the stress and the couple stress resultant tensors may be rep- Russia resented as a sum of two terms. The first term is the volume stress resultant while the second one is determined by the surface stresses and the shell geometry. This means that the stress resul- ON TRANSFORMATION SURFACES CONSTRUCTION tants for the shell with surface stresses can be written as follows FOR PHASE TRANSITIONS IN DEFORMABLE SOLIDS

∗ ∗ M.A.ANTIMONOV A.V.CHERKAEV A.B.FREIDIN T = T + TS , M = M + MS , (1) where T and M are the classical stress and couple stress resultant We consider stress-induced phase transitions of martencite tensors given for example in [7–9], while TS and MS are the type and develop a procedure for transformation surfaces con- resultant tensors induced by the surface stresses, see [4–6] for struction in a strain space. We discuss and use together two ap- details. proaches. The first one is based on a local thermodynamic equi- The surface stresses have an influence on the effective stiff- librium condition. The second one is based on finding an energy ness properties of the elastic plates and shells. In the linear case lower bound and showing two phase structures at which the lower this modification results in new terms added to the elastic stiff- bound is reached. ness parameters. We show that the surface elasticity makes a This work was supported by RFBR (grant No 10-01-00670-a) shell more stiffer in comparison with the shell without surface and National Science Foundation, DMS-0707974. stresses. The numerical examples show that the influence of the surface stresses is negligible for the plate thickness more then 20 Mikhail A. Antimonov, Alexander B. Freidin, Institute for prob- nm. The effect of the surface stresses may be more significant lems in mechanical engineering RAS, Bolshoy pr., V.O., 61, for multilayered plates and shells and for plates and shells with Saint-Petersburg, 199178 fractal-like surface. The second author was supported by the RFBR with the grant Andrej V. Cherkaev, University of Utah, Salt Lake City, Utah No. 09-01-00459 and by the DFG with the grant No. AL 84112 U.S.A.

20 Alexey A. Tikhonov, Universitetsky prospekt, 28, Peterhof, St. TENSOR APPROACH FOR MATHEMATICAL Petersburg, 198504, Russia MODELLING OF GEOMAGNETIC FIELD

K.A.ANTIPOV A.A.TIKHONOV CFD-ANALYSIS OF HYDROGENERATOR THRUST antipov [email protected] [email protected] BEARING CAPACITY

To develop a mathematical model of rotational motion of an ar- O.V.ANTONOVA A.I.BOROVKOV tificial satellite about its center of mass under the action of various A.A.MICHAILOV I.B.VOYNOV forces (magnetic, Lorentz, etc.) caused by the geomagnetic field, [email protected] it is necessary to know the induction of the Earth’s magnetic field (EMF) as a function of the radius vector of a given point in the The thrust bearing is one of the most critical hydrogenerator near-Earth space. Because the EMF possesses a complex struc- nodes, because it undergoes axial loads caused by weight of ro- ture and the above-mentioned functional dependence is unavail- tating parts and reaction of water to the turbine wheels. It is very able in explicit analytic form, a set of approximate models of the important to properly estimate the carrying capacity of the thrust EMF should be used. The simplest and well known such model – bearing. a right dipole (aligned with the axis of rotation) allowed to solve To study the thrust bearing carrying capacity the mathematical a lot of problems, but it does not enable one to reveal in detail models were developed. Using these models and numerical sim- the influence of diurnal EMF rotation on the rotation motion of ulation approach (CFD-system ANSYS / CFX) series of results a satellite. The next and more realistic EMF approximation – were obtained. The finite-volume hexahedral mesh was created an inclined magnetic dipole – does not suffer from the above- for this analysis by the software ANSYS ICEM CFD. mentioned drawback. However, it is shown in [1] that not all cor- Hydrodynamic models of thrust bearing workspace filled by rections to the magnetic induction of the EMF of the same order liquid lubricant were developed. The problem was solved with of magnitude are taken into account in the course of transforma- taking into account the cyclic symmetry and the isothermal for- tion from the model of aligned dipole to the model of inclined mulation. dipole. So, to develop the EMF model accurately accounting for For comparison of engineering methods for evaluation of the the absence of axial symmetry of the EMF with respect to the axial gap with the numerical approach, the series of hydrody- axis of diurnal rotation of the Earth, in general, the effect of the namics problems for various values of minimum thickness of the quadrupole component of the geomagnetic potential on the EMF oil-film wedge and different pad angles were solved. Using the induction (and, probably, even the components of higher orders) obtained distribution of pressure over the pad surface axial force must be taken into consideration. The mathematical ground for corresponding to carrying capacity of thrust bearing as a whole the construction of the EMF induction in quadrupole approxima- was determined. tion was suggested in [2]. The development of this mathematical ground given in [3], allowed to construct first, second and third Olga Antonova, 29 Polytechnicheskaya st., St. Petersburg, multipole components of the EMF induction. 195251, Russia In this paper the multipol tensor of the n-th rank was introduced and studied and its components were obtained analytically by using the International Geomagnetic Reference Field IGRF- NOVEL HIERARCHICAL CERAMIC-POLYMER 2005. As a result, the EMF induction was written in convenient COMPOSITES WITH EXTREMELY SMALL AMOUNTS and concise multipole form. Thus, the solution of the problem, OF POLYMER started in [2] and continued in [3], was led to its logical completion. The multipole structure of the EMF is proved. The algo- S. ANTONYUK V. SALIKOV K. BRANDT rithm for the multipole representation of the EMF induction with M.WOLFF S. HEINRICH G.SCHNEIDER an arbitrary degree of precision is elaborated. [email protected] [email protected] [email protected] [email protected] REFERENCES [email protected] [email protected] [1] A. A. Tikhonov. Refinement of the Oblique Dipole Model in the Evolution of Rotary Motion of a Charged Body in the Hierarchical-structured natural composites, like enamel and Geomagnetic Field // Cosmic Research 2002, vol. 40(2) pp. nacre, offer excellent mechanical properties: high damage tol- 171-177 erance and strength. The lightweight of these materials are of enormous interest for materials science. [2] A. A. Tikhonov, K. G. Petrov. The moment of Lorentz forces In order to reproduce these by nature created strengthening and acting on a charged satellite in the Earth’s magnetic field. toughening mechanisms a novel class of hierarchical ceramic- Part 1: The Earth’s magnetic field strength in the orbital co- polymer composite materials is developed. The required struc- ordinate system //Vestnik of St.-Petersburg State University ture will be composed of three hierarchical levels from nm to mm 1999, vol. 1 (1), pp. 92-100 length scales. Furthermore the content of soft (polymer) phase [3] A. A. Tikhonov, K. G. Petrov. Multipole Models of the should be, analogous to the above-mentioned biological materi- Earth’s Magnetic Field // Cosmic Research 2002, vol. 40(3) als, extremely small (< 20 vol.-%). The hierarchical structure pp. 219-229 allows a high freedom of design by variation of ceramics and

21 polymer binder as well as strength of interfaces in and between EVOLUTION MECHANISMS OF PORES AND levels for adaptation of the novel material on required mechanical MICROPIPES DURING SIC GROWTH STUDIED BY and functional properties. SINCHROTRON RADIATION PHASE CONTRAST In this contribution the obtained results of the most impor- IMAGING tant first hierarchical nano-level will be presented. The ceramic- polymer composite was produced from titanium dioxide nanopar- T.S.ARGUNOVA M.YU.GUTKIN J.H.JE ticles and polymethylmethacrylate by the use of colloidal chem- E.N.MOKHOV S.S.NAGALYUK Y. HWU istry techniques. The porosity and mechanical properties, like hardness, strength and fracture toughness of the produced ma- [email protected] terial, are investigated by variation of particle size distribution, polymer content, compacting pressure and further numerous pro- Modeling the structure and properties of highly porous solids cessing parameters. Further used characterization methods are is among the hot topics in advanced mechanics. In particu- TEM, SEM, TGA, DSC, IR-Spectroscopy and nanoindentation. lar, much attention in recent years has been attracted to mechanisms of nucleation and development of rather complex defect To predict the particle size distribution which can give the min- structures in bulk SiC crystals which is one of the most promis- imal packing structure the Discrete Element Simulation of these ing semiconductor materials for high-power electronics. These structures were performed. By introducing a solid bridge model defect structures are composed of pores, hollow-core superdis- in DEM, the binding properties of polymer between ceramic- locations called ‘micropipes’ (MPs), inclusions of foreign SiC particles were calculated. The influence of the packing structure polytypes, full-core dislocations, etc. During the crystal growth, and amount of polymer on the strength and hardness of compos- they strongly interact with each other: dislocations and MPs are ites was studied. largely induced by foreign polytype inclusions (FPIs) [1]; MPs, An application of the modern spouted fluidized bed technology when accumulated at the boundaries of FPIs, may coalesce each is planned to realize the further hierarchical levels. other, forming pores [2 4]; pores can cause the formation of MPs [5]. Mechanical modeling of nucleation, propagation, interaction and conversion of these defects must be based on data of detailed Sergiy Antonyuk, Institute of Solids Process Engineering and experimental investigations. Particle Technology, Denickestr. 15, Hamburg, 21073, Germany The purpose of the paper is to investigate the evolution of pores and MPs during SiC growth using synchrotron radiation white beam phase contrast imaging. The measurements were performed at the 7B2 X-ray microscopy beamline of Pohang Light PRECIPITATION KINETICS OF THE HARDENING Source, Korea. The sublimation sandwich method [6] was used PHASE IN ALMGSI ALLOYS to grow a 6H-SiC boule in [0001] direction at the growth temperature 2200o C and with the growth rate 300 µm h−1. Six on- Y. AOUABDIA A.BOUBERTAKH K.TALEB · C.SERRAR AND S.HAMAMDA axis wafers were cut off from the boule, nearly perpendicular to the growth direction. Defect evolution from wafer to wafer was [email protected] investigated on the region of interest from the matched overlap between the wafers (established by measuring Laue patterns). A Light-weight alloys recently received increased attention for photoluminescence was applied to identify the FPIs. energy-hungry applications, since they would allow energy sav- It is found that, in the early growth stage, FPIs not only induce ings consequent to the gain in weight. In automotive applications, massive generation of full-core dislocations and MPs but also aluminium alloys containing Mg and Si as major additions, were attract them, forming slit-type pores at the boundaries of FPIs. chosen for an interesting combination of properties. The hard- In the intermediate stage, when FPIs stop to grow and become ening phase in these alloys has recently been identified to be the overgrown by the matrix, the pore density significantly reduces, coherent metastable phase. An understanding of the kinetics of which is attributed to their transformation into new MPs. In the the precipitation of this phase is valuable since it would allow later stage, the MP density decreases, providing evidence for their an optimization of the paint-bake process that these alloys would partial annihilation and healing. undergo as part of the manufacturing of car bodies. The following models are discussed: generation of pores at Precipitation reactions are the most frequently studied solid- the boundaries of FPIs; generation of MPs from the pores; and state reactions in metallic alloys. However, the theoretical treat- the evolution and healing of MPs. This study suggests that the ment of such reactions is surprisingly summary, to say the key point for the elimination of defects from such crystals is the least. The best-known model is the Johnson-Mehl-Avrami-Kol- suppression of FPI nucleation. mogorov, which is used to describe the above solid-state reactions. REFERENCES The purpose of this work is to study the kinetics of the precipitation of the phase in two Al-Mg-Si alloys. The kinetic param- [1] N. Ohtani, M. Katsuno, H. Tsuge, et al., J. Cryst. Growth 286 eters are determined from DSC thermograms, using a modified (2006) 55. JMAK model for solid-state precipitation. [2] M. Yu. Gutkin, A. G. Sheinerman, T. S. Argunova, et al., J. Appl. Phys. 100 (2006) 093518. Smail Hamamda, Route ain El-Bey, Constantine, 25000, Algeria [3] M. Yu. Gutkin, A. G. Sheinerman, T. S. Argunova, et al., Phys. Rev. B 76 (2007) 064117.

22 [4] M. Yu. Gutkin, A. G. Sheinerman, M. A. Smirnov, T.S. Ar- fabricated and occurred free of cracks. These results may be ap- gunova, et al. J. Appl. Phys. 106 (2009) 123515. plicable to wafer bonding of other materials inﬂuenced by high thermal stress such as a GaAs and Si. [5] T. Kuhr, E. Sanchez, M. Skowronski, et al. J. Appl. Phys. 89 (2001) 4625.

[6] Yu. A. Vodakov, A. D. Roenkov, M. G. Ramm, et al., Phys. REFERENCES Status Solidi B 202 (1997) 177. [1] H. Kanbe, M. Komatsu, M. Miyaji, Jap. J. Appl. Phys. 45 (2006) L644. Mikhail Y. Gutkin, Bolshoj 61, Vasil. Ostrov, St. Petersburg, [2] S. W.Bedell, A. Reznicek, K. Fogel, et al. , Mater. Sci. Semi- 199178, Russia cond. Proc. 9 (2006) 423. [3] T. S. Argunova, I. V.Grekhov, M. Yu. Gutkin, et al., Phys. CRACK PREVENTION IN WAFER BONDED GE/SI BY Solid State 38 (1996) 1832. GROOVED PATTERNS AT THE INTERFACE [4] T. S. Argunova, M. Yu. Gutkin, L. S. Kostina, et al. Scr. Mater. 62 (2010) 407. T.S.ARGUNOVA M.YU.GUTKIN L.S.KOSTINA I.V.GREKHOV E.I.BELYAKOVA J.H.JE [5] S. V.Bobylev, M. Yu. Gutkin, I. A. Ovid’ko, J. Phys.: Con- dens. Matter. 15 (2003) 7925. [email protected]

Thin film heterostructures composed of alternative germanium Mikhail Y. Gutkin, Bolshoj 61, Vasil. Ostrov, St. Petersburg, and silicon layers are of great importance for contemporary tech- 199178, Russia nologies in micro-, opto- and nanoelectronic devices such as pho- todetectors, solar cells, and high mobility channel field effect transistors (FETs). However, their wide use is still hampered DYNAMICS OF SNAP-THROUGH OF BI-STABLE by the problem of misfit of the crystalline lattice parameters and COMPOSITE PLATES coefficients of thermal expansion of these materials, which in- A.F.ARRIETA S.A.NEILD D.J.WAGG duce strong residual misfit and thermal stresses capable of relax- P. HAGEDORN ing through crack and dislocation generation. To cope with this problem in epitaxial growth of Ge layer onto Si substrates, they [email protected] use special SiGe buffers to accommodate the 4.2% lattice misfit between Ge and Si. Alternatively, the direct wafer bonding Multi-stable composites are a new type of composites consid- technology has been developed and incorporated into the man- ered for adaptive structures applications as this material can attain ufacture of photodiodes [1] and FETs [2]. Different from the two statically stable configurations [1]. The multi-stability prop- epitaxy, the wafer bonding has no limitations caused by thread- erty arises from internal thermal stresses induced during the man- ing dislocations, however, both the technologies are challenged ufacturing process. Most of the research carried out on this com- by high thermal stresses. In the present work we investigate ther- posite material has focused on studying and modelling the static mal stresses in wafer bonded Ge/Si and propose a patterned Si properties of bi-stable composites using low order models and fi- substrate as a mean to reduce them and prevent crack generation nite element techniques [2]. However, the operation conditions in the system. for real applications incorporating bi-stable, or in general multi- The Ge/Si structures were formed by bonding [3] of mirror stable, composites would include dynamic loads. Thus, studying polished n-Ge (111) and n-Si (111) wafers (30 mm in diameter the dynamic response of such structures is very important in the and ≈ 0.5 mm thick). Scanning electron microscopy and syn- design of these structures. Few studies have addressed the dy- chrotron radiation phase contrast imaging were employed to test namic behaviour of bi-stable composites. The dynamic response the quality of the structure and observe cracking on the wafer confined to one stable configuration has been modelled with a low interfaces. The experiments were performed at 7B2 x-ray mi- order model showing good agreement with experimental results croscopy beamline of the Pohang Light Source, Korea. [3]. A study on the oscillations after a snap-through instability We observed that in the case of smooth Si wafer, cracking was triggered by a quasi-static load of bi-stable composites was starts to occur at hGe 0.5hSi, where h is wafer thickness, presented [4], however it did not treated the response of the com- by the thermal stresses.≤ To reduce stresses in the bonded Ge/Si posite to a dynamic excitation. The dynamics of snap-through are structure, we tried to apply patterned Si substrates. However, first very important in the development of these structures as through we theoretically analyzed thermal stresses in Ge/Si bonded struc- this mechanism dominates the changes of configuration. tures in detail. The grooved patterns at the Ge/Si interface were In this paper a low order model to capture the main dynamic modeled through arrays of continuously distributed virtual edge features of the snap-through instability triggered by dynamic dislocations whose density corresponded to the thermal misfit in loads of a bi-stable composite plate with unrestricted boundaries the system and the array periodicity to that of the grooved pat- is presented. Simulations for snap-through triggered with dy- tern [4,5]. It was shown that grooves effectively reduce thermal namic oscillations are obtained for several levels of forcing. A bi- stresses, the decrease varied from 8 to 50% depending on the dis- stable plate specimen is experimentally tested, revealing complex tance to the grooves and the interface. Based on these results non-repeating oscillations for levels of forcing triggering con- and using the grooved wafer bonding technology, a Ge/Si struc- stant snap-through between stable states. Evidence of a chaotic ture with the thickness ratio hGe/hSi 0.5 was successfully response is found for the response at this forcing regime. The ≈

23 obtained results are compared to numerical simulations showing shafts with the rubber or polyurethane cover and others. Taking good agreement. In particular, the model captures the complex into account the modified Maxwell equation and the first law of oscillations showing similar evidence pointing to a chaotic re- thermodynamics the creep fracture criterion for incompressible, sponse for high levels of forcing. aged polymer materials is formulated. Financial support of the Russian Foundation for Basic Re- search (Grant N 08-08-00201) is gratefully acknowledged. REFERENCES Robert A. Arutyunyan, Universitetskii pr., 28, Faculty of Mathe- [1] Dano, M.-L., and Hyer, M. W. 1998. “Thermally-induced de- matics and Mechanics Sankt-Petersburg State University, Sankt- formation behavior of unsymmetric laminates”. International Petersburg, Petrodvoretz, 198504, Russia. Journal of Solids and Structures, 35:2101–2120. [2] Dano, M.-L., and Hyer, M. W. 2002. “Snap-through of un- Kira S. Yakimova, Institute of Problems in Mechanical Engineer- symmetric fiberreinforced composite laminates”. Interna- ing, Russian Academy of Sciences, V.O., Bolshoy pr., 61, Sankt- tional Journal of Solids and Structures, 39:175–198. Petersburg, 199178, Russia. [3] Arrieta, A. F., Neild, S. A., andWagg, D.J. 2009 “Nonlinear dynamic response and modelling of a bi-stable composite PARTICLE DYNAMICS MODEL OF THE PERCUSSIVE plate for applications to adaptive structures”. Nonlinear Dy- DRILLING PROCESS namics, 58:259–272. [4] Diaconu, C. G., Weaver, P.M., and F. Mattioni, F. 2008. I.E.ASONOV I.E.BERINSKIY A.M.KRIVTSOV “Concepts for morphing airfoil sections using bi-stable S.A.LE-ZAKHAROV laminated composite structures”. Thin-Walled Structures, [email protected] 46:689–701. This work is devoted to modelling of percussive drilling using Andres Arrieta, 4 Magdalenenstrasse , Darmstadt, 64289, Ger- particle dynamics (PD) method. many Percussive drilling has different advantages in comparison with rotary drilling: higher drilling speed, reduced weight on bit, possibility to drill through different formations with the same bit. FATIGUE AND CREEP FRACTURE OF AGING A PD model of percussive drilling process is created. Material ELASTIC VISCOUS MATERIALS SUBJECTED TO removal rate is investigated as a function of static, dynamic and CYCLIC COMPRESSION shear force applied to bit where shear force simulates rotational torque. Comparison of numerical results with analytical results R.A.ARUTYUNYAN K.S.YAKIMOVA [1] is provided. [email protected]

The problem of fatigue and creep fracture of aging elastic- REFERENCES viscous materials subjected to cyclic compression is considered. [1] Krivtsov A.M., Wiercigroch M. Penetration rate prediction Investigations of the mechanical properties of polyurethane in the for percussive drilling via dry friction model. Chaos,Solitons experiments on pressure and stress relaxation for two values of and Fractals 11 (2000) 2479-2485. rate loading were carried out. In the experiments the specimens without aging and with climate and deformation aging (more then ten years) were used. The deformation pressing curves of steel Igor E. Asonov, Polytechnicheskaya st., b.29, Saint-Petersburg, cylinders segments and the segments made of polyurethane are 195251, Russia constructed and the contact strips for the two force level are defined. The curves of cyclic creep accumulation were studied in the experiments with the rotating rollers with the polyurethane INVESTIGATION OF INFLUENCE OF NONEQUIAXIAL covering under different loading conditions. The significant fre- COMPRESSION ON GENERAL FEATURES OF quency dependence of the cyclic creep accumulation and the MECHANICAL RESPONSE OF SHEARING FAULT number of cycles to fracture is observed. The problem of thin ZONES layer made of incompressible viscous material subjected to com- S.V.ASTAFUROV E.V.SHILKO V. V. SERGEEV pression by two long parallel plates is considered. The stress di- S.G.PSAKHIE agrams for the unaged and aged material are constructed. It is shown, that on the surface and inside of the pressed layer the [email protected] [email protected] state of hydrostatic pressure is occurred. The value of this state [email protected] [email protected] is increased with the increasing of aging effect and the loading rate. Superposition of these effects is the main cause of degra- It is well known that stress state of geological media is nonuni- dation and fracture of material in the process of cyclic pressure. form and complex and one of its main features are constrained The received results can be applied to estimate the fatigue life of conditions. Degree of constraint, determined through a value different elastic viscous materials (natural and artificial rubbers, of applied stresses, greatly determines deformation mechanisms polyurethane, asphalt) and the items where these materials are and fracture regime of the medium. Significant parts of mas- used as working elements, for example, roads, tires, the pressing sifs, in particular, zones of active faults, are in very complex

24 conditions, including, as compressive and significant shear com- DEVELOPMENT OF MOVABLE CELLULAR ponents. These loading conditions can be characterized by the AUTOMATON METHOD FORMALISM FOR scheme “shear+compression”. Therefore investigation of an in- COMPUTER-AIDED SIMULATION OF fluence of the degree of constraint on the character of deforma- ELASTIC-PLASTIC RESPONSE OF tion and fracture of block-structured geological media is the ac- MONOCRYSTALLINE METAL MATERIALS. tual problem. This paper is devoted to theoretical investigation of the influence of this factor on the shear strength, deformation pa- S.V.ASTAFUROV E.V.SHILKO A.YU.SMOLIN rameters and other characteristics of response of block-structured S.G.PSAKHIE medium under constrained shear loading. The investigation was carried out on the base of computer-aided simulation by movable [email protected] [email protected] cellular automaton method. [email protected] [email protected]

To achieve the objectives of the paper 2D structural model Design of new submicrocrystalline and nanocrystalline mate- of fragment of block-structured medium was developed. In this rials is an important task of modern materials science. In this model medium was represented as an ensemble of blocks sep- connection, experimental and theoretical investigations of defor- arated by interfaces. Mechanical properties of the blocks cor- mation and fracture of such structures in which the grain size responded to granite; properties of the interfaces were imitated less than 100 microns are widely carried out for the last two by corresponding characteristics of cataclastic material. Initial decades. For understanding of mechanisms of deformation pro- stress state of the specimen was set by biaxial compression with cesses which take place on the micro- and nanoscale levels in forces in vertical and horizontal directions. After this procedure grains and subgrains single-crystal metal materials often use as the specimen was subjected to a shear deformation at a constant model objects of investigations. But it is necessary to mention speed. Influence of the degree of constraint in the direction of that the results of experiments often do not allow to draw a com- shear deformation on main features of processes of strain local- prehensive conclusion about the contribution of various factors in ization and damages formation under constrained shear loading the mechanical response of material to external loading. There- was investigated. fore, important information about the processes of deformation and fracture of single crystal materials could be obtained using It was shown that under shear loading the degree of constraint computer-aided simulation method. In this case most common is important factor determining relative contribution of differ- method in use is molecular dynamic approach, which allows to ent deformation mechanisms to integrated response of block- obtain sufficiently reliable information about the operating defor- structured media. Increasing of compressive stresses in the shear mation mechanisms. But application of molecular dynamics is direction leads to reducing of contribution of deformation mech- limited by considering of the relatively small size of crystallites anisms of low scales, leading to the accumulation of irreversible due to restrictions imposed by the capacity of modern computing deformation in the interblock interfaces. Main reason of this systems. So, in recent years are widely used models of plastic- effect is increasing of degradation degree of medium in initial ity of single crystals applied in the framework of “macroscopic” stress-strain state, which leads to rapid formation of discontinu- methods of continuum mechanics to describe the response of de- ities in most weak interfaces under shear loading. At high degrees formable solids. However, the use of continual approach also has of constraint formation of these “mesoscopic” discontinuities and several disadvantages, in particular, the complexity of modeling their association in interblock cracks becomes the dominant de- of the processes of destruction, mass transfer and mixing of the formation mechanism. Changing of the deformation mechanism masses. Therefore seems a promising development of models is manifested as change of a trend, and sometimes of a sign of the of the single crystal plasticity in the framework of methods of integral characteristics of deformation response of the medium discrete elements. This work is devoted to the development of (e.g. shear strength, value of ultimate strain, changing of linear the formalism of movable cellular automaton method for simu- dimensions specimen (this parameter is connected with dilatancy lation of elastic-plastic response of single-crystal materials. The of the medium), etc.). In particular, at high degrees of constraint rate-independent single crystal plasticity model proposed in the takes place increasing of width of fault zones fragments. This papers of L. Anand and collaborators [1] was used as the base effect is connected with decreasing of role of the low-scales de- approach. In this paper presented two-dimensional realization of formation mechanisms and their substitution by higher structural this model in the approximation of plane-strain state. It is nec- level mechanisms, which promote the involvement of dilatancy essary to mention that this model is based on the hypothesis that processes associated with slip on the interblock interfaces. plastic flow is realized by slipping along pre-defined active slip systems. The choice of this approach is due to a good agreement This work was supported by grant of the President of Rus- of obtained with its use results with experimental data for a wide sian Federation (MK-130.2010.5), RFBR grant No.09-05-00968- variety of polycrystalline materials, as well as its relatively easy a and project of program of the Presidium of RAS 16.8. adaptation to the movable cellular automaton method formalism. Developed approach allows to calculate the stress-strain state of a single crystal specimen and the evolution of its “texture” (the orientation of slip systems in deformed body) over all loading process. Some results of test simulation of deformation and fracture processes of FCC nickel specimens and their comparison with results of theoretical and experimental data obtained by other au- Sergey V. Astafurov, Akademicheskii ave., 2/4, Tomsk, 634021, thors are also presented in this paper. This work was supported Russia by grant of the President of Russian Federation (MK-130.2010.5)

25 and RFBR grant No. 09-08-00498-a. Kirill Belov, 37 Sodrugestvo st., 84, Rostov-on-Don, 344103, Russia

REFERENCES TWO-CONTINUUM MODELS OF THE DISSOLVED [1] Anand L., Kothari M. // J. Mech. Phys. Solids, 1996, V.44, HYDROGEN INFLUENCE ON STRENGTH OF No.4, pp.525–558. MATERIALS

A.BELYAEV D.INDEITSEV YU.YAKOVLEV Sergey V. Astafurov, Akademicheskii ave., 2/4, Tomsk, 634021, V. POLYANSKIY Russia [email protected] [email protected]

DISPERSION RELATIONS IN THE COUPLED The influence of dissolved hydrogen concentrations on mate- PROBLEM OF THERMOELASTICITY rial properties is very considerable. It becomes observable at the level of about a single atom per 50000 atoms of matrix of the M.BABENKOV matter. [email protected] On one hand, this influence of hydrogen can not be ignored since, in many cases, it is the only explanation of changes in ma- The dispersion curves, envelope and phase velocity curves terial properties in time and under external loading. On the other were obtained for the equations of the related problem of ther- hand, this influence is difficult to describe due to smallness of this moelasticity which includes the heat equation of hyperbolic type. factor and variety of consequences. For example, under change The asymptotic analysis of the dispersion curves was carried out. in the hydrogen concentrations by three-four times at the level of The obtained dispersion curves were compared with the ones one atom per 10000 matrix atoms the mechanical properties of found in the framework of classical thermoelasticity. the material can change from brittleness to superplasticity. An important role is played by the hydrogen stage and bond- Mikhail B. Babenkov, 29 Polytechnicheskaya str., Saint- ing energy in material. The hydrogen with large bonding energy Petersburg, 195251, Russia has low diffusion mobility and interacts with material more intensive than the diffuse hydrogen with low bonding energy. In some cases (for example inside the gas pipe wall) the pressure CONTACT OPTIMIZATION PROBLEMS WITH of the diffusely hydrogen blows up the material. However very UNCERTAINTIES often the hydrogen concentration is so small that its pressure is practically equal to zero. N.V.BANICHUK S.YU.IVANOVA E. V. MAKEEV The main hydrogen influence mechanism in that case is dif- fusive redistribution of the hydrogen in tensile stresses zones (Gorsky’s effect). The hydrogen material interaction increases In this study problems of optimal shape design of punch in- the hydrogen bonding energy in those zones and decreases local teracted with elastic medium under conditions of incomplete in- mechanical properties of the material. formation regarding the external actions are investigated. In for- These effects can be described by a two-continuum model. mulating and solving problems with incomplete data the proba- The presentation is concerned with the model itself and its most bilistic approach is applied. As a result the optimal designs of the important properties. The presence of the second continuum al- punch with circular base are obtained and presented. lows one to describe simultaneously the hydrogen embrittlement and superplastic transformations in the stress-strain diagram in the form of certain structural instability. ON THE FLOWS IN A FREE-SUSPENDED By using two-continuum model we constructed time- VISOELASTIC MICROPOLAR FLUID FILM dependence of the stress relaxation under increasing natural concentration of hydrogen. As an example we demonstrate the stress K.L.BELOV V.A.EREMEYEV distribution for a flange of the gas pipeline. Under the redistribu- [email protected] tion of hydrogen due to the tensile stresses on the pipeline surface the concentrators of these tensile stresses were shown to lead to We present the results of modeling of the slow flows in the increasing accumulation of hydrogen in the concentration zones, thin layer of a viscoelastic micropolar fluid bounded by two free appearance of hydrogen embrittlement and destruction. surfaces. The surface tension is taken into account. We used the Application of two-continuum models for describing the mul- constitutive equations of the viscoelastic micropolar fluid of dif- tiscale materials, e.g. steels with nanostructures and/or nanos- ferential type. For slow flow we apply the averaging procedure tructural films allows obtaining adequate results which describe to derivation of the two-dimensional motion equations from the SMART-influence on properties of these structures. In contrast to three-dimensional ones. The procedure bases on the through-the- the homogeneous materials, the nanocrystal materials in the pres- thickness integration technique of the 3D balance laws which are ence of hydrogen demonstrate stabilization of their properties as written in the integral form. The reduction procedure is discussed hydrogen decreases the surface energy of crystals in material and in details. The possible applications of proposed model are dis- prevent their increasing up to the microsize. cussed for description of flows in free-suspended films made of Not only the average hydrogen influence on the material prop- liquid crystals. erties can be described by the model equations. They also de-

26 scribe the hydrogen redistribution within the volume and bound- DEVELOPMENT AND STABILITY OF A CONVECTIVE ing energy levels under the mechanical and thermo mechanical MASS TRANSFER INDUCED BY LOCAL loading. HETEROGENEITY OF SURFACE-ACTIVE SUBSTANCE Summarising, we suggest a novel approach to modeling the DISTRIBUTION NEAR A FREE LIQUID SURFACE solids with account for the influence of hydrogen on the properties of free surface on monocrystals of various scales. R.V.BIRIKH A.I.MIZEV M.O.DENISOVA [email protected] alex [email protected] Alexander K. Belyaev, Bolshoy pr. V.O. 61, St.-Petersburg, [email protected] 199178, Russia The results of experimental and theoretical study of development and stability of solutal (buoyant and capillary) convection initiated by presence of local heterogeneity of surface-active sub- MULTYBODY POTENTIALS TO DETERMINE ELASTIC stance distribution near a free liquid surface are presented in the PROPERTIES OF DIAMOND AND GRAPHENE paper. It is shown from laboratory and numerical experiments that oscillatory or stationary flow can be obtained in the system I.E.BERINSKIY A.M.KRIVTSOV depending on the relative intensity of these two convective mech- [email protected] anisms. The experiments have been carried out in Hele-Shaw geom- The properties of two-atomic crystal lattices are investigated. etry that allowed us to realize interferometric methods for con- The multybody interaction potentials for covalent systems by Ter- centration distribution investigations. Trace particles technique soff and Brenner are considered. A new approach to evaluate the was used to study the structure of a convective flow. In addi- elastic characteristics of the crystall lattices using the parameters tion we can measure a flow velocity in a certain point of the liq- of such potentials is proposed. A linear elastic deformation of uid with the help of miniature anemometer. A small dissolving crystall lattice is considered. A general form of the multybody drop of an aliphatic alcohol held on a tip of a thin needle and interaction is obtained using this consideration. It is shown that placed at a certain depth under a free surface was used to create Tersoff and Brenner potentials could be adjusted to such form as the local surface-active substance heterogeneity. Triply deion- a result of the series expansion. The elastic properties of diamond ized water was used as a surrounding liquid. Use of a series and graphene are found using the parameters of these potentials. of aliphatic alcohols from n-butanol to n-hexanol allowed us to A conclusion about the adequacy of these potentials to determine study the system in a wide range of Marangoni and Grasshoff the elastic properties is made. numbers. We found out that depending on concentration dynamic Bond number, which shows the relation of buoyant and solutocapillary forces and which was introduced by us on the analogy Igor E. Berinskiy, Bolshoy pr. V. O., 61, St. Petersburg, 199178, of thermal case, one can obtain an oscillatory or stationary flow. Russia The oscillation regime is observed under relatively small values of dynamic Bond number. A period of the oscillations strongly depends on the ratio of the drop size and its position relative to the free surface. With increase of the dynamic Bond number the CALCULATION OF THE SPATIAL STRUCTURES WITH period decreases and under a certain value the oscillatory regime A GLANCE OF RHEOLOGICAL AND changes to the stationary one. HETEROGENEOUS BEHAVIORS OF THE MATERIALS On the basis of experimental results the theoretical model is E.BEZOYAN proposed. We present a simple mathematical model, which shows that a competition between the solutal gravitational convection [email protected] and the Marangoni convection at large values of the Schmidt number can lead to the onset of the oscillatory mode of convec- The calculation apparatus was built for exposure of the influ- tion. The finite difference method has been used to simulate nu- ence of rheological behaviors of material on the tense-deformed merically the development of the free convection over a localized condition of heterogeneous spatial structures, like plates and source of light surfactant placed at the bottom of a rectangular shells. horizontally elongated cavity. Two types of surface-active sub- Topicality of these tasks conditioned by the majority of the stance source, described with step-function distribution or with real structures (reinforced concrete shell-like plates, tunnel lin- normal one, were studied: It is shown that the last way describes ings, etc.) during calculation of which it is necessary to count the the experimental results better. It has been found that the oscil- influence of the creep, that have different deformation behaviors lation period is strongly affected by the Grashof number, with in different points. the increase of which the oscillation period decreases. The os- It was provided with calculation samples of the elongated cillation period also depends on the depth of the source submer- single-layered and triple-layered shells and plates, during pro- gence. At low submergence depth one can observe a transition to longed action of the loadings. double-period oscillations. The calculated distributions of surfactant concentration at different phases of oscillations qualitatively agree with the results of the experiment. Eduard Bezoyan, 24 Acharyan Str. First Drive, Yerevan, 0044, The work was supported by the RFBR projects 08-01-00503 Armenia and by the Integration project of SB, UB and F-EB of RAS

27 (116/09-C-1-1005). breathers have been obtained. The special cases of collision processes have been discussed: (i) the interaction of a kink and a Maria O. Denisova, Academician Korolev str., 1, Perm, 614013, linear wave; (ii) the interaction of discrete breather and a linear Russia waves; (iii) the collision of two identical breathers at the centre of mass framework. It has been shown that found expressions can be used for the quantitative description of the discrete shock ACCOMODATION OF GRAIN BOUNDARY SLIDING BY wave and breather reflection from the free and fixed boundaries GRAIN BOUNDARY MIGRATION IN DEFORMED of the crystal. The reflection from a free boundary is described NANOMATERIALS by the two-kink and two-breather counterpart solutions while the reflection from a fixed boundary is described by the kink-antikink S.V.BOBYLEV N. F. MOROZOV I.A.OVID’KO and “breather-antibreather” solutions, respectively. [email protected] [email protected]

New mechanism of grain boundary sliding accomodation in REFERENCES deformed nanomaterials, namely stress-induced grain boundary [1] R. Hirota, J. Phys. Soc. Jap., V.35, 289, 1973. migration, is suggested and theoretically analyzed. Grain boundary sliding and migration provide mutual accomodation that leads [2] M. M. Bogdan, G. A. Maugin, Proc. Estonian Acad. Sci. to the intensiﬁcation of both processes and increases nanomate- Phys. Math., V.52, 76, 2003. rial’s plasticity. Conditions for simultaneous realization of both processes in nanoscale grains are calculated. Denis Laptev, B. I. Verkin Institute for Low Temperature Physics and Engineering NAS of Ukraine 47 Lenin Ave., 61103 Kharkov, Sergey V. Bobylev, Bolshoj 61, Vas. Ostrov, St. Petersburg, Ukraine 199178, Russia

ON THE MAGNETIC SUSCEPTIBILITY OF FRACTAL EXACT DESCRIPTION OF INTERACTION AND FERRITE DIELECTRIC REFLECTION OF DISCRETE BREATHERS IN ANHARMONIC LATTICE S.B.BOGDANOVA S.O.GLADKOV [email protected] [email protected] M.M.BOGDAN D.V.LAPTEV [email protected] The dependence of longitudinal magnetic susceptibility of fractal ferrite dielectrics on frequency field, temperature, expo- The dynamics of strongly excited low-dimensional crystals is nent and other characteristics of structure were calculated. The described by nonlinear equations, but a few of them can be solved unconventional behavior of longitudinal susceptibility as function exactly using the soliton theory. R. Hirota has proposed the exact of temperature was proved. It was demonstrated, that anisotropy integrable equation for description of the oscillations of the an- of magnon spectrum has a very strong effect on all physical dissi- harmonic 1D crystal with the tangent interaction force between pative coefficient such as magnetic structure due to exponent. The nearest atoms and has found its multi-soliton solution [1]. A sin- calculations are based on using of quasiclassical kinetic equation gle soliton (kink) in the Hirota nonlinear lattice corresponds to for distribution function of magnons. the shock wave. Recently the exact discrete breather solutions for the 1D lattice have been found [2]. The discrete breather is Sofja B. Bogdanova, Radio str, 10a, Moscow, 105005, Russia the self-localized oscillation which describes the intrinsic localized mode in a highly-localized limit and a dynamical bound state of the soliton-antisoliton pair in the case of the weak localization. MICROCRYSTAL MATERIAL DYNAMICS AT The existence of the multi-soliton solution and the explicit ex- IRRADIATION BY POWERFUL BEAMS OF CHARGED pressions for the moving discrete breathers allows to describe ex- PARTICLES actly all interaction processes of the excitations. It is known that I.N.BORODIN V.S.KRASNIKOV A.E.MAYER in integrable equations solitons keep their individuality during a motion, and their interaction results in the center-of-mass shifts [email protected] in the case of kinks and additionally to the phase shifts of time oscillations in the case of breathers. The interaction of the arbi- Plasticity is an important property of metals influencing on trary number of solitons can be described by the sequence of their their response to the shock-wave loading, powerful electron and pair collisions. ion irradiation. In coarse-grained materials plastic deformations In the present work the pair interaction processes of the are defined by kinetics of dislocations. In micro- and nanocrys- breathers with each other as well as with shock waves and lin- talline metals the grain boundaries volume fraction can achieve ear waves (phonons) are investigated. By means of the explicit 50 expressions for the three- and four-soliton solutions obtained for This work is supported by RFBR grant project number 09-08- the first time we completely analyse the collision process of 00521. the kink and breather and a scattering of two moving breathers. The analytical expressions for the scattering data, the center-of- Ilia N. Borodin, Geroev Tankograda, Chelyabinsk, 454071, Rus- mass shifts of solitons and phase shifts of internal oscillations of sia

28 [1]. A distinguishing feature of the system is that the layer rup- INVESTIGATION OF SOME RAPIDLY SOLIDIFIED ture in the absence of any external force has the shape of a circle, inside which the fluid of the lower layer comes into contact with A.BOUBERTAKH K.TALEB C.SERRAR gas. Due to the absence of contact with solid surfaces the rup- S.HAMAMDA ture is highly sensitive to the external effects and therefore can [email protected] be deformed and move even under the action of arbitrarily small forces. Thus, in the case when the upper layer is formed by a fer- Melt-spun ribbons and atomised powders, of Al-5wt.%Cr- rofluid, an exposure of the system to the external magnetic fields 2wt.%Zr alloys, have been studied in as-cast conditions and after can cause a displacement of the rupture and a change of its shape. heat treatments using optical microscopy, x-ray diffraction ant In this case the degree of changes is proportional to the field in- transmission electron microscopy. Optical microscopy observa- tensity. tions revealed the presence of two distinct zones in both materi- In the experiment with a two-fluid system a layer of the als. A variety of solidification microstructures has been observed ferrofluid was applied on a substrate made of perfluorooctane by transmission electron microscopy and this includes the obser- C8F18, which had higher density and smaller surface tension vation of chromium-containing precipitates which are spherulite- compared to the basic fluid [2]. The fluid system was placed in a like in shape with an order of magnitude of about one micrometer. cuvette in the form of Petri dish. Selected area diffraction patterns of these spherulites exhibited As one could expect, the use of the magnetic field applied par- five-fold symmetry. The obtained quasi-crystals have been ob- allel to the layer surface caused redistribution of the ferrofluid served even in heat treated samples at high temperatures, which along the field direction. This led to a displacement of the rup- suggests that this type of particles is thermally stable. EDX ana- ture with respect to the cuvette center which was accompanied by lytical data of these particles revealed that the average composti- the change of its shape of a circle to that of an ellipse. With a fur- tion of the spherulite-like particles appeared to contain 8.1at.%Cr ther increase of the field intensity the rupture rapidly approached and 0.26 at.%Zr with aluminium balance. the cuvette wall and took the form of a trapezoid provided that the initial diameter of the rupture was larger than the cuvette ra- Abdelhamid Boubertakh, Ain El-Bey, Constantine, 25000, Alge- dius. In the opposite case, with increase of the field intensity the ria rupture took the form of a “drop” and then closed. It was found that the non-uniform magnetic field perpendic- NUMERICAL IMPLEMENTATION OF THE ular to the layer surface could have a twofold effect: it could INCUBATION TIME FRACTURE CRITERION deform the stable rupture by increasing its diameter or provoke rupture formation in the case of layer integrity. In this situation, V. BRATOV switching on the magnetic field caused first redistribution of the ferrofluid in the upper layer. Then, as the filed intensity increased [email protected] to the critical value, the layer deformation spread deep into the layer until it reached the surface of the liquid substrate. At this The paper is discussing problems connected with embedment point there occurred a threshold rupture of the layer in the form of of the incubation time criterion for brittle fracture into finite ele- a circle, the diameter of which is determined by the initial layer ment computational schemes. Incubation time fracture criterion thickness, the ferrofluid density and the difference in the surface is reviewed, practical questions of its numerical implementation tension of the fluids. It should be noted that switching on the are extensively discussed. Several examples of how the incuba- uniform magnetic field directed perpendicular to the continuous tion time fracture criterion can be used as fracture condition in fi- layer surface leads to a transformation of the layer into an ordered nite element computations are given. The examples include sim- system of drops, which then changes into a system of cones sub- ulations of dynamic crack propagation and arrest, impact crater mersed into the fluid of the substrate. During our investigation formation (i.e. fracture in initially intact media), spall fracture we determined the dependence of different characteristics of rup- in plates, propagation of cracks in pipelines. Applicability of the tures and drops on the field intensity and the value of ferrofluid approach to model initiation, development and arrest of dynamic magnetization. fracture is claimed. The results of our studies support the view that the magnetic Vladimir Bratov, IPME RAS, Bolshoy pr. V.O., 61, 199178, St.- field depending on its orientation with respect to the surface of Petersburg, Russia the ferrofluid layer can cause both a layer rupture and its closure. The examined two-layers system represents a new class of problems on statics and dynamics of fluids and can have considerable DYNAMICS OF THE FERROFLUID LAYER WITH A promise for technological applications. STABLE RUPTURE OF THE SURFACE UNDER THE The work was supported by the grants of the research pro- ACTION OF EXTERNAL MAGNETIC FIELD gram of the RAS No 09-T-1-1005 and the RFBR projects No 10-01-00337 and 10-02-96022 and grant of UB RAS for young C.A.BUSHUEVA K.G.KOSTAREV A.V.LEBEDEV researchers. [email protected] [email protected] [email protected] REFERENCES We investigate the behavior of a two-layer liquid system with a stable rupture of the upper layer caused by the action of pondero- [1] Yu. Yu. Stoylov. Oscillation of evaporating liquids and ispala- motive forces. Such a system has only recently been explored tor paradoxes // Phys.-Usp. 2000. V. 170, No 1. P. 41-56.

29 [2] Yu. K. Bratukhin, A. L. Zuev, K. G. Kostarev, A. V.Shmyrov. the results of terrestrial modeling of surfactant diffusion into the Stability of stationary rupture of the liquid layer on an im- drop. miscible ﬂuid surface // Fluid Dynamics. 2009. V.41, No3. P. The work was supported by the RFBR project No 10-01-96028 514-520 and the program of the Department of Power Engineering, Me- chanical Engineering, Mechanics and Control Processes of RAS No 09-T-1-1005. Christina A. Bushueva, Academician Korolev str., 1, Perm, 614013, Russia

REFERENCES TERRESTRIAL MODELING OF SURFACTANT DIFFUSION THROUGH THE DROP-SURROUNDING [1] K. G. Kostarev, V.L. Levtov, V.V.Romanov, A. V.Shmyrov, FLUID INTERFACE A. Viviani. Experimental study of surfactant transfer in fluid systems in microgravity conditions // Acta Astronautica, C.A.BUSHUEVA M.O.DENISOVA 2010, Vol. 66, Is. 3-4, pp. 427-433. K.G.KOSTAREV [email protected] [email protected] [email protected] Christina A. Bushueva, Academician Korolev str., 1, Perm, 614013, Russia Considerable progress in the development of the space materials science is impossible without understanding the specific behavior of many of the known physical-chemical mechanisms in KINETIC MODEL OF WATER CLUSTER FORMATION microgravity conditions. However the possibilities of conduct- AND GROWTH IN RAREFIED GAS FLOWS ing purely scientific experiments on board the space vehicle are strongly restricted due to high cost of delivery of the goods to the N.Y.BYKOV D.V.LESHCHEV space station, low power of the solar batteries, etc. This problem [email protected] [email protected] can be partially solved by attempting a terrestrial modeling of the examined processes under conditions of maximum suppression Water clusters formation and growth phenomena accompany of the effects caused by the gravitational force. Regarding hydro- many processes in the nature and are of great interest for large dynamics and heat/mass transfer phenomenon one can refer to number of applications in science and technology. In particu- such effects the gravitational convection, the intensity of which is lar, water condensation takes place in the inner rarefied atmo- determined by the characteristic vertical dimension of the cavity. spheres of comets under expansion of evaporated from nucleus In this paper we discuss the results of terrestrial modeling water molecules in vacuum. The one of most adequate methods of mass transfer caused by diffusion of a surfactant through the of nucleation modeling in rarefied gas flows is the direct simula- fluid-fluid interface. In the examined two-fluid system one of the tion Monte Carlo (DSMC) technique. The method requires de- fluids forms a thin layer of 1 mm thick and the other is shaped as velopment of cluster formation and growth models which should a vertical cylindrical drop with a free lateral boundary, which is describes in details kinetics of condensation process. surrounded by the first fluid. Visualization of the concentration The presented kinetic model describes water dimer formation fields showed that surfactant diffusion through the lateral drop by both triple and binary collisions of water molecules. Fur- surface is accompanied by further development of the capillary ther cluster growth is supposed by molecule-cluster and cluster- effects and formation of the gravitational convection. The main cluster collisions. The decay of water cluster is considered as reason for the onset of the gravitational convection is large char- one monomer evaporation from the cluster surface. The key fea- acteristic times of surfactant diffusion, which are responsible for ture of the presented kinetic model is in description of association generation of density gradients at both sides of the interface. In rates. The probabilities of association under mutual particles col- the case of surfactant diffusion from the drop, co-existence of lision were obtained with the use of data on corresponding reac- two competitive mechanisms of motion leads to development of tion rates. The comparison with values of probabilities predicted oscillatory flow modes and in the case of diffusion into the drop by molecular dynamics simulation was fulfilled. The model takes - to suppression of the capillary motion. However, in the latter into account rotation and vibration degrees of freedom and cor- case due to creation of the non-uniform distribution of the surfac- responding internal energies. The energy exchange under elastic tant along the lateral surface of the drop the capillary forces may and inelastic collisions is described by Larsen-Borgnakke model. exert much stronger effect than the gravitational forces. In such The frequency of monomers evaporation from the water cluster situations formation of a large-scale capillary flow occurs around is defined by RRK theory. the drop in the plane of the layer. Development of the capillary The kinetic model of water cluster formation was used for motion in the orthogonal direction to gravity allows us to suppose modeling of 1D rarefied flow of water molecules from spheri- that a similar flow can exist in weightlessness. cal surface by DSMC method. The simulations were carried out This hypothesis is built upon a comparison between terres- for the range of Knudsen numbers. The peculiarities of the model trial modeling of surfactant diffusion from a drop and the related application for the simulations of rarefied flows with water nucle- space experiment. From the results obtained it may be concluded ation processes were analyzed. that terrestrial modeling, despite the existence of the gravitational flow, can give proper estimations of the characteristic times of capillary motion and its structure inside the drop. We also give Nikolay Y. Bykov, Polytechnicheskaya 29, Saint-Petersburg, interpretation of some other results of comparison and discuss 195251, Russia

30 expansion of pressure in terms of full system of harmonical func- BOUNDS FOR THE OVERALL PROPERTIES OF tions. The structure of this system depends on the topology of INELASTIC COMPOSITES the region. The pressure is the projection of generalized function with support on the boundary on the subspace of harmonical A.CARINI functions. Using the pressure distribution, we calculate the ve- [email protected] locity field on the boundary. We investigate the evolution of free surface for some initial We consider composites with inelastic constitutive laws for shapes of surface. The behavior of large perturbation of free some phases. Of course, for this kind of composites it is impos- boundary like hole depends on relationship between depth and sible to produce bounds for the overall properties, based on the width of the perturbation. The wide and shallow hole relaxed to classical energy principles. The bounds produced in the proposed the flat surface. In opposite case, the narrow and deep hole, the approach are based on the transformation of the inelastic consti- edges of hole are closed i in the course of time. As result, the tutive laws, characterized by the lack of symmetry of the relevant bubble near the surface is formed. operators, into constitutive laws characterized by a potential operator. We show that in this way it is possible to generalize the well The strict approximations for evolution of bubbles system known bounds of Reuss and Voigt to the inelastic composites. A were obtained. We calculated the upper bound of bubbles col- hint on the generalization of the Hashin-Shtrikman bounds is also lapse time. The influence of pressure inside of bubbles was in- given. Some applications to viscoelastic composites are shown. vestigated.

Angelo Carini, Via Branze, 43, Brescia, 25123, Italy Sergey A. Chivilikhin, Kronverkskii 49, St. Petersburg, 197101, Russia RELAY CONTROL IN NOISY VIBRATION SYSTEMS

M.CHERKAY S.VAKULENKO

[email protected] [email protected] THE STRESS-STRAIN STATE OF POROUS BURNER UNDER VARIOUS REGIMES OF GAS FILTRATION D.A. Tomchin and A.L. Fradkov proposed and investigated BURNING new control methods for some systems modeling rotor electro- motor on a platform. This sophisticated relay feedback allows YU.CHUMAKOV to avoid resonance effects in system functioning, the important [email protected] advantage of this method is the energy minimization. The need energy is defined by a feedback coefficient M. We have studied rotor systems where the platform is under The stress-strain state of hollow porous cylindrical burner a noise. The noise is simulated by a time dicsrete Ornstein- based on conversation energy releasing from lean mixture com- Uhlenbeck process. This process is defined by two parameters, bustion of combustible gas with air is investigated. The gas burn- memory parameter and noise amplitude. ing regimes is studied for analyzing of different variant of tensor To study problem, we have used numerical simulations by stress and strains components distribution in porous two-layer MATLAB 2009. We applied a slightly modified relay feedback burner by varying of conditions at the entrance to the porous control. body and that of heat exchange with the gas phase surrounding The results are as follows. Even for sufficiently large noise the burner and with the heat exchanger. The mechanicals stresses levels the relay control allows to avoid resonances. The feedback and strains caused by inhomogeneity of thermal field have been coefficient is an increasing function of the noise amplitude and estimated from thermoelasticity solution problem of equilibrium decreasing function of the memory parameter. hollow coaxial cylinder free from influence of external forces. The noise influence is negative: to stabilize the system func- Analysis of analytical solution for extreme cases of model al- tioning under the noise, we must use a larger feedback coefficient. lows determining the process-dependent parameters are satisfiable from practice viewpoint. The optimal variant of burner for practice is that of internal layer made from ceramic and exter- Mikhail V. Cherkay, V.O., Bolshoj PR.,61, St. Petersburg, nal layer made from intermetallic with equal layer thickness and 199178, Russia pore size is shown by detailed numerical investigation. This construction results in the highest radiation heat release for heating under condition the lowest stresses during operation (to compare EVOLUTION OF FREE SURFACE OF HIGH VISCOUS with other considered cases of thickness and position layers and LIQUID DUE TO CAPILLARY FORCES materials porosity). The regime of gas burning and character that S.A.CHIVILIKHIN of influence on stress-strain state is shown to depend on ratio of layers size and ratio of internal and external radiuses of burner. [email protected]

We are describing the ﬂow of incompressible high viscous liquid in Stokes approximation due to capillary forces. The Stokes Yurii A. Chumakov, Akademicheskii 2/4, Tomsk, 634021, Russia equations were solved. The method of calculating based on the

31 mmax nmax INFLUENCE OF FIBER WINDING ANGLE ON w = Bmnsin(λmα nθ)sin(ωmnt), − m=1 n=1 INTRINSIC FREQUENCIES OF FREE VIBRATION OF X X CYLINDRICAL SHELL UNDER SYMMETRY where u, v and w - displacements of a median surface of a shell BOUNDARY CONDITIONS to a direction of axes of coordinates, θ - angular coordinate in a district direction, α - coordinate in a direction forming, ex- V. CHZHVAI S.M.BOSIAKOV mπR pressed in radius shares, λm = l , R - radius of a median [email protected] surface of a shell, l - length of shell, m, n - wave formation parameters, ωmn - intrinsic frequencies, Amn, Bmn and Cmn - any constants. The characteristic equation relatively free vibration in- Definition of intrinsic frequency and shape of small vibra- trinsic frequency of shell was obtained. Solution of equation was tions is necessary element in research of shell dynamics, fre- done for different approximations. Expressions for first frequen- quencies from lower spectrum is more interesting for applica- cies of free vibrations depend on elastic constants, geometrical tions. Using polymeric material, particularly glass-fiber materi- shape of shell and fiber winding angle were obtained. The nu- als, in engineering make necessary consideration of elastic prop- merical calculations of intrinsic frequencies by the example of erty anisotropy when lower frequency vibrations of shells are re- glass-fiber material were realized. searched. The present work shows the results of definition of free vibration intrinsic frequencies of cylindrical shells from oriented Sergey M. Bosiakov, Nezavisimosti avenue, 4,, Minsk, 220030, glass - fiber material with boundary conditions for shell end cor- Belarus responding to the (1110) group by P. E. Tovstik classification. It is assumed basic elastic directions constitute ϕ angle with coordinate lines. Then Hooke’s law can be written in the following NUMERICAL ANALYSIS OF THRUST ON way: STRUCTURES FOR ACTIVE DEFENSE AGAINST σ11 = B1111ε11 + B1122ε22 + 2B2111 ε12, AVALANCHES

σ22 = B1122ε11 + B2222ε22 + 2B1222 ε12, S.CRESSERI F. GENNA σ12 = B2111ε11 + B1222ε22 + 2B1212 ε12. [email protected] Here εij - deformations of shell middle surface, i, j = 1, 2. Elas- [email protected] tic constants depends on fiber winding angle ϕ (orientation angle of fiber relative to glass fabric base) for , , , B1111 B2222 B1122 An elastic-viscoplastic constitutive model for the description B , B and B can be written in the following way: 1212 2111 1222 of the mechanical behavior of dry metamorphosed snow is pre- 4 4 sented. This model has a purely phenomenological basis, and B1111 = b1111cos (ϕ)+ b2222sin (ϕ)+ is developed with the aim of simulating situations with practi- 1 +(b + b )sin2(2ϕ), cal engineering interest, such as the stability of mountain slopes 1212 2 1122 4 4 over long periods of time, or the effects of snow interacting with B2222 = b1111sin (ϕ)+ b2222cos (ϕ)+ structures or other products of human activities. Here, we test the 1 2 performance of this model with reference to the analysis of the +(b + b )sin (2ϕ), 1212 2 1122 thrust exerted by snowpacks on mountain slopes against works 2 2 B1122 =(b1111 + b2222 2b1212)sin (ϕ)cos (ϕ)+ for defense against avalanches. First we validate the model by − comparison with published results; next, we study some practical + b (sin4(ϕ)+ cos4(ϕ)), 1122 application case, comparing the results with those provided by 2 2 B1212 =(b1111 + b2222 2b1122)sin (ϕ)cos (ϕ)+ the design specifications of the Swiss Technical Guidelines. Itis − 2 found that, in some cases, the thrust values computed according + b1212cos (2ϕ) to the Guidelines are smaller than those predicted by the numeri- 1 2 2 B1222 = (b2222cos (ϕ) b1111sin (ϕ))sin(2ϕ) cal analyses, which raises some questions about the safety margin 2 − − involved by the currently holding legislation. 1 (b + 2b )sin(4ϕ) 4 1122 1212 − Francesco Genna, Via Branze 43, Brescia, 25123, Italy 1 2 2 B2111 = (b2222sin (ϕ) b1111cos (ϕ))sin(2ϕ) 2 − − 1 (b1122 + 2b1212)sin(4ϕ) MECHANICAL STRENGTH OF THE CONCRETE − 4 DETERMINED BY THE POROSITY OBTAINED BY where b1111, b2222, b1122, b1212 - basic elastic modules of or- X-RAY MICROTOMOGRAPHY thotropic material. Solution of system equation of motion for cylindrical shell with boundary conditions from (1110) group of J. T. DE ASSIS G.CARVALHO R.EINSFELD P. E. Tovstik classification can be written in the following way: J.R.C.PESSOA R.L.B.BREDER I.LIMA

mmax nmax [email protected]

u = Amncos(λmα nθ)sin(ωmnt), − m=1 n=1 The mechanical strength is one of the main properties of the X X concrete. This resistance is directly connected to the quality of mmax nmax the hydrated cement paste (matrix) and the connection between v = Bmncos(λmα nθ)sin(ωmnt), − this matrix and the aggregates which are presented in the mix. m=1 n=1 X X

32 In this context, the matrix porosity becomes an important param- minum alloys on the base of computer modeling of macroscopic eter and it is possible to evaluate the mechanical strength. The experiments for one-axis tension – compression of usual cylin- investigation of empiric formulas that relate the porosity to the drical specimens. Known from literature deformation diagrams mechanical strength and the automation of non-destructive meth- found for different loading rates and different temperatures are ods which determines this porosity is the objective of recently used as experimental information. Known Orowan’s relation per- researches. The main goal of this research is to calculate the forms as interlink between dislocation motion characteristics and porosity value of the hardened concrete from x-ray microtomog- macroscopic approach. The problem of kinetic parameters calcu- raphy images and relate it trough empiric formulas available in lation is examined by two different formulations: using the model the literature with the mechanical strength of the material. These of elastic plastic medium and on the base of model of nonlinear procedures are performed by using techniques of pixel process- Maxwell medium. ing and mathematic mor phology. The microtomography images The procedure of kinetic parameters determination is based on of the concrete samples were obtained from a Skyscan 1172 sys- the reverse problem solution and leads to the functional mini- tem with a grayscale of 256 levels, adjusted in contrast, binarized mization for quadratic deviation of experimental and calculating and ﬁltered. In the last step, the morphological operations of ero- diagrams. Each step of the iteration procedure starts from numer- sion and dilatation with circular structuring element were used in ical tension - compression experiment for uniform rood and come order to remove isolated points, interpreted as noise. In the pro- to calculation deformation curve. This curve is compared by cessed images the empty spaces obtained were counted in order mean-square norm with corresponding curve found from physical to determine the matrix porosity. experiment. Then in compliance with certain strategy, the choice of new values of kinetic parameters is carried out to assert the best Joaquim T. de Assis, Rua Alberto Rangel sn, Nova Friburgo, coincidence of calculating and experimental diagrams. Described 28630-050, Brazil process has been repeated to receiving the best mean-square approximation. The globallness of found extreme has been controlled by choice of different initial approximations for kinetic APPLICATION OF X-RAY TENSOMETRY FOR STUDY parameters. OF SUPERFICIAL STRESSES WITH STRONG GRADIENT The comparison of numerical values of kinetic parameters with their theoretical evaluation shows that they can differs from J. T. DE ASSIS V. MONIN S. MARRERO IGLESIAS above by several digits. At the same time the calculating diagrams describe well the experimental curves. The possible causes [email protected] for such disagreements are discussed in this paper.

Study of superficial stress gradients arising after different Valerii Demidov, Lenin’s str., 30, Tomsk, 634034, Russia modern treatments of materials like material modifications by ion beam technologies or surface treatments by laser is very important problem both for theoretical and experimental mechanics ARISING OF SOLUTOCAPILLARY CONVECTION IN and physics. X-ray diffraction method is widely used method WATER-ALCOHOL SYSTEMS for stress measurements because of its accuracy, facility and non- destructivity. But there are some difficulties with using of X-ray M.O.DENISOVA K.G.KOSTAREV tensometry for study of stresses with strong gradient related with incompleteness of theoretical and experimental analysis of this [email protected] [email protected] problem. Computer simulation of diffraction profile formed by superficial layers subjected with stress gradient gives possibility A threshold nature of the evolution of capillary flows was ex- to resolve the problem of determination of stresses characterized plained quite recently owing to application of the advanced visu- with strong gradient and allows restoring stress distribution in su- alization techniques. This effect frequently observed in experi- perficial layers of material. The objective of the presented paper ments contradicts a traditional view on the initiation of motion in is to develop the methodology of determination X-ray of stress Newtonian fluids. It was found that formation of the surface ten- gradient parameters and application of this methodology for mea- sion gradient capable of initiating the Marangoni convection was surements of mechanical stresses with strong gradient. closely related to the layer of uncontrolled impurities adsorbed on the free surface of the fluid [1]. The structure of this layer can fail Vladimir Monin, Rua Alberto Rangel, Nova Friburgo - RJ, CEP only at sufficiently high shear stresses, which, in the solutal case, 28630-050, Brazil are obtained by creating the surfactant concentration difference of several weight percents. In this paper we discuss the results of experimental investiga- CALCULATION OF THE MICRO PARAMETERS OF tion of the solutal Marangoni convection developed on the free DISLOCATION KINETICS OF PLASTIC surface of water due to a local injection of aqueous solutions of a DEFORMATION FOR ALUMINUM ALLOYS USING surfactant. For surfactants we used monatomic alcohols forming THE RESULTS OF COMPUTER MODELING homologous series. Such a choice allowed us to determine the dependence of the concentration threshold on the physical-chemical V.N.DEMIDOV properties of the alcohols including their surface activity. It was vn [email protected] found that the value of the threshold decreases with an increase in the length of the surfactant molecule and characteristic dimen- In this paper the problem on the micro structure parameters sions of the free surface. The dependence of the flow structure of dislocation kinetics of plastic deformation is studied for alu- on the degree of alcohol solubility was analyzed. The obtained

33 relationships allowed us to obtain estimates of the threshold tem- needed separation of the problem to two successive stages be- perature gradients in the case of thermocapillary convection. comes available when using the specific feature of the dual reci- The work was supported by the project of RFBR No 09-01- procity method (DRM) on the first stage: a solution of a diffusion 00484; the joint project of SB, UB and F-E B of RAS No 116/ problem is represented as the sum of (i) a “quasi-steady” part sat- 09-C-1-1005 and grant of UB RAS for young researchers. isfying the Laplace equation and accounting for the complicating factors (blocky structure, discontinuities and singular points), and (ii) a linear combination of smooth radial basis functions (RBF) REFERENCES with time-dependent coefficients. For the “quasi-steady” part, the conventional BEM of the steady thermo-(poro-) elasticity is ap- [1] M. O. Denisova and K. G. Kostarev. Development of capil- plicable. For the linear combination of RBF, a particular solu- lary motion at the free surface of a fluid// Proc. 7th World tion of the equations for displacements and stresses is given by Conference on Experimental Heat Transfer, Fluid Mechan- an analogous linear combination of particular solutions, corre- ics and Thermodynamics (ExHFT-7) Krakow, Poland, 2009. sponding to each of the chosen RBF. We present these standard In Electronic Conference Proceedings CD FM-30, P. 1159- particular solutions in a simple analytical form for the Gaussian 1164. RBF. Finally, the displacements, stresses and stress intensity factors are found by using the matrix of a conventional BEM, while the vector of the r. h. s is the sum of easily evaluated vectors, Maria O. Denisova, Academician Korolev str., 1, Perm, 614013, corresponding to the parts (i) and (ii) of the diffusion solution. Russia The method is obviously applicable both to 3D and 2D problems. Its efficiency and accuracy are illustrated by numerical results for problems concerning with thermal shock in plane regions with THE INFLUENCE OF COPPER ON TRIBOLOGICAL cracks. The results are obtained by using the code of the com- CHARACTERISTICS OF AUTOMOTIVE PAD-DISC plex variable DRM on the first stage and the complex variable INTERFACE. RESULTS OF NANO-SCALE MODELING. hypersingular BEM on the second stage.

A.I.DMITRIEV W. OESTERLE H.KLOSS Anastasia Dobroskok, Box 30001 MSC 3450, Las Cruces, NM, [email protected] [email protected] 88001, USA [email protected]

In spite of many fundamental studies have been conducted to DEM STUDY OF THE BREAKAGE PROCESS IN A understand the processes taking place in brake system this still FLUIDISED BED APPARATUS remains a challenging area of research. Modern pad materials are polymer matrix composites which usually contain more than 10 M.DOSTA S.ANTONYUK L.FRIES ingredients, such as steel fibres, brass and copper chips, as well S.HEINRICH as various fillers, abrasives and solid lubricants. The main aim of [email protected] [email protected] the paper is to show by modelling how nanocrystalline copper at [email protected] the surfaces of brake pads behave during tribologiocal stressing in [email protected] comparison to primary contact sites provided by steel fibers. The boundary conditions of the model were adjusted to experimen- During agglomeration in a fluidised bed apparatus the parti- tal observations obtained by Transmission Electron Microscopy cles and agglomerates are mechanically stressed due to particle- (TEM). Theoretical investigation of copper or brass as a pad in- particle and particle-wall impacts, which leads to breakage of the gredient was carried out on the basis of modeling by the method particles. In this contribution the simulation of the particle break- of movable cellular automata (MCA). age process during fluidised bed agglomeration is presented. For this purpose the three-way coupling of different simulation tools Andrey I. Dmitriev, Akademicheski 2/4, Tomsk, 634021, Russia has been used. By using the Discrete-Element-Method (DEM), the mechanical behaviour of solid particles in a fluidised bed apparatus was described, where the contact forces in each impact of TWO-STAGE BEM FOR EFFICIENT SOLVING particles were calculated. TRANSIENT THERMO-(PORO-) ELASTIC PROBLEMS The actual particle positions obtained from the DEM were transmitted after each synchronization step to a CFD simulation A.DOBROSKOK A.LINKOV system, which describes the complex gas dynamics in the flu- [email protected] idised bed. With help of the CFD simulation the drag forces on [email protected] the particles and the fluid pressure field were calculated. The information about the particle stress state calculated in the A two-stage method to efficiently solve uncoupled thermo- first and second tool were used for the description of the breakage (poro-) elastic problems for blocky systems with pores, inclu- process of particles on impact. When the energy based breakage sions and open surfaces of discontinuity (cracks, thin inclusions) criteria during impact of particles were fulfilled, the old particle is suggested. It is based on the fact that the equations for dis- was replaced by a set of a new particles with the same total mass, placements and stresses to be found on the second stage, do not i.e. fragments were generated in the apparatus. contain the time explicitly, while the temperature (pressure) may The new simulation concept allows taking into account the be found independently on the first stage. We show that the micro-properties of particles and apparatus geometry in order

34 to get kinetic parameters which afterwards will be used during practical implementation of the system. Examples of the software macroscopic simulations the process. For example the breakage application to underwater vehicle and robot manipulator are given kernel for a population balance model can be obtained. and discussed.

Sergiy Antonyuk, Denickestr. 15, Hamburg, 21073, Germany E. Lyubimov, Neybuta 57, 68, Vladivostok, 690010, Russia

ON THE DESIGN OF EXTENDED INERTIAL THE STRAIN-STRESSED STATE OF THE BAND NAVIGATION SYSTEM WINDED UP ON THE RIDGED BODY

N.DVAS V. KUZKIN P. DYATLOVA V. LUTOV V. KOROVKIN [email protected] [email protected] [email protected] [email protected]

Usage of inertial navigation is limited because of a well-known The causes of unevenness of strain-stressed state of compos- effect of integration drift. However, if one needs to know both the ite materials reinforced by dint of winding of textile bands are rotational and the transitional motions of the body, it is necessary analyzed. Inﬂuence of geometric shapes of reinforced articles on to use inertial navigation. The extended inertial navigation sys- localization and character of the unevenness is considered. Meth- tem uses acceleration and gyro velocities sensors as any INS sys- ods of strains and stresses calculation in the process of reinforc- tem does, but in addition it uses the measurements of a magnetic ing of complex shape articles by elastic textile materials are pro- ﬁeld vector. This data, combined with data obtained via GPS posed. As example of the problem of reinforcing articles by mate- system, allows to calculate trajectory and orientation history of rials with hereditary properties, a problem of winding on a cylin- the system with relatively high precision. An effective yet simple drical tube of a band having canvas structure and corresponding algorithm of bootstrapping the data is presented in the work. to constitutive equations of differential type is investigated.

Nikolay Dvas, V.O. Bolshoj pr., 61, St. Petersburg, 199178, Rus- Vasiliy Korovkin, Bolshaya Morskaya, 18, Saint-Petersburg, sia 191186, Russia

ON THE PROBLEM OF INTERACTION BETWEEN THE THE ROLE OF ELECTRICAL FORCES IN THE SOFT SHELL AND THE SOLID COVERED BY IT FLUIDIZATION OF FINE POWDERS

P. DYATLOVA A. MAJOROV V. LUTOV M.J.ESPIN M.A.S.QUINTANILLA A.CASTELLANOS J.M.VALVERDE [email protected] [email protected] [email protected] [email protected] [email protected] The variational method for investigation of interaction be- [email protected] tween elastic soft shell and surface of rigid body covered by the shell is developed. The method is based on principle of mini- It is well known that electric fields affect the dynamics of gas- mization of strain energy functional of the shell in its equilibrium fluidized beds. Our experimental results, and those of other au- state. Extremals of the functional are sought in form of linear thors, show that electrical forces may hamper or enhance the combinations of coordinate functions which are chosen in accor- fluidization characteristics depending on the size and electrical dance with boundary conditions that are prescribed for the shell’s properties of particles, and on the amplitude and frequency of edges. The nonlinear equations set for determination of coeffi- the electric field. In this communication we analyse the motion cients of these linear combinations are investigated by numerical and possible aggregation of particles as a guide to understand the methods for various cases of solid’s and shell’s shapes. macroscopic behavior of the fluidized bed.

Polina Dyatlova, Bolshaya Morskaya, 18, Saint-Petersburg, Antonio Castellanos Mata, Avda. Reina Mercedes s/n, Sevilla, 191186, Russia 41012, Spain

SYMBOLIC SYNTHESIS OF CONTROL LAWS FOR JAMMING TRANSITION AND TENSILE STRENGTH UNDERWATER VEHICLE AND ROBOT MANIPULATOR OF MAGNETIC STABILIZED BEDS

A.A.DYDA E.V.LYUBIMOV M.J.ESPIN M.A.S.QUINTANILLA J.M.VALVERDE A.CASTELLANOS [email protected] [email protected] [email protected] [email protected] This paper deals with the problem of the automation of de- [email protected] sign procedures for nonlinear control systems. As an example the procedure has been realized for the well-known technique A magnetic stabilized fluidized bed consists on a bed of mag- consisting of the exact linearization via nonlinear feedback. The netizable particles agitated by a gas flow in the presence on a developed software achieves the synthesis of control laws in sym- magnetic field. In the absence of magnetic field, there is a given bolical form that is convenient for further analysis, modeling, and gas velocity at which the particles can form a permanent network

35 capable of sustaining small stresses. This gas velocity marks the effects. All models are coupled ones. jamming transition of the bed. For gas velocities above the jam- The problems are solved numerically. ming transition, the bed resembles a liquid. Below the jamming Each time the problem consists of two parts. At the first stage transition, the bed behaves as a weak solid and, in the case of the heat conduction and chemical kinetics equations are solved using particles we have used in our studies, it shows a tensile strength alternating direction method. Then equations of chamber sub- when the gas flow is increased. The tensile strength has its ori- stance movement are solved with the help of successive overre- gin in non-magnetic attractive forces acting between particles. In laxation method. Change of variables is used in the tapered part the presence of a magnetic field, the gas velocity corresponding of the chamber so that we can examine the entire chamber in uni- to the jamming transition and the tensile strength of the bed deform cylindrical coordinate system. pend on the field strength. Both facts are a consequence of the The program was tested. Grid convergence of finite-difference magnetic attraction between particles arising from their magne- scheme was investigated. Our solution of heat conduction prob- tization. However, the tensile strength and the jamming transi- lem was compared with the solution of Elcut software package tion of particles with similar magnetic susceptibility but different according to the developed algorithm, deviation was less than 1%. strength of the non-magnetic forces show a different response to Particular parametric investigation of the problem was carried the magnetic field. The differences can be explained if the non- out. Difference in distributions of stresses and strains in a sample magnetic forces influence the network of contacts, which in turn corresponding to different rheological models was demonstrated. act as a magnetic circuit for the field lines. This view is in con- Influence of coupling effect on the process passing and transfor- trast with the customary approach of considering that magnetized mation modes was revealed. Influence of chamber geometry on particles behave as point dipoles that arrange themselves in chain- the product deformation level at various loading rates was shown like structures along field lines. qualitatively.

Miguel A. Sanchez Quintanilla, Avenida Reina Mercedes s/n, Nikolay K. Evstigneev, 36, Lenina pr., Tomsk, 634050, Russia Sevilla, 41012, Spain

ANALYSIS AND ACCURATE DETERMINATION OF SIMULATION OF INTERMETALLIC COMPOUND EIGENVALUES IN A NONLINEAR EIGENVALUE SYNTHESIS DURING SOLID-PHASE EXTRUSION PROBLEM FOR MODE II CRACK PROBLEMS UNDER THROUGH TAPERED DIE PLANE STRAIN CONDITIONS IN A POWER-LAW MATERIAL N.K.EVSTIGNEEV A.G.KNYAZEVA M.E.FEDINA I.A.KUZMINA [email protected] [email protected] The simulation concerns modes of intermetallic compound synthesis in a powder pressing. A cylindrical chamber with ta- The nonlinear eigenvalue problem arising from fracture me- pered outlet is considered. 3Ni+Al powder was chosen as a re- chanics analysis is considered. The present study gives an ap- action mixture. Exothermic chemical reaction is initiated in the proach to find the whole eigenspectrum of eigenvalue problems. chamber by heating chamber walls for a certain period of time The underling idea of the method is to consider the eigenvalue and the reaction product forms. Then the extrusion of chamber to be a difference between the eigenvalue corresponding to the matter composition by press starts. Values of press velocities are linear problem and a small parameter which quantitatively de- about 1-2 cm/s, so that we may consider loading conditions to scribes the nearness the eigenvalues as the exponent n is chang- be quasi-static. After completion of chemical reaction the sys- ing. The required function and the exponent n are represented by tem starts to cool, which is described by Newton law. The model asymptotic expansions as a power series of this small parameter. takes into account that internal stresses, which influence the reac- This leads to a series of boundary value problems for nonhomo- tion together with external loading, appear in the system during geneous linear ordinary differential equations. Their solutions do the process of heating and chemical conversion. The problem is not exist unless the solvability equations are valid. Thus the solv- axially symmetric and it is formulated in 2D in a cylindrical co- ability equations result in coefficients of the exponent n what in ordinate system. The aim of the work is to investigate the fields turn leads to the estimation of the eigenvalue corresponding to the of temperature, conversion degree and mechanical stresses and nonlinear problem. strains which appear during the reaction, explore the influence of coupling between chemical and physical phenomena on the Maria E. Fedina, str. Acad. Pavlov,1, Samara, 443011, Russia process, estimate the influence of geometry on product structure formation. Several rheological models with different correlations between TOWARDS THE MICRO AND NANO PARTICLE stress and strain tensor components are examined. In hypoelastic IGNITION AND BURNING THEORY model increments of stress tensor components are connected with A.V.FEDOROV A.V.SHULGIN increments of strain tensor components, temperature and conversion degree in a linear way. In the model of viscous media well- [email protected] [email protected] known analogy of correlations between stress and strain tensor components is extended to internal stresses which are caused by Submitted by a short overview of the some work of ITAM nonuniform heating and changing the matter properties during the SB RAS in the field of mathematical modeling of ignition and conversion. Viscoelastic model takes into account all the named combustion of regular (>1 mkm) and nanosized metal particles

36 (1-100 nm). The distributed 1D and 2D mathematical model of QUANTUM MECHANICS AT DIFFERENT SCALES. ignition and combustion of metal particles with allowance for the PART 2: LOCALIZATION, MULTISCALES AND heterogeneous chemical reaction and for the region of the ther- COMPLEX QUANTUM PATTERNS mal inﬂuence of the particle on the gas is developed. Problems of particle ignition under the action of a uniform and a nonuniform A.N.FEDOROVA M.G.ZEITLIN thermal ﬁeld in a rectangular microchannel are solved. a point mathematical model of the aluminum nanoparticle combustion is [email protected] [email protected] presented. It allows to satisfactory predict the combustion time in the initial pressure range of p0 (8, 32) atm and temperature We present a family of methods which can describe complex ∈ range of T0 (1200, 2200) K with the oxygen fraction in the behaviour in quantum ensembles. We demonstrate the creation mixture varied∈ from 10% to 50%. of nontrivial (meta) stable states (patterns), localized, chaotic, en- tangled or decoherent, from the basic localized modes in various collective models arising from the quantum hierarchy described by Wigner-like equations. Alexandr V. Fedorov, Institutskaya str. 4/1, Novosibirsk, 630090, Russia Antonina Fedorova, V.O. Bolshoj pr., 61, St. Petersburg, 199178, Russia

MATHEMATICAL MODELLING OF SUPESONIC FLOW QUANTUM MECHANICS AT DIFFERENT SCALES. OVER OPEN CAVITY WITH MASS SUPPLY PART 1: RESOLUTION OF A (QUANTUM) POINT

ANTONINA N.FEDOROVA MICHAEL G.ZEITLIN N.N.FEDOROVA I.A.FEDORCHENKO M.A.GOLDFELD YU.V.ZAKHAROVA [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] We consider some generalization of the theory of quantum states, which is based on the analysis of long standing problems and unsatisfactory situation with the possible interpretations of In the paper, turbulent supersonic flows in a plane channel with quantum mechanics. We demonstrate that the consideration of a cavity presenting a model combustion chamber are studied. The quantum states as sheaves can provide, in principle, more deep work focuses on the fundamental investigations of channel flow- understanding of some phenomena. The key ingredients of the field over an open cavity with aft wall ramp. The flows are inves- proposed construction are the families of sections of sheaves with tigated experimentally and numerically with / without taking into values in the category of the functional realizations of infinite- account the mass supply into the supersonic turbulent boundary dimensional Hilbert spaces with special (multiscale) filtration. layer before the cavity. Several schemes of sonic fuel injection are investigated. Michael Zeitlin, V.O., Bolshoj pr., 61, St. Petersburg, 199178, The experiments were performed at M=2, 2.5 and 3 in the IT- Russia 302M hotshot wind tunnel (ITAM SD RAS). Installation of in- jectors, pressure taps, thermocouples, and gas sampling for mea- SIMULATION OF THERMAL INTERACTION IN A surements were available at the top and bottom wall of the cavity. ZONE OF EXTRACTIVE WELL IN PERMAFROST In the tests, pressure distributions in cavity and along channel and flow visualization were obtained. M.YU.FILIMONOV A.G.KRAVETS As an instrument for the numerical study, RANS model com- N.A.VAGANOVA plemented with several two-equation turbulence models has been [email protected] chosen. An in-house 2D code developed in ITAM, Novosibirsk, along with ANSYS CFD 12.0 facilities is applied. Validation of A new 3D mathematical model of thermal interaction in a zone the results is provided by comparison with experimental data. of extractive well in permafrost is constructed. An algorithm of The parametric study with changing flow Mach number, the numerical solution of 3D Stephan’s problem and numerical re- cavity length to depth ratio and the aft ramp angle as well as the sults of permafrost defrosting in a ground surface with and with- fuel injection schemes was conducted numerically to get the bet- out of thermal isolation are presented. Using the elaborated pro- ter mixing characteristics. The particle traces of the gas injected gram complex a number of computations of permafrost defrost- into the main flow were constructed and time of their staying in ing in a zone of influence of extractive well is carried out with the cavity is evaluated. The results obtained with several popular taking into account of real soils characteristics. In contrast to two-equation turbulence models are compared. another, not only different engineering construction of wells but The work has been financially supported by Russian Founda- also seasons, solar activity and temperature changing are included tion for Basic Research in a frame of grant #09-08-01001-a. into this model. This work is partially supported by Russian Foundation for Ba- sic Re-search - URAL 10-08-96014, by the Program of the Pre- sidium of the Russian Academy of Sciences and the Program of Natalya N. Fedorova, 4/1 Institutskaya, Novosibirsk, 630090, Integrative Projects between Ural Branch, Siberian Branch, and Russia Far Eastern Branch of Russian Academy of Sciences.

37 Mikhail Y. Filimonov, S.Kovalevskaya str., 16, Ekaterinburg, CAVITATIONAL SYNTHESIS OF CARBON 620219, Russia NANOSTRUCTURES

THE VIBRATIONS OF THE BUILT-UP E.M.GALIMOV CONSTRUCTIONS PARTIALLY SUBMERGED INTO [email protected] THE LIQUID The possibility of synthesis of the diamond phase when excit- G.FILIPPENKO ing cavitation in a carbon-containing fluid has been experimen- [email protected] tally corroborated by means of experimental setup with Venturi tube (collaboration with the group of Prof.Skorobogatski V.N., The problem of oscillations of elastic constructions partially Bauman State Technical University). Benzene is used as the submerged into the water is one of the actual problems of modern working substance. The saturated vapor of this low-viscosity techniques. Ships, oil platforms, sea airports are the examples fluid has high pressure under normal conditions. In addition, it of such bodies. However the exact calculation of such bodies is characterize by a high carbon-to-hydrogen ratio (C6H6). In vibrations is rather complicated. So it is useful to explore the the experiments, we aim to reproduce the process of adiabatic possible oscillations in these objects taking as an example more collapse of cavitation bubbles by means of specialized values of simple mechanical systems. pressure and velocity of benzene. This experimental work with The aim of this work is to build the exact solution of free os- excitation of cavitations bubbles and its following adiabatic col- cillations problem of the built-up construction: the cylinder with lapse results in hard aggregates of diamond crystals with size object on the top of it. The cylinder shell is partially submerged equal to 1-10 M at the benzene. These results support the idea into the liquid. The free oscillations problem of this system is that the thermodynamic conditions inside of cavitations bubbles considered in the rigorous mathematical statement. The disper- are appropriate for the diamond phase stability. By means of ana- sion equation and corresponding forms are analyzed. lytical and numerical methods it was established that the thermodynamic parameters required for the synthesis of diamond (min. George V. Filippenko, Vasilievsky Ostrov, Bolshoy Prospect 61, pressure 40 kbar, temperature 1000 C), may occur in adiabatic St.-Petersburg, 199178, Russia collapse of the cavitating bubbles, with an allowance for heat conduction, diffusion and thermal dissociation of benzene. The first EXACT ANALYTICAL SOLUTIONS TO PROBLEMS ON estimation of the proper interaction potentials of carbon clusters EQUILIBRIUM STATE OF THICK-WALLED HEAVY at the generated gas mixture was made in collaboration with the ANISOTROPIC AXIAL- AND CENTRAL-SYMMETRIC group of Prof. Krivtsov A.M., IPME Russian Academy of Sci- BODIES AND THEIR APPLICATIONS ence. Development of the proper mathematical and analytical methods by means of the molecular dynamics approach is fol- A.A.FUKALOV A.V.KUTERGIN A.V.ZAITSEV lowed as well. [email protected] Erik Mikhailovich Galimov, Institution of the Russian Academy of Sciences V.I. Vernadsky Institute of Geochemistry and Analyt- Using decomposition of hoop and radial components of dis- ical Chemistry RAS, 19 Kosygina str., Moscow, Russia placement vector to the trigonometrical and generalized power series, the new exact analytical solutions to problems on equilibrium state of thick-walled heavy transversally-isotropic central- POLYMER/SILICATE NANOCOMPOSITES. symmetric and orthotropic axial-symmetric body, which are fixed NONLINEAR ELASTOPLASTIC MODEL WITH on the interior surfaces and are subject to the action of uniform DAMAGE ACCUMULATION UNDER DEFORMATION external lateral pressure, are obtained. This can set a pattern for exact solutions in particular cases of the relations for displace- O.K.GARISHIN ments, stresses and strains at the points inside thick-walled heavy isotropic cylinder and sphere, the interior surfaces of which are [email protected] fixed, while the exterior one being under the uniform pressures. The estimation of an initial strength of cylindrical and spher- Computer simulation of process of a nonlinear elastic-plastic ical solid-cast mine working supports, which are used for long- straining polymer/silicate nano-composite with allowance for its term storage of high-corrosive and high-reactive media, is carried volume modification caused by internal damage accumulation is out on the basis of a multicriteria approach taking into account accomplished. The main subject of research is nano-composites real damage mechanisms (i.e. damage from tension or compres- on a basis of polyolefin binder and lamellar silicate minerals as sion in radial, hoop and axial directions, and from transversal and filler. antiplane shear) of anisotropic axial- and central-symmetric bod- These materials represent a mix of assemblage of ultrathin sil- ies. icate plates which are arbitrarily located in the polyolefin matrix. The authors acknowledge the support of the Russian Founda- Single filler particles have width about 1-2 nanometers and dition for Basic Research (Grant RFBR–Urals No 07–01–96056). ameter in the range from 20 nanometers to micrometer. Often enough they can form separate crystallites (tactoids) from several Aleksey V.Zaitsev, Mechanics of Composite Materials and Struc- (order of tens) parallel lamellae (intercalated filler). tures Dept., Perm State Technical University, Komsomolskiy It is experimentally established that these nanocomposite sys- Ave., 29, 614990, Perm, Russia tems can change essentially their volume under deformation

38 due to accumulation of internal defects (pores, cracks, inter- conducted to estimate the influence of structural factors on resis- phase debondings). A relative volume change in such systems tance to high strain deformation and spall fracture. The measure- can reach 20-25%, and ultimate deformations 700-800%. This ments of dynamic yield strength and ultimate strength at the spall phenomenon has been described by means of a structurally- conditions under loading by plane shock waves with intensity up phenomenological model of the heterogeneous solid continuum, to 10 GPa were carried out. Part of investigations with alloy capable to carry significant nonlinear elastic and plastic deforma- D16T samples has been conducted over wide temperature range. tions accompanied with appreciable volume change. The samples of alloy 6063T6 with an average grain size of 100 To obtain the constitutive equations, the differential approach mkm were subjected to repetitive equal channel angular extrusion based on interpretation of the mechanical behavior of material (ECAE) at room temperature. After 2 and 8 extrusions heteroge- with the help of symbolical circuits and the additive decomposi- neous and homogeneous ultra-fine grain structure with average tion of the rate of deformation tensor in elastic and plastic parts size 1.2 mkm and 1.5 mkm correspondingly have been formed. is used. The alloy D16T samples in the plane normal to the rolling direc- The computational scheme of the model consists of two el- tion had equiaxed grains 1-20 mkm in size. Part of the samples ◦ ements connected in tandem: elastic and plastic. Mechanical were annealed at 450 C for 2 h and slowly cooled. After anneal- properties of elastic element are described by conventional equa- ing, the grain size increased mainly due to the displacement of the tions of the theory of elasticity for compressible media. It is small fraction of grains. To form a ultra-fine grained structure in ◦ assumed that only the elastic element answers for the volume the alloy multiple forging in the temperature range of 450-200 C change, which completely corresponds to the standard hypothe- was used. After multiple forging, the samples had nonequiaxed ses of the plasticity theory. The elastic potential for compressible grains with sizes of 0.2-0.8 mkm. The character of deformation continuum is written down as the sum of two parts responsible and fracture, as well as quantitative parameters of the process for the shape and volume changes separately. The plastic ele- - peak pressure of shock compression, a dynamic yield stress, ment behavior is defined as the basic Prandtl-Reuss equations of a critical fracture stresses under spall fracture, a strain rate be- the classical plastic flow theory. The proportional association before fracture, thickness of spall plate were determined from the tween the deviator of a total media tensor of strain rates and the analysis of free surface velocities profiles. The motion of the deviator of a strain rates tensor of a plastic element is used for free surface of the sample was continuously recorded with a closure of constitutive equations system. VISAR Doppler laser velocimeter. The conducted experiments Numerical dependences between stresses, relative volume have shown that the change of initial inner structure of alloys change and deformation of pure polyethylene and polyethy- 6063T6 and D16T samples as a result of heat treatment and se- lene/clay nanocomposite material were constructed with the help vere plastic deformation influence on elastic-plastic and fracture of this approach. The assaying of the gained outcomes testifies to properties at very high strain rates. Annealing of as-received al- availability, at least, two structural factors generating mechanical loy D16T decreased the dynamic strength up to 30% and dy- properties formation: namic yield in 4 times. SPD at high temperature of alloy D16T 1) On the one hand filler promotes interior structural damage ac- samples leads to decrease of dynamic yield with the factor 2.5 cumulation, i.e. the material in the course of a straining becomes increases of dynamic strength insignificantly as compared with softer and is easier gives in to plastic flow. initial samples. SPD at normal temperature by ECAE of alloy 2) On the other hand the same corpuscles hinder reorientation 6063T6 resulted in increase of dynamic strength up to 10% and of polymeric molecules and permolecular structural objects (like dynamic elastic limit was increased from 0.3 GPa to 0.5 GPa after spherulites and so on) in a polyethylene matrix, which, on the 8 extrusions. The quantity of times of ECAC-procedures prac- contrary, should hinder development of plastic flowages. tically do not influence the mechanical properties of this alloy. This work is executed at financial support of Russian fund of Using of SPD method in combination with heating for forming basic researches (Grants 09-08-00339, 09-06-00402) and Integra- of a ultra-fine grained structure in alloy was not resulted in in- tion project of fundamental scientific investigations executable crease of their mechanical properties. It has been established that jointly by scientists UB RAS, SB RAS and FEB RAS “Modeling the dynamic yield stress of the alloy D16T in as-received state of deformation processes with allowance for multifactor contin- decreases with the temperature increasing, whereas the dynamic uum behavior and exterior effects”. yield stress of the annealed alloy increases with temperature increasing. The dynamic strength of the alloy in the as-received Oleg K. Garishin, 1, Academic Korolev st., Perm, 614013, Russia state decreases significantly under heating, while the strength of the annealed material varies only slightly with temperature.

Gennady V. Garkushin, Semenova 1, Chernogolovka, 142432, INFLUENCE OF INTERNAL STRUCTURE OF Russia ALUMINUM ALLOYS ON FRACTURE PROPERTIES UNDER SHOCK WAVE LOADING THERMODYNAMICS OF HYPOPLASTICITY G.V.GARKUSHIN S.V.RAZORENOV G.I.KANEL L. MEYER M.HOCKAUF S.N.GAVRILOV E.F.GREKOVA E.V.SHISHKINA A.SAVITSKI [email protected] [email protected] [email protected] [email protected]

In this work, the shock-wave tests on aluminum alloys 6063T6 In 1955 Truesdell suggested theory of hypoelasticity. This the- and D16T samples with different initial inner structure have been ory, being a generalization of classical isotropic elasticity, has

39 incremental formulation. The behavior of a hypoelastic mate- to these complications in comparison with gas dynamics, shock- rial is rate independent but generally path depended. During the fitting algorithms have not been considered in solid dynamics long time this theory was a subject of mathematical interest, but it thus far and computations have been based on various shock- still does not have applications in engineering. In 80th the group capturing schemes. When the nonstationarity of a problem is es- of experimenters from Karlsruhe, working with sands, suggested pecially important, shock-capturing schemes are not effective and new theory called hypoplasticity, which is in fact a generalization shock fitting becomes of primary importance. In [1], asymptotic of hypoelasticity. Hypoplasticity describes some peculiarities in expansions around wavefronts were proposed for this goal. One the mechanical behavior of sands. of the most effective techniques for constructing such expansions Thermodynamic formulation for both hypoelasticity and hy- is the ray method. This method is based on a recurrence formu- poplasticity is not straightforward, because the theories does not lation of the geometric and kinematic consistency conditions for operate with strains, but operate with strain rates. In series of discontinuities of the derivatives of functions that are discontinu- studies by Bernstein with co-authors the possibility of a thermo- ous on a moving surface. The ray method became popular in the dynamic formulation for hypoelasticity is investigated. Recently 1960s–1980s in the theory of plastic flows, hyperbolic heat trans- (2008), the necessary and sufficient conditions for hypoelastic fer, dynamic elasticity, and the study of weak waves in nonlinear equations to be compatible with laws of thermodynamics were elastic media. In [2], the method was extended to shock load- found. ing. It was found that the near-front expansions thus constructed The aim of this work is to suggest a thermodynamic formula- make it possible to trace the position of wavefronts in the course tion for hypoplasticity. It may be noted that thermo-mechanical of numerically solving shock-loading boundary value problems. problems in the framework of hypoplasticity have important en- This approach was implemented for plane shock waves in [1]. In gineering applications. this report, we propose a shock-fitting algorithm as for one – dimensional shock-loading problems as for multidimensional prob- Serge Gavrilov, V.O., Bolshoy pr. 61, St. Petersburg, 199178, lems with curvilinear and diverging rays. The algorithm is illus- Russia trated by solving one-dimensional and two-dimensional problems concerning the anti-plane motion of an incompressible nonlinear elastic medium. ON NEW PHASE NUCLEATION DUE TO SHOCK This work was supported by the Russian Federation Presi- WAVES COLLISION dent’s Grant (MK-2436.2009.1).

S.N.GAVRILOV E.V.SHISHKINA shishkina [email protected] REFERENCES

The dynamic processes in an straight elastic bar with ﬁnite [1] A. A. Burenin and P.V.Zinov’ev. “Detection of Discontinuity length, composed of a material, which is capable to undergo Surfaces in Numerical Methods as Applied to Solid Mechan- ics” in Problems in Mechanics: Colleted Papers Dedicated phase transitions, are under consideration. At t = 0 the ends of the bar begin to move at a given speed in directions outside to the 90th Birthday of A.Yu. Ishlinskii (Fizmatlit, Moscow, from the middle of the bar. The problem is symmetric with re- 2003), pp. 146–155 [in Russian]. spect to the middle of the bar. At a certain moment of time two [2] A. A. Burenin and Yu. A. Rossikhin. “Ray Method for Solv- shock waves running from the ends collide at the middle of the ing One-Dimensional Problems in Nonlinear Dynamic Elas- bar. This can lead to new phase nucleation in a small neighbor- ticity with Plane Discontinuity Surfaces” in Applications hood of the middle of the bar. The aim of this paper is describe in Solid Mechanics (DVO Akad. Nauk SSSR, Vladivostok, the dynamical process of the new phase nucleation. 1991), pp. 129–137 [in Russian].

Ekaterina V. Shishkina, V.O. Bolshoy pr. 61, St. Petersburg, Ekaterina A. Gerasimenko, Radio Str., 5, Vladivostok, 690041, 199178, Russia Russia

THE USE OF NEAR-FRONT EXPANSIONS FOR SHOCK COMPARISON OF THERMOELASTIC STRESS LOADING SIMULATION ANALYSIS AND PHOTOACOUSTIC MICROSCOPY APPROACHES TO THE PROBLEM OF RESIDUAL E.A.GERASIMENKO A.V.ZAVERTAN STRESS DETECTION [email protected] A.L.GLAZOV K.L.MURATIKOV Shock-fitting techniques have been well developed and their [email protected] descriptions comprise whole sections in summarizing editions. However, these techniques are hardly applicable in solid dynam- Various experimental evidences on the possibility of residual ics. The cause is that the propagation of boundary perturbations stress detection and imaging by photoacoustic (PA) microscopy in solids actually involves two interacting processes: the propa- have been presented by us during the last years. The experimen- gation of strains related to variations in the volume (as in gas dy- tal results of PA detection of residual stress fields for Vickers namics) and in the shape. Relations associated with shear strain indented ceramics and metals were demonstrated and analyzed propagation are fundamentally different from those associated [1,2]. The theoretical model of the PA effect in stressed materials with volume strain propagation, which are better studied. Due was also developed by us [3]. It relates the influence of stresses

40 on the PA signal with non-linear elastic and thermoelastic prop- amorphous component of a two-component system on the stress- erties of materials. strain state of crystallizing polymer. Additional kinetic equations On the other hand the thermoelastic stress analysis (TSA) is are derived to characterize the energy evolution of the segmen- actively developed at present for residual stress detection and tal mobility of a polymer network caused by the bonds of inter- imaging [4]. The two new techniques such as Stress Pattern molecular interactions. Analysis by measurement of Thermal Emission (SPATE) and The proposed system of constitutive equations can be used to DELTATHERM have been already developed based on TSA re- describe the peculiarities of the thermo-mechanical behavior of sults. The main purpose of this work is to compare scientific the amorphous component of polymers and to study the influence foundations of both TSA and PA approaches. Based on the sys- of these specific features on the behavior of the two-component tem of thermoelastic equations it is shown that both techniques system during the process of crystallization and after its comple- are based on the dependence of the thermoelastic parameter on tion. stress. The results of numerical experiments carried out to investigate The difference between these techniques consists in using dif- different uniform stress-strain states demonstrate the potential ferent working cycles. The TSA technique is based on the tem- of the developed constitutive equations for description of some perature cycle under periodic external loading of a sample while thermo-mechanical effects observed in materials under relaxation PA technique uses the deformation cycle produced in a sample and phase transitions. by a modulated laser radiation. Some examples of the PA images This study was supported by RFBR under grant 10-01-96032. of stressed materials are presented. It is also shown that these images can be used for extracting important relations between stress intensity factors of cracks characterizing external and resid- Lyudmila A. Golotina, 1, Ak. Korolev str., Perm, 614013, Russia ual stresses. It is pointed out that the main advantage of the PA technique over TSA one consists in the high spatial resolution. This research was supported by the RFBR under grant number 06-02-17148. INVESTIGATION OF THE DEPENDENCE OF POTENTIAL ENERGY DENSITY ON STRUCTURAL PARAMETERS IN MODELING THE MECHANICAL REFERENCES BEHAVIOR OF COARSE-GRAINED ELASTOMERIC COMPOSITES [1] K. L. Muratikov and A. L. Glasov. Proc. of the 6th Int. Congress on Thermal Stresses, 26 – 29 May 2005, Vienna, L.A.GOLOTINA L.L.KOZHEVNIKOVA Austria, pp.297-300. [email protected] [email protected] [2] K. L. Muratikov and A. L. Glasov. Mater. Sci. Forum, 524- 525, pp.471-476 (2006). Dispersion-filled elastomeric composite materials have found wide application in different engineering areas. A specified unit [3] K. L. Muratikov, A. L. Glasov, R. N. Rose, and D. E. Dumar. cell approach is capable to model fairly well the complicated J. Appl. Phys., 88, 2948-2955 (2000). macroscopic properties of particulate polymer composites. These [4] G. Pitarresi and E. Patterson. J. Strain Analysis, 38, 405-417 materials are the systems in which it is possible to stand out the (2003). representative structural cells. This allows one to quantitatively estimate the mechanical behavior of cells depending on the governing structural parameters. In this case, it is also possible to Kyrill L. Muratikov, Polytehnicheskaya 26, St. Petersburg, investigate the evolution of cells on all stages since the moment 194021, Russia of their formation to fracture. The transformation of the stress and strain fields existing inside a cell into the corresponding macroscopic characteristics of the cell is the step in describing a con- INVESTIGATION OF THE INFLUENCE OF tinuous medium. The approach to the macroscopic continuum RELAXATION TRANSITION IN THE AMORPHOUS description of the behavior of particulate polymeric composites COMPONENT OF AMORPHOUS-CRYSTALLINE was proposed. It consists in prescribing the constitutive relations POLYMER ON DEFORMATION PROCESSES for tested materials as the dependence of the density of potential strain energy on the finite strain tensor invariants, obtained in L.A.GOLOTINA I.N.SHARDAKOV studies of deformation and fracture in specialized cells. This rela- [email protected] tion is represented not analytically but as electronic spreadsheets that involve the results of solving the boundary-value problems This paper is concerned with amorphous-crystalline polymers of deformation of structural cells at each stage of their life cycle. in which amorphous and crystalline phases may coexist in stable To create the database containing as many combinations of the equilibrium. For modeling the thermo-mechanical behavior of invariants implemented during deformation as possible, different these materi-als, the system of constitutive equations describing types of loading were analyzed. Also we explore the dependence the relationship between the stress-strain state and the processes of the density of deformation potential energy of structural cells of crystallization and glass transition is constructed using the ap- and mesoelements averaging the mechanical behavior of the en- proaches proposed in earlier research. The distinctive feature of semble of cells on such parameters as the volumetric filling of the developed constitutive equations is that they take into account the composite with a solid phase, the sizes of rigid inclusions, the the influence of relaxation transitions (glass-transition type) in the filler–matrix adhesion strength, etc.

41 This study was supported by the Fundamental research proof the bodies. The approaches of contact mechanics are used to gram of the Department of Power Engineering, Mechanical Engi- evaluate the evolution of the nominal contact characteristics and neering, Mechanics and Control Processes of RAS “Tribological macroshape of contacting bodies in wear process (macroscale). and strength properties of structured materials and surface layers” Based on the methods developed the evolution of the wheel pro- No 09-T-1-1006. ﬁles in wear process and the contact fatigue defect formation is analyzed. The effect of surface wear on defect formation in the Ludmila Kozhevnikova, 1, Ak. Korolev Str., Perm, 614013, Rus- wheels of heavy-haul vehicles is discussed. sia The work is supported by RFBR (grants NN 09-08-12135 oﬁ m and 10-08-90047-Bel a). INFLUENCE OF ELASTIC DEFORMATION ON SET OF MAGNETIC CHARACTERISTICS FOR STRUCTURAL STEELS Irina G. Goryacheva, Ishlinskii Institute for Problems in Mechan- ics of Russian Academy of Sciences, Moscow, Russia E.S.GORKUNOV S.M.ZADVORKIN A.N.MUSHNIKOV E.I.YAKUSHENKO [email protected] [email protected] MULTISCALE MODELLING FRICTION OF ROUGH [email protected] SURFACES

In this paper influence of elastic deformation by compression, I.G.GORYACHEVA A.V.SAKHAROV tension, torsion and internal pressure upon set of magnetic char- [email protected] aleksandr.s [email protected] acteristics of high-strength low alloy structural steels 15CrNi4Cu and 11CrNi3Cu is studied. Following magnetic parametres were Work is devoted research of joint influence of the form of sep- investigated: coercive force, residual induction, the maximum arate roughness and density of their distribution to size of force of magnetic permeability, distribution of critical magnetic fields. a friction at some allocated scale level. The periodic contact prob- Consideration of these characteristics in aggregate allows receiv- lem for the system of stamps sliding on border of the viscoelastic ing unequivocal interrelation of tension and magnetic parameters basis which is modeled by Calvin’s one-dimensional body is with of a material. Besides, the separate attention is given to distri- that end in view put and solved in flat statement. The stamps bution of critical magnetic fields which allow receiving data con- modelling roughnesses of a surface, are considered in the form cerning volume material which changed the condition in a mag- of a trapeze which is interesting to that asymptotically it can be netic field with given intensity and pressure. reduced to various figures, for example, to a triangle. The problem decision has allowed to calculate contact pres- Alexandr N. Mushnikov, Komsomolskaya, 34, Ekaterinburg, sure, the sizes of area of contact and force of a friction at various 620219, Russia values of speed of the sliding, the enclosed loading, parameters of the viscoelastic environment, geometry of roughnesses. The anal- MODELING OF WEAR AND CONTACT FATIGUE ysis of dependence of factor of a friction from speed is carried out at various forms of roughness and distances between them. It is I.G.GORYACHEVA shown that at increase in an angle of slope of a lateral surface of roughness to the basis the friction factor increases. [email protected]

Wear of surfaces leads to the continuous irreversible changes Alexander V. Sakharov, prosp. Vernadskogo 101, Moscow, of the surface micro- and macroshape in time. Calculation of 119526, Russia the wear kinetics for different junctions is a necessary condition for design of long-life machines. In this study wear of two bod- SOILS IN TERMS OF NONLINEAR ELASTIC REDUCED ies in contact is studied based on the methods and approaches of COSSERAT MEDIUM: SMALL PERTURBATIONS NEAR contact and fracture mechanics. Two scales analysis is used to calculate the wear process. The basic steps in wear modelling AXISYMMETRIC PRESTRESSED STATE at microscale include the evaluation of internal stresses and tem- E.GREKOVA perature distribution in subsurface layer taking into account its speciﬁc mechanical properties and roughness, determination of [email protected] the onset of failure based on the fracture criterion, and modelling of the particle detachment. Both analytical and numerical (step Various authors (Askes, Besdo, de Borst, Harris, Metrikine, by step procedure) methods are used to calculate the stress and Suiker, Sulem, Vardoulakis etc.) suggest that rotational degrees temperature distribution at the subsurface layer in wear process. of freedom are important for modelling of granular materials and The wear equation which describes the dependence of the wear soils. There are also experimental evidences of grain rotation in rate on the nominal contact pressure, velocity and other character- shear bands. Most of scientists consider these materials in terms istics of the wear process follows from this analysis. The damage of linear full Cosserat media. accumulation function in cyclic loading of the subsurface layers Shwartz, Johnson and Feng have proposed in 1984 the linear is also calculated based on the determined internal stress and tem- elastic isotropic reduced Cosserat theory to model soils and gran- perature distributions. It is shown that localisation of the maxi- ular media. In such a theory, couple stresses are zero, though mum damage within the contact region leads to the fatigue wear rotations and translations are kinematically independent. The

42 Cauchy stress tensor, generally speaking, is not symmetric. Re- The appropriate wave solutions have been found that deter- duced Cosserat theory describes better those media where there mine the velocities of wave propagation, the indices of refraction is no ordered structure of particle rotations (present, e.g. in mag- and absorption in each of the mentioned media. The wave struc- netic media), though rotations are important. Grekova, Herman, ture and the features of mutual connection of the elastic waves and Kulesh investigated wave propagation in the linear elastic re- and the defects field waves determining the plastic deformation duced Cosserat continuum. In this work we formulate a physi- have been investigated. The study of structure and connection of cally linear and geometrically nonlinear elastic reduced Cosserat various types of waves is important for a number of reasons. One theory in order to model consolidated soils, rocks, and granular of them is that the results of this investigation are necessary for materials. We consider a prestressed state, a nonlinear homo- the analysis of propagation of defect field waves through the in- geneous stress state with axial symmetry, reached quasistatically terface boundary between two media. Various media in the nature from the natural configuration. This problem is the first step to usually contain interfaces and there is extensive evidence that in- model the triaxial test and similar experiments. terface boundaries play an important role in deformation process. In statics and zero external torques, the reduced Cosserat the- Propagation of waves through the interface boundary between ory yields in symmetric stress tensor. Therefore it has to be two viscoplastic media has been considered in more detail. For almost symmetric for the triaxial test while the critical force is this case the reflection and refraction coefficients as well as the not reached. However, rotational degrees of freedom can be im- reflection and transmission capacities of interfaces have been de- portant when we consider small deviations from this prestressed termined. First results on propagation of plastic deformation state. In experiments it is shown that when the material fails in across the boundary between viscoelastic media have been ob- the triaxial test under quasi-static loading, there are fast dynamic tained, which can be used in non-destructive testing of materials processes where particles rotate strongly. When the material does in seismic interpretation, where the dynamic effects of inelastic not fail, particle rotation may be important for wave propagation. deformation determined by the defect structure of materials are In this work, we consider small perturbations from this nonlin- important. ear state for some particular cases. For instance, when the stress The investigation is supported by the Russian Foundation for tensor in the prestressed state is spherically symmetric, the wave Basic Research project No 09-01-00264a. propagation near this state is similar to that one for the isotropic linear elastic reduced Cosserat medium: it has a prohibited fre- Nadezhda V. Chertova, Pr. Academicheskii 2/4, Tomsk, 634021, quency band for the shear wave, where localisation is possible, Russia and strong dispersion near this zone is present. The author is grateful to Prof. Antonio Castellanos, Dr. Serge Gavrilov, and Dr. Ekaterina Shishkina for valuable discussions. MICROMECHANICS OF NANOTUBE GLIDE IN This research is supported by Spanish Government (Project FIS DEFORMED CERAMIC NANOCOMPOSITES 2006-03745) and by Shell International E. & P. (Project “Ad- vanced modelling of mechanical behaviour of rocks and soils”). M.YU.GUTKIN I.A.OVID’KO [email protected] [email protected] Elena F. Grekova, Bolshoy pr. V.O., 61 / Facultad de Fisica, Avda. Reina Mercedes s/n, St. Petersburg / Seville, 199178 / 41012, Ceramic nanocomposites are greatly attractive for wide use in Russia / Spain advanced technologies due to their superstrength, superhardness and good wear resistance at both room and high temperatures [1]. The general lack of the most of such materials is still their WAVE PROCESS IN MEDIA WITH DEFECTS low toughness. However, rather high toughness was shown by several ceramic nanocomposites, in particular those reinforced YU.V.GRINYAEV N.V.CHERTOVA by carbon nanotubes [2-4]. These examples stimulate interest [email protected] in understanding micromechanisms which hamper the growth of cracks and are specific for nanostructures. The well-known re- For many years the studying of inelastic behavior of deformed inforcement mechanisms of nanotubes are their pulling-out, rup- bodies is the topical problem of mechanics and physics deformed ture, bridging and crack deflection [5]. In recent years, special solids. It can be explained by the fact that the region of elastic attention has been paid to multidirectional nanotube bridges and deformation is quite limited and many processes important with anchors, nanotube/ceramic interface and Y-junction nanotubes respect to creation and usage of new materials and their products [6]. The latter are representatives of large variety of branched take place beyond the elastic limit. In this work, we study features nanotubes (C, CNx, Al2O3, etc.) which are produced mainly of inelastic deformation due to dynamics of translational defects for nanodevice applications. At the same time, it seems that Y- on the basis of wave solutions of defects field theory. junction nanotubes might be exploited as strengthening elements The equation system of field theory of defects includes the of nanocomposites. In doing so, due to their complicated shape, kinematical identities of continual theory of defects and the dy- such nanotubes are expected to be more effective, compared to namic equations of gauge theory taking into account the dissipa- their conventional straight counterparts. tion of energy. This system describes the dynamics of disloca- This paper is aimed at the elaboration of theoretical models tion ensemble characterized by defect density tensor and defect describing the glide of straight and Y-junction nanotubes through flux density tensor. As source terms for the defect field the con- generation and glide of nanoscale dislocation loops, providing ef- sidered equations contain the effective impulse and the effective fective strain energy relaxation near bridging cracks and strength- stress tensor, which, introducing certain assumptions, can define ening effect of Y-junction nanotubes in ceramic nanocomposites. viscoelastic, viscoplastic and elasto-viscoplastic media. First, we consider the generation and slip of prismatic dislocation

43 loops from the crack surface along the nanotube/matrix interface. structure in the process of their fabrication and use. In particular, It is shown that these processes can be energetically favorable the mechanisms of relaxation of residual stresses in pentagonal and barrier-less under high but still real local shear stress acting nanowires are of great interest [1,3]. The relaxation mechanisms in the interface near the crack surface. The critical stress value in- are well known in the case of pentagonal nanowires of homoge- creases with rising ratio of the matrix and fiber shear moduli. Ina neous composition. However, a large part of these nanowires is wide interval of values of this ratio, it also grows with the thick- prepared from metallic alloys and semiconductor solid solutions ness of the fiber wall. The dislocation mechanism provides sig- [2]. The inhomogeneity of their atomic composition leaves room nificant relaxation of the strain energy near the cracks in ceramic for other possible relaxation mechanisms such as nucleation of nanocomposites and therefore can increase their toughness. second-phase precipitates (quantum dots) having the elastic fields Second, we investigate the strengthening effect of Y-junction which partly compensate the residual stresses in the pentagonal nanotubes in nanocomposites. Within the model, the nanotube nanowires. The present work is devoted to investigation of con- glides along the nanotube/matrix interface through slip of pris- ditions necessary for nucleation of quantum dots (finite-length matic dislocation loops. When such a loop meets a Y-junction inclusions) in pentagonal nanowires of inhomogeneous composi- of nanotubes, it needs a critical shear stress to bypass the junction. tion by a mechanism of loop transformation. The critical stress A pentagonal nanowire is modeled as an infinite elastically is calculated and analyzed in detail. In particular, it is shown isotropic cylinder which contains a positive partial wedge discli- that its value increases with decreasing radius and wall thickness nation along its axis [1,3]. It is assumed that the inclusion is of the nanotubes. Therefore, the thinnest nanotubes like single- finite in length and cylindrical in shape, and the inclusion and wall carbon nanotubes are expected to provide the most effective cylinder axes coincide. The lattice misfit is represented through a strengthening and toughening of such nanocomposites. This con- homogeneous continuous distribution of virtual circular prismatic clusion is in agreement with the experimental results reported by dislocation loops over the inclusion side, which provides a homo- Mukherjee and co-workers [2-4]. geneous axial tensile eigenstrain inside the inclusion. Using the Our general conclusion is that the finest branched nanotubes earlier found elastic stresses of the inclusion [4], we have calcu- must provide the best toughness and strengthening when embed- lated and analyzed in detail the energy gain due to the inclusion ded to ceramic nanocomposites. nucleation. It is shown that depending on the system parameters, the nucleation of inclusions of different shapes can be energetically favorable. REFERENCES For example, if the specific energy of the inclusion/matrix interface is small enough, the inclusion nucleation can occur in a [1] C. C. Koch, I. A. Ovid’ko, S. Seal, S. Veprek, Structural barrier-less spontaneous mode and the optimal shape of the in- ∗ Nanocrystalline Materials: Fundamentals and Applications, clusion is easy to find. At the inclusion eigenstrain ε = 0.02, Cambridge University Press, Cambridge, 2007. its optimal shape is a thin disc. When ε∗ increases from 0.02 [2] G. D. Zhan, J. D. Kuntz, J. Wan, A. K. Mukherjee, Nature to 0.05, the growth curve in the space of normalized radius and Mater. 2 (2003) 38. length changes its character. At the early stage of the nucleation process, the inclusion grows synchronously in radius and length, [3] J. D. Kuntz, G. D. Zhan, A. K. Mukherjee, MRS Bull. 29 while at the late stage, three different ways are possible when (i) (2004) 22. ∗ the inclusion grows in length but shrinks in radius for ε = 0.02, [4] G. D. Zhan, A. K. Mukherjee, Rev. Adv. Mater. Sci. 10 (ii) it proceeds to grow synchronously both in radius and length ∗ (2005) 185. for ε = 0.03, and (iii) it grows in radius but shrinks in length for ∗ [5] J. Ning, J. Zhang, Y.Pan, J. Guo, Mater. Sci. Eng. A 357 ε = 0.05. The optimal shape of the inclusion strongly depends on ε∗: if ε∗ is relatively small (about of 0.02) then the elongation (2003) 392. ∗ prevails over the radial extension, if ε is relatively large (about of [6] K. Balani, T. Zhang, A. Karakoti, W. Z. Li, S. Seal, 0.1) then conversely, the radial extension prevails over the elon- A. Agarwal, Acta Mater. 56 (2008) 571 gation. In a wide range of ε∗, the optimal radius of the inclusion almost does not change, while its optimal length changes in an Mikhail Y. Gutkin, Bolshoj 61, Vasil. Ostrov, St. Petersburg, order of magnitude. 199178, Russia If the specific energy of the inclusion/matrix interface is large enough, an energy barrier for the inclusion nucleation appears and the situation needs to be considered within a special kinetic SPONTANEOUS FORMATION OF QUANTUM DOTS IN approach. PENTAGONAL NANOWIRES OF INHOMOGENEOUS COMPOSITION REFERENCES M.YU.GUTKIN S.V.PANPURIN [1] V.G. Gryaznov, J. Heydenreich, A. M. Kaprelov, et al., Cryst. [email protected] ser [email protected] Res. Technol. 34 (1999) 1091.

Pentagonal nanoparticles and nanowires, which clearly [2] H. Hofmeister, In: H. S. Nalwa (ed.), Encyclopedia of demonstrate the difference between nanoscale and conventional Nanoscience and Nanotechnology, Vol. 3, Stevenson Ranch, solids, have remained the subject of extended studies during some American Scientiﬁc Publishers, 2004, p. 431. decades [1,2]. Nevertheless, it is still very far from total un- [3] A. L. Kolesnikova and A. E. Romanov, Tech. Phys. Lett. 33 derstanding of the physical mechanisms of development of their (2007) 886.

44 [4] K. V.Kuz’min, M. Yu. Gutkin and A. G. Sheinerman, In: Ma- of magnesium to aluminum as the main potential agents of im- terials of the Best Presentations at the All-Russian Interuni- provement largely mechanical properties, low density (∼ 2 6) re- versity Research Conference of Students and PhD Students, sults in equal volumes of documents about three times less severe 24-29 November 2008, St. Petersburg, Izd-vo politechn. un- than if they were made of steel or copper, combined with very ta, 2009, p. 53, in Russian. good corrosion resistance and low melting temperature (660oC) facilitating its development in all casting processes. These al- loying elements come into solution and may also be present as Mikhail Y. Gutkin, Bolshoj 61, Vasil. Ostrov, St. Petersburg, intermetallic phases. The composition of these phases, but above 199178, Russia all their delicacy, their distribution, their consistency vis-`a-vis the aluminum matrix, their frailties, their stability as a function of heat treatments are also decisive for the properties of the alloy. INFLUENCE OF INDUSTRIAL PROCESSES OF Experimental procedure: MOLDING AND TIME OF MATURATION ON THE To determine the behavior of the material deal with various CHARACTERISTICS OF RESISTANCE OF DUCTILITY stresses it may encounter during use, these solicitations are repro- AND STRUCTURAL ALLOY HYPOEUTECTIC duced using static or dynamic tests, usually conducted on stan- ALSI7MG dard specimens in order to know the characteristics Figures of the material. Four techniques are used, namely traction to iden- A.HAKEM Y. BOUAFIA A.SALHI tify the various constraints, the Brinell hardness HB for the stress a [email protected] bouafia [email protected] ﬁeld, resilience Kcv us about the mode of fracture, fragility and [email protected] resistance to shock and metallographic to identify structures.

This study aims to determine the influence of the chemical composition of industrial sand casting and metal shell cast by gravity at room temperature or under pressure both mechanical REFERENCES and manual metal parts for the various achievements SNVI (Alu- minum Foundry Unit Rouiba) and ENEL (Unit engine Freha in [1] A. Hakem, Y.Bouafia, S. Naili, A. Bouhaci, Development Tizi-Ouzou) Algeria and heat treatments on mechanical behavior of industrial aluminum alloys foundry AlSi7Mg, AlSi10Mg of fracture, quasi-static axial tensile united, Brinell hardness, al- and AlSi13, International Symposium - Characterization and most dynamic resilience and microstructure of the alloy foundry Modeling of Materials and Structures 16, 17 and18 Novem- chemical designation AlSi7Mg and digital 42000, The addition of ber, 2008 - University Mouloud Mammeri Tizi-Ouzou, Al- 7% silicon and magnesium percentage ( 1% Mg) in aluminum geria. are the principal agents of the improvement≤ of mechanical properties in addition to specific heat treatments which show different [2] A. Hakem, memory magister, Microstructure and Mechan- types of precipitates that hinder the movement of dislocations. ical Properties of Alloy hypoeutectic Al7SiMg, 2005, De- The physical characterization, chemical and general engineer- partment of Engineering - Mechanical Engineering, Faculty ing in particular is crucial for the design of various metal parts of Engineering Building, University Mouloud Mammeri Tizi subject to external forces varied constituents various mechanisms - Ouzou . Algeria. in motion a mechanical component. The designer can neither calculate nor resize these parts without identifying and quantifying [3] M. Colombia and Coll, Industrial Materials: Metal Materials, their characteristics. To determine them, we reproduce these so- p.867, Dunod, Paris (2000). licitations using static and dynamic tests, usually conducted on [4] Jean-Paul Baillon, Jean-Marie Dorlot, Materials, p.729, 3rd standard specimens. edition, Ecole Polytechnique de Montreal, Montreal (2000). The alloy AlSi7Mg, which governs our study, is a shade containing some magnesium added in small amounts (0.20 to 0.40) % [5] Jean Baralis Gerard Maeder, Handbook of Metallurgy: Mg alloy to allow hardening and efficient use in applications with (Preparation, structure, properties, normalization), p.232, high mechanical properties in T6 condition. This alloy contains 1ereédition, AFNOR-Nathan, Paris (1997). 7% of silicon, which gives very good properties implementation in foundry (average flow ability, low volumetric contraction dur- [6] Bruno Barlas, study of behavior and fatigue damage in alu- ing solidification, reduction of withdrawal to the solid state and minum alloy casting, thesis submitted and publicly defended the expansion coefficient.). It is used for complex shapes, to the 5 ’in February 2004, Ecole des Mines de Paris, p228 requirements of mechanical strength and maximum thicknesses which are very low (∼ 3 mm). [7] Asserin-Lebert Experimental study and prediction of failure The unalloyed aluminum with mechanical properties very re- mechanisms of plates and joints butt welded aluminum alloy duced, leading to add two items of very low density with 7% sil- 6056, and presented publicly defended February 18, 2005, icon and traces of magnesium less than 1% ( 1% Mg) which is Ecole des Mines de Paris, p190 the lightest of all metals capable of stable industrial≤ employment to improve their properties and obtain an alloy AlSi7Mg super lightweight. The alloy foundry ultimate AlSi7Mg governing our study is an alloy with a set of properties that in many circum- Ahmed Hakem, Department of Engineering - Mechanical Engi- stances make it an irreplaceable material. Among these include neering Faculty of Construction University Mouloud Mammeri the addition of a high percentage of silicon and a low percentage Tizi - Ouzou Algeria., Tizi-Ouzou, 15000, Algeria

45 motion equations to include both transverse and longitudinal tem- CONTACT PROBLEM FOR ELASTIC WEDGE peratures. To do so we partition the contributions of the heat STRENGTHENED BY RIGID INCLUSION transfer, the work done, and the heat flux vector between the longitudinal and transverse temperatures. With shockwave bound- V. HAKOBYAN ary conditions time-dependent solutions of these equations con- [email protected] verge to give stationary shockwave profiles. The profiles include anisotropic temperature and can be fitted to molecular dynam- The contact problem for two same symmetrically located ics results, demonstrating the utility and simplicity of a two- smooth punches with elastic wedge, which is strengthened by temperature description of far-from-equilibrium states. absolutely rigid thin inclusion on some section of its midline, is considered. The interaction of punches with inclusion subject to distance of punches and inclusion from wedge vertex is researched. In general case, this problem mathematically is for- Ruby Valley Research Institute, Highway Contract 60, P.O. Box mulated as the system of two singular integral first-kind equa- 598, Ruby Valley, 89833 Nevada, USA tions, which is solved by the Chebyshev orthogonal polynomial method. The special cases of this problem are considered. In two cases the close solutions are obtained. ON THE COUPLING OF NONLINEAR DISCRETE Vahram Hakobyan, Baghramyan 24B, Yerevan, 0019, Armenia ELEMENT MODELS WITH CONTINUA IN THE TIME DOMAIN SMOOTH PARTICLE APPLIED MECHANICS: J.S.HOVING A. V. METRIKINE THE METHOD, WITH THREE EXAMPLE PROBLEMS [email protected] C.G.HOOVER W.G.HOOVER

[email protected], When the behaviour of a material is governed by nonlinear [email protected] phenomena, such as dry friction, discrete elements are often used to model that material. From a numerical point of view, it is de- Smooth Particle Applied Mechanics (SPAM) is a numerical sirable to minimize the number of degrees of freedom considered method for solving the Lagrangian (comoving) equations of con- in the discrete element models. To minimize the number of de- tinuum mechanics: the continuity equation, the equation of mo- grees of freedom, the discrete model is coupled to a continuum tion, and the energy equation. The basis of the SPAM algorithm of the same dimension, allowing for the description of nonlin- is a smoothed continuum representation of the particles’ density. ear phenomena near a load source using discrete elements, while With this representation the partial differential equations of con- the linear behaviour far from the source is described by a continuum mechanics become a set of ordinary differential equations tinuum. To execute the coupling between the discrete and con- for the particles’ motions and energies. This simple method is tinuum segments of the model, their interaction is described such extremely valuable for solving problems which would result in that the wave reflection is minimized at the discrete-continuum mesh tangling if Lagrangian finite-element techniques were used. interface. This interaction is described by a force-displacement We illustrate the algorithm with three example problems: (i) free relation known as the dynamic stiffness that follows from the expansion of a gas, (ii) collapse of a column of water, falling Green’s function of the continuum. under the action of gravity, and (iii) convective conducting com- Sometimes the Green’s function can be obtained in the time pressible fluid flow, with gravity, the Rayleigh-Bénard problem. domain analytically by applying the inverse Laplace transform For more details, see Wm. G. Hoover, Smooth Particle Ap- to the frequency-domain Green’s function. Often it is the case plied Mechanics, the State of the Art (World Scientific, Singa- however that the time-domain Green’s function has to be calcu- pore, 2006). lated numerically by truncating the originally infinite limits of Carol G. Hoover and William G. Hoover, Ruby Valley Research the inverse Laplace transform. This may lead to instability of the Institute, Highway Contract 60 Box 598, Ruby Valley, Nevada numerical scheme, should the truncation be chosen improperly. 89833, USA This contribution shows how to deal with this problem. The featured solution method is explained by evaluating the coupling of the nonlinear discrete and linear continuous segments WELL-POSED TWO-TEMPERATURE CONSTITUTIVE of a one-dimensional model. The kinematic behaviour of the dis- EQUATIONS FOR STABLE DENSE FLUID SHOCK crete elements in this material is described using springs, dash- WAVES USING MOLECULAR DYNAMICS AND pots and dry friction elements, to account for wave propagation, GENERALIZATIONS OF NAVIER-STOKES-FOURIER energy dissipation and stick-slip behaviour respectively. The re- CONTINUUM MECHANICS sult of the coupling in the time domain is illustrated by showing WM.G.HOOVER C.G.HOOVER the response of the discrete-continuous model due to a pulse load at several moments in time. [email protected]

Guided by molecular dynamics simulations, we generalize the Navier-Stokes-Fourier constitutive equations and the continuum Jeroen S. Hoving, Stevinweg 1, Delft, 2628 CN, Netherlands

46 The experimental technique and experimental results are de- STABILITY OF SOLITON-ANEURYSMS IN A scribed in the report. We discuss and model the hydrogen redis- THIN-WALLED HYPERELASTIC TUBE tribution during fatigue loading. In this case the two-continuum model describes the dynamical accumulation of the hydrogen A.T.IL’ICHEV content in the critical zones. [email protected] The financial support of the Sandia Corporation, contract 891092, and the Russian Foundation for Basic Research, grants Flow of an inviscid fluid in an elastic cylindric tube is under 08-08-00737 and 08-01-12017, is acknowledged. analysis. The tube is modelled by an axisymmetric membrane shell. It is considered in biological literature, that provided the state equation (elastic potential) is properly chosen, the flow of REFERENCES blood in arteries can be modelled by such a motion. Dynamic instability of standing solitary waves, having the [1] D. C. Ahn, P.Sofronis, R. Dodds Jr. “Modeling of hydrogen- form of an aneurysm, is proved with the help of construction of assisted ductile crack propagation in metals and alloys” Int. the Evans function. This function has at least one zero in the J. Fract. pp.145:135–157 (2007) right half of the complex plane of the spectral parameter which [2] D. A. Indeitsev, B. N. Semenov. “About a model of structural- coincides with the unstable eigenvalue of the associated spectral phase transformations under hydrogen influence” Acta Me- problem. chanica Vol.195, pp.295-304 (2008).

Andrej T. Il’ichev, Gubkina str. 8, Moscow, 119991, Russia Yuriy A.Yakovlev, 195220, p/b 162, St.-Petersburg, Russia

THE HYDROGEN MATERIAL INTERACTION DURING CYCLIC MECHANICAL LOADING PARTICLE DYNAMICS TO MODEL BRITTLE FRACTURE IN ROCKS D.A.INDEITSEV V.A.POLYANSKIY YU.A.YAKOVLEV J.ING E.PAVLOVSKAIA M.WIERCIGROCH [email protected] [email protected]

The hydrogen is contained in any material. Its concentration Particle dynamics can be used to model materials on the inside the materials such as steels, aluminum alloys leads to me- macroscale. It is particularly powerful when modelling mate- chanical properties degradation. Usually the hydrogen is con- rial damage accumulation, since FE and analytical methods do tained in the traps of different nature. This nature determines the not cope well in this domain. For modelling rocks, a polycrys- hydrogen material interaction character. taline material of specified grain size is simulated by controlling We propose to use the hydrogen binding energy as the univer- the cooling rate of a fluid of Lennard-Jones particles. Modifica- sal parameter that determines this interaction. For example traps tion of the standard 6-12 potential is made in order to tune the with binding energy 1.5-3 eV (hydrides) may enclose high hy- ultimate strain of the material, which is also shown to be depen- drogen content without hydrogen embrittlement. In metals with- dent upon grain size. Impact tests conducted on such a material out hydrides such content leads to the destruction. The hydrogen show that micro crack formation occurs along grain boundaries. with low binding energy has very high diffusion mobility and its These cracks eventually join together and result in failure of the redistribution inside the material gives local hydrogen accumula- material. tion effect. Such a local process leads to local embrittlement. The thermo-mechanical loading is known to change the hydro- James Ing, Fraser Noble Bld, King’s College, University of Ab- gen traps volumes. We can describe this process as a process of erdeen, Aberdeen, AB24 3UE, United Kingdom hydrogen redistribution under external stress. This redistribution can occur both “in the volume” and “among the energy levels”. A number of papers were concerned with the influence of MODELING OF ACTIVE, PASSIVE AND SOFTENING hydrogen on the mechanical properties of metals, see e.g. BEHAVIOR OF SOFT BIOLOGICAL TISSUES BASED [1].Nearly all papers use phenomenological models and the redis- ON POLYCONVEX STRAIN ENERGIES tribution of hydrogen over the traps is not described. The degradation of mechanical properties is carried out by means of some M.ITSKOV empirical dependences. [email protected] The two-continuum model of solid [2] allows one to describe the influence of small concentration of hydrogen on the mechan- A thermodynamically consistent anisotropic inelastic model is ical properties of materials in terms of changing the bonding en- proposed in order to describe both passive and active responses ergy of the second continuum, the latter being responsible for the of soft biological tissues. The model is based on a polyconvex hydrogen concentration. The results obtained with the help of and coercive strain energy function so that the material stability simple models should be generalized in order to account for the and the existence of a global minimizer of the total elastic energy effect of small hydrogen concentration on the mechanical prop- are guaranteed in the elastic domain. Due to the generalized se- erties of materials available for the engineering practice. ries form the model is applicable to a wide variety of anisotropic

47 materials at large strains and demonstrates excellent agreement medium allows us to describe the internal damping and the heat with experimental data on the passive response of soft biologi- conduction mechanism. In special cases the mathematical de- cal tissues. The inelastic anisotropic behavior is considered by scription of proposed model is proved to reduce to the well- internal variables governing both anisotropic properties and the known equations such as the heat conduction equation, the self- stress-free conﬁguration of the material. In order to describe ac- diffusion equation and the equations of the coupled problem of tive materials such as skeletal muscle tissue the internal variables thermoelasticity. The mathematical description of the proposed are associated with the muscle activation. Thus, tissue contrac- mechanical model includes not only the classical formulation of tion and active stress response are captured in a straightforward coupled problem of thermoelasticity but also the formulation of manner. the coupled problem of thermoelasticity with the hyperbolic type heat conduction equation. In the context of proposed theory we consider the original model of internal damping and give inter- Mikhail Itskov, LF Kontinuumsmechanik, Dept. of Continuum pretation of the volume (acoustic) and shear viscosities. Just due Mechanics RWTH Aachen University, Eilfschornsteinstr. 18, to our interpretation of the volume and shear viscosities we treat 52062, Aachen, Germany proposed theory as a theory of thermoviscoelasticity. The work was supported by the grant of RFBR N 09-01-00623-a.

VIBRATIONAL TRANSLATION DUE TO Elena A. Ivanova, Bolshoy pr. V.O., 61, St. Petersburg, 199178, TRANSVERSAL STRAIN IN A CHELOMEI Russia PENDULUM-LIKE SYSTEM

K.IVANOV ON THE STABILITY OF COMPRESSIBLE ELASTIC [email protected] BODIES WITH MICROSTRUCTURE AT TENSION

M.KARYAKIN O.PUSTOVALOVA The vibrational translation phenomenon in a Chelomei pendulum-like system was studied with putting forward hypothe- [email protected] sis of alternative mechanism of its possible origin. The new case [email protected] of the translation which doesn’t require transversal deformation of a rod was physically described and mathematically explained. The paper presents various statements of a problem about the The complete mathematical explanation led to numerical algo- instability of the stretching process of the nonlinearly elastic rithm developing, which allows to predict some average charac- cylinder. The base model for characterization of material proper- teristics in the translation. Using the algorithm it was obtained ties is Blatz and Ko [1] material. The characteristic feature of this some possible conditions under which the translation would oc- material model is the presence of a point of a maximum on the cur. stretching diagram after which the falling region follows. This falling region is the reason of the material instability of the given Kirill S. Ivanov, V.O., Bolshoj pr., 61, St. Petersburg, 199178, model at the very large tensions. At the same time it is possi- Russia ble to show that though infinitely long sample loses stabilityina point of a maximum of the tension diagram [2], there are some parameters of the constitutive relation and geometrical sizes of ON ONE MODEL OF INTERNAL DAMPING AND the sample when the deformation along the whole falling region THERMAL CONDUCTIVITY is stable. To take into account the influence of microstructure the E.IVANOVA Cosserat model of elastic media extended to the case of large [email protected] strains was used. In this case the finite displacement of the point is describe by its radius-vector R in the actual configuration and We consider the mechanical model of a two-component the proper orthogonal microrotation tensor H. The strain-energy medium whose first component is the classical continuum and the W becomes a function of strain measure Y and tensor of flexural other component is the continuum having only the rotational de- strain L. The dependence of W from L is usually assumed to grees of freedom. We show that proposed model can be used for be quadratic, while the form of the dependence of W from Y has description of thermal and dissipative phenomena. In our opinion been received as generalization of Blatz and Ko model to the case the internal damping and heat conduction should be considered of non-symmetrical argument. As a result following expression as a result of interaction of atoms with the infinite surrounding for strain energy function was obtained medium which can be called the “thermal field” or the “thermal 2 −1 ether”. We propose the mechanical model the “thermal ether” µ I1 I3 W = + 2√I3 5 which is the continuum of particles having the translational and 2 2 − − 1 rotational degrees of freedom and interacting by the elastic mo- (µ + σ)Gˆ Gˆ +(µ σ)Gˆ Gˆ T + ments. We consider two problems of elastic interaction of the − 8I3 ⊙ − ⊙ h i “thermal ether” with the particle having a special structure. As a γ + η γ η + δ tr2 L + tr L2 + − tr(L LT ), result of analysis of the problems we show that the influence of 2 2 · the “thermal ether” on the particle can be modeled by the damp- 2 ing moment proportional to the angular moment of the particle. where Gˆ Y , Ik – principal invariants of Gˆ ; coefficients µ, δ, ≡ Using of the damping moment in the model of a two-component γ, η, σ are material parameters of Cosserat media.

48 The analysis of the stability of the deformation at the falling where the functions and variables are dimensionless. The func- region shows that material constants, or more exact – relations tion q(x,t) is proportional to the contact pressure, P (t), to the between them, make appreciable quantitative and qualitative im- applied force, M(t), to the moment of application of this force, pact on a stability zone and its size. By means of selection of δ(t), to the punch settlement, α(t), to the punch tilt angle, and these constants it is possible to achieve both essential expansion g(x), to the function of the punch base shape which can be of this zone and its elimination. rapidly oscillating; I is the identity operator, Vk (k = 1, 2) are The influence of the isolated defect in the cylinder on the pro- Volterra integral operators, and F is a Fredholm integral operator. cess of its tension is studied. The main feature of the Blatz and The solution of the integral equation with the additional condi- Ko model is the existence of the solutions featuring formation of tions must be sought in the class of functions that are continuous a cavity along the defect’s axis. For the isolated wedge disclina- in time t in the Hilbert space L2[ 1, 1]. To this end, we first − tion two cases of a tension of the cylinder were analyzed. In the construct a special function system pk(x) that is orthonormal { } first case the cylinder remains solid, in the second – the tension is in L2[ 1, 1]. The generalized projection method allows one to accompanied by the cavitation. obtain− the solution that has the structure This work was supported by Federal Program “Scientific and pedagogical cadre of innovative Russia for 2009–2013 years” q(x,t)= z0(t)p0(x)+ z1(t)p1(x)+ y(t)g(x)+ ..., and Russian Foundation for Basic Research (Project No. 09-01- where zk(t) and y(t) are functions of time t, and hence to sep- 00459). arate the function of the punch base shape in the solution, i.e, to find the fine structure of the solution. This permits obtaining effective solutions of problems of contact interaction for coated REFERENCES foundations in the case, where the punch base shape is described [1] P.J. Blatz and W. L. Ko (1962). Application of finite elastic by complicated functions, in particular, by rapidly oscillating theory to the deformation of rubbery materials, Soc. Rheol- functions. ogy, 6, 223-251. The author thanks A.V. Manzhirov for help in stating the problem and useful discussions of the problem. [2] N. Owen (1990). Some remarks on the stability of the homo- This research was financially supported by the Russian Foun- geneous deformation for an elastic bar, Journal of Elasticity, dation for Basic Research (projects Nos. 08-01-00003, 08-01- 23, 113-125. 00553, and 09-08-01180), and by the Basic Research Program [3] L. M. Zubov (1997). Nonlinear theory of dislocations and No. 13 of the Department of Power Engineering, Mechanical disclinations in elastic bodies, Springer. Engineering, Mechanics, and Control Processes of the Russian Academy of Sciences. Mikhail I. Karyakin, Milchakova 8a, Rostov-on-Don, 344090, Kirill E. Kazakov, Vernadskogo ave. 101 bldg 1, Moscow, Russia 119526, Russia

ON A NEW SOLUTION OF THE CONTACT PROBLEM FOR FOUNDATIONS WITH COATINGS MOLECULAR DYNAMICS OF NANODROPS

K.KAZAKOV G.V.KHARLAMOV A.A.ONISCHUK S.V.VOSEL P.A.PURTOV [email protected] g [email protected] [email protected] This work is devoted to constructing a new solution of the [email protected] well-known problem of contact interaction between a viscoelastic foundation with a thin coating and a rigid punch. Previously An effective method of nanoparticles production is the con- the solution of this problem was constructed as a series in normed densation of nanodrops from supersaturated vapor. Nucleation, Legendre polynomials. But obviously that, in the cases where the formation of critical nuclei, is the first stage of such condensa- punch base shape or the coating thickness the solution representation. The surface tension of small drops is the most important tion as a series in classical orthogonal polynomials is inefficient. parameter in the nucleation theory, as it determines the work of Therefore, a demand arose for new approaches and methods for the critical nuclei formation. solving the problems with parameters described by such func- In our report the results of molecular dynamics calculations tions. One of such methods is the Manzhirov generalized projec- of nanodrops are presented. The equilibrium states of Lennard– tion method. Jones molecular system containing one nanodrop of 50 – 4000 The problem of action of a rigid punch on a foundation with molecules and the saturated vapor have been prepared. The den- a coating that is thin compared with the punch width is reduced sity profiles, the Irving–Kirkwood pressure tensors, the chemical to solving the mixed integral equation with additional conditions potentials of the systems, the equimolar radii of the drops and which, in the plane strain case, have the form the radii of tension, the mechanical and thermodynamic surface tensions have been calculated. c(t)(I V1)q(x,t)+(I V2)Fq(x,t)= The dependence of the surface tension of Lennard–Jones liq- − − = δ(t)+ α(t)x g(x), x [ 1, 1], uid nanodrops on the equimolar radius and temperature has been − ∈ − 1 1 calculated by the molecular dynamics method. It was found that q(ξ,t) dξ = P (t), ξq(ξ,t) dξ = M(t), the mechanical surface tension is different from the thermody- − − Z 1 Z 1 namic surface tension for drops of 50 – 2000 molecules and

49 is equal for drops of more than 2000 molecules. It is shown Macroscopic equation of elastoplastic deformation of ran- that both the mechanical and thermodynamic surface tensions de- domly reinforced composite material has a representation: crease with the decrease of the equimolar radius of the drop, and ∗ reach zero at the same R0 depending on temperature. The radii sij = 2µ (Λm, Λ1,..., Λn) eij , (3) of tension also reach zero. The dependence of the ratio of the sur- h i ∗ h i σpp = 3K (Λm, Λ1,..., Λn) εpp . (4) face tension of the drop to the surface tension of the flat surface h i h i liquid-vapor on the ratio of the equimolar radius of the drop to Effective plasticity tensors calculated by the formulas: R0 is a universal function [1, 2]. The authors are grateful to Novosibirsk State Technical Uni- n versity, Ministry of Science and Education (“Development of [µs] csαs high school scientific potential (2009-2010)”, Project 2.1.1/1958) ∗ s=1 µ = µm + , (5) for financial support. P n cm + csαs s=1 P n REFERENCES [K s] csγs ∗ K = K + s=1 . (6) m P n [1] G. V.Kharlamov, A. A. Onischuk, P.A. Purtov, S. V.Vosel, cm + csγs A. V.Bolesta. Computation of surface tension of small s=1 droplets by the molecular dynamics method// Atmospheric P αs and Oceanic Optics. vol. 21, 2008, No.09, p.679-683. Λs = n e, (7) cm + csαs [2] G. V.Kharlamov, A. A. Onischuk, S. V.Vosel, A. V.Bolesta s=1 and P.A. Purtov. On the problem of surface tension defini- P tion for droplets. In: Nonequilibrium Phenomena. Plasma, 1 Λm = n e. (8) Combustion. Atmosphere. Eds. G.Roy, S.Frolov, A. Starik, cm + csαs Moscow, Torus Press, 2009, p. 444-450. s=1 P These relations can be applied to finding effective mechanical properties of multicomponent matrix of randomly reinforced Georgy V. Kharlamov, 20 K. Marx Prospekt, Novosibirsk, composite materials weakened by various defects - voids, pores, 630092, Russia cracks and ellipsoidal forms. If we consider the pores in the form of ellipsoids of rotation, then, if the parameter ω tends to zero, we obtain a composite material, weakened by a circular uniformly MODELING OF EFFECTIVE CHARACTERISTICS OF oriented planar cracks. MULTICOMPONENT RANDOMLY REINFORCED Effective modulus of the composite material are calculated by COMPOSITE MATERIALS WEAKENED BY CRACKS formulas: NONINTERACTING

YU.KHOKHRYAKOVA n [µ s] csαs [email protected] ∗ s=1 µ = µm + Pn − cm + csαs + c0α0 In a scientific paper calculates the macroscopic characteristics s=1 (9) of the elastic-plastic deformation of multicomponent matrix com- Pµmc0α0 , posites randomly reinforced with ellipsoidal inclusions of various − n cm + csαs + c0α0 configurations and materials weakened noninteracting cracks.. s=1 Let a composite material formed components, interconnected P with the ideal adhesion. The first component is the role of the n matrix - Vm, while others n - the role of individual randomly [K s] csγs ∗ s=1 distributed in the matrix of inclusions of different materials in the K = Km + n (s) (s) (s) P − shape of ellipsoids with principal axes a1 ,a2 ,a3 . Moreover cm + csγs + c0γ0 n s=1 (10) Vf = Vs, ( Vf - the volume of all the inclusions, Vs - the PK c γ s=1 m 0 0 , volumePs component). − n cm + csγs + c0γ0 Initial local equation of elastoplastic deformation: s=1 P c - concentration of cracks. That is, at low concentration c s = 2µ (√e e ) e , (1) 0 0 ij m,s kl kl ij obtained formulas for determining the effective moduli of a body σpp = 3Km,s εpp. (2) weakened by cracks noninteracting.

µm,s(√eklekl) - nonlinear shear modulus of plasticity, Km,s - 1 1 volume modulus, sij = σij δij σkk, eij = εij δij εkk Yulia Khokhryakova, Samara State University, Faculty of Econ- − 3 − 3 - deviator of the stress and strain tensors, σij ,εij - stress tensor omy and Management, 1, Academic Pavlov, Samara, 443011, and small elastoplastic strains. Russia

50 A-B transition, DNA melting, base opening, as well as to study CAPE CANAILLE CLIFF FALLING: HYPOTHETIC mechanical properties of single DNA molecule. Firstly, simpli- TSUNAMI CONSEQUENCES ESTIMATION fied (as compared with all-atomic one) description enables to increase considerably molecular dynamics trajectories time and O.E.KHVOSTOVA E.L.AVERBUKH A.A.KURKIN length scale what is important because time scales of many phys- [email protected] ical events (such as A-B transition) cannot be reached by state-of- the-art unrestrained all-atomic simulation. Secondly, within the Cassis, one of the top places in the south France, is a nice model an analytical description is possible, so nonlinear localized touristy town located near Marseille, France. One of its sightsee- phenomena can be thoroughly investigated both numerically and ing’s, the cliff of Cape Canaille is 390 m high, and is the high- analytically. est coastal cliff in Europe. Due to global warming a sea level will rise and will probably enhance the slow erosion of the marl In our model every nucleotide consists of 6 grains: three – for masses located at the foot of the cliff, causing the overhanging of backbone and three – for base. The first version of our model the cliff. The estimated volume of the slide is from 3 000 000 m3 proved to simulate stable B-form of DNA and to obtain MD tra- to 12 000 000 m3 of sandstones. This may cause a local tsunami jectories for rather long sequences (up to several hundreds of nu- which may become a hazard to the citizens of Cassis Different cleotides). But it also proved to have several shortcomings con- scenarios of this hypothetic event was modeled and analyzed. cerning stacking interactions, backbone electrostatics and van- The result and estimation of the tsunami consequences for Cassis der-Waals interaction parameters. and its vicinities are presented. In the framework of the parm99 AMBER force-field, the For the first scenario an axisymmetric, parabolic disturbance stacking interactions between the base pairs were investigated. A of water surface was used. This formula has been used by many very strong sequence dependence of these interactions was dis- researchers to describe the waves generated by explosions in wa- covered so that choosing of the potentials between the grains of ter, explosive eruptions of underwater volcanoes and asteroids CG model which would simulate the ’mean’ interaction between striking the ocean. Also the collapse of several blocks was ana- the base pairs failed. So for calculation of interaction between lyzed. Seven identical blocks was considered. The third scenario neighbouring bases the AMBER force-field was kept. Such ap- is cliff falling in a form of an avalanche of rock debris. In all proach enables to keep sequence dependence of these interactions studies it was considered average initial volume of 6 000 000 m3. and slightly increases computational demands. According to obtained results all three scenarios give us approximately the same picture. Only in the case of several blocks falling The parameters for electrostatics and van-der-Waals interac- the generated wave propagates a little faster. In all cases the wave tions between grains were chosen so as to simulate all-atomic amplitude in the port of Cassis fluctuating between 2.7-3.1 m. interaction between all atoms of these grains. Additional analy- Time of the first wave is 3-4 minutes. For all cases significant sis of electrostatics has shown that in our model the Generalized surging (up to 0,5-0,8 m) of sea level will last more than 10 min- Born approximation of salted solution fails to reproduce effects utes and then will decrease slowly. responsible for stability of the A-form. More thorough modeling For the next step the first scenario was studied with different of specific interaction between water molecules, counterions and volume of initial slide. It was estimated that the maximum wave DNA is necessary. height will be in a Cassis harbor if initial slide is average volume (6-7 millions of cubic meters). For the CG model with preliminary parameters, we have built It should be noted that in all cases wave heights will be enough the dispersion curve. It includes 36 branches. Only two of them to suspect flooding of the Cassis harbor. Also the first wave time are the acoustic ones and describe the dispersion of the longitu- is very short (3-4 minutes after cliff falling) and the wave height dinal and torsion waves of the double helix. The branches allow is significant. So this local tsunami that can be generated by the determination of the two sound velocities: for longitudinal (1923 collapse of Cape Canaille can be dangerous for the Cassis and m/s) and torsional (855 m/s) long-wavelength phonons. The ob- people there. The wave can reach Cassis and flood the port where tained results are in good accordance with the experimental val- a lot of yachts are lying alongside and a lot of people have rest in ues. a great number of cafes and restaurants. A program was elaborated for simulating of the DNA within Olga E. Khvostova, Minina St, 24, Nighny Novgorod, 603950, the model. Its calculations are 30 times faster than the most ef- Russia ficient programs for all-atomic simulations. Also it enables to simulate rather long sequences. The further investigations are aimed at developing of electrostatics interactions and investigat- NEW COARSE-GRAIN MODEL FOR ANALYSIS OF ing the electrostatic origin of A-form stability and at study of the LOCALIZED NONLINEAR EXCITATIONS IN DNA essentially nonlinear effects.

I.P.KIKOT E.A.ZUBOVA M.A.MAZO The research was partially supported by Division of Chemistry N.A.KOVALEVA A.V.SAVIN L.I.MANEVICH and Materials Technologies of RAS (grant program N3), RFBR A.ONUFRIEV (grant 08-04-91118-a) and CRDF (grant RUB2-2920-MO-07). [email protected]

New coarse-grained model of DNA is proposed which is built on the analysis of intramolecular mobility. Such approach enables to get reliable data about multiple physical phenomena – Irina P. Kikot, Kosygina, 4, Moscow, 119991, Russia

51 COULOMB LAW IN THE GENERALIZED THE EXPERIMENTAL DETERMINATION OF THE DIFFERENTIAL FORM IN PROBLEM ABOUT COEFFICIENTS OF THE COUPLED TORSION OSCILLATIONS OF THE CYLINDRICAL TWO-DIMENSIONAL MODEL OF THE COMBINED ROD ON THE INFINITE BELT DRY FRICTION

A.A.KIREENKOV A.KIREENKOV S.SEMENDYAEV [email protected] [email protected] [email protected]

It is proposed the coupled exact integral model of the sliding It is presented dynamically coupled dry friction model which and spinning friction built on the supposing of the validity of the is described dynamics of the heavy disk on the rough plane under generalized Coulomb law in differential form for elementary area conditions of combined kinematics. inside of contact spot. It is shown that as in case of using classical The procedure of the models constructing is based on the well Coulomb law in differential form, the normal component of fric- known results from the theory of elasticity that tangent stresses tion force is zero. The friction force is direct opposite the relative lead to shift in the symmetric diagram of the normal contact velocities of sliding, but additional polynomials terms appear in stresses in the direction of sliding. The subsequent integration on expressions for friction force and torque. To escape using of the the contact spot of the net vector and torque differentials yields integral expressions in the equation of motion, the model based the exact couple integral model of the sliding and spinning fric- on Pade expansions of the first and second order are constructed. tion. To escape the double integrals calculation in the motion Additional polynomials terms which are appeared in expressions equations, the exact integral expressions are replaced by appro- for friction force and torque lead to arising self-exited torsion os- priate Pade expansions. cillations in the systems consist of the spring-supported cylindri- It is shown that the distortion of the symmetry in the distribu- cal rod one end of that is bearing to the infinite belt moved with tion of normal contact stresses in the case of circular contact sites the constant velocity. results in the appearance of the friction force component directed along the normal to the trajectory of the mass center of the rubbed Institute for Problems in Mechanics RAS, Prosp. Vernadskogo solids. 101, korp.1, 119526 Moscow, Russia Investigation of the sliding of the heavy rotating disk along the rough plane in assumption that friction laws are defined by the THE EXPERIMENTAL DETERMINATION OF THE models based on the Pade expansions shows that the disk mass COEFFICIENTS OF THE COUPLED center trajectory is declined from the straight line. TWO-DIMENSIONAL MODEL OF THE COMBINED DRY FRICTION Alexey A. Kireenkov, Prospekt Vernadskogo 101, korp.1, Moscow, 119526, Russia A.KIREENKOV S.SEMENDYAEV [email protected] STRUCTURED ELASTIC WAVES [email protected] A.P.KISELEV For the purpose of the adequacy checking of the offered theoretical models the experimental installation has been developed [email protected] for experimental definition of the dry friction force at rectilinear sliding of rubbing solids at the presence of spining. The talk presents a review of results generalizing traditional Installation allows the coefficients defining of the dry fiction plane waves which have a constant amplitude along the wave models based on Pade expansions for the various contact areas front. The most simple generalization in the scalar case in i(kx−ωt) of the rubbing solids. Moreover, the unit parameters give the the time-harmonic regime is Φ = ye , satisfying the 2 chance to make experiments for a wide range of changing of the Helmholtz equation Φxx + Φyy + k Φ = 0. Further, assum- kinematics parameters defining a power state at the various values ing that k = ω/a, where a is the velocity of longitudinal waves, of the normal pressure. we get a wavefield u = gradΦ for isotropic elasticity. It has the Experimental measurement of the friction was spent for the velocity of propagation typical to longitudinal waves but its am- steel disk consistently displaced from the centre of the rotating plitude varies along its wavefront and its polarization is anoma- steel platform to its edge, at various variants of loading. lous. Results of the spent experiments brightly demonstrate that at Generalizations of this elementary example to body waves in the conditions of the combined kinematics the real friction laws arbitrarily anisotropic elastic body, to surface and interface waves have principle differences from the classical Coulomb law. This etc. is discussed. The talk is based on the papers [1-6]. difference is especially strong when the instant centre of veloci- A support from the RFBR grant 08-01-00511 is acknowl- ties lies inside of contact spot. The behavior of the friction force edged. in the region of small velocities is similar to the viscous friction. The rest friction is escaping and any infinitesimal force parallel directed to the contact spot cause to sliding. REFERENCES

Sergey V. Semendyaev, Institutskii per. 9, Dolgoprudny, 141700, [1] A. P.Kiselev. Proc. Roy. Soc. London, Ser. A 460 3059-64 Russia (2004)

52 [2] A. P.Kiselev, A. M. Tagirdzhanov. Vestnik SPb Univ. Math. depends on the cell division, growth and displacements. Meso- 41(3) 278-81 (2008) scopic models consider formation of cell clusters, pattern forma- [3] A. P.Kiselev. Doklady Physics 53(1) 48-50 (2008) tion in the extracellular matter and development of the conducting system providing the mass and heat exchange. Macroscopic [4] D. F. Parker, A. P.Kiselev. Quart. J. Mech. Appl. Math. 62 models describe the growth of continuous media. The problem of 19-31 (2009) interconnection between the models is not fully solved yet. [5] A. P.Kiselev, G. A. Rogerson. Wave Motion 46 539-47 Here a comparative study of the growth curves and the sta- (2009) tistical data generalizing the growth patterns and rules in plants [6] A. P.Kiselev, D. F. Parker. Proc. Roy. Soc. London, Ser. A and animals are summarized. Some special features of the mi- (2010). croscopic and macroscopic growth models and their interconnec- tions (from a single-cell growth model to multicellular models) are discussed. A survey of the mathematical models describing Aleksei Kiselev, Steklov Mathematical Institute, St.Petersburg the surface and volumetric growth and remodeling (inner growth) Department and IPME RAS, Bolshoi 61, SPb, 199178, Russia is given. A mixture model of the growing body including the solid skeleton, intracellular, extracellular and transporting liquids is proposed. Growth of the skeleton is provided by the mass delivery with a liquid transported through the elements of the con- DERIVATION OF REDUCED ODE MODELS ductive system and therefore is considered as a phase transition. DESCRIBING COARSENING DYNAMICS IN THIN The body is considered as an open thermodynamic system. Heat LIQUID FILMS transfer (solar radiation, heat transfer, latent heat) and mass transfer (delivery of water and nutrients, water evaporation, gas ab- G.KITAVTSEV B.WAGNER sorption) are taken into consideration. The model is applied to [email protected] the plant shoot (1D model) and plant leaf (2D model) growth and the theoretical results are compared to the experimental data. In this talk the topic of reduced ODE models corresponding to a family of onedimensional lubrication equations derived by Munch et al. 06’ is addressed. This family describes the dewet- Natalya Kizilova, Svobody sq., 4, Kharkov, 61077, Ukraine ting process of nanoscopic thin liquid films on hydrophobized polymer substrates due to the presence of several intermolecular forces and takes account of different ranges of slip-lengths at the CONVECTION IN STEADY THERMAL STRATIFIED polymer substrate interface. Reduced ODE models derived from MEDIUM underlying lubrication equations allow for an efficient analytical and numerical investigation of the latest stage of the dewetting L.S.KLIMENKO D.V.LYUBIMOV T. P. LYUBIMOVA process: coarsening dynamics of the remaining droplets. We first give an asymptotical derivation of these models and use them to [email protected] investigate the influence of slip-length on the coarsening dynamics. In a so called strong-slip case we find a unique critical slip- Convective flows near bodies immersed in steady thermal length at which the direction for migration of droplets changes. In stratified fluid were investigated. When equilibrium is incredi- the second part of the talk we present a new geometric approach ble, i.e. when thermal conductivity of a body does not equal to which can be used for an alternative derivation and justification thermal conductivity of fluid, convective flow is expected to have of above reduced ODE models and is based on a center-manifold specific structure: horizontally oriented jet stream. The reason reduction recently applied by Mielke and Zelik 08’ to a certain for this phenomenon is damping effect of steady stratification on class of semilinear parabolic equations. vertical motion. It is expected that the motion will be weak when the tempera- Georgy Kitavtsev, Mohrenstr. 39, Berlin, 10117, Germany ture gradient is small. However an attempt to find the solution using Raleigh number series decomposition was unsuccessful. This paradox is similar to Whitehead one that exists in the problems of THERMODYNAMICS OF BIOLOGICAL GROWTH: homogeneous flow around a spherical body. Also it is impossible EXPERIMENTAL DATA, GENERAL REGULARITIES to find analytical solution as power series of a small parameter, AND MATHEMATICAL MODELS which is dimensionless difference between body heat conduction and liquid heat conduction. Thus the flow structure near the rigid N.KIZILOVA body was found numerically. [email protected] To generalize results we consider the convection near motionless isothermal horizontal cylinder, surrounded by fluid heated Biological growth has been studied in numerous experiments from above, and near cylinder of finite heat conduction, differ- and observations at different external mechanical and chemical ent from that of surrounding fluid. Using Fluent package it was conditions. Different mathematical models are proposed for un- shown that in both cases near the body convective flow has antic- derstanding biomechanics of growth, explanation of the mea- ipated structure: horizontally oriented jet stream. sured growth curves and growth control (cancer treatment, tissue engineering). Microscopic models consider growing bodies as complex nonuniform systems, whose development and properties Lyudmila S. Klimenko, Byukireva str. 15, Perm, 614990, Russia

53 In the present work, we demonstrate how deformation of the SELF-SIMILAR REGIMES OF SOLID PHASE layer in the space between filler particles leads to the appearance CONVERSIONS IN MEDIA OF VARIOUS RHEOLOGY of elongated fibers. Formation of the fiber is simulated using the finite element method by means of isotropic models known in the A.G.KNYAZEVA S.N.SOROKOVA literature (Proni series, the neo-Hookean description of the elastic [email protected] s [email protected] contribution of Koshi’s stress tensor, neo-Hookean binding). We consider the fragment of nanocomposites consisting of two It was shown by authors that coupling character of various rigid particles assuming that the particles are circular and hud- physical and chemical phenomena is principal for detecting of dled together that the adjacent near-surface layers coalesce. The conversion regimes in deformed media. Analytical estimations material’s fragment was undergoing large deformation. Finally were received for the rates and product temperature of some the solving of the problem showed that if the distance between regimes and numerical solutions of the problems on the propa- the centers of inclusions will increase twice, then the spacing be- gation of low temperature radical reactions, heavy metals aside tween the surfaces of particles can increase tenfold. Herein, form decomposition, metal-thermal reactions etc. It was ascertained of nanolayer will greatly change forming elongated fiber. that low conversion regimes (solid phase combustion) and quick The research was supported by the RFBR within the limits regime (solid phase detonation) are connected with the preva- of the project 09-08-00530-a and Fundamental research program lence of different energy transfer mechanism and various proof the Department of Power engineering, Mechanical Engineer- cesses interrelation. In this paper the rheology properties influ- ing, Mechanics and Control Processes of RAS “Tribological and ence are analyzed on the characteristics and existence conditions strength properties of structured materials and surface layers”. of low and quick regimes. The joint expansion method is used for problem solutions. The practical employments of found result are Lyudmila A. Komar, 1 Str. Acad. Korolev, Perm, 614013, Russia discussed.

Anna Knyazeva, pr. Academichaskii, Tomsk, 634021, Russia INVESTIGATION OF STRUCTURE-PHASE CHANGING IN TITANIUM CRYSTALLITE UNDER DIFFERENT QUANTITIES AVAILABLE FROM TWO-MICROFONE RATE MECHANICAL LOADING MEASUREMENT OF ACOUSTICAL INTENSITY. IV.S.KONOVALENKO D.S.KRYZHEVICH V. KOLYKHALIN S.NEGANOV K.P.ZOLNIKOV S.G.PSAKHIE [email protected] [email protected] [email protected] [email protected] [email protected] Measurement of energy flow in acoustical and vibrational fields on the detection of one linear field quantity (sound pres- Initiation and evolution of plastic deformation in hexagonal sure) and its spatial gradient, using two transducers. This paper close-packed phase of a titanium crystallite with grain boundary first rewiews the quantities available from two-microfone mea- under uniaxial tension is carried out on the base of molecular dy- surement of acoustical intensity:intensity and potential energy namics simulation. Influence of loading rate on structure-phase density. Then analogous, but approximate for structural intensity changing is studied. A correlation between dependences of atom are diven.Examples of measured energy densities and intensity in number with slip vector more than 0.06 lattice parameter and po- cylindrical oil tank are also given. tential energy changing on deformation is revealed. It is shown that increasing of loading rate leads to a growth of threshold de- Vitaliy Kolykhalin, Leninskiy prospeck, 95/2, 403 , Sankt- formation value under which local atomic displacements are gen- Petersburg, 198302, Russia erated in simulated crystallite.

Ivan S. Konovalenko, pr. Akademicheskii , Tomsk, 634021, Rus- FORMATION OF POLYMER NANOFIBER IN THE GAP sia BETWEEN FILLER PARTICLES IN DEFORMED ELASTOMERIC NANOCOMPOSITES ON STRENGTH PROPERTIES ESTIMATION OF L.A.KOMAR S.N.LEBEDEV POROUS CERAMICS COATING [email protected] IG.S.KONOVALENKO A.I.DMITRIEV A.YU.SMOLIN S.G.PSAKHIE Carbon black reinforced elastomers are able to withstand large deformation not suffering from any macroscopic damages. In this [email protected] [email protected] case, nanofibers with specific properties different from the prop- [email protected] [email protected] erties of an elastomeric matrix are formed. One explanation for this difference in the properties is a supposition concerning the A numerical approach to strength properties estimate of porous formation of nanolayers near the filler particle surface. When the ceramics coating was developed on the basis of movable cellular material experiences large deformations, just these layers endure automaton method. It was shown that strength properties of a the heavy load, and their structure and shape are changed in some coating depend on pore size and pore laying depth. In particu- way. lar, the increase of pore length and decrease of pore laying depth

54 resulted in loss of coating strength properties. It was revealed COMPUTER SIMULATION OF NANOTUBE BUCKLING that this dependence was a power one. The pore size range for AND POST-BUCKLING DEFORMATION WITH THE correctness of the revealed tendency was found. The approach VAN DER WAALS FORCES TAKEN INTO ACCOUNT developed can be used for investigation of strength properties of material surface layers on the basis of such parameters as pore S.N.KOROBEYNIKOV B.D.ANNIN V. V. ALEKHIN laing depth and pore size. Thus, the obtained result could be A.V.BABICHEV I.A.TSUKANOV helpful also in friction pair analysis. [email protected]

Igor Konovalenko, Akademicheski 2/4, Tomsk, 634021, Russia Numerical solution procedures based on time discretization of nonlinear equations of molecular dynamics/mechanics are developed for problems of quasi-static/dynamic deformation and buck- IDENTIFICATION OF STRAIN HARDENING CURVE ling of graphene-like nanostructures. The matrices and vectors of BY PORTABLE HARDNESS TESTER the molecular dynamics/mechanics equations are determined using the Morse potential of covalent (chemical) atomic bonds, fic- D.A.KONOVALOV E.N.PERUNOV S.V.SMIRNOV titious elastic bar elements for modeling bond-angle bending, and [email protected] [email protected] the Lennard-Jones potential for modeling the long-range van der [email protected] Waals forces. A stability criterion for solutions of nonlinear ordinary differential equations in a finite time interval is employed The mechanical properties of metals such as yield point and in the determination of the critical deformation parameters and strain hardening curve are the important characteristics of a ma- buckling modes of nanostructures using the equations of molec- terial for account of technological process of constructional ele- ular mechanics. The critical forces/parameters of the quasi-static ment manufacturing. On their basis it is also possible to make motion of nanostructures correspond to turning/bifurcation points evaluation of product maintenance exhaustion resource. of the integral curve of the fundamental solution (studied for sta- The evaluation of material properties of many constructions bility), and the critical forces/times of dynamics motion corre- can be carried out only with the help of non-destroying test meth- spond to quasi-bifurcation points of this curve. In both cases, the ods. The method of kinetic indentation concerns to one of such buckling modes of nanostructures are found by solving the gen- methods, when during introduction indentation in metal there is eralized eigenvalue/eigenvector problem. recording the indentation diagram. For constructions or details The developed procedures are implemented in the PIONER which are taking place in exploitation, it is impossible to apply code, which employs the molecular mechanics (MM) method to the equipment with the large loading efforts. For this purpose it solve nonlinear problems of deformation of graphene-like nanos- is necessary to use portable devices. tructures. This code was used to solve the problem of buckling In this connection the work was directed to the decision of an of a twisted carbon nanotube under quasi-static deformation. The actual problem of the operative quantitative evaluation of metal post-buckling equilibrium modes are determined by solving the material mechanical properties in conditions of long-term loading same problem in a dynamic formulation. It is shown that the without manufacturing from them of special samples for research modes of equilibrium configurations of the nanotube in the ini- realization. tial post-buckling deformation correspond to the buckling mode In the Institute of Engineering Science of Urals Branch of Rus- obtained both at the bifurcation point of the solution of the quasi- sian Academy of Sciences the new technique of definition of the static problem and at the quasi-bifurcation point of the solution of the dynamic problem. With a further increase in the twisting hardening curve, which was approximated by three-parametrical power function, and the initial yield point by the test results of angle, the scenarios of nanotube deformation can be different, three conic indenters with various cone angle indentation was de- depending on whether the van der Waals forces between the opposite walls of the nanotube are taken into account or not. If the veloped. The technique allows to investigate the metal materials with the expressed yield point. The technique was passed van der Waals forces are ignored, the deformed nanotube config- verification in laboratory conditions and was adapted to portable uration has the shape of a twisted candy wrapper with the axis close to a straight line. If these forces are taken into account, the hardness-testing machine TEST - MINI-(UT) with the maximal effort of indentation equals 10H. The software for test result pro- deformed nanotube configuration has the shape of a twisted rib- cessing was created. An experimental pattern of the precision bon whose axis is also twisted. This problem is solved also by the molecular dynamics (MD) method, and the results of com- mobile measuring and computing complex of non-destroying control of objects by the kinetic indentation method with the max- puter simulation of the post-buckling nanotube deformation are imal effort equals 25H was also created. close to the results obtained by the MM method. However, in the MD method, there is no indicator for determining the criti- The developed method was used at two enterprises for the eval- cal forces/times and the buckling modes of the nanotube can be uation of the degradation strength properties of shell construction obtained only by direct numerical calculations. materials and welded connections in conditions of a long storage and operation. The MD method was used to solve the problem of buckling of a compressed two-walled axial nanotube. In this case, the van This work was performed together with Institute of Organic der Waals forces serve not only to prevent the self-intersection of Synthesis Ural branch of Russian Academy of Sciences. the nanotube walls during developed post-buckling deformation of the nanotube, but also to join the nanotube walls during their Dmitry A. Konovalov, 34, Komsomolskaya, Ekaterinburg, simultaneous deformation. 620219, Russia Acknowledgements. The supports from the Russian Founda-

55 tions for Basic Research (Grant No. 09-08-00684) and the Pro- demonstrate analytically and numerically that in a system of two gram of Basic Research of the Russian Acad. of Sci. (Project No. weakly coupled pendulums or oscillators, linear or nonlinear, an 2.13.5) are gratefully acknowledged. efficient irreversible transfer of vibration energy from one subsystem to another occurs when the coupled subsystems pass through Sergey N. Korobeynikov, Lavrentyev avenue, 15, Novosibirsk, the internal resonance. The internal resonance takes place due to 630090, Russia parametric drive when the length, mass or spring stiffness of at least one of the pendulums or oscillators varies in the course of FEM SIMULATION OF DAMAGE ACCUMULATION vibrations. Nonlinear effects result in a separatrix mode of vi- AND FATIGUE CRACK GROWTH IN ELASTIC bration energy transfer, in the vicinity of which the irreversible MATERIALS character of the energy transfer is enhanced. Our simulations also show that the modified Landau-Zener-like evolution equa- I.K.KOROLEV S.V.PETINOV A.B.FREIDIN tions can give quantitative prediction for the part of initial vibration energy, which is retained asymptotically in the protected [email protected] subsystem. This means that surprisingly the regularities of quantum Landau-Zener tunneling can be experimentally studied with A FEM-based procedure of fatigue crack growth simulation the use of classical systems. in the field of progressive damage is developed. A new finite element grid is suggested in this procedure to consider fatigue damage accumulation and crack growth based on a unified point Yuriy A. Kosevich, Kosygin str. 4, Moscow, 119991, Russia approach. The grid design suggested allows saving information on accumulated damage and “natural” tracing of growing crack, which is timely and laborious using the standard FE-procedures. With application of the assumed grid fatigue behavior under NUMERICAL EVALUATION OF ASYMPTOTE cyclic loading of a thin rectangular plate with initial randomly EXPONENTS AT SINGULAR POINTS IN ANISOTROPIC distributed defects is analyzed. Defects in material structure are COMPOSITES modeled by elements with negligibly small stiffness. The fatigue V. KOSHELEV properties are described by the Basquin equation. The trajectories of growing cracks and the damage accumulation in the plate [email protected] are presented at different phases of fatigue life. The interest in accounting for the internal structure of a ma- Igor Korolev, V.O., Bolshoj pr., 61 , St. Petersburg, 199178 , terial stimulates developing methods for the accurate evaluation Russia of fields near singular points in structural elements. They are corners of compound grains or inclusions, crack tips, kink and MD SIMULATION OF HIGH SPEED ROTATING notch points etc. Generally, in 2D case the common vertex of a PARTICLE COLLISIONS system of inhomogeneous wedges serves as a proper model of the singular points. Formerly (Blinova and Linkov, 1995; Linkov S.YU.KOROSTELEV A.YU.SMOLIN N.V.ROMAN and Koshelev, 2006), the method of stress exponent’s evaluation [email protected] [email protected] has been worked out for an arbitrary system of isotropic elastic wedges under anti-plane or plane strain/stress conditions. Now Collision of nanosized copper particles under various loading we suggest the generalized approach for numerical analysis of conditions are investigated using molecular dynamics simulation. elastic asymptote behaviour at joints of anisotropic wedges. Both Special attention was paid to the influence of particle rotation on the plane and anti-plane deformation cases are under considera- the crystalline structure change. It is shown that as a result of tion. Anti-plane elastic equations are also applicable in the prob- the interaction within the particles there is a change of the ini- lem of 2D stationary potential flow of fluid, electricity or heat in tial ideal crystalline structure, and the shape of the final particle composites with inhomogeneous inclusions. As a starting point is not symmetric. Change of potential energy of nanoparticles, of the analysis we utilize a well-known (Bogy, 1972) solution for structural transitions from one type of a crystalline lattice to an- the single infinite wedge under prescribed tractions (or fluxes) other, and formation of fragments of the material which is being at the boundaries. It is presented in the form of relations be- in a quasi-amorphous state are analyzed in details. tween Mellin’s transforms of tractions and displacements at the boundaries. Thereafter, successive consideration of the compati- Sergey Y. Korostelev, Pr. Academichesky 2/4, Tomsk, 634021, bility and equilibrium conditions at the boundaries of the wedges Russia leads to the system of three-point matrix difference equations. The physically acceptable roots of the determinant of the system present the asymptote exponents at the singular point. The ap- EFFICIENT IRREVERSIBLE TRANSFER OF proach has been implemented in computational codes. Numerical VIBRATION ENERGY IN LINEAR AND NONLINEAR results for the case of orthotropic wedges illustrate the potential COUPLED PARAMETRIC SYSTEMS of the method. For the two-wedge systems they are compared YU.A.KOSEVICH L.I.MANEVITCH with the data published by other authors. [email protected] Vadim Koshelev, 61, Bolshoi Prospect, Saint-Petersburg, 199178, We present a novel principle of vibration energy trapping. We Russia

56 average surface is equal to zero the equations of motion and ther- ON THE INFLUENCE OF GRAVITY FORCE ON mal conductivity have solutions generated by scalar potentials. ELASTIC WAVE PROCESS IN AN IDEAL GAS A special class of boundary conditions that permit one to re- D.P.KOUZOV duce the considered boundary value problem to a system of linear algebraic equations and the auxiliary problem for the wave equa- [email protected] tion is found. Analytical solutions of the latter are well known for the canonical bodies and can be represented in the terms of The influence of gravity force on elastic wave process in an tabulated special functions. Solution of the considered inhomo- ideal gas is considered. The simplest linear barotropic energet- geneous problem corresponding to transient oscillation can be ob- ically closed model is used to describe such process. The wave tained in the form of spectral expansions in biorthogonal system equation acquires a linear correction term which takes into ac- of eigenfunctions. count both the gravity force and the variation of gas density with A numerical example for an elliptical thermoviscoelastic height caused by it. Taking typical wave process (plane harmonic, three-layered plate is considered. The distribution of complex- spherical harmonic and non stationary spherical wave) as exam- valued eigenvalues are studied. Negative real parts of eigenval- ples it is shown what changes with correction introduced into the ues characterize the damping of natural vibrations caused by the analytical representation of this process and how this is reflected intrinsic and thermal dissipation. The expansion of the dynamic in their character. response of the plate under the impulsive loading is obtained. The vibrations and stresses caused by impulses of various profiles are Daniil Kouzov, Institute of Mechanical Engineering, Vasilievsky numerically studied. Ostrov, Bolshoy Prospect 61, St.Petersburg, 199178, Russia The research was supported by the RFBR (08-01-00553-a, 09-08-01180-a, 09-08-01194-a).

TRANSIENT OSCILLATION OF THE ELLIPTIC THERMOVISCOELASTIC PLATES Vladimir A. Kovalev, Sretenka st. .28, Moscow, 107045, Russia

V.A.KOVALEV vlad [email protected] LOCALIZED AND PERIODIC SOLUTIONS OF THE OREGONATOR MODEL OF BZ REACTION Development of the analytical methods for determining the M.KOVALEVA L. MANEVITCH stress strain state and transient vibrations of three-layer thin- walled structures are of theoretical and practical importance. In [email protected] point of fact the multilayered thin-walled structures have less weight for given rigidity in comparison with homogeneous struc- Belousov-Zhabotinsky Reaction is one of the well studied non- tures. In addition the middle layer can perform structural feature, linear phenomena. Many different regimes for this system were not related to hardness, such as temperature insulation. seen in experiments, found numerically, but not properly analyt- We consider three-layer thin-walled structures under the as- ically studied. In this work we present an asymptotical study of sumption that the thickness of the inner layer is sufficiently the localized and space periodic solutions for the system. greater than the thickness of the outer layers. The kinematic rela- We study the classical model of the Belousov-Zhabotinsky Re- tions for the inner layer are in the form of Mindlin-Reissner shell action named Oregonator for the unstirred system. Diffusion of theory, for the outer layers are in the form of membrane theory. the components is taken in account. The stoichiometric factor is The deformations of the whole layered structure obey the poly- used as a controlling parameter. Using the smallness of reaction line hypothesis. The material of outer layers is suppose to be rates and constants of the system we modify the original system thermoelastic isotropic, whereas that of inner one being isotropic and reduce it to a one representative equation. It describes the thermoviscoelastic. The constitutive law for the material of the evolution of the concentration of one of the reaction species with middle layer is in the common form of integral equation. time in one space dimension. To find stationary in time solutions A variational principle of convolution type for abovemen- we make a time derivative equal to zero and obtain an equation of tioned three-layer structures is proposed. It can be proved that the nonlinear oscillator with a second derivative in space variable. the three dimensional equations of motion and heat conduction A phase plane for this equation in a wide range of parameters generate the differential operators, which are self-adjoint with re- was studied, quite various behavior of the system was found. For spect to the convolution bilinear form. This permits one to ob- a narrow range of the controlling parameter we found the exis- tain a well-defined variational statement for linear dynamic ther- tence of a localized soliton-type solution corresponding the sepa- moviscoelasticity of the media with internal constraints, partic- ratrix trajectory on the phase plane. It separates the region on the ularly plates and shells. It is important to note, that in the con- phase plane corresponding to periodical in space time-stationary trast with conventional variational principles the initial conditions solutions and the region of unperiodical solutions. yield from the convolution variational statement in the same way To prove the stability of the solutions we made a series of nu- as boundary conditions. merical simulations of the effective evolution equation and the The variational technique is utilized to derive the equations of original system. motion and heat conduction as well as appropriate boundary and initial conditions. The boundary conditions are considered in the most general form, corresponding to elastic fixing of thin-walled Margarita A. Kovaleva, GSP-1,Leninskie Gory,1, Moscow, structures on the boundary. It is shown that if the curvature of the 119991, Russia

57 In this paper, the nonlinear stresses are constructed by averag- NONLINEAR VIBRATIONS OF THE DISTRIBUTED ing stresses in the micro-model over the nonequilibrium angular ROTOR FROM A MATERIAL WITH distribution, which is determined by the solution of the kinetic NONLINEAR-HEREDITARY PROPERTIES Fokker - Plank equation. Stresses σ[ij] are of relaxation nature A.U.KOVERGA E.P.KUBYSHKIN and disappear in the limit of small relaxation times of orientational order. Nonlinear moments Γ are constructed directly from [email protected] the kinetic equation. Relationship between moments and stresses Γi = (σij σji)nj enables in addition to define the coefficient − The equations of the ideal distributed rotor (shaft) from a mate- of rotational viscosity γ1. rial with nonlinear-hereditary properties (J.N.Rabotnov’s model) It is shown that nonlinear relaxation effects should be taken which ends lean against bearings are received. Stability of the into account in calculating acoustic flows in anisotropic liquids, longitudinal form of balance of a rotor is investigated. In space exposed to sound compression and shear in viscous waves of the of parameters areas of stability of a rotor which can have rather same frequency. We analyze situations in which the contribution difficult appearance are constructed. Type of instability is inves- of nonlinear relaxation effects to the formation of streams is large tigated. It is shown that at stability loss can arise or periodic solu- compared with convective stresses σij = ρ vi vj , responsible tions (precession), or invariant torus (beating modes). Influence for acoustical flow in isotropic liquids. Nonlinear− hydrodynamic of periodic vibration of a support on dynamics of a rotor is stud- equation constructed here were also used to analyze the deforma- ied. It is shown that periodic vibration of a support can lead to tion of homeotropic nematic structure under ultrasound. occurrence of chaotic fluctuations of a rotor (strange attractors).

Alexander Y. Koverga, Sovetskaya st., 14, Yaroslavl, 150000, REFERENCES Russia [1] E. N. Kozhevnikov // Akust. Journal. 1994. V.40. P.613-618. NONLINEAR STATIONARY HYDRODYNAMICS OF [2] A. I. Semenov // Jetf. 1983. V.85. No (8).P.549-460. ANISOTROPIC LIQUIDS

E.KOZHEVNIKOV Evgeny N. Kozhevnikov, Ac. Pavlov st., 1, Samara, 443011, Rus- sia [email protected]

Molecular-kinetic approach to the construction of nonlinear SPATIALLY MODULATED STRUCTURE IN A NEMATIC hydrodynamic description of liquid crystals in the field of sound LIQUID CRYSTAL LAYER and viscous waves is proposed. The basis of the constructions is E.N.KOZHEVNIKOV YA.V.KUCHERENKO the molecular model of the nematic [1], in which all processes in a liquid medium is described on the conception of a single elon- [email protected] gated molecule behavior with its long axis orientation L. The interaction of particles with the environment is modeled by viscous Periodic shear of boundary plate of homeotropic nematic liq- stresses in the fluid surrounding particles, and viscous moments uid crystal (NLC) layer leads to formation of a spatially modu- acting on the particle from the environment. Forceful interaction lated structure, i.e. domains. The appearance of domains in a of particles is described by a self-consistent field with a poten- homeotropic NLC layer was first observed and studied experi- tial E = E(Ln). Macroscopic stresses and moments are con- mentally for frequency range 100 Hz-40 kHz and for layer thick- structed by averaging microscopic values over the angular dis- nesses h=20-200 mkm [1]. In these experiment the moment of tribution of molecules orientations. Evolution of the density of the domain formation was detected by the appearance of light molecules axis angular distribution under the action of periodic diffraction. At frequencies above 1000 Hz the threshold ampli- compression and shear is described by the Fokker-Planck equa- tude uth was found to be inversely proportional to the frequency tion. Averaging micro-moments and micro-stresses on the equi- ω of the shearing action. librium angular distribution was discussed earlier and allowed to Theoretical picture of the effect is the following. Periodic build the viscoelastic stresses in nematic liquid crystals [1] and shear deformation of a medium in a layer causes the deflection suspensions (for ex.[2]). Viscosity coefficients in those strains of molecules from its initial orientation and leads to occurrence (Leslie coefficients) do not depend on the deformation degree of vortex oscillating stream. Delay of vortex velocity from shear the medium. flow results in stationary stresses and moments which increase an In fact, values of the viscosity coefficients Leslie αk(k = 1..6) initial deflection of molecules. At the threshold of the effect sta- are determined by the degree of orientational ordering of a ne- tionary moment are stabilized by Frank elastic moments. Above matic crystal. Change in orientational order under compression the effect threshold spatially modulated distortions of structure and shear deformation results in changes of values αk. The re- arise of a crystal. laxation delay of structural changes and accordingly, changes in Vortex motion delay from shear motion play the main role in viscosity coefficients of δα due to deformations in the acoustic domain formation. It has the different nature at different frequen- and viscous waves should lead to a stationary nonlinear stresses cies. On high frequencies the delay is defined by inertia of media. of the type σij Re εpq(δα) (δεpq) vij quadratic in the In this case dependence of threshold amplitude on frequency and ∼ · 2 −1 amplitude of the impact. Here εpq are components of the strain a thickness of a layer is the following uth (ωh ) [2] and ∼ tensor, vpq =ε ˙pq, the line above denotes averaging over the pe- the size of domains is d h. With frequency decrease the inertia riod of sound oscillations. of the medium become∼ small and does not prevail in the effect.

58 In this frequency range the formation of domains in homeotropic rotation axis becomes oriented along the cavity one. The transi- NLC layer is caused by boundary effects arising in orientation tions on the boundaries of the resonant area are analogous to [1]. waves near the bounding plates. In these case, the dependence of Inconstancy of the sphere rotation speed may, however, persist the threshold amplitude on frequency and layer thickness changes in conditions of the vibrational resonance; in this case the speed 2 −1/4 an is the following u0 (ωh ) . In both above cases the variation range substantially narrows. ∼ angle of molecular deflection is small and the description of the The liquid between the body and the cavity end-walls moves effect on the base of the linear hydrodynamic is valid. [3]. At as a continuation of the sphere, forming a liquid cylinder, similar further frequency decrease the time delay of votex is caused by to the Taylor–Proudman column [2]. Light-diffusing particles in elastic delay of molecules deflection angle from shear strain. In liquid make it possible to see the boundary of the column. On this this range of frequencies the angle of molecules deflection in os- boundary, an azimuthal wave propagates, the crests of which are cillating flow is not small θ > 1 and linear approach is not valid elongated from one of the cavity end-walls to the other. When for the description of the effect. Theoretical analyze of the effect the sphere rotation axis deflects from the cavity axis, the liquid at such low frequencies is not yet carried out. column deflects as well. The present paper is devoted to numerical analyze of the do- The research is supported by RFBR (grant 09-01-00665a) and main structure formation in a nematic liquid crystal layer under administration of PSPU (grant 2010). low-frequency periodic Couette flow. The analyze is based on nonlinear hydrodynamic equations of nematics. It is shown that nonlinear range of the effect the threshold of the effect become REFERENCES independent on the shear frequency. The threshold amplitude of shear is uth 2h, and the size of domains is estimated by size [1] V.G. Kozlov, N. V.Kozlov. Vibrational Dynamics of a Light ∼ d h. Body in a Liquid-Filled Rotating Cylinder // Fluid Dynamics. ∼ 2008. Vol. 43. No 1. P. 9–19. [2] H. P.Greenspan. The Theory of Rotating Fluids. Cambridge, REFERENCES University Press, 1968.

[1] G. N. Belova, E. I. Remizova //Acoustic journal. 1985. V.31. P.298-295. Nikolay V. Kozlov, 24, Sibirskaya av., Perm, 614990, Russia [2] E. N. Kozhevnikov, Ya. V.Kucherenko //Acoustic journal. 2006. V. 52. No 6. P.799-804. HEAT TRANSFER AND OVERCRITICAL [3] E. N. Kozhevnikov //Crystallography. 2005. No 5. V.50. CONVECTION IN ROTATING HORIZONTAL ANNULUS P.908-914 V.G.KOZLOV A.A.VJATKIN R.R.SABIROV [email protected] [email protected] Yana V. Kucherenko, Ak. Pavlova, 1, Samara, 443011, Russia Thermal convection of liquid in the thin horizontal annulus uniformly rotating about its axis is experimentally studied. The VIBRATIONAL DYNAMICS OF A SPHERICAL BODY IN heat transfer and the structure of convective flows depending A ROTATING CYLINDER WITH LIQUID on the rotation velocity and temperature difference between the boundaries of the annulus is investigated. N.V.KOZLOV S.V.SUBBOTIN It is revealed [1] that thermal convection in the rotating hor- [email protected] izontal annulus excites at the higher temperature of the exterior boundary and interior one, too. In the case of the exterior heating The behaviour of a light spherical body is experimentally the convection appears due to centrifugal force and is determined investigated in a horizontal cylindrical cavity, filled with low- by centrifugal Rayleigh number. In the case of the interior heat- viscosity liquid and rotating around its axis. The axis of vibra- ing the centrifugal force plays stabilizing role, nevertheless the tions is normal to the cavity rotation axis, frequency and ampli- thermal convection arises. It is determined by thermovibrational tude of vibrations are varied. mechanism [2,3]; the average convection is caused by oscillations In the absence of vibrations at rotation the body lags behind the of nonisothermal fluid in the cavity frame induced by the gravity cavity. Its behaviour is unstable: the body rotation speed changes force. with time in a complex manner. At the rotation frequency in- The scope of this research is to study the case of interior heat- crease the body average speed relative to the cavity decreases, ing when the centrifugal force plays stabilizing role. The con- the limits of the speed variation extend. At certain conditions the vection is excited in a threshold manner at decreasing of rotation rotation axis of the sphere deflects from the cavity rotation axis, velocity and is followed by critical growth of a heat flux through the new position of the body is stationary. the annulus. Different regimes of convection are observed. At The vibrations influence the body behaviour in resonant areas, rather fast rotation the spatially periodic system of thermovibra- where the amplitude of its oscillations significantly increases. In tional convective rolls parallel to the axis of rotation is excited. the experiment a resonant excitation of the outstripping motion is The wavelength of this rolls is large, it exceeds the layer thick- observed, which is analogous to the one found in the experiments ness few times. At lower rotation (in overcritical domain) the with cylindrical body [1]. Vibrations stabilize the behaviour of system of the rolls oriented perpendicularly to the axis of rotation the sphere: it displaces in the center of the cavity length and its appears, it is followed by the critical change of heat transfer, as

59 well. The spatial period of these transversal rolls is comparable This way of modeling is reasonable because of the physical back- with the annulus thickness and is small compared to the longwise ground of the basic elements. One major problem especially in rolls size. At very slow rotation the system of crescent rolls elon- creep problems is the large number of these elements needed to gated in a vertical plane appears. These rolls are similar to the cover a wide relaxation spectrum. Introduction of fractional ele- gravitational convective structures in the immovable horizontal ments leads to significant reduction of parameters to model sev- annulus. eral time decades. The substitution of dashpots through fractional This work is supported by the Russian Foundation for Basic elements changes the order of the resulting differential equations Research and Academic council of PSPU (the grant of 2010). from whole numbers to fractional ones. This change in the underlying differential equations leads to changes of properties which in turn influence techniques to gain a solution. Here the rheo- REFERENCES logical standard model is used as an example to demonstrate the disposal of a fractional element to simulate a one-dimensional [1] A. A. Vjatkin, A. A. Ivanova, V.G. Kozlov. Convective stabil- tensile creep experiment. The results of the fractional model are ity of a nonisothermal fluid in a rotating horizontal coaxial compared to those of a generalized Kelvin-Voigt model for Ultra- gap // Fluid Dynamics. 2010. V. 45. N 1. P. 10–18. mid A3K. [2] A. A. Ivanova, V.G. Kozlov, V.V.Rylova. Thermal convection in a plane layer rotating about a horizontal axis // Fluid Carsten Kroener, Kurt-Mothes-Straße 1, Halle (Saale), 06120, Dynamics. 2003. V. 38. N 1. P. 9–17. Germany [3] V.G. Kozlov. Thermal vibrational convection in rotating cavities // Fluid Dynamics. 2004. V. 39. N 1. P. 3–11. NONLINEAR OSCILLATIONS OF THE DISTRIBUTED ROTOR FROM A MATERIAL WITH Alexei Vjatkin, Sibirskaya, 24, Perm, 614099, Russia NONLINEAR-HEREDITARY PROPERTIES.

E.P.KUBYSHKIN A.YU.KOVERGA MODELLING OF SOLIDS WITH MICROSTRUCTURE [email protected] [email protected] AT DIFFERENT SCALE USING PARTICLES The equations of the ideal distributed rotor (shaft) from a mate- A.M.KRIVTSOV rial with nonlinear-hereditary properties (J.N.Rabotnov’s model) [email protected] which ends lean against bearings are received. Stability of the longitudinal form of balance of a rotor is investigated. In space Materials and processes that fail to satisfy the continuity hy- of parameters areas of stability of a rotor which can have rather pothesis are discussed and analyzed. Method based on analyti- difficult appearance are constructed. Type of instability is inves- cal and computational consideration of dynamics of interacting tigated. It is shown that either periodic solutions (precession) or particles is considered as an effective alternative or complemen- invariant torus (beating modes) can arise at loss of stability. Influ- tation for the traditional continuum mechanics for such problems ence of periodic vibration of a support on dynamics of a rotor is investigation. Application of the particles dynamics for solution studied. It is shown that periodic vibration of a support can lead to of thermomechanical problems at different scale levels is con- occurrence of chaotic fluctuations of a rotor (strange attractors). sidered, starting from nanolevel (description of nanomechanical systems using molecular dynamics), mesolevel (deformation and Evgeny P. Kubyshkin, Sverdlova str. 43-A-61., Yaroslavl, fracture of solids with microstructure), macrolevel (astrophysical 150040, Russia processes modeling). The basic concepts allowing establish the link between microscopic parameters of the mathematical model and macroscopic parameters of the object under consideration are THE INFLUENCE OF EXTERNAL STRAINS AND outlined. Application of the presented methodology for the solu- MATERIAL PARAMETERS ON THE KINETICS OF tion of practical problems is shown. MOVING INTERFACES

D.O.KUCHEYEVA A.B.FREIDIN Anton M. Krivtsov, Institute for problems in mechanical engineering, Bolshoy pr., 61, V.O., St. Petersburg, 199178, Russia [email protected] [email protected]

We study plane stationary moving interfaces in elastic GENERALIZED FRACTIONAL ELEMENTS IN isotropic solids in a case of axially-symmetric deformations. RHEOLOGICAL MODELING OF POLYMER CREEP First, we construct phase transition zones (PTZ) formed by all strains which can exist on the stationary moving interfaces with a C.KROENER H.ALTENBACH K.NAUMENKO given interface velocity. We focus on the PTZ boundaries. Every A.KUTSCHKE point of the PTZ boundary corresponds to some interface spread- [email protected] ing with a given velocity, and the orientation of the interface depends on the strain state (the relationship between eigenstrains). Viscoelasticity of polymers ist often modeled with rheologi- We construct PTZ boundaries for various interface velocities and cal elements such as Newtonian dashpots and Hookean springs. various material parameters. This in turn allow us to study how

60 material parameters and external strain affect the interface veloc- REFERENCES ity and orientation. In the context of biomechanical applications we discuss how external loading and volume growth parameter [1] A. Gray. A Treatise on Gyrostatics and Rotational Motion. affect the orientation and kinetics of a growth plate in long bones. London. MacMilland and Co. 1918. [2] F. Hess. Boomerangs, Aerodynamics and Motion. PhD The- sis, Groningen, 1975. Daria Kucheyeva, V.O., Bolshoj pr., 61 , St. Petersburg, 199178, [3] F. Hess. The aerodynamics of boomerangs. Scientific Amer- Russia ican, 219(5), 124-136, 1968. [4] M. Hubbard and A. L. Moore. How to throw a boomerang that returns. Proceedings of IX International Symposium on REGULARIZATION OF NUMERICAL SOLUTIONS OF Computer Simulation in Biomechanics, Sydney, Australia, ISOTHERMAL EHL PROBLEMS. STABLE SOLUTIONS 2-4 July, 2003. I.I.KUDISH [5] G. T. Walker. On Boomerangs. Philosophical Transactions of [email protected] the Royal Society of London. Series A, 190, 23-41, 1898. [6] J. Walker. Amateur Scientist – Boomerangs! How to make To obtain stable solutions of asymptotic equations in the inlet them and also now they fly. Scientific American, 240, 162- and exit zones of a heavily loaded isothermal EHL contact a regu- 172, March 1979. larization technique is employed. The approach follows from the [7] J. Walker. Amateur Scientist – More on Boomerangs includ- asymptotic analysis of a thermal EHL problem with high slide- ing their connection with the dimpled golf ball. Scientific to-roll ratio. This regularization provides stable numerical solu- American, 240, 180-189, April 1979. tions in a very wide range of parameters which covers the whole practical range of EHL parameters. The success in regularization of asymptotic equations is extended on the original equa- Alexander S. Kuleshov, Leninskie Gory, Moscow, 119991, Rus- tions of the isothermal EHL problem. The numerical scheme for sia regularized original equations of the isothermal EHL problem is developed. Some numerical examples of regularized solutions DESCRIPTION OF EQUIVALENT ELASTIC are presented. It is shown that the regularization approach works CONTINUUM FOR GRAPHENE LATTICE well and produces stable numerical solutions for very large dimensionless pressure viscosity coefficients up to 50. A.M.KUDAROVA A.M.KRIVTSOV [email protected] Ilya Kudish, 1700 University Avenue, Flint, 48504, USA The work contributes to the study of equivalent continuum model that is able to describe the elastic moduli of graphene — monolayer of carbon atoms. The moment interaction between the MATHEMATICAL MODEL OF THE BOOMERANG particles that form the lattice is described by four force constants and depends on the position of particles and their orientation in A.S.KULESHOV space. An approach is applied to link the macroscopic moduli [email protected] and the parameters of interaction at microlevel based on the comparison of the energy of the microstructure and the energy of the Most of people are familiar with the boomerang, the wonderful equivalent continuous plate. The comparison with structural me- stick that returns when you throw it. There is a whole number of chanics approach is carried out and the elastic moduli are derived publications devoted to the study of motion and aerodynamics of for the parameters of interaction taken from AMBER force field. a boomerang. The first serious study of a boomerang dynamics have been made by G.T. Walker [5]. In 1975 Felix Hess in his Asiya M. Kudarova, Anton M. Krivtsov, Institute for problems PhD thesis [2] (see also [3]) presented the most comprehensive in mechanical engineering, Bolshoy pr., 61, V.O., St. Petersburg, analysis of a boomerang motion. There are some other studies on 199178, Russia a boomerang flight [1,4,6,7]. However the most part of the cited papers used some heuris- tic considerations about the nature and character of the aerody- ON THE USE OF GRADIENT CONTINUA FOR namic forces acting on a boomerang during its flight. Therefore MINIMIZING THE WAVE REFLECTION AT A the problem of construction of an exact and rigorous mathemati- DISCRETE-CONTINUUM INTERFACE cal model of a boomerang motion is still important problem. A.M.KUDAROVA A. V. METRIKINE In this presentation we propose more exact mathematical model of a boomerang in comparison with the previous investi- [email protected], gations. We derive equations of motion of a boomerang using the [email protected] general theorems of dynamics and give the analytical and numerical investigation of these equations. We discuss also the influence This development contributes to the studies on the dynamic of various parameters of the problem on its dynamics. coupling of discrete systems and continua. The approach based

61 on the use of the Green’s function of the continua is employed. THE APPLICATION OF NEUBER’S RULE FOR The novelty of the presented research is in the use of a gradi- LIFETIME PREDICTION OF NOTCHED STRUCTURES ent model of the continuum instead of the classical one. The SUBJECTED TO COMPLEX LOADING enhanced continuum model is chosen for its ability to mimic the wave dispersion properties of discrete systems — a property A.KUTSCHKE K.NAUMENKO H.ALTENBACH that can not be granted by the classical model. The wave re- C.KROENER flection coefficient is studied in this paper at the interface of a one-dimensional chain of masses and springs and a correspond- [email protected] ing continuum. The continuum is modelled using both the classical and a gradient equations of motion. It is shown that the Neuber’s rule is an easy way to estimate the plastic deforma- gradient continuum with properly chosen parameters reduces the tion in the notch root with a simple elastic calculation. This con- reflection significantly. This is also confirmed by numerical sim- cept has been established in every day worklife of an engineer ulations. allready many years ago. But it is also known that Neuber him- self limited his concept for certain stress states, [1]. For a reliable lifetime prediction of sensitive structures for example with the Manson-Coffin approach engineers use Neuber’s Asiya M. Kudarova, Institute for problems in mechanical engi- rule to estimate the plastic deformation in the notch root, [2]. But neering, Bolshoy pr., 61, V.O., St. Petersburg, 199178, Russia as mentioned this is valid only for certain stress states. To generalize Neuber’s rule the assumption that the elastic work is equal to the plastic has to be extended. Andrei V. Metrikine, Faculty of Civil Engineering and Geo- Therefore the theoretical background of Neuber’s rule is pre- sciences, TU Delft, Netherlands sented and the equality for certain stress states is shown. The ideas of extension will be explained and their application to notched components with complex loading and a temperature NONLINEAR RESPONSE OF A FLEXIBLE MEMBER gradient will be presented. SUPPORTED NON-IDEALLY AT ONE END

S.KUMAR I.SHARMA P. WAHI REFERENCES [email protected] [1] Theory of Stress Concentration for Shear-Strained Prismat- ical Bodies With Arbitrary Nonlinear Stress-Strain Law. Some engineering structures are supported in the vertical di- Trans. ASME, 1961. rection on hydrostatic bearings to prevent metal-to-metal contact. [2] A constitutive model for anisotropic materials based on Neu- Such structures might undergo large vibrations in the transverse ber’s rule. Comput. Methods Appl. Mech. Engrg., 1992. direction when subjected to base excitation due to seismic activity. These structures can be modeled using a flexible member which is free at one end and has a non-ideal possibly non-smooth Andreas Kutschke, Kurt-Mothes-Straße 1, Halle (Saale), 06120, support at the other end. In this work, we analyze the response of Germany such system under base excitation. The reaction forces and moments applied by the hydrostatic bearing on the flexible member FINITE-ELEMENT SIMULATION OF ELASTIC are modeled using nonlinear springs distributed along the length INTERACTION OF POINT DEFECTS NEARBY of the member inside the bearing. The equations of motion are STRESS-FREE SURFACE IN CRYSTAL OF SILICON formulated using the Lagrangian formulation with assumed mode CARBIDE shapes. To validate the equations of motion obtained using the approximate formulation, the case of linear springs has been consid- S.KUKUSHKIN A.V.OSIPOV S.KUZMICHEV ered and the results for natural frequencies and mode shapes are compared against those obtained using a modal analysis. Three kukushkin [email protected], different sets of assumed modes have been used viz., the set of [email protected] polynomials, the Fourier series and the cantilever modes in con- junction with displacement and rotation at the support. The last The paper is devoted to the problem of elastic interaction of set of assumed modes has been found to be the best in the senses point defects nearby stress-free surface in crystal of silicon car- that the overall error in both the natural frequencies as well as the bide. The purpose of the paper is to investigate influence of mode shapes is the least. The equations of motion using these as- stress-free surface on elastic interaction between vacancy of sili- sumed mode shapes have been used for analyzing the nonlinear con and molecule of silicon carbide. response and interesting features have been obtained. The effect In last decade the concept of the elastic dipole has been devel- of the nonlinearity in the support springs on the amplitude of the oped by several authors, particularly Eshelby, Kroner and Nowick response of the flexible member is under consideration at the mo- and Heller. ment and will be discussed at length during the conference. In the context of this paper a stress and strain fields have been discovered using finite-element method. The elastic interaction energy between vacancy of silicon and molecule of silicon car- Pankaj Wahi, Indian Institute of Technology-Kanpur Campus, bide is derived and used to estimate the kinetics of migration of Kanpur, 208016, India these defects in the stress field in crystal of silicon carbide. The

62 influence of the stress-free surface which is of central importance constitutive relation for the Piola stress tensor: there exist the de- due to the long-range of the elastic interaction is included. vices in which the uniform deformation within the bounds men- In addition influence of crtystalographic directions on elastic tioned is realizable, while beyond the bounds there can be no sta- interaction of point defects is analyzed. It is shown that the elas- bility (the Basic Hadamard stability theorem). Thus, the moduli tic interaction energy is changed subject to crtystalographic di- tensors involved in the softening condition and in the Hadamard rections and the distance from stress-free surface. inequality, are different, but the relation between them is well- The obtained results are important for future research of known, and the current values of stress enter it. The achievable growth of thin silicon carbide films on silicon substrate. states of softening are the ones, for which the condition of softening (for corresponding moduli tensor) is compatible with the Sergey V. Kuzmichev, V.O., Bolshoj pr., 61 , St.Petersburg, Hadamard inequality (for another moduli tensor). 199178, Russia When analyzing the achievable extent of softening for the Tresca elastic-plastic materials, one should take into account the existence of edges of the yield surface. Note that the normals to SELECTIVE LAMB MODE EXCITATION BY COAXIAL the surface (in the strain space) at the points on the faces are the RING PIEZO-ACTUATORS symmetrized dyads, whereas the edges are characterized by that the normals to adjacent faces are co-axial symmetrized dyads. On O.V.KVASHA E.V.GLUSHKOV N.V.GLUSHKOVA the edges of yield surface there are four regimes of the material R.LAMMERING response: elastic unloading, two regimes of partial plastic load- [email protected] ing and that of full plastic loading. It is proved in the work that softening is impossible on the faces of the yield surface, the same An omnidirectional multi-element transducer composed of a being true for the regimes of partial plastic loading on the edges. system of coaxial ring-shaped piezoelectric elements actuated by It is proved also that there is an achievable finite extent of soft- n-cycled sine tone bursts is proposed. The transducer is capa- ening that corresponds to the regime of full plastic loading. The ble of generating a single required Lamb wave mode in an elas- achievable extent of softening mentioned is found analytically; it tic plate, suppressing all undesired ones. The mode selection is appears to depend on the quantities characterizing the full plastic achieved through a special choice of the amplitudes and time de- low on the edge. lays of input driving signals. The method for the determination of these parameters is based on the maximization of a certain functional expressing the contrast of the desired radiation in the limiting case of infinite number of excitation cycles. The efficiency Tatyana Borisovna Lavrova, acad. Pavlova, 1, Samara, 443011, of the approach proposed is illustrated by numerical examples. Russia

Oleg V. Kvasha, Stavropolskaya, 149, Krasnodar, 350040, Russia

DISCRETE MODEL OF NANOCRYSTAL WITH COMPLEX CRYSTAL LATTICE THE EXTENT OF SOFTENING, THAT CAN BE ACHIEVED UNDER STABLE DEFORMATION WITH O.S.LOBODA A.M.KRIVTSOV THE TRESCA ELASTIC-PLASTIC MATERIAL loboda [email protected] T.B.LAVROVA [email protected] The discrete model of nanocrystal with complex crystal lattice is considered. This lattices is not close-packed and is typical The question of realizability of the states of softening for the for elements with covalent bonds, for example diamond, graphite. Tresca elastic-plastic material under deformation testing is stud- Traditionally many-particle interaction potentials are used for de- ied theoretically. Realizability is identified with stability of the scription of such kind of lattices. The our alternative approach corresponding uniform deformation process. By hardening and considers rotatory degree of freedom and allows to take into ac- softening are meant some possible generalization of these notions count the moment contribution to interatomic interaction. The (introduced first by Drucker in the framework of linear solid me- particles interact using forces and moments. The characteristics chanics, where the initial stresses are considered negligible) for of interatomic bonds for crystals with diamond structure (carbon, the case of non-zero initial stresses. Specifically, the state for silicon, germanium) and zinc blende are determinated. The rela- which the quadratic form for the elastic-plastic moduli tensor set- tions between the elastic constants of cubic crystals are obtained. ting the Jaumann increment of the Cauchy stress, is positive defi- A comparison with the two-parameter power Bourne’s model is nite (can take negative values), is called that of hardening (soften- considered. Variation of the elastic properties of crystals, depending). Hardening is equivalent to positivity of all the six genuine ing on the bonds metallization and ionization is analyzed. (according to Rychlewski) moduli of such a forth-order tensor, whereas softening is equivalent to existence of at least one negative. It is known that the bounds of stable deformation of a solid are determined by the bounds of validity of the Hadamard inequal- Olga S. Loboda, Institute for problems in mechanical engineering ity for the tensor of elastic-plastic moduli setting the incremental RAS, Bolshoy pr., V.O., 61, Saint-Petersburg, 199178

63 EXPERIMENTAL STUDY AND THEORETICAL GENERALIZED SELF-CONSISTENCY ESHELBY DESCRIPTION OF THERMOVISCOHYPERELASTIC SOLUTION OF GRADIENT THERMOMECHANICS OF BEHAVIOR OF ELASTOMERS FILLED WITH STRUCTURED MATERIALS ( FILLED COMPOSITES, NANOPARTICLES CERAMICS)

E.V.LOMAKIN YU.P.ZEZIN T.A.BELIAKOVA S.LURIE D.VOLKOV-BOGORODSKII N.TUCHKOVA [email protected] [email protected] [email protected] [email protected] The filled elastomers are used widely in many branches of modern technique. The rubber members of different construc- The consistent and correct theory of interphase layer taking tions and machines can serve under the stressed state for a long into account scale effects (cohesion and adhesive interactions) is time. On many occasions the normal operation of the construc- used to model the effective characteristics for the structured medi- tion members of the filled elastomers is defined by the stress ums across the length-scale effects[1-5]. In the present work the level in the members that can change with the time as a result generalized methods of composite materials mechanics are elabo- of the relaxation processes. The intensity of the stress relaxation rated for estimation of the effective mechanical and thermophys- in the filled elastomers depends substantially on the temperature. ical properties of the structured materials with high density of the Many elastomers filled with nanoparticles exhibit the anomalous internal boundaries. dependence of the relaxation behavior on the temperature that Generalization of a method of the self-consistency averaging is displayed in non-monotonic dependence of relaxation modu- method on the gradient elasticity and heat conductivity models is lus on the temperature. Thus the traditional method of the time- given. The integral equation for an increment of energy is con- temperature superposition can not be used. The generalized de- structed at transition from a homogeneous cell to a cell with inclu- pendence of the relaxation modulus on the reduced time was ob- sion in frameworks gradient models under condition of a homo- tained. geneous field on infinity. Thus, the method of the self-consistency averaging for gradient models which gives the effective tool of The experimental studies of non-linear elastic and relaxation an estimation of properties of composites with inclusions of any properties of filled elastomers were carried out under different form is constructed. values of constant strain rate up to certain strain levels and subse- For spherical generally layered inclusion the algorithm of an quent relaxation. The results of experimental studies of hypere- analytical estimation of elasticity modulus in frameworks of the lastic and relaxation properties of filled elastomers under the con- gradient models, generalizing corresponding result of classical ditions of tension and compression at different strain levels were model is constructed. Thereby it is received gradient solutions analyzed, and the dependence of relaxation properties of elas- and the effective technique of an estimation of properties for the tomers on the strain level was discovered. This effect is caused wide class of the nanocomposites is offered. The analysis of de- by material damage that depends on the value of strain. gree of influence the gradient effects is given. An approach to the formulation of non-linear theory of ther- The problem about calculation of effective heat conductivity moviscoelasticity for the characterization of elastomers behavior in frameworks the gradient thermomechanics is separately con- under the finite deformation is proposed allowing taking into ac- sidered. It has been shown that in this case the closing integral count the dependence of the relaxation functions on the strain equation of Eshelby’s type is equivalent to a condition of unifor- level in the frame of the endochronic non-linear approach. The mity of a field in the effective environment as a whole. It has method for the determination of interrelated hyperelastic and rhe- allowed to formulate simple algorithm for an estimation of effec- ological characteristics of deformation of filled polymeric mate- tive characteristics on a method of three phases, reduced to the rials is proposed, based on the long-term values of stresses during solution of system of the linear equations of a small order (7x7). the process of relaxation. Using the generalized gradient model with specific adhesion scale The modification of finite element program is proposed that properties allow to model the Kapitsa types thermo-resistance ef- gives the possibility to describe geometrical and physical non- fects in crystal structures. linearity of viscoelastic materials more precisely without consid- Supported by the RFBR 09-01-00060, 09-01-13533, Program erable change in the certificated programs. As an example of of the RAS PR21. application and verification of proposed theory and numerical algorithm, the calculation of rubber absorber is performed which have the form of disk inserted into iron cylinder, and correspond- REFERENCES ing tests were carried out too, under similar conditions of the confined compression with constant strain rate and subsequent [1] A. A. Gusev, S. A. Lurie. Advanced Engineering Materials, stress relaxation under maximum level of strain. Quite satisfac- 2010 (in print) tory correspondence between the calculated values of axial load, [2] S. Lurie, D. Volkov-Bogorodsky, V.Zubov, N. Tuchkova. geometrical parameters of the sample and the results of experi- Comp. Mat. Science, V. 45, 3, P. 709-714, 2009. mental study is shown. [3] S. Lurie, P.Belov. Int.J.of Fracture, 50,(1-2), pp. 181-194, This work was supported by RFBR (grant 09-08-00502). 2008. [4] S. Lurie, P.Belov, D. Volkov-Bogorodsky, N. Tuchkova. J. of Mat. Scs.41:20, 6693-6707, 2006. Evgeny V. Lomakin, Leninskie Gory, Moscow State Lomonosov [5] S. Lurie, P.Belov, N. Tuchkova. Int. J. Comp Mater Sci, A., University, Moscow, 119991, Russia vol. 36(2), 145-152, 2005.

64 Sergey A. Lurie, Leninskii pr, 32a, Moscow, 119991, Russia gas compressibility. Darcy law in the form derived by Buckley- Leverett is used to model ﬁltration of liquid and gas. It is assumed that the phase change between the gas-hydrate and gas- EFFECT OF INERTIA TERMS ON QUALITATIVE water states occurs in thin intermediate layer. The water is con- BEHAVIOUR OF RIGID PLASTIC SOLUTIONS IN THE sidered to be incompressible. Physical parameters of the model VICINITY OF MAXIMUM FRICTION SURFACES correspond to the conditions of the permafrost. One of the goals E.LYAMINA S.ALEXANDROV of the study is to ﬁnd out the situation where directional motion of the methane bubbles occurs that subsequently leads to the ejec- [email protected] tion of gas into the environment. sergei [email protected] The work was supported by RFBR (grant 09-01-92505).

In the case of quasi-static flow of ductile materials a narrow Andrey O. Ivantsov, Academ. Koroleva, 1, Perm, 614013, Russia layer of intensive plastic deformation occurs in the very vicinity of frictional interfaces. A theoretical description of this feature of plastic deformation is possible by adopting a model of rigid NUMERICAL INVESTIGATION OF A DILUTION AND perfectly/plastic solids and the maximum friction law at the frictional interface. The maximum friction law states that the friction TRANSPORT OF HEAVY POLLUTANTS IN WATER stress at sliding is equal to the shear yield stress of the material. BODIES TAKING INTO ACCOUNT DENSITY STRATIFICATION EFFECTS It is known that the velocity field is singular in the vicinity of maximum friction surfaces. In particular, the equivalent strain T. P. LYUBIMOVA YA.N.PARSHAKOVA rate (second invariant of the strain rate tensor) approaches infinity. Thus such theoretical solutions are in qualitative agreement [email protected] [email protected] with the aforementioned experimental results. It is also known that there are other rigid plastic models (double shearing model, The paper deals with the 3-D numerical simulation of a di- double slip and rotation model, a class of viscoplastic models) lution and transport of highly mineralized brines in large water which lead to the same qualitative behaviour of solutions in the bodies taking into account density stratification effects. The prob- vicinity of maximum friction surfaces. On the other hand, there lem has an exceptional importance for ecological safety because are such rigid plastic models, for example the coaxial model of of the extension of salt depositions exploitation. Numerical sim- pressure-dependent plasticity, which do not lead to the same qual- ulation is carried out in the frame of non-stationary approach. itative solutions in the vicinity of maximum friction surfaces as K-epsilon model is used for the description of turbulent flows. rigid perfectly plastic solutions. Moreover, all of these studies Pollution distribution in the domain which contains one pollu- have been restricted to quasi-static flow. Therefore, it is of inter- tion source is studied. At the first step the concentration field is est to reveal an effect of inertia terms on the qualitative behaviour calculated in a small domain near admixture source. The data on of solutions in the vicinity of maximum friction surfaces. In the concentration and velocity fields obtained at the first step are used present paper, a generalization of the famous Prandtl’s problem to set the boundary conditions for the calculations for larger do- for compression of a plastic layer between two plates is given main. As the result the energy of turbulent pulsations was shown assuming that inertia terms cannot be neglected. The maximum to be decreased with density stratification decrease. For long dis- friction law is adopted at the frictional interfaces. Qualitative be- tances the pollutant is concentrated near the bottom just slightly haviour of the solution is discussed and compared to the quasi- changing its concentration. static case. The work was supported by Ural Brunch of RAS from the In- terdisciplinary project of fundamental research carried out in the Elena A. Lyamina, pr. Vernadskogo, Moscow, 119526, Russia Institutes of Ural Brunch of RAS in 2009-2011 and innovation project of Ural Brunch of RAS for young researchers. NUMERICAL MODELING OF GAS RELEASE PROCESS FROM METHANE HYDRATES Yanina Nikolaevna Parshakova, Koroleva, 1, Perm, 614013, Rus- sia D.V.LYUBIMOV T. P. LYUBIMOVA K.B.TSIBERKIN A.O.IVANTSOV NUMERICAL SOLUTION OF PROBLEM FUSION [email protected] [email protected] PENETRATION METAL, WHEN GLASS WELDS WITH [email protected] [email protected] METAL BY DIFFUSION WELDING

O.N.LYUBIMOVA K.N.PESTOV E.A.GRIDASOVA Methane hydrates are believed to reside throughout the globe in sea-floor sediments and permafrost that makes hydrates to be a [email protected] promising natural gas resource. The understanding how methane hydrates behave will help to develop the methods of harvesting This work will address the characteristic analysis of welding methane from hydrates. The motion and evolution (growth/decay and fusing glass-metal composite material particularly in relation and interface deformation) of a single bubble in porous medium to thermal conduction during intensive heating. Thermal field saturated by hydrate and the dynamics of chimney-like methane- data analysis is required to predict such things as material de- saturated channels penetrating the hydrate stability zone are stud- formations, potential stress points, diffusion process, structural ied taking into account the effects of porosity, gravity, viscos- changes in the heat-affected zone and ultimately to refine the op- ity, capillarity, hydrate formation/dissociation at the interface and erating the practices of welding. This article will demonstrate

65 a mathematical model which can take into account the conju- LIMITING PHASE TRAJECTORIES AND “WAVE – gate character of heat exchange in the system, latent heat ab- PARTICLE” DUALITY IN CLASSICAL PHYSICS sorption of melting during phase transformation and the temperature dependence of the thermophysical properties of the base and L.I.MANEVITCH weld material. Calculations and graphic representations results of the suggested model’s conclusions are compared with the quasi- [email protected] stationary solution of Leybenzona. It is well-known that the Classical Mechanics is based on the concept of point-like particles. If interparticle interaction is ab- Olga N. Lyubimova, Far East State Technical University, Neybuta sent (with the exception of collisions) such a concept allows not 57, 68, Vladivostok, 690010, Russia only to derive the equations of motion but also to solve them efficiently with use of the conservation laws. If the interparticle interaction is strong (e.g., in crystalline lattice) the language ON EXISTENCE AND UNIQUNESS OF SOLUTIONS IN of particles is not efficient when analyzing the equations of mo- SEVERAL BOUNDARY-VALUE PROBLEMS FOR THE tion. The reason is that in this case not the particles but collective EULER EQUATIONS (wave–like) motions – normal modes, solitons, and breathers can be considered as non-interacting or weakly interacting ones. So, A.E.MAMONTOV in Classical Physics two limiting cases corresponding to particle- mamontov [email protected] like or wave-like representation can be distinguished. However, it was not recognized that even in the framework of Classical Physics it is necessary to single out the important class of the We consider several boundary-value problems for the Euler systems in which both languages have to be used. Namely in this equations describing flows of an ideal incompressible fluid in a sense one can say about “wave – particle” duality outside Quan- bounded domain: the problem with non-penetration condition tum Mechanics. (the problem NP) at the boundary as well as so called through- flow problems describing flows of the fluid through the domain, The simplest example: two coupled identical nonlinear oscil- with different types of boundary conditions. Namely, three types lators. If the coupling is absent, they can be considered as the of through-flow problems are considered. In the first problem particles interacting only by collisions. In the case of strong cou- (denoted as TF-I) the vorticity is prescribed at the entrance (i.e., pling, the normal modes (in-phase and out-of-phase) do not in- at the part of the boundary where the fluid enters the domain). teract or interact weakly because their frequencies are strongly More exactly, in the three-dimensional flows we should prescribe different. Weakening of the coupling is accompanied by intensi- only the tangent components of the vorticity, and, in any dimen- fication of the intermodal interaction. For linear systems it seems sions, the normal component of the velocity is prescribed at the unimportant because of the superposition principle validness that whole boundary (except the exit, in some versions). In the prob- allows predicting the system’s motion for arbitrary coupling and lems TF-II and TF-III the whole velocity vector is prescribed at arbitrary initial conditions on the basis normal modes. Therefore, the entrance and the pressure of the normal component of the ve- the class of weakly-interacting particles was not really singled locity, respectively, are specified at the exit. We are interested in out although the superposition principle is absent for nonlinear global (i.e., “in the whole” in time and input data) existence and systems. In fact, the duality “wave – particle” can be revealed uniqueness theorems for these problems formulated in classes of just in this case. solutions as wide as it is possible. Such interest if stimulated by Really, the wave language is valid as before if the initial condi- well-known global existence problem for three-dimensional Eu- tions correspond to any of modes. However, the modes strongly ler equations, where solutions become nonsmooth even if they interact now because their frequencies are close and the reso- are smooth at the initial moment of the time, so the named prob- nance relations appear between them. As a consequence, both lem seems to find its solution only in the classes of extremely modes have to be taken into account if the initial conditions are irregular solutions. In the other words, we have to study nons- far from the normal modes and wave description becomes to be mooth solutions of the Euler equations and prove their existence inefficient one. At that, the particle description becomes more and uniqueness. We present two main results. The first result natural even in the case of linear system, all the more for weakly consists in uniqueness of solutions of the problems NP, TF-II coupled nonlinear oscillators. The appropriate independent vari- and TF-III (for any dimensions of the flow) in the classes with ables here are not displacements and velocities but variables char- non-bounded vorticity. These classes are presented using the Or- acterizing the energies relation of oscillators and phase shift be- licz classes and seem to be easy to verify in applications since tween them. Then the adequate description of energy exchange they are formulated in a rather clear form as against well-known between weakly interacting oscillators is attained in the frame- results. The second result consists in global existence theorem work of Limiting Phase Trajectories (LPT) concept. This concept for the two-dimensional problem TF-I in the classes of solutions is equally general as that of normal modes but in contrast to latter with non-bounded vorticity that belongs to the Lebesgue spaces describes not energy conservation but complete energy exchange Lp with p>4/3. Our methods discover curious relations of the (CEE) between the oscillators. Using LPT concept we describe named problems with the theory of integral transforms and rep- two dynamical transitions taking place when nonlinearity grows. resentations of functions. First of system is due to bifurcation of normal mode caused by its instability and appearance two additional normal modes. As this take place, the complete energy exchange describing by LPT is Alexandr E. Mamontov, Lavrentyev Prospekt 15, Novosibirsk, possible as before. The second dynamical transition occurs when 630090, Russia LPT coincides with separatrix, and after that the CEE becomes

66 impossible. Consequently the transition “CEE – energy localiza- description). In contrast, Bloch oscillations are not excitations tion” on the initially excited oscillator takes place. This example of ground state and, similarly CEE in oscillatory chains are de- confirms distinctly the existence of “wave – particle” duality. scribed by LPTs. Further we consider its extension in different direction. First Finally, we discuss an analogy between arising effectively of them is transient resonant traction of weakly forced nonlinear weak coupling between groups of particles in high-frequency do- oscillator. In such a case we deal with energy exchange between main of oscillatory chain and necessity of quantum description the oscillator and source of energy. It is shown that wave and in the same domain (below Debye temperature) that manifests particle languages have to be also used dependent on initial con- “wave – particle” duality in both cases. ditions. At that, two dynamical transitions take place again. First of them is caused by bifurcation of stationary state with appear- Leonid I. Manevitch, Kosygina str., 4, Moscow, 119991, Russia ance of new stationary states and separatrix. Second transition is a consequence of coincidence of LPT with separatrix and prohi- bition of CEE. FINITE DEFORMATIONS OF ACCRETED SOLIDS If the force has a variable frequency, two qualitatively new A. V. MANZHIROV S.A.LYCHEV types of energy transitions appear. Both ones are caused by the slow frequency change during the vibration and slow passage [email protected] [email protected] through resonance. First of them is denoted by us as Landau- Zener (LZ) type transition by analogy with known quantum phe- Basic fundamentals of the mathematical theory of growing nomenon, second is well-known autoresonance. The LZ-type solids with finite deformations are under consideration. The clas- process is possible in linear system also; it leads to irreversible sification of various methods of solids accretion is presented. energy transfer from energy source to oscillator due its frequency Special attention is paid to the accretion of 3D solids by 2D sur- drift from resonance. As for autoresonance, existing in nonlinear faces. The centrally symmetric problem for an accreted elastic systems only, it consists in continuum increase of the oscillator globe is considered. The deformations are supposed to be finite energy. In both cases LPT changes continuously during the en- and the material is to be incompressible. The constitutive equa- ergy exchange which can be characterized as wandering on LPT tions are formulated with respect to complete distortion tensor family. The irreversible energy exchange is a consequence of which may be representing as the composition of initial distor- such wandering, in the first case – because of inevitable leav- tion and compatible deformation gradient. The initial distortion ing the resonance and in the second one (autoresonance) – due induces the linear connection on the material manifold which be- to continuous synchronization of the force frequency and chang- comes a flat space of affine connectivity with nontrivial torsion. ing (because of nonlinearity) frequency of oscillator. Besides, the transitions from one type of energy exchange to other one caused Sergey A. Lychev, Vernadsky Ave 101 Bldg 1, Moscow, 119526, by change of the system’s parameters can be observed. Russia Unexpectedly, the “wave – particle” duality manifests also in periodic nonlinear oscillatory chains modeling, in particular, the oligomers molecules. The analysis of such chains is more com- VARIATIONAL FORMULATION FOR SOLVING plex than that for the system of two weakly coupled oscillators PROBLEMS OF COMPLEX INCOMPRESSIBLE MEDIA and requires extension of the beating notion on the systems con- MECHANICS sisting of many particles. P. V. MARKOV A.L.SVISTKOV While use the initial variables, the “weak” links are absent here at all. Therefore the wave representation (in modal variables) can [email protected] be naturally used in the systems with small number of particles. It corresponds to “strong coupling” between the particles but weak A variational formulation for the solution of the problems interaction between the normal modes. However, when the num- of complex incompressible media mechanics under finite defor- ber of particles grows, the frequencies of different modes become mation is presented. A certain set of elastic, viscous, plastic close that leads to manifestation of intermodal interaction. First and transmission elements connected together in a special way of all, it relates to the highest frequency modes. Then the lan- is modeled. We consider simple nonlinear continuum models, guage of particles becomes adequate if initial conditions corre- in particular, hyperelastic, viscoelastic, and elastoplastic bodies. spond to excitation of the system playing the role of oscillator Use of the so-called transmission elements makes it possible to in the system with two degrees of freedom. As a result, all pro- take into account the influence of the microstructure of the ma- cesses in such oscillatory chains turn out to be similar to those terial on its macroproperties. The present approach allows us in the mentioned 2 DoF system (bifurcation due to instability of to generalize the algorithm for construction of matrix equations highest-frequency mode, beating with energy exchange between and offers good prospects for numerical studying the deformation different parts of the chain), transition to coincidence of LPT and state of structures made of different materials. With the proposed separatrix and so on. technique, the complexity of the model under consideration can The discussed duality manifests also in a wide class of physical be increased arbitrarily by adding other elements and rearranging problems relating to Solid State Physics (SSP), Nonlinear Optics variational equations. and Photonics. They are united by common phenomenon denoted This work was supported by the Russian Foundation for Basic by term “Bloch oscillations” and have unusual features. Thus, in Research, grant 09-08-00530-a, and by the Presidium of Russian SSP it means output from basic paradigm of SSP connecting dy- Academy of Sciences, projects 09-T-1-1006, 09-C-1-1008. namic processes in crystalline lattice with wave-type excitations of the ground state (i.e., phonons, magnons, polarons in quantum Pavel V. Markov, 1 Korolev Str., Perm, 614013, Russia

67 developed. The external boundary is modeled using dislocation MODULATION OF GRAVITY IN PROBLEM OF FREE dipole elements, each consisting of four edge dislocations, ina CONVECTION boundary element approach. The developed model is used to study the crack growth of a microstructurally short edge crack A. MELENTYEV E.TARUNIN subjected to fatigue loading. The crack is assumed to be located [email protected] within one grain in a semi-infinite body and an external fatigue load is applied at infinity, parallel to the free edge of the body. Influence of gravity modulations on free convection in a close The crack grows due to shearing and opening between the crack cavity was investigated in [1-3]. It was shown that modulation surfaces as a consequence of nucleation, glide and annihilation of changes critical Raley’s number and leads to periodical oscilla- discrete dislocations in the material. The model has been used to tion regimes. As usual, modulations were symmetrical (not sym- evaluate the influence on the crack growth behaviour for a num- metrical modulation was considered only in [3]). In this article ber of different parameters such as grain size, external load, over- we were interested in resonance effects of modulation in differ- loads, crack length, initial crack angle, different descriptions of ent cases of free convection: heating was from below and from the grain boundaries, etc. side. We used two-field method (stream function and vorticity variables) to solve equations of free convection. At first, reso- Solveig Petersson, Ole Romers v 1, Lund, 22100, Sweden nance effects were considered in symmetrical cases and then we investigated influence of asymmetry. Parameter of asymmetry ξ was induced by periodical movement of a cavity up and down. MULTIMODEL ANALYSIS OF ELASTO-PLASTIC Dependence of nonlinear solution on parameters of modulation PROBLEM UNDER COMPLEX LOADING PROGRAMS (a – amplitude, ω – frequency, ξ – parameter of asymmetry) was found. B.E.MELNIKOV A.S.SEMENOV [email protected] [email protected] REFERENCES

[1] G. I. Burde. Numeric Research of Convection in the External Requires for reliability and durability of structures and their Forces Periodical Modulation Case. USSR AS Proceedings, elements with simultaneous material economy have stimulated MFG, 1970, No 2. improvement of constitutive equations for description of elastoplastic deformation processes. This has led to the develop- [2] G. Z. Gershuni, E. M. Zhuhovickij. Convection Stability of ment of phenomenological modelling of complex phenomena of Incompressible Fluid. M: Fizmatgiz, 1972. – 392p. irreversible deformation including history-dependent and rate- [3] L. A. Kashina. Convective Flows Induced by Periodical but dependent effects. During the last several decades many works Unharmonic Oscillation of Cavities. Abstract of Joint X Eu- have been devoted to the development of elasto-plastic models, ropean and VI Russian Symposium “Physical Sc. In Micro- in order to better predict the material behaviour under combined gravity”, St. Petersb., 1997, p.15. variable thermo-mechanical loading. The increase of accuracy of stress analysis and safety factors for complex structures with the help of modern finite-element Anatoly Melentyev, Genkelya str. 7, Perm, 614990, Russia packages can be provided only by use of complex and special variants of plasticity theories, which are adequate for the considered loading conditions and based on authentic information ON THE UNPREDICTIBILITY OF SHORT FATIGUE about properties of materials. The areas of application of the CRACK GROWTH. A STUDY USING A DISLOCATION various theories (models) are as a rule unknown to the users of BASED MODEL finite-element packages at the existing variety loading condition sin machine-building designs. At the moment a universal theory S. MELIN P. HANSSON of inelasticity is absent and even the most accomplished theories [email protected] can not guarantee adequate description of deformation processes [email protected] for arbitrary structure under wide range of loading programs. The classifier of materials, loading conditions, effects (phe- It is well known that the behaviour of microstructurally short nomena) and list of basic experiments are developed by the au- cracks subjected to fatigue loading is strongly influenced by the thors. Use of these classifiers for an establishment of hierarchy surrounding microstructure in the material. Such short cracks are of models is a first step for introduction of the multimodel anal- known to grow in a single shear mechanism due to nucleation, ysis into computational practice. The set of the classic and mod- glide and annihilation of dislocations in the material. The crack ern inelasticity theories is considered, so that they are applicable grows along specific slip planes in the material, instead of per- for stress analysis of structures under complex loading programs. pendicular to the loading axis, as typically observed on a global Among them there are plastic flow theories with linear and non- scale for long cracks. Due to the low growth rates and sensitiv- linear isotropic and kinematic hardening, multisurface theories, ity to the microstructure it is important to take into account the endochronic theory, holonomic theory, rheologic models, theory individual dislocations created during the fatigue process, when of elasto-plastic processes, slip theory, physical theories (single considering the growth behaviour of short fatigue cracks. crystal and polycrystalline models) and others. To this end, a discrete dislocation model, describing both the A unified general form of constitutive equations is presented geometry and the plasticity by discrete dislocations, have been for all used material models based upon the concept of internal

68 state variables. The wide range of mentioned above inelastic ma- Andrei V. Metrikine, Faculty of Civil Engineering and Geo- terial models has been implemented into finite element program sciences, TU Delft, Netherlands. PANTOCRATOR developed by authors. Application possibility of different material models is considered both for material element and for complex structures subjected to complex non- INFLUENCE MECHANISMS OF STRAINS AND proportional loading. STRESSES TO DIFFUSION IN LOADED PLATE This work was supported by RFBR grant N 09-01-00506-a. M.A.MIKOLAYCHUK A.G.KNYAZEVA [email protected] [email protected] B.E. Melnikov, A.S. Semenov, 29 Polytechnicheskaya st. St. Pe- tersburg, 195251 Russia The model of diffusion saturation of loaded plate by admixture from environment was suggested and investigated. Two possible influence mechanisms of starins and stresses on diffusion were THE SYNCHRONIZATION PHENOMENON IN analyzed. First of them related with diffusion activation energy MODERN STRUCTURAL DYNAMICS: VORTEX-, change by lattice deformation. To relate activation energy with PEDESTRIAN- AND ICE- INDUCED VIBRATIONS OF stress and strains, which presence in the system, we need intro- STRUCTURES duce some notion, such as activation volume. Activation volume is difference between local volumes of the system in the ground A. V. METRIKINE and activated states. Ultima analysi, we can say that the work [email protected] of stresses, which are presence in the local volume, is explicitly influence to diffusivity. We have used this diffusivity in the our In a classical context, synchronization (from Greek syn = the model. Second mechanism consist in a mass transfer of impu- same, common and chronos = time) means adjustment of rhythms rity under stresses. It like pressure diffusion mass transfer in a of self-sustained periodic oscillators due to their interaction. The liquids. Interference of first and second mechanisms on the be- history of synchronization goes back to the 17th century when havior of system relation to physical and geometrical parameters the famous Dutch scientist Christiaan Huygens reported on his was investigated. observation of synchronization of two pendulum clocks which he had invented shortly before. : “... It is quite worth noting Mikhail A. Mikolaychuk, pr. Akademichesky, Tomsk, 634021, that when we suspended two clocks from hooks embedded in the Russia same wooden beam, the motions of each pendulum in opposite swings were so much in agreement that they never receded the least bit from each other and the sound of each was always heard INFLUENCE OF WETABILITY OF BOUNDING WALLS simultaneously...”. Being, probably, the oldest scientifically stud- ON FLUID FLOW IN NANOSCOPIC TUBES ied nonlinear dynamical effect, synchronization was understood only in 1920th when E. V.Appleton and B. van der Pol systemat- L. V. MIRANTSEV A.K.ABRAMYAN ically – theoretically and experimentally – studied synchroniza- [email protected] tion of triode electronic generators. Nowadays, this phenomenon is relatively well known and has been observed and used in nearly Numerous experiments that have been performed for the last all fields of physics and engineering. two decades show significant differences in behavior of fluids in In this presentation the role of synchronization phenomenon confined nanoscopic volumes (∼ 10 and less of molecular diam- in modern structural dynamics will be addressed. Three chal- eters) from behavior predicted by classical continual theories. In lenging engineering problems will be discussed, in which the some of these experiments, a significant increase in an effective synchronization phenomenon plays a governing role, namely the viscosity compared to its bulk macroscopic value and occurrence vortex-induced vibrations (VOR IV) of deep-water risers, ice- of layering structures near the bounding solid walls have been ob- induced vibrations (ICE IV) of bottom-founded flexible offshore served. At the same time, other experiments on a flow of various platforms and human-induced vibrations (HUM IV) of pedes- fluids through carbon nanotubes revealed a very strong excess of trian bridges. The mentioned structures – risers, platforms and an average fluid flow velocity over that given by the classic Hagen bridges – may exhibit sustained self-excited vibration of percep- - Poiseuille equation that describes a flow of the Newtonian fluid tible amplitude. Though different in great deal of details and par- through tubes of the conventional macrocopic size. These contra- ticipating physical mechanisms, the self-induced vibration of all dicting to each other results suggest that the structure of bounding these structures seem to occur because of one phenomenon – syn- walls and a character of their interactions with fluid atoms should chronization of a large ensemble of initially uncorrelated oscilla- have a strong influence on properties of fluids confined to such tors by the structure. These oscillators are vortices shed from a nanoscopic volumes. The classic continual theories do not take section of the riser in case of VOR IV, meso-cracks in an ice – into account an atomic (molecular) structure of fluids, and, hence, sheet crushing against a platform in case of ICE IV, and humans these theories cannot give an adequate description of these differ- in case of HUM IV. ences. As a result, computer simulation methods become very A critical discussion will be presented of the models, known important in a such descriptions because these methods are orig- to the speaker, that are used to predict the above-mentioned self- inally based on accounting atomic structure of fluids. Therefore, excited vibrations. The focus of the discussion will be placed on using 3-D molecular dynamics (MD) simulations, we have in- the drawbacks and possible improvements of the models, as well vestigated a fluid flow through nanoscopic tubes with amorphous as the gaps in the available experimental data. solid walls under external force. The interatomic interactions in

69 fluids as well as interactions of fluid atoms to those of the walls controlled by dislocation emission from crack tips. Within our were modeled by means of well known Lennard - Jones (LJ) po- description, lattice dislocations emitted from cracks are stopped tential. We considered two possible scenarios of fluid flow. In at GBs and produce blunting of cracks. Both crack blunting and the first scenario, fluid - wall interactions are assumed to be at- the stress fields of the stopped dislocations hamper further dis- tractive, i.e. the walls are well wettable or hydrofilic. According location emission from cracks in nanocrystalline materials. As to the second scenario, fluid - wall interactions are assumed to a result, crack blunting is suppressed, while crack growth is en- be repulsive, i.e. the walls are not wettable or hydrofobic. It has hanced. The dependences of the maximum number of disloca- been shown that, in the first case, an average fluid flow velocity tions, emitted by a crack, and the critical stress intensity factor may be lower than that given by the Hagen - Poiseuille equa- on grain size in Al and α-Fe are calculated. It is demonstrated tion, whereas, in the second case, this velocity may be tens times that a decrease in grain size from 300 to 10 nm in Al and α-Fe higher than value obtained for the Newtonian fluid flow. decreases their critical stress intensity factors by a factor of 2 to 3 and thereby dramatically reduces the toughness/ductility of these Leonid V. Mirantsev, V.O. Bolshoi 61, St. Petersburg, 199178, materials. Russia In parallel with the study of dislocation emission and crack blunting on fracture toughness of nanocrystalline materials, we have theoretically analyzed the influence of alternative deforma- FATIGUE FAILURE WEAKENED ZONE OF THE tion mechanisms (specific to nanocrystalline solids) on fracture COMPOSITE DISC toughness. We considered such deformation mechanism as rotational deformation and GB migration. Within the model, special V. MIRONOV A.YAKUSHEV rotational deformation (driven by the external stress concentrated [email protected] near a crack tip) occurs in a nanograin through formation of immobile disclinations whose strengths gradually increase during A coupled boundary value problem has been developed for a the formation process conducted by grain boundary sliding and geometrically linear body, with allowance made for the descend- diffusion. GB migration occurs under the action of high shear ing branch of the quasi-static deformation diagram. A variant of stresses acting near crack tips and is also accompanied by the problem extension to the instance of high-cycle loading is pro- formation of the quadrupoles of wedge disclinations. Both spe- posed. It is supposed that, at a constant and originally safe value cial rotational deformation and GB migration release, in part, of external effects, the fracture of a solid is caused by the degener- high local stresses near crack tips, thus serving as toughening ation of material properties. This is substantiated by some results mechanisms in nanocrystalline materials. It is shown that each of of nonstandard tests on single specimens with constructing full these deformation mechanisms can increase fracture toughness of deformation diagrams. nanocrystalline materials by 10-15%. Experiments with structural materials have shown that cyclic loading changes the parameters of the static diagram nonmono- tonically, though eventually leads to its degeneration. This prop- Alexander G. Sheinerman, Bolshoy 61, Vasil. Ostrov, St. Peters- erty has been exploited to calculate the durability of the depressed burg, 199178, Russia zone of a compound disc. The constitutive concept of interaction between the static and cyclic material properties is implemented in the problem through FRAGMENTATION OF A PROJECTILE ON DISCRETE establishing a representative parameter of the cyclic state, de- BUMPERS scribing the kinetics of the fatigue process, developing methods for calculating permanent strains, stating the problem on the dura- N.N.MYAGKOV T.A.SHUMIKHIN bility and longevity of a structure with an alternating stress level L.N.BEZRUKOV and offering solution methods. nn [email protected]

Alexey V. Yakushev, St. Petersburg State Railway University, A high-velocity impact interaction of a polyethylene projec- Moscow pr., 10-12, Saint-Petersburg, 190031, Russia tile and aluminium projectile with string and mesh bumpers was investigated experimentally and numerically. The study showed that the projectile fragmentation on string and mesh bumpers is EFFECTS OF GRAIN SIZE AND ROTATIONAL especially distinguished by the accompanying formation of jets DEFORMATION ON FRACTURE TOUGHNESS OF of fragments which are ejected from the front part of the projec- NANOCRYSTALLINE MATERIALS tile along and across its movement direction. The jets intensity N. F. MOROZOV I.A.OVID’KO A.G.SHEINERMAN depends on impact velocity and geometrical parameters of the E.C.AIFANTIS bumper-projectile system. The frontal fragmentation dominates at larger cell apertures. However, when the aperture lessens the [email protected] part of projectile mass fragmented due to the jets diminishes sig- niﬁcantly and the shock-wave destruction of the projectile pre- Aspects of the ductile vs brittle response of nanocrystalline vails. materials are theoretically examined. The focuses are placed on the combined effects of grain boundaries (GBs) and blunting of cracks on the fracture toughness of nanocrystalline materials in a Nikolay Myagkov, 32A Leninsky prospect, Moscow, 119991, typical situation where crack blunting and growth processes are Russia

70 INVESTIGATION OF SURFACE ACTIVITY DYNAMICS PROPAGATION OF LOW FREQUENCY WAVES IN OF MULTICOMPONENT SURFACTANT SOLUTIONS POROUS BIOT LAYER SURROUNDED BY TWO ELASTIC HALF-SPACES A.I.MIZEV A.I.LUTSIK L.A.MOLOTKOV alex [email protected] [email protected] [email protected]

We present the results of experimental investigation of a sur- The porous Biot [1] layer surrounded by two elastic half- face phase formation and dynamics of a surface activity in multi- spaces is considered. For this medium, two dispersion equations component solutions of high-molecular surface-active substances are established. These equations correspond to symmetric and by the methods of tensiometry. During formation of a new free antisymmetric part of the medium. The investigation of these surface of the solution, it takes some time for a surfactant to equations is fulfilled in region of low frequencies. In this region emerge to the surface. This time depends on the intensity of the dispersion equations transfer to algebraic equations; in addi- the diffusion and adsorption processes. For solutions containing tion, the roots of these equations are the velocities of the normal several surfactants the processes of the surface phase formation waves on zero frequency. The symmetric dispersion equation de- depend on the surface properties of the components and on the termines two velocities which coincide with the velocities in free possible interaction of their molecules both in the bulk and at the porous Biot layer with closed pores on boundaries [2]. These surface of the solvent. In this case, the dependence of the surface two velocities and the velocities of the volume waves in the Biot tension on time (or on the age of the surface) will represent the medium satisfy some equalities. The root of antisymmetric dis- peculiarities of the surface layer formation. persion equation is the velocity of the Rayleigh wave on the free To investigate surface properties of the solutions we used the surface of an elastic medium. Thus, at low frequency three nor- method of maximum pressure in a bubble (BPA-800P, KSV Ltd., mal waves propagate. Finland). Formation of the bubbles at different rates allows us The work was supported by RFBR (grant 08-01-00511). to obtain interfaces, which have different lifetimes at the instant of approaching the pressure maximum. For measurements it will suffice to use a rather small volume of a liquid (of order of one REFERENCES milliliter). The specific convenience of the method is that there is no need to measure the wetting angle and to keep control of the [1] L. A. Molokov. Investigation of wave propagation in porous meniscus, which makes it possible to take measurements even in fractured media on the basis of effective models of Biot and opaque media. The experiments were made with the solutions layered media. S-Petersburg. Nauka, (2001), 348, (Rus). of potassium salts of caprylic and lauric acids in an aqueous mo- [2] L. A. Molotkov. Wave propagation in an isolated Biot layer lar solution of potassium alkali. These surfactants have differ- with closed pores on the boundaries. Zap. Nauch. Semin. ent surface activity and rate of adsorption. During experiments POMI, 354, (2008), 173-189. we changed only the relative content of the laurate and capry- late defined by the molar fraction of the surfactant in the mixture, whereas the total molar concentration of the solutions being kept constant. Lev A. Molotkov, Fontanka 27, St-Petersburg, 191023, Russia It was shown that the contribution of each surfactant to formation of the surface phase is of additive character only in the INVESTIGATION OF THE INFLUENCE OF case of low concentration of both surfactants in the solution. At BLEACHING ON HUMAN TEETH ENAMEL higher concentrations (starting approximately with 1/10 CMC for MICROSTRUCTURE the most surface-active component) the contribution of the surfactants becomes non-additive: a surfactant with shorter time of I.A.MOROZOV A.L.SVISTKOV O.K.GARISHIN adsorption and higher surface activity plays a decisive role. For O.S.GILEVA E.S.EROFEEVA a two-component mixture of the unknown concentration a new technique of determining the concentration of each surfactant has [email protected] [email protected] been proposed. The method is based solely on the surface tension [email protected] measurements, that is, there is no need of applying the analytical methods of investigation. This work focuses on the quantitative analysis of the enamel surface microrelief and the effect of bleaching on the microstruc- The work was supported by the RFBR (project No 10-01- ture of enamel surface. The AFM images (15 x 15 micrometers) 96009) and by the Department of Power Engineering, Mechanical that exhibited the enamel surface of the unbleached tooth, the Engineering , Mechanics and Control Processes of RAS (project tooth bleached once and the tooth bleached thrice were examined. No 09-T-1-1005). To clarify the general pattern of the enamel microrelief alteration, we applied the so-called cartographic method showing flat areas, cavities and elevations on the surface of a tooth. We present a method for structural analysis of such parameters as density, size, perimeter, area, volume, and fractal dimensions. Anastasia I. Lutsik , Academician Korolev str., 1, Perm, 614013, The obtained results show that the first bleaching removes pel- Russia licles (adsorbed salivary protein film) from the surface of teeth

71 enamel, and further bleaching causes enamel wear, especially in therefore the wave length L. But in a linear approximation the the cavity areas of the relief. Elevations and depressions become buckling mode which is described by the relation between r1 and more round in shape. The structures observed on enamel surface r2 is still indefinite. In [1,2] the post-buckling behavior of the are found to be related by fractal relationships, which is indica- compressed plate on the elastic foundation is investigated and it is tive of their self-similarity over the entire range of sizes observed shown that the minimal elastic energy density corresponds to the in this study. case with r1 = r2. Namely we have a chessboard-like buckling This study was supported by the Ministry of industry, innova- mode. In [4] it is indicated that the chessboard-like distribution tions and science of Perm Kray (grant No 09-08-99102-r ofi). of strains (namely, with r1 = r2) is typical for many interfacial problems. Ilya A. Morozov, 1, Academic Korolev st., Perm, 614013, Russia In all these problems, especially in the cases without a plate, the initial critical strains are large enough. They may be of the order of 0.1–0.3. Here and in [5] influence of the pre-buckling strains on the critical stresses and on the buckling modes is inves- VOLUME AND SURFACE STABILITY A OF tigated and the non-linear approach is used. It is established that TRANSVERSELY ISOTROPIC MATERIAL neglecting the pre-buckling deformations essentially changes the critical stresses both for volume and for surface loss of stability. N. F. MOROZOV P.E.TOVSTIK Only in the case of strong anisotropy with small preliminary de- [email protected] formations the difference between the linear and the non-linear [email protected] approach is inessential. The paper is written with the financial support of Russian A compressed elastic material loses volume stability since for Foundation of Basic Investigations, grants 09.01.92002 HHC-a, the large enough compression the Hadamard condition is not sat- and 10-01.00244-a. isfied [1]. An elastic half-space loses surface stability under large enough compression with the buckling waves localized near the free plane with amplitude exponentially decreasing as the dis- REFERENCES tance to the free plane is increased [2]. A compressed elastic plate lying on an elastic foundation also may lose stability [3]. [1] Morozov N. F., and Tovstik P.E. On the modes of the sur- For these problems we find the critical compression and the buck- face stability loss. Problems of nonlinear mechanics. 2009. ling modes for a transversely isotropic material. Kazan, pp. 270-273. We assume that the constant initial horizontal compressions 0 0 [2] Morozov N.F., Paukshto M.V., and Tovstik P.E. Stability ∂u1/∂x1 = ∂u2/∂x2 = e are given, and the initial ver- − 0 of the surface layer under thermal loading. Izv. Russ. Akad. tical stresses are equal to zero (σ33 = 0). We assume als0 that the elastic modules may depend only on the vertical co- Nauk. MTT. 1998. No. 1. pp. 130-139. ordinate z. We seek solution of the bifurcation equations in the [3] Tovstik P.E. Vibrations and stability of a pre-stressed plate on form uk(x1,x2,x3)= uk(z)exp i(r1x1 + r2x2) , i = √ 1, an elastic foundation. Pricl. Math. Mehh. Vol.73. No. 1. pp. { } − k = 1, 2, 3, where uk are the additional displacements at bifurca- 106-120; 2009. tion and rj are the unknown wave numbers. It is possible to find [4] Panin L.E., and Panin V.E. Effect of the “chessboard” and only relation between the wave numbers rj , therefore the wave mass transfer in interfacial media of organic and inorganic lengths Lj = 2π/rj remain indefinite. Moreover the bifurca- nature. Physical Nanomechanics. 2007. Vol. 10. No. 6. pp. 2 2 tion equations contain only r = r1 + r2 but r1 and r2 are not 5-20. contained in these equations separately. p It is established that two kinds of buckling modes for volume [5] Tovstik P.E. The volume and the surface stability of a stability loss are possible. The kind 1 is the rotation mode with transversely isotropic material under compression. Vestnik St.Petersburg Univ. Ser. 1. 2010, No 1. displacements in the plane of isotropy (u1, u2 = 0, u3 = 0). For the kind 2 displacements in all the directions are6 not equal to zero (u1, u2, u3 = 0), Nikita F. Morozov, Petr E. Tovstik, SPbGU. Universitetsky str., 6 If the material is homogeneous one then the buckling mode oc- 28. St.Petersburg, Russia cupies the entire space. In this case the critical compression does not depend on the wave number r. If the material is heterogeneous one then there is the weakest plane z = z∗ near which the ROLE OF RELAXATION IN FORMATION OF CYCLIC buckling mode is localized, and the critical compression depends RESISTANCE OF ELASTOMERIC NANOCOMPOSITES on the wave number r. The cases of continuous modules and of piecewise continuous modules are studied. V. V. MOSHEV S.E.EVLAMPIEVA For the surface stability loss only the kind 2 of the buckling [email protected] mode is possible. In the same way the interfacial stability is investigated. As in the case of volume stability the bifurcation A mathematical model of cyclic testing of time-dependent 2 2 equations contain only r = r1 + r2 but the critical compres- elastometic nanocomposites that takes into account the peculiari- sion e(s) does not depend on r. ties of their relaxation behavior is presented. p The buckling problem of the compressed plate lying on the A model cell is represented as two substrates (filler particles) soft elastic foundation has a much more definite solution. From separated from one another by a space specifying the filler con- the linear equation of bifurcation we find the definite value r and centration. An elastic spring used to model a rubber molecule

72 sorbed by the surface of particles is placed on the substrates. FINITE ELEMENT MODELING OF PIEZOCOMPOSITE Sorption of rubber with nanosized filler particles excludes the MATERIALS WITH DIFFERENT CONNECTIVITY: possibility of the normal detachment of rubber molecules from MODELS, RESULTS AND APPLICATIONS the surface of particles (substrates), but allows the slippage of rubber molecules (spring) along the surface of particles (sub- A.NASEDKIN M.SHEVTSOVA strates). Hysteresis loops are formed by separation and contraction of [email protected] the substrates. On extension, the gap between the substrates increases, and the spring in the space begins to resist elastically. Porous and polycrystalline piezoceramic materials have re- This causes friction slipping of the spring from the substrates into ceived considerable attention due to their application in ultrasonic the gap between them. The contraction of the model leads to the transducers, hydrophones, sensors, actuator and other piezoelec- return of the part of the spring from the gap to the substrates. The tric devices. The classification of piezoelectric composites was friction character of the spring motion along the substrates gives initiated by Newnham’s connectivity theory. In compliance with rise to the relaxation effect. this theory and polycrystalline piezoceramics can be classified as two or three phase composites with different connectivity. As is Experimental hysteresis curves are obtained for different well known, the 3–0, 3–3, 1–3 and other piezocomposites in the cyclic loading conditions. form of porous PZT materials show considerably improved trans- The research was supported by the Fundamental research pro- ducer characteristics. Porous piezocomposites have great poten- gram of the Department of Power Engineering, Mechanical Engi- tial for low acoustic impedance and higher efficiency compared to neering, Mechanics and Control Processes of RAS “Tribological conventional dense PZT piezoceramic materials. However, their and strength properties of structured materials and surface layers” depth-handling capability and the stability to hydrostatic pressure No 09-T-1-1006. have yet to be proved, particularly for 3–3 porous structures. The mechanical strength of the devices can be improved by filling with polymer material as the second phase (or as the third phase Svetlana E. Evlampieva, 1, Ak. Korolev Str., Perm, 614013, Rus- for porous polycrystalline piezoceramic material). sia Porous or polycrystalline composite piezoceramic having pore sizes, lesser that 100 mcm may be accepted as a quasi homogeneous medium with some effective moduli for most applications. A NEW APPROACH TO THE PROBLEM OF RAYLEIGH In present work we have developed the effective moduli WAVE PROPAGATION IN DISCRETE LATTICES method and finite element technique. Theoretical aspects of the effective moduli method for inhomogeneous piezoelectric media V. G. MOZHAEV were examined. Four static piezoelectric problems for a repre- [email protected] sentative volume that allow finding the effective moduli of an inhomogeneous body were specified. Respective equations for The problem of Rayleigh wave propagation in continuous me- calculation of effective moduli of piezoelectric media with arbi- dia can be solved by direct integrating equations of motion [1]. trary anisotropy were derived. The present study is devoted to the generalization of this direct Based on these equations the full set of effective moduli for integrating approach in the case of discrete systems like crystal porous or polycrystalline composite piezoceramics having wide lattices. A simplified discrete difference model of orthorhombic porosity or/and injection range was calculated with help of finite crystals is considered. In the frames of this model, the equations element method. For the modeling of porous piezoceramics with of motion couple displacements of three neighboring discrete par- 3-3 connectivity the representative volume having skeleton struc- ticles. A procedure is found to reduce these equations to the dif- ture was considered. For taking into consideration inhomoge- ference of neighboring values of a functional which is defined neous or incomplete ceramics polarization the preliminary mod- only at two neighboring sites. Further application of the discrete- eling of polarization process was performed. To determine the form boundary conditions in an arrangement of this functional zones that have different polarization the finite element calcula- yields the secular equation of the problem. Limiting transition tions of the electrostatic problem were executed. for this equation to the case of continuum media is fulfilled. For polycrystalline piezoceramics the crystallites of sapphire (α-corundum) Al2O3 were considered as material of inclusions. The effective moduli for inclusions were calculated as the average moduli of monophase polycrystallite of trigonal system. REFERENCES The results of FEM modeling were compared with the experimental dates for different porous ceramics in the relative porosity [1] V.G. Mozhaev (1995) Some new ideas in the theory of sur- range of 0–70%. We note that for modeling porous and poly- face acoustic waves in anisotropic media. In: Anisotropy, In- crystalline material the calculation of inhomogeneous ceramics homogeneity and Nonlinearity in Solid Mechanics. Ed. by polarization is very important. D.F. Parker and A.H. England, Kluwer Academic Publish- This work is partially supported by grants of the Russian Foun- ers, Dordrecht, pp. 455-462. dation of Basic Researches.

Andrey Victorovitch Nasedkin, Southern Federal University, Vladimir G. Mozhaev, Leninskie Gory, Moscow, 119991 GSP-1, Faculty of Mathematics, Mechanics & Computer Science, Russia Miltchakova str., 8a, Rostov on Don, 344090, Russia

73 REFERENCES MODELING OF FAILURE PROCESSES IN PROBLEMS OF GEOMECHANICS OF COAL BEDS [1] A. A. Nasedkina, V.N. Trufanov. Numerical modeling of hy- drofracturing in a multilayer coal seam. Journal of Mining A.NASEDKINA Science, Springer, 2006, Vol. 42, No. 1, pp. 54-62. Translated [email protected] from Fiziko-Tekhnicheskie Problemy Razrabotki Poleznykh Iskopaemykh, No. 1, pp. 61–70, January–February, 2006. DOI 10.1007/s10913-006-0029-4 The project deals with the problem of methane extraction from [2] A. A. Nasedkina, A. V.Nasedkin, G. Iovane. A model for hy- coal seams. The research objectives of the project are mathemati- drodynamic influence on a multi-layer deformable coal seam. cal modelling and computer simulation of degassing processes in Computational Mechanics, Springer, vol. 41, no. 3, 2008. P. coal seams which occur during methane extraction. As methane 379-389. DOI 10.1007/s00466-007-0194-6 in coal matter is generally found in a bound state, the purpose of applying methane extraction methods is to destroy coal-gas system and get free methane. The destruction of the system can be Anna A. Nasedkina, Miltchakova str., 8a, Rostov on Don, achieved by water pumping into the borehole as used in hydro- 344090, Russia dynamic influence degassing method. The general approach to create a model for hydrodynamic influence method is to develop a number of simplified mathematical models for different pro- STATISTICAL APPROACH TO CONSTRUCTING cesses, such as fluid filtration, porous body deformation and frac- EQUIVALENT MODELS OF ROCKS BASED ON THE ture phenomena, and their interrelations. The theory of porous INVERSE PROBLEM SOLUTION media, rock and fracture mechanics form the scientific basis for the project. The coal seam is presented as multiphase porous L.A.NAZAROV A.I.ARTEMOVA L.A.NAZAROVA medium. Computer simulations are carried out on the base of [email protected] [email protected] finite element method with the help of finite element software [email protected] (ANSYS). Equivalent models of rocks put up a bridge between the inten- The model of water filtration in porous media is the first step tion to having an exact description of a geomechanical body and towards the stated goal. A nonlinear transient 2D axisymmetric the shortage of in-situ data on the body’s structure and properties problem for simulating the hydrodynamic disjointing method for as well as the limited computational resource. a three-layer coal seam was considered in [1]. The next step is to A new approach to describing an inhomogeneous medium in- extend pure filtration model which presented one-phase porous cludes the notion of a physical volume by analogy with an el- medium, into a coupled filtration-deformation model for a two– ementary volume in general use in continuum mechanics. The phase saturated porous medium. Mathematical model is based on physical volume W of an inhomogeneous medium is a small ele- transient poroelastic coupled equations and an analogy between ment with the representative content of the medium structure pe- poroelasric and thermoelastic equations that allowed performing culiarities. Parameters of the equivalent medium equation of state thermo-structural coupled analysis. The results of the simulation are found by statistical processing of solutions (derived with ana- of nonlinear coupled poroelastic problem for a three-layer coal lytical or numerical methods) of inverse problems about loading seam can be found in [2]. tests on a physical volume. Equivalent model for porous-elastic media. Let W be a rectan- In order to take into account the processes of coal destruction gular region of an orthotropic elastic medium (A and B are Young another step is to incorporate failure criteria into the problem. To moduli) with equally distributed inhomogeneities with density m compute the stress intensity factors at the stage of post-processing (qA is Young modulus, q is characteristic of the contrast range the results the differences in the displacements were calculated at of inclusions). Moduli of the equivalent medium were found the corresponding nodes at the crack edges. The energy integral from condition of equality of average strains in the initial and (J-integral) for axisymmetric poroelastic and thermoelastic prob- equivalent media. Numerical calculations showed that the known lems was also considered as another fracture criterion. Special estimation formula for elastic characteristics of porous media, macros for computing the energy integral was written in APDL E(m, q)=(1-m+qm)A, is satisfiable only with low m and with q ANSYS programming language. of the order of 1; otherwise error can reach 30%. Equivalent model for media with discontinuities. The physi- The results obtained so far were directly applied in geology cal volume W contains a randomly oriented, thin, inclined elastic engineering. The computations were performed for coal seams isotropic layer. Parameters of the equivalent model were deter- in the East Donbass Krasnodonetsk and Kuzbass Taldinsk coal- mined by the analogous procedure. fields. The results of the computations for the poroelastic prob- Testing of the model involved the problem about uniaxial load- lems permit to obtain the pictures of the pressure, displacements, ing applied to a rectilinear region D with an arbitrary amount of strain and stress distribution in the cylindrical layer. They also discontinuities. Enclosing medium is an isotropic elastic medium allow to determine the values of the pore pressure function at dif- with Young modulus E and shear modulus M. Discontinuities ferent time moments and in different points of the computation were of the same thickness h and had equal deformation prop- zone and to analyze the concentration of the stresses and the pos- erties. Normal and shear rigidities of the discontinuities were sibilities of the crack propagation. In the thermoelastic problems calculated as N=qE/h and T=qM/h, respectively; parameter q was the analoguos results are obtained for the thermoelastic variables. varied in a wide range.

74 It was found that the equivalent model offers nearly an exact The study was carried out with financial support from the description of stress-strain state of the structured medium. When Siberian Branch of the Russian Academy of Sciences, Integra- q>1 deformation of a rock block depends mostly on the proper- tion Project No. 60, and from the Russian Foundation for Basic ties of enclosing rocks, whereas with q<1 (though total thickness Research, Project No. 10-05-00736. of discontinuities was under or equal to 1% of the linear sizes of D) the deformation of D is governed by the inter-block disconti- Larisa A. Nazarova, Krasny prospekt 54, Novosibirsk, 630091, nuities. Russia The proposed procedure is applicable to the construction of equivalent models for any media, based on the equivalency con- ON THE 2D EQUATIONS OF MATHEMATICAL dition that depends on the key physical field parameters involved PHYSICS into description of the behavior of a medium under external action. I.NEYGEBAUER The study was carried out with financial support from the [email protected] Siberian Branch of the Russian Academy of Sciences, Integra- tion Project No. 61, and from the Russian Foundation for Basic The classical equations of mathematical physics do not give Research, Project No. 09-05-00975. the physically accepted solutions in some cases of point boundary conditions. For example: the given finite temperature inside Leonid A. Nazarov, Krasny Prospekt 54, Novosibirsk, 630091, the 2D domain for the heat conduction equation or the given dis- Russia placement of one internal point for elastic membrane create the situations where the solution of the boundary value problem does not exist. Another wave equation and heat conduction equation in ORIGINATION AND EVOLUTION OF FAILURE ZONES 2D case are introduced to accept the point boundary conditions. IN THE VICINITY OF INHOMOGENEITIES IN THE The method to create the above equations is presented. ROCK MASS Igor Neygebauer, University of Dodoma, Tanzania L.A.NAZAROVA I.N.ELTSOV L.A.NAZAROV

[email protected] [email protected] STABILITY OF ELASTOPLASTIC PRESSED TO [email protected] FOUNDATION GIRDER

The irreversible deformation zones may arise in the vicinity of L.V.NIKITIN rock mass heterogeneities under definite combination of geome- [email protected] chanical and geotechnical factors (natural stress magnitude, rock strength properties, shape of underground object and so on). Itis necessary to determine the dimensions and configuration of such Stability of the process of longitudinal compressing of an zones e.g. for evaluation of mine workings and oil wells stability elastoplastic girder pressed to a smooth foundation is studied.The or planning in situ stress measurement experiment. The similar girder face lying on the foundation and adjusting it faces are zones initiates in Earth’s bowels in the vicinity of future seismic pressed to rigid smooth walls along which they may freely slide event focus. Just irreversible deformation and failure processes with possibility to tear off. On the reminder six face the uni- are the probable cause of postseismic activity. formely distributed compressing normal load is acting. The upper and law bounds of the longitudinal compressing stress are Based on failure criteria and state equations of incremental obtained that turned out of the same order as the modulas of the type we established the dependence of geometry of possible fail- material hardening and not at all depending on geometrical sizes ure region in the vicinity of different kind inclusions (opening, of the construction.The obtained results testify the surface nature deep oil well, anomalous section of fault) on the rock mass of instability and unexeptableness the hypothesis of plane normal strength parameters (internal friction angle, cohesion, dilatation cross sections. coefficient L) as well as ratio of tectonic and lithostatic stresses in the natural field. Lev V. Nikitin, 10 B.Gruzinskaya street, Moscow, 123995, Rus- The numerical experiments have shown, in particular, that the sia disintegration zones F (where elongation takes place in radial direction and contraction – in circumferential one) originates in the vicinity of openings and holes. The F length depends linearly on APPLICATION OF THE ACOUSTOELASTIC EFFECT coefficient L (at –0.1¡L¡0.1) and comparatively small decrease of TO EVALUATION OF STRESSES IN TECHNOLOGICAL cohesion leads to more than fivefold raise of F dimension. PIPELINES Due to failure nearby deep oil wells the circumferential per- N.YE.NIKITINA meability may increase whereas radial permeability – decrease (up to 10-15%) which stipulate the redistribution of seepages and [email protected] flow rate variation. Anomalous parts of faults in Earth crust (which arise owing Opportunities of the nondestructive ultrasonic method for eval- to local edges slipping or fluid infiltration) cause nucleation of uation of stresses are shown by an example of measuring of non- failure zones extending orthogonal to fault plane. Such zones are design axial stresses in technological pipelines. The results of ap- the possible aftershock focal region. plication of acoustoelasticity effect realized in device IN-5101A

75 are given for carrying out direct measurements of mechanical which such flows do exist. The linear stability of vertical station- stresses “in situ” in pipeline sole timbers of the main equipment ary states is considered. It is shown that considering of the com- of compressive stations. pressibility of steam increases the area of stability in the space of Features of application of the acoustoelasticity method for physical parameters of the system and changes the type of tran- measuring stresses in these objects and prospects of its applica- sition to instability. There are two types of transition in case of tion for solving problem of the periodic testing of a stress state compressible vapor: with the most unstable mode correspond- of pipeline systems are considered. The opportunity of applica- ing to the zero wave number or the finite nonzero wave number. tion of the “non-zero” ultrasonic tensometry for determination of As distinct to the case of incompressible vapor two possible po- a stress state of objects in action has been experimentally con- sitions of stationary phase transition interface do not coincide on firmed by an example of measuring of axial stresses in techno- the margin of stability corresponding to the first type. The second logical pipelines of compressive stations. type replaces the transition to instability with the most unstable The method has good prospects for measuring stresses in mode corresponding to the infinite wave number. The influence weldless pipes, for which manufacture the rolled sheet is not used of temperature, thickness of the porous layer, pressure on its mar- that frequently possesses a high degree of an intrinsic acoustical gins, and capillary effects on the location of stationary phase tran- anisotropy. The proposed manner can be also used for other ob- sition interface and its stability is also described in the paper. jects, which material has the intrinsic acoustic anisotropy not exceeding 0,1%, and there is an opportunity of testing an analogous Vera E. Odintsova, prosp. Vernadskogo 101, block 1, Moscow, material in the unstressed state. 119526, Russia The research was made possible due to a partial financial support by RFBR (projects N 09-08-97032-r povolzhye a, N 10-08- 01108-a). NUCLEATION OF NANOGRAINS NEAR CRACKS AS A NEW TOUGHENING MECHANISM IN DEFORMED NANOCRYSTALLINE MATERIALS Nadezhda Y. Nikitina, 85 Belinskogo, Nizhny Novgorod, 603024, Russia I.A.OVID’KO N.V.SKIBA A.K.MUKHERJEE [email protected] skiba [email protected] STUDY OF THE INFLUENCE OF THE STRUCTURE OF GRAIN BOUNDARIES ON THEIR BEHAVIOR UNDER Nucleation of nanograins near crack tips and its role as a spe- SHEAR LOADING. cial (new) toughening mechanism in deformed nanocrystalline materials are theoretically described. Within our description, A.YU.NIKONOV A.I.DMITRIEV stress-driven splitting and migration of grain boundaries occur near crack tips. These stress-driven processes result in nucleation [email protected] [email protected] of nanograins and cause a partial stress relaxation near cracks. It is shown that the nucleation of nanograins near crack tips is en- Molecular dynamics simulation of behavior of grain bound- ergetically favorable and effective in toughness enhancement in aries in copper bicrystal under shear loading were carried out. nanocrystalline ceramic Si-Al2O3. The possibility of movement of the grain boundary in a direction perpendicular to the applied load was shown. The influence of Nikolay V. Skiba, V.O., Bolshoj pr., 61, Saint-Petersburg, grain boundaries structure on such behavior of boundaries was 199178, Russia studied. The dependencies of the velocity of grain boundaries on the applied load are analysed. Estimates value of critical stress initiating the movement of the grain boundary was calculated. MECHANICAL SYSTEM SIMULATION VIA This behavior of grain boundaries can have a significant influ- AUTOMATIC DIFFERENTIATION ence on the change of microstructure of the material and, as a result of its properties and behaviors under stress. S.PALIS F. PALIS [email protected] Anton Y. Nikonov, Akademicheskii 2/4, Tomsk, 634021, Russia A standard approach of modeling mechanical systems is through Hamilton’s Principle leading to the Euler-Lagrange equa- THE INFLUENCE OF VAPOR COMPRESSIBILITY ON tions of motion or, applying Legendre transformation, to the THE STABILITY OF PHASE TRANSITION INTERFACE Hamilton equations of motion. The main structural difference is IN GEOTHERMAL SYSTEMS that Euler-Lagrange equations are, depending on the Lagrangian, implicit differential equations of second order, whereas Hamil- V. ODINTSOVA ton equations are explicit differential equations of first order, de- [email protected] pending only on the partial derivatives of the Hamiltonian. These structural advantages of the Hamilton equations of motion can The influence of compressibility of vapor on stationary verti- be used in combination with automatic differentiation to derive cal flows in geothermal systems consisting of a water layer over- a numerical integration scheme using only the Hamiltonian of laying the vapor one is considered in isothermal approximation. a system. Automatic differentiation is an algorithm calculating Capillary pressure is taken into account. It is shown that com- derivatives by exploiting the fact that any computation can be de- pressibility increases the range of parameters of the system for composed in a sequence of elementary operations, which can be

76 differentiated separately applying the chain rule. For efficiency of solid-liquid interface. They prevent the pit formation, lead to a reason here the reverse mode of automatic differentiation is used. significant decrease of radial dopant segregation and substantially In this paper the combination of automatic differentiation and increase the overall homogeneity of grown crystal. However, at Hamilton equations of motion will be used to simulate the load high enough vibration intensity crystallization front becomes un- motion of a crane providing only the system’s Hamiltonian, i.e. stable to 3-D perturbations: critical parameters for 3-D instability the system’s energy in phase space. Using the presented method are obtained. Numerical modeling allows to determine parameter laborious and error-prone symbolic derivation of the equations range where the overall vibration effect is stabilizing. of motion can be avoided. A comparison in Matlab programming The work was partially supported by Ural Brunch of RAS environment shows that automatic differentiation outperforms the (grant for young researchers). alternative using computer algebra for derivative calculation. Yanina N. Parshakova, Koroleva, 1, Perm, 614013, Russia Stefan Palis, Universitetsplatz 2, Magdeburg, 39104, Germany THE STRESS-STRAIN STATE OF A CYLINDRICAL SOME CONSTITUTIVE MODELLING ISSUES IN WIRE VAULT BUILT BY USING A CENTERING DRAWING ANALYSIS D.A.PARSHIN A.PANTEGHINI F. GENNA [email protected] [email protected] [email protected] It is considered a process of gradual building a cylindrical vault on a smooth rigid base. Building is implemented by means of The numerical simulations of wire drawing processes deal continuous attaching layers of additional material to the formed with strong nonlinearities, due to the presence of contact with vault external surface. The vault internal surface is supported by friction, large displacements/rotations/strains, and nonlinear plas- a rigid circular centering during the whole building process. The tic response of the wire material. It is well known that during a material used for building manifests properties of creep and aging multi-pass industrial wire drawing process the material is sub- (in particular, concrete can be considered). jected to loading-unloading cycles, when passing from a die to It is stated a quasistatic initial-boundary problem of accreted the next one. Moreover, it must be observed that plastic loading- solids mechanics that describes the process of the vault deforma- unloading occurs also in a single-pass drawing, because of the tion under gravity action from the start of building to the instant very geometry of the process. For all these reasons, the numerical of centering removing from the completed vault some time after simulation of a wire drawing process, both single-pass and multi- the end of building. The case of plain strain state is considered. pass, requires special attention in the definition of the constitutive Strains are supposed to be small. model to be adopted. Since the materials usually employed in The analytical solution of the stated non-classical mathemat- drawing exhibit kinematic hardening, there are at least 3 issues to ically two-dimensional problem is built in trigonometric series consider accurately: (i) adequacy of hardening models differing and quadratures. A proposition about the structure of technologi- from kinematic ones, as often done in simulations of single-pass cal residual stresses taking place in the completed vault for an ar- processes; (ii) implementation of complex kinematic hardening bitrary long time after centering removing is proved. This propo- models in the presence of large strains and rotations; (iii) devel- sition allows to determine efficiently the stresses mentioned. Ac- opment of a constitutive model able to describe the quite different cording to the proposition these stresses can be determined as dif- tension/compression behavior of steel under extremely large plas- ference between the stresses corresponding to the solved problem tic strains. In this paper we attempt a preliminary discussion of of the vault accretion on a centering and stresses corresponding these issues, and illustrate some results obtained adopting a new to a certain classical problem of elasticity theory with zero mass constitutive model for the simulation of the plastic cyclic behav- forces. ior of steel at large strains, showing the residual stress profiles The work is supported by the Russian Foundation for after a single-pass wire drawing process. The new results are Basic Research (under grants Nos. 08-01-00553-a, 08- compared with literature numerical and experimental results. 01-91302-IND a, 09-08-01180-a, 09-08-01194-a, and 10-01- 92653-IND a), by the Department of Energetic, Mechanical Francesco Genna, Via Branze 43, Brescia, 25123, Italy Engineering, Mechanics and Control Processes of the Russian Academy of Sciences (Program No. 13 OE), and by the Russian 3-D NUMERICAL MODELING OF A DIRECTIONAL Science Support Foundation. SOLIDIFICATION OF BINARY ALLOYS UNDER THE Dmitry A. Parshin, Vernadsky Ave 101 Bldg 1, Moscow, 119526, ACTION OF ROTATIONAL VIBRATIONS Russia YA.N.PARSHAKOVA T. P. LYUBIMOVA [email protected] [email protected] BRIDGED CRACKS GROWTH CRITERION

M.PERELMUTER Numerical simulation of a directional solidiﬁcation of binary alloys in the presence of rotational vibrations is performed in [email protected] the frame of three-dimensional (3-D) approach taking into account the existence of mushy zone. It is shown that the vibra- The nonlocal fracture criterion of bridged cracks is proposed. tions make strong stabilizing effect of morphological instability In the frames of the model of bridged crack it is assumed that the

77 mechanical bonds or fibers constraining the crack opening exist FATIGUE ASSESSMENT OF STRUCTURES IN between the crack surfaces. For composites and nanocomposites HIGH-CYCLE SEGMENT: TECHNIQUE AND based on polymers or ceramics matrix these mechanical bonds PROBLEMS are chains of molecules, nanotubes and nanoparticles. The size of the zone filled with the bonds (the bridged zone) can be compa- S.V.PETINOV I.M.AFANASYEVA rable to the size of the whole crack. The energetic characteristics of the crack such as the deformation energy release rate (ERR) [email protected] associated with the crack tip stress intensity factors (SIF) and the [email protected] rate of the energy dissipation by the bonds (RED) associated with the energy necessary to create a unit of new surface of the bridged Common rules for fatigue assessment of marine welded struc- zone are considered. In this model the ERR and the RED depend tures in formats “Hot–spot – Life” or “Notch stress – Life” are on the bridged zone size and the bonds characteristics. The non- based on implication of elastic behavior of material at critical local fracture criterion of bridged cracks consists of the following locations and one may expect a conservative estimation of fa- two conditions: 1) the necessary energetic condition of the crack tigue resistance. A more physically reliable evaluation of fatigue tip limit equilibrium (this condition is based on the equality of the damage may be provided by applying the “Strain–life” criterion; ERR and the RED values); 2) the sufficient condition of the bond however, uncertainties imbedded into current procedures remain limit stretching at the trailing edge of the bridge zone. On the and affect the results as shown briefly in an example. The main basis of these two fracture conditions the regimes of the bridged sources of uncertainties are indicated and the directions of further zone and the crack tip equilibrium and growth are considered: development of procedures are discussed. 1) the crack tip propagates and the size of the bridged zone increases without rupture of the bonds; 2) the size of the bridged Sergei V. Petinov, V.O. Bolshoy Pr.61, St.Petersburg, 199178, zone decreases due to rupture of the bond without the crack tip Russia propagation; 3) the crack tip propagates with simultaneous bond rupture at the trailing edge of the bridged zone. FORCED QUASI 2D TURBULENCE WITH CHEMICAL The last case corresponds to quasistatic crack growth. The REACTION: ANALYTICAL THEORY equilibrium size of the bridged zone and the critical external stress are determined from two abovementioned fracture condi- V.E.PETROV tions. Analytical application of the proposed criterion was per- [email protected] formed for the problem of the straight crack in homogeneous plane with the rectilinear law of bond stresses. The applications of the proposed model for different problems of fracture The experimental studies (Wells M.G. at el.,2008) led to the from macro-scale to nanoscale are presented. In the general case partial approvement of the hypotesis forced 2D turbulence with of the bond deformation law the problem of the bond stresses chemical reactions, proposed by the author (Petrov,2001,2006). determination is transformed to the system of singular integral- In the new paper we establish a series of exact results for this differential equations and all steps of the fracture criterion appli- model. cation are performed numerically. The estimations of the equilibrium size of the bridged zone, the adhesion fracture energy and REFERENCES the external critical stress (the interface strength) depend on the crack size are found. Some steps of the problem solution also are [1] M. G. Wells, H. J. H. Clercx and G. J. F. Van Heijst presented in the papers [1-3]. (2008):“Dlsperslon and Mixlng in Quasi-two-dimensional Rotating Flows.” Transport and Mixing in Geophysical Flows, Lect. Notes in Phys.744,pp119-136. Springer-Verlag Berlin Heidelberg. REFERENCES [2] V.E. Petrov (2006): “The forced model of the 2D turbulent chemical reacting flow(ln Russ.)”. An. of paper, 9th Russ. [1] R. V.Goldstein, M. N. Perelmuter. Modeling of bonding at Cong. on Theor. and App. Mech.,V.2. NNSU, Nizhny Nov- the interface crack, Int. J. of Fracture, 1999, vol.99, N 1-2, gorod. p.53-79

[2] M. Perelmuter. Growth of cracks bridged by nanoﬁbers, In: Vladimir E. Petrov, Ac. Lavrentiev str.,1, Novosibirsk, 630090, The abstract book of 11th International Conference on Frac- Russia ture, March, 20-25, 2005, Turin, Italy

[3] M. Perelmuter. A criterion for the growth of cracks with COMPLEX DYNAMIC SYSTEMS SYNTHESIS, BASED bonds in the end zone, Journal of Applied Mechanics and ONMETHOD OF LYAPUNOV’S EDGE CONDITIONS Mathematics (PMM), 2007, vol. 71, N1, pp.152-171 V.N.PILISHKIN V. V. SOSNOVTSEV [email protected] [email protected]

Mikhail N. Perelmuter, prospect Vernadskogo 101-1, Moscow, One of the main problems in complex system control is high 119526, Russia dimension of dynamic objects. It severely complicates the usage

78 of many known methods. Also different limitations (structural, has arbitrary given shape. These expressions and presupposed parametric, etc.) can be applied to the system; the disturbances forms of constitutive equations of the shell material make it pos- with incomplete information about their properties can inﬂuence sible the equations set for determination of sought characteristics the system, and the simulator itself becomes more complex. of the shell’s initial shape and its material be obtained. To insure The abovementioned factors severely complicate forming of unique solution of the equations set it is necessary some addi- required control laws, because during solving the problem of ro- tional conditions are arbitrary presupposed. The rational basis bust regulator synthesis given limitations are usually taken into for option of these conditions is not deﬁned. consideration marginally. So, with the problem of high dimension there is a problem of consideration of limitations and incom- Vitaly Lutov, Bolshaya Morskaya,18, Saint-Petersburg, 191186, plete information about disturbances at the same time. Russia To meet various conditions, applied to the system, to formal- ize their description and to form effective control algorithms, according all the abovementioned, the straight regulator synthesis “SMART HYDROGEN” AND ITS DETERMINATION IN method is offered. It is based on the Phase Limitations Variation MICRO AND NANOSTRUCTURES IN MATERIALS Method (PLVM) and simple enough algebraic transformations. This method allows to form the required control law, using quite A.POLYANSKIY V. POLYANSKIY E.CHERNYAEVA simple correlations, being, in fact, a standardized method. The YU.YAKOVLEV usage of this method for space satellites is offered in this work. [email protected] [email protected]

Vladimir V. Sosnovtsev, 1-st Schemilovskiy pereulok, bld. 16. , Hydrogen in small concentrations is containing in all mate- Moscow, 127473, Russia rials and its influence on materials’ properties is of crucial importance. Even for concentrations of about 1 hydrogen atom per DIRECT TENSOR CALCULUS IN THE PROBLEM OF 10000 atoms of the matrix one can observe the hydrogen embrit- MECHANICAL ENERGY DISSIPATION IN FLUIDS tlement. This influence can not be explained by means of the hypothesis M.PODOLSKY of the uniform distribution of hydrogen in the matter. Only the localization of hydrogen in microdefects, on the grains’ boundaries [email protected] and on the surface of nanostructures in the material can explain In well known educational hydrodynamic literature the prob- the considerable change in mechanical properties. lem of mechanical energy dissipation in fluids is considered as The investigation of the micro and nanostructures – hydro- a problem for fluids which have shear viscosity µ only. Volume gen interaction gives as new information about nano, micro and meso-scale of the material properties. The presentation reports viscosity µV is supposed to be equal to zero. Besides, even for some experimental proof of this interacton on example of high- the case µV = 0 traditional consideration of problem requires awkward and not obvious algebraic transformations. temperature extraction of hydrogen from aluminum alloys, sili- Direct tensor calculus gives a possibility to receive the solution con plates and nanomaterials. in a very simple way and to consider the general case when all When microdefects and nanostructures are present, the hydro- viscosities (both shear and volume) are taken into account. gen is localized on the free surface and decreases the free sur- In this work compact formulae for power of inner viscous face energy. This decrease results in termination of growth of forces are obtained and their physical sense is clarified. Numeri- micro- and nanocrystals. So the hydrogen leads to nanoscale sta- cal example illustrating the role of volume viscosity is done. bilization and we suggest to refer to such effects as the “smart hydrogen”. On the other hand, this leads to reduction of crack Marlen E. Podolsky, 3, Lotsmanskaya Str., Saint Petersburg, resistance and, in turn, to brittle destruction of the material. 190008, Russia The highly sensitive hydrogen analyzer AV-1 allows one to observe the hydrogen emission from microcracks and dislocations on the surface of specimen. The emission occurs during the spec- ON THE PROBLEM OF SOFT TOROIDAL SHELLS imen heating in vacuum and is registered in the form of small DESIGNING peaks on the extraction curve. The results of experimental investigations of aluminum alloys and steels, including zinc coated E.POLYAKOVA V. LUTOV V. CHAIKIN steels, steels with nanostructures are shown. The size of defects [email protected] [email protected] determined by means of microscopy is in a good agreement with v [email protected] the volume of hydrogen extracted under the atmospheric pressure. This allows us to assume that the microcracks and defects The designing of the soft toroidal shells that have prescribed are filled by hydrogen. mechanical properties consist of two interconnected problems, These assumptions were proved on monocrystals of silicon. i.e. the problem of choosing the initial parameters of the un- The microdefects on the surface having the characteristic size of stressed shell’s shape and the problem of determination the de- 30 micron were produced by using the infrared pulse laser. Small formational characteristics of material of which the shell would peaks on the curve of high-temperature hydrogen extraction curve be fabricated. were observed when these specimens were tested. The method of dealing with these problems that is proposed It became clear that for the materials which have no stable in this work is based on solution of auxiliary task that gives ex- chemical link with hydrogen, nearly whole hydrogen is con- plicit expressions of stresses in the soft toroidal inflated shell that centrated in microdefects and on the surface of micro- and

79 nanograins. Aluminum alloys, some steels, ceramic and semi- Sergey V. Polyanskiy, Politekhnicheskaya, 29 , St.-Petersburg, conductors eject hydrogen. Thus, the general amount of hydro- 195251, Russia gen can be used for estimation of the volume concentration of microcracks and dislocations. The diagnostics in terms of the amount of extracted hydrogen turns out to be efficient and less effort-consuming than other SOLITONS AND FLOW IN NANOTUBE methods of investigations. For instance, in order to determine an average concentration of microdefects by means of microto- I.YU.POPOV O.A.RODYGINA S.A.CHIVILIKHIN mography one needs to conduct measurements of about 12 hours V. V. GUSAROV whereas the hydrogen analysis takes 45-60 min. Summarising we can say that we suggest a new approach to [email protected] experimental investigation and diagnostics of microdefects and nanostructure in materials. For the case of the liquid flows in the nano-channel a nontrivial effect can play an important role. Namely, it is necessary to Vladimir Anatolievich Polyanskiy, p/b 162, St.-Petersburg, take into account vibration and waves in molecular chains form- 195220, Russia ing the nanotube wall. The most interesting is solitary wave. It is known that there exists stable solution in long molecular chain (so-called “Davidov soliton”). Geometrically, this solution seems as a local perturbation of the tube, which moves with some veloc- THE FOURIER SERIES EXPANSION IN THE THERMAL ity. This motion causes the motion of the liquid which is in this STRESS PROBLEM ON ROTATING WORK ROLLS perturbation. It is the mechanism of the flow in the case of very DURING HOT ROLLING narrow nanotube. S.POLYANSKIY Consider the soliton influence on the flow. Soliton moves in [email protected] nanotube wall and looks like a moving local extension (or constriction) of the tube. Our calculation shows that solution causes the increasing of the flow velocity in two cases: a) fast soliton Rolling is the main product in metallurgy. The main problem corresponds to the extension of the nanotube; b) slow soliton cor- of hot rolling is the fragile destruction of the surface of mill rolls responds to the constriction of the nanotube. In other cases the since it results in decrease of the quality of rolling. soliton causes the decreasing of the flow velocity. The mill rolls made of high strength steel. The rolls are heavy- loaded because of the direct contact interaction with hot rolling. The mill rolls are rotating and the contact patch moves on the surface of mill which leads to a cyclic loading. Evaluation of mechanical stress caused by heating and pres- I. Yu. Popov, Kronverkskii 49, St. Petersburg, 197101, Russia sure in the contact patch shows that thermal stresses considerably exceed the pressure. Immediately after contact with surface of the rolling the mill is subjected to cooling by with air-water mixture. Because of the rapid rotation the mill experiences a thermal APPLICATION OF NONLINEAR WAVES FOR THE shock which means that the thermal stresses are localized in a PROBLEM OF GROWTH OF LONG BONES thin surface layer. The temperature gradient in the body of mill is of the order of thousands of degrees Celsius per centimeter. A.V.PORUBOV The present report describes the model of rolling mill which [email protected] has contact patch moving on the surface. With the help of complex variable the problem is reduced to the equations of plain- stress states with a static load. Solution of this problem by Finite A nonlinear dynamic model is developed to account for mate- Element Methods is difficult because of the high temperature gra- rial inhomogeneities in an articular cartilage that corresponds to a dient in very thin surface layer of rolling mill. growth plate in long bones. The governing equations are obtained We suggest a traditional method of the expansion in Fourier to account for nonlinear dispersive, viscoelastic and inhomoge- series for the numerical solution of thermal problem. This ap- neous features of the growth plate. The evolution of nonlinear proach makes it possible to sum up the series with high accuracy strain waves over the material inhomogeneities is obtained via and take the solution with a minimal number of the harmonics. In the asymptotic solutions. It is shown that variations in the ampli- other hand we have the analytical solution. Using the temperature tude and the width of both the bell-shaped and kink-shaped waves fields in closed-form enables obtaining the thermoelastic poten- reflect the position and the size of the inhomogeneity. This may tial which in turn delivers the thermal stresses. The Airy function be used for a detection of the growing plate features and in the allows us to satisfy any boundary conditions on the mill’s surface. development of the reaction-diffusion equation for the stimulus We derive the equations for calculating stresses in terms of the of the growth of long bones. thermoelastic potential and the Airy function. Simulations show that the temperature difference on the mill’s surface (of order 500 degrees) is localized in a thin layer of about 1-2 millimeters. This leads to considerable cyclic stresses and then to a fragile destruc- Alexey V. Porubov, Bolshoy av. 61, V.O., Saint-Petersburg, tion of the rolling mill’s surface 199178, Russia

80 solids and the path length of such a particle is comparable with its IMPROVEMENT OF THE DYNAMIC BEHAVIOURS OF initial width, then the classical wave description is unsatisfactory THE PLANT WITH SELFSYNCHRONIZING for electrons a priory. VIBROEXCITERS BY MEANS OF ADDITIONAL The research was carried out under the financial sup- DEGREES OF FREEDOM port of the RFBR (projects N 09-08-97032-r povolzhie and M.A.POTAPENKO 09-02-97053-r povolzhie). [email protected] Igor S. Pavlov, Belinsky str., 85, Nizhny Novgorod, 603024, Rus- The question about possibility of provision of synphase rota- sia tion of two unbalanced rotors of vibration plant in selfsynchronizing regime is discussed.In this case under consideration only the antiphase rotating is “natural stable”. The possibility of using MOLECULAR MECHANISMS OF POLYMER for this aim the vibroexciters with additional degree of freedom COMPOSITES WEAR AS REVEALED BY of rotors or the changing the number of the degrees of freedom MASS-SPECTROMETRY of the platform is studied. Here we consider one of arrangements A.POZDNYAKOV B.GINZBURG with self-synchronized vibroexciters which can be used in various cases when both horizontal and vertical oscillations of work- [email protected] ing tool are required. Installations used in foundry manufacture can exemplify this fact. We discuss the results of the systematic study of thermal decomposition (TD) and mechanical decomposition (MD) Maria Potapenko, Bolshoy pr, 61 V.O., SPb, 197191, Russia MS analysis of polymers, polymer-fullerene C60 composites and polymer-fullerene covalently bound star-shaped compounds. The results were obtained with the aid of time-of-flight mass- DESCRIPTION OF NANOSTRUCTURED MEDIA BY spectrometer which enables the immediate kinetic analysis of the MEANS OF QUANTUM AND CLASSICAL number of volatile products, temperature of the sample and total APPROACHES pressure inside MS. The monomer and fullerene desorption from the sample into vacuum chamber of MS system are discussed in A.I.POTAPOV I.S.PAVLOV terms of the characteristic experimental parameters (pumping re- [email protected] laxation time, heat diffusion and diffusion relaxation times etc.) with an emphasis on the possibility of extraction of the correct Nanostructures are examples of objects in which classical rep- kinetic rate constants of the decomposition elementary processes resentations start to conflict to the quantum nature of the physi- (depolymerization, initiation activation energies etc.). cal processes occurring in a substance. The accurate approach to studying of nanostructures is assumed to be based on the quan- Aleksei Pozdnyakov, Novostroek 20/86, S-Petersburg, 198188, tum theory laws, and mathematical modeling consists in transi- Russia tion from models of an atomic level to mesoscale models. How- ever, criteria are still unknown when methods of classical physics are permissible for studying of dynamic processes in nanostruc- INFLUENCE THE DISPERSION IN NONEQUILIBRIUM tures, and it is known that the more tightly the microscopic object MODELS FOR CONTINUOUS MECHANICS is bound to classical surroundings, the its behaviour is closer to classical one. In nanostructures, physical phenomena can be di- E.V.PROZOROVA vided into the following: [email protected] – pure quantum phenomena missing in a classical limit, but playing an essential role in macroscopic behaviour of a substance; We discuss the problems that can be appearing to considerate – pure classical effects; the angular momentum variation in elementary volume and cross – effects of the mixed type occuring both in the quantum, and in flux through the sides of an elementary volume. The equilibrium the classical theories. conditions of forces are the special case of more common condi- The goal of the given work is detection of boundary between tions of equilibrium of angular momentum and are followed from quantum and classical effects in nanostructures. Such opposite these.The aim of this paper is to find the reason for appearing the (from the viewpoint of classical physics) concepts as the particle second viscosity for cross flow at large gradients of the physi- and the wave, are unified at a level of microeffects and one ob- cal values. For normal shock waves angular momentum does not ject arises that is called “microparticle”. Classical concepts of a work but the self- diffusion give additional diffusion. The con- space location and a pulse quantity are applicable to a microparti- vective flow has the same value as diffusion flow for the Mach cle only within the scope of the Heisenberg uncertainty relations. number M=1. It will be received in our work. The boundary con- Since the localized particle is described by a wave train occupy- ditions for rarefied gas near the surface make more accurate than ing a small area of space, there is a precise quantitative analogy early. between the Heisenberg uncertainty relations and the wave properties. Due to that, the evaluations are performed, which show that if time of applicability of the classical theory to a particle Evelina V. Prozorova, Mathematics & Mechanics Faculty, St. Pe- with atom mass is comparable with time of “life” of phonons in terburg State University av. 28 , Peterhof, 198904, Russia

81 axe direction periodic boundary conditions were applied, in oth- INFLUENCE OF POROSITY PERCOLATION ON ers directions free surfaces were simulated. Opposite to load- STRENGTH AND ELASTIC PROPERTIES OF CERAMIC ing surface was ﬁxed in loading direction to avoid specimen dis- MATERIALS. 3D SIMULATION USING MOVABLE placement as a hole. Indenter had form of half-cylinder, contain- CELLULAR AUTOMATA ing copper atoms. Cylinder axe was parallel to loading surface. Loading speed varied from 5 mps upto 50 mps. Such indenter S.G.PSAKHIE N.V.ROMAN A.YU.SMOLIN shape allows to simplify of structural defects formation and its [email protected] evolution particularizes analysis in comparison with sphere indenter form. It is well known that porosity dependence of strength and elas- Structural changes analysis was performed on the base of tic properties of porous materials is determined by pore morphol- method taking into account nearest neighbors relative space po- ogy. If porosity is small, each pore is closed and isolated form the sition for each of atoms. others. With porosity increasing the number of pores becomes Calculations have shown that with indenter approaching the greater and the distance among pores decreases. When porosity surface local structural defects (called protodefects) began to ap- reaches percolation threshold the pores are not closed anymore pear which then spread in atomic planes [110] and [11¯0]. The and consequently form new morphology. Thus, porosity depen- development of defects generation leads to staking fault forma- dence of strength and elastic properties should be changed when tion which are the partial dislocation carriers. With defect system percolation threshold is exceeded. development both types of staking fault, intrinsic and extrinsic, In this paper 3D computer simulation of mechanical behavior appeared as well as their combination. Staking fault analysis had of brittle porous material under uniaxial compression is consid- shown that the sum of partial dislocation carried on it could be ered. Movable cellular automaton method, which is a representa- equal to one or two perfect dislocations. As staking fault reachs tive of particle methods in solid mechanics, is used for computa- the free surface step appears on it. Size of step depends on count tion. In initial structure automata are positioned in FCC packing. of perfect dislocations carried on current staking fault. Pores are set up explicitly by removing single automata from ini- Thus received results demonstrate the process of local struc- tial structure. At that when the next automaton is being chosen to tural defects formation during nanoindentation which appear to be removed it is checked if there are no close neighboring pores be of some interest from both scientiﬁc and practical points of around this automaton. If porosity is small that algorithm works, view. but when porosity approach to percolation threshold we have to allow merging some pores. The computational results show that the curve of porosity de- Dmitrij S. Kryzhevich, Academicheskii pr. 2/4, Tomsk, 634021, pendence of strength and elastic properties of the modeled spec- Russia imens has a break at the porosity about 20%, i.e. percolation threshold. It has to be noted that the curves are plotted in loga- rithmic coordinates The obtained results are in good agreement ON SIMPLE DYNAMICAL SYSTEMS WITH with available experimental data. HYSTERESIS OPERATORS

Nikita V. Roman, pr. Akademichesky 2/4, Tomsk, 634021, Rus- G.RADONS sia [email protected]

PLASTIC DEFORMATION NUCLEATION ON INITIAL Complex hysteresis is a well-known phenomenon in many STAGE OF COPPER CRYSTALLINE branches of science [1]. The most prominent examples come NANOINDENTATION INVESTIGATION from materials with a complex microscopic structure such as magnetic materials, shape-memory alloys, or, porous materials. S.G.PSAKHIE K.P.ZOLNIKOV D.S.KRYZHEVICH Their hysteretic behaviour is characterized by the existence of [email protected] [email protected] multiple internal system states for a given external parameter [email protected] and by a non-local memory. The input-output behaviour of such systems is well studied and in a standard phenomenological ap- Studies of material behavior on nano-scale level under contact proach described by the so-called Preisach operator. loadings are known as a good instrument for incipient plastic- What is not well understood, are situations, where such a hys- ity investigation. With experimental investigation methods last teretic system is dynamically coupled to its environment. Since decades computer simulation of nanostructural materials behav- the hysteretic sub-system provides a complicated form of nonlin- ior during nanoindentation is under heavy development. This earity, one expects non-trivial, possibly chaotic behaviour of the process is induced by not only appearance of supercomputers but combined dynamical system. We study such combined dynami- also by development of interatomic potentials which can describe cal systems with hysteretic nonlinearity for four cases of increas- physical and mechanical properties with a good accuracy. ing complexity: In present work the investigation of plastic deformation nu- 1) In the simplest class the hysteretic nonlinearity simply acts cleation in thin monocristalline copper ﬁlm under initial stage of as a transducer, which transforms input time series into output indentation ( 0.2nm) of perfectly rigid half-cylinder is carried time-series. The case of i.i.d. random variables as input time out. ∼ series is considered and analytical results can be obtained. Itis Computational experiments were carried out under copper shown that under quite general circumstances the output time- monocrystalline ﬁlm containing 330 000 atoms. In cylinder series exhibits long-time tails and 1/f-noise [2-4]. ∼

82 2) Situations similar to 1) are studied, but with more structured Upon recalling a few basic concepts of dynamical systems the- input time-series. Especially input from nonlinear dynamical sys- ory (invariant manifolds, homoclinic/heteroclinic bifurcations, tems, which may be periodic, fractal, or chaotic is considered, safe attractor and/or basin), along with their theoretical and prac- and the corresponding output of the hysteretic system is charac- tical relevance, some main issues concerned with the concept of terized [5]. dynamic integrity are presented and discussed. 3) Output of a hysteretic system is transformed by a nonlinear a) The proper definition of “safe basin”, which depends on con- map and fed back as a new input to the hysteresis operator. The sidering the solely steady or also transient system dynamics, on properties of such a hysteretic feedback system are described. accepting fractality of the relevant basins of attraction or requir- We find, for instance, new kinds of attractors and a fractal de- ing their strict compactness, as well as on specific mechanical and pendence of the asymptotic behaviour as function of the starting dynamical system features. values [5]. b) The most suitable measures of safe basin integrity, which are 4) A simple differential-operator equation with hysteretic crucial for a reliable quantitative assessment of system overall damping, which describes a magnetic pendulum is considered. safety. Differences and similarities to case 3) are emphasized. The sen- c) The investigation of the evolution of attractor-basin portraits sitivity of the system to perturbations is investigated [6]. due to variations of system parameters, which is evaluated via In summary, we show that the coupling of simple dynamical the construction of integrity profiles of great practical interest. systems with a typical model for hysteresis, the Preisach model, d) The unwanted erosion of safe basins and the ensuing need for leads to interesting new scenarios in nonlinear dynamics. its reduction. e) The theoretical formulation and implementation of a nearly “universal” technique for controlling homoclinic or heteroclinic REFERENCES bifurcations of a hilltop saddle, as main triggering events for the onset of transient, and possibly steady, complex phenomena. [1] G. Bertotti and I. D. Mayergoyz (eds.) 2006 The Science of Hysteresis, Vol.1-3, Academic Press. The main focus is on the practically most interesting cases in which the control parameter is the excitation amplitude and the [2] G. Radons. 2008 Hysteresis induced long-time tails, Phys. system integrity reduces as it increases. With the aim of reducing Rev. Lett. 100, 240602. erosion, i.e. of shifting the erosion profiles toward larger excita- [3] G. Radons. 2008 Spectral properties of the Preisach hystere- tion amplitudes, a control method based on optimally choosing sis model with random input. I. General results, Phys. Rev. the shape of the periodic excitation is applied. E. 77, 061133. [4] G. Radons. 2008 Spectral properties of the Preisach hystere- Referring to single-well or multi-well potential systems, ero- sis model with random input. II. Universality classes for sym- sion of in-well dynamics from the out-of-well unbounded (es- metric elementary loops, Phys. Rev. E. 77, 061134. cape) one, and the competing integrity of coexisting in-well attractors, or of in-well vs out-of-well attractors, are addressed by [5] G. Radons, F. Heße, R. Lange and S. Schubert. 2008 In Ver- numerically constructing and systematically comparing the ero- netzte Wissenschaften, Crosslinks in Natural and Social Sci- sion/integrity curves of different classes of oscillators, selected ences (eds. Plath, P.J. and Haß, E.-Ch.) pp. 271-280, Logos, with the aim of highlighting the effects of some main mechani- On the dynamics of nonlinear hysteretic systems. cal, dynamical and control features. [6] G. Radons, A. Zienert. 2009 In Book of Abstracts, 2nd In- ternational Conference on Recent Advances in Nonlinear In particular, attention is devoted to a number of archety- Mechanics (RANM), 24-27 August 2009, Kuala Lumpur, pal single-degree-of-freedom oscillators representing either natu- Malaysia, (eds. Ho,J.-H., Wiercigroch, M. and Woo, K.-Ch.) rally discrete systems or first-order Galerkin-based approxima- pp. 92-93, Nonlinear dynamics of a simple system with com- tions of the finite-amplitude dynamics of infinite-dimensional plex hysteresis. systems. They include the Helmholtz and Duffing oscillators, a piecewise-smooth oscillator describing the rocking dynamics of a rigid block, an escape-like oscillator suitable to analyze the Guenter Radons, Reichenhainer Str. 70, Chemnitz, D-09107, dynamic pull-in phenomenon in an electrically actuated thermo- Germany elastic micro-beam for MEMS applications, and a parametrically excited pendulum.

DYNAMICAL INTEGRITY AND CONTROL OF Then, a few systems with two degrees-of-freedom are consid- NONLINEAR MECHANICAL AND STRUCTURAL ered, including the Augusti inverted pendulum, a model of rigid SYSTEMS inverted pendulum constrained by three springs - to be considered as a ﬁrst approximation of a stayed cable telecommunication an- G.REGA tenna - and a reduced order model describing the post-buckling [email protected] coupled dynamics of a parametrically excited cylindrical shell.

Dynamic integrity of nonlinear oscillators and its control are The extension to multi-dimensional systems of the numeri- topics of considerable interest in view of properly pursuing and cal techniques developed for evaluating the dynamic integrity of guaranteeing the global safety of mechanical systems and struc- basins of attraction in a one-dimensional framework shows to be tures within an evolutionary context. quite involved due to computational and topological difﬁculties.

83 REFERENCES the start-up of the machine. We also represent some new results of the numerical modeling of VTM dynamics. [1] Lenci, S., Rega, G., A unified control framework of the non- It was stated that the motion of the working element (WE) in regular dynamics of mechanical oscillators, J. Sound Vibra- the case of three VE is not progressive. Changing the VE position tion, 278, 1051–1080, 2004. and eccentric moments makes different points of the WE move by [2] Rega, G., Lenci, S., Dynamical integrity and control of non- different trajectories. linear mechanical oscillators, J. Vibration and Control, 14, We have found VE position and eccentric moments leading to 159–179, 2008. such distribution of vibration types on the charge and discharge [3] Orlando, D., Gonçalves, P.B., Rega, G., Lenci, S., The use of ends of the machine’s WE which will let us intensify the bulk analytical dynamics in the global analysis of systems liable materials dispersion from the bunker on the charge end and also to unstable post-buckling behavior, 20th Int. Congr. Mechan- intensify the fractures separation (the sieving) on the discharge ical Engineering, Gramado, CD-Rom, 2009. end of the machine. [4] Gonçalves, P.B., Silva, F.M.A., Rega, G., Lenci, S., Global dynamics and integrity of a two-d.o.f. model of a parametri- Sergey A. Rumyantsev, 66, Kolmogorova st., Ekaterinburg , cally excited cylindrical shell, Nonlinear Dynamics, 2010 (to 620034, Russia appear).

Giuseppe Rega, Dipartimento di Ingegneria Strutturale e Geotec- EVER-TOPICAL QUESTIONS REGARDING THE nica, Sapienza Universit´adi Roma via A. Gramsci 53, 00197, “EVER-HOOKEAN” MATERIAL Roma, Italy E.I.RYZHAK

INVARIANT INTEGRALS IN LINEAR ELASTICITY E I [email protected] THEORY FOR BODIES WITH RIGID INCLUSIONS AND CRACKS An old question of whether the Jaumann stress-rate applies for expressing incremental constitutive relations in the case of finite E.RUDOY strains and rotations is discussed. Analyzed is the well-known [email protected] example of the deformation process of monotonic simple shearing of the “ever-Hookean” material. By this is meant a one for which the Jaumann rate of the Cauchy stress is related linearly The equilibrium problem of the elastic body, containing a rigid to strain-rate tensor by virtue of a constant (depending neither on inclusion is considered. There exists a crack between the rigid strains, nor on rotations) fourth-order moduli tensor identical to inclusion and the body. The volume forces are defined in the that of Hooke’s law. whole domain, including the rigid inclusion. The displacements of the rigid inclusion are non-zero in a general case. It is stated that such a choice of an incremental moduli ten- Using the formula of the derivative of the potential energy with sor of a solid is absolutely unnatural, and it is just the reason respect to a parameter of a domain perturbation a common form causing strange behavior of the material in hand under simple of invariant integrals is constructed. In particular, examples of in- shear, whereas by no means it is caused by usage of the Jaumann variant integrals for different types of perturbations and different stress-rate, legality of which follows directly from the principle geometry of the domains are given. of material frame-indifference. Invariant integrals are widely used in the fracture mechanics With regard for known (obtained earlier) dependence of the for defining of energy release rate, SIF. Moreover, invariant in- Cauchy stress on a parameter γ (tangent of the angle of shear) tegrals are applied in problems of an identification of defects in the Hadamard inequality is recorded for the incremental moduli elastic bodies. tensor associated with the Piola stress. Assertion of the Basic Hadamard stability theorem reads that just this inequality estab- Evgeny M. Rudoy, pr. Lavrent’eva, 15, Novosibirsk, 630090, lishes the universal local necessary condition of stability; note Russia that it involves not only moduli, but also current value of the Cauchy stress. Analysis of the Hadamard inequality shows that it is violated NUMERICAL SIMULATION OF NON-LINEAR immediately after the value γ = π/3, when the shear stress is still DYNAMICS OF VIBRATION TRANSPORT MACHINES increasing (it begins to decrease after the value γ = π/2). Thus, IN CASES OF FOUR AND THREE INDEPENDENTLY there will be no oscillation for the shear stress, which is usually ROTATING VIBRATION EXCITERS “incriminated” to the Jaumann stress-rate. Actually there will be something much worse: the material will collapse at γ = π/3+0, S.A.RUMYANTSEV D.Y.TARASOV that point pertaining to the rising part of the diagram of shear. [email protected] To conclude, it is the absurd ever-Hookean incremental constitutive relation that is guilty for all of this, not the innocent Jau- In this report we present some new results on dynamics of vi- mann stress-rate! bration transport machines (VTM) of different types with independently rotating vibration exciters (VE). The principal objects of this research are transient dynamic processes accompanying Evgeniy I. Ryzhak, 10 B.Gruzinskaya, Moscow, 123995, Russia

84 MATHEMATICAL MODELS OF THERMOELASTICITY KINETIC FORMULATION FOR THE DARCY–STEFAN OF MICROPOLAR THIN ELASTIC SHELLS, PLATES PROBLEM ON PHASE TRANSITIONS IN A AND BARS SATURATED POROUS GROUND

S.SARGSYAN S.SAZHENKOV [email protected] [email protected]

Presently, thin bars, plates and shells on the basis micropolar In 1999 J. F. Rodrigues and J. M. Urbano proposed a rather (asymmetrical, momental) theory of elasticity (i.e. with the ac- generic multi-dimensional Darcy–Stefan model to describe the count of the momental contribution in interatomic interaction) are processes of freezing (thawing) of a viscous liquid in a motion- considered from the point of mathematical models of thin-walled less porous soil with identical values of density in the liquid and nano-sized objects (as nano bars, biological-cell, membranes or frozen phases and with allowance for the buoyancy force de- nano-composites). From the point of weightiness of applications pending nonlinearly on temperature [1]. In the present talk we the construction of continual models of micropolar thin elastic formulate and study a Cauchy problem for the model, supple- bars, plates and shells has become rather actual. mented with periodic initial data. We explain the physical sense and the framework of applicability for the model. The Darcy– The basic problem of constructing general models of thermoe- Stefan problem is studied in terms of an unknown specific inter- lastic micropolar shells, plates and bars consists in approximated nal energy rather than temperature, which makes the mathemati- but adequate reduction of the three-dimensional problem of mi- cal formulation much more difficult, because the energy balance cropolar theory of elasticity to two or one-dimensional boundary equation with such a choice of the sought function is a degen- problem. In order to achieve this the asymptotic method or the erate parabolic-hyperbolic equation, the degeneration proceeding received qualitative results of asymptotic integration of bound- on a segment of values of the specific internal energy. A def- ary problem in the micropolar theory of thermoelasticity in thin inition of the entropy solution of the Darcy–Stefan problem is spheres can be applied. introduced. This solution is more restrictive than the standard Paper [1] demonstrates the general equations and boundary definition of the weak generalized solution. All possible entropy conditions of thermoelasticity of micropolar thin shells (with in- solutions are found to satisfy the second law of thermodynamics dependent fields of transition and rotation; with constraint rota- postulating non-negative production of entropy. From the physi- tion and with small shift rigidity) received with the help of the cal viewpoint, this is an advantage of the entropy solution over the asymptotic method based on values of sizeless physical parame- standard weak generalized solution. A theorem of the existence ters. In this case shift deformations are not factually taken into of the entropy solution is formulated; the proof of this theorem is account. based on the results of the Antontsev–Monakhov theory and on In the present paper on the basis of qualitative results of initial the recently created method of the kinetic equation, which allows approximation of the asymptotic method of integration of bound- the original entropy formulation to be interpreted in terms of the ary problem of the three-dimensional micropolar thermoelastic- linear equation of the type of the Boltzmann equation in the ki- ity in the thin sphere of the shell, there are formulated hypotheses netic theory of gases. The complete text of the work is published (assumptions), which enable the construction of general specified in [2]. models of thermoelasticity of microolar shells with independent rotation; with constraint rotation and with small shift rigidity. In the constructed theories all the rotation and shift deformations REFERENCES are taken into account. It should be noted that if through the constructed theories we switch to the classical theory of shells, [1] J. F. Rodrigues, J. M. Urbano. On a Darcy–Stefan problem then we shall obtain the specified theory of thermoelastic shells arising in freezing and thawing of saturated porous media // of Timoshenko-Reissner type. Contin. Mech. Thermodyn. 1999. V. 11. No 3. P. 181-191. In the present paper, as special cases of the theory of shells [2] S. A. Sazhenkov. Studying the Darcy–Stefan problem on there are formulated the general equations and boundary condi- phase transition in a saturated porous soil // J. Appl. Mech. tions for micropolar thin elastic plates and shells. Technical Phys. 2008. V. 49. No 4. P. 81-93.

Sergey A. Sazhenkov, Prospekt Lavrentyeva, 15, Novosibirsk, 630090, Russia REFERENCES

[1] S. H. Sargsyan. Thermoelasticity of Thin Shells on the Basis VIBRATIONAL LIFT FORCE ACTING THE CYLINDER of Asymmetrical Theory of Elasticity// Journal of Thermal IN VISCOUS LIQUID Stresses. 2009. Vol. 32. N8. pp-791-818. V.D.SCHIPITSYN A.A.IVANOVA [email protected] [email protected]

The lift force acting on a cylinder in a cavity ﬁlled with a vis- Samvel Sargsyan, Paruir Sevak 4, Gyumri, 3101, Armenia cous liquid and subjected to high-frequency horizontal vibration

85 is experimentally investigated. Amplitude and frequency of vi- sidered as the second streamlined aerofoil with extremely small brations, liquid viscosity, relative density of the body and its di- (vanishing) extent but with non-zero circulation around it. ameter are varying. The experiments are made with the use of a The set of parameter’s values (oncoming flow velocity and cir- high-speed camera of high resolution. culations around aerofoil and point vortex) when the flow scheme The horizontal translational vibrations of the cavity generate a has a form of adjoining to aerofoil cavity is determined. The force repulsive force which is able to keep the heavy body at some dis- on aerofoil is calculated. In particular it turns out that for cavity tance from the cavity bottom and the light body at some distance formation the circulation around point vortex (vortex strength) from the ceiling. In both situations the body performs oscillations must be in definite limits depending on vortex position and on- without touching the wall of the cavity. The vibrational lift force coming flow velocity. At that when the circulation is enough large is significant at a distance comparable with the thickness of vis- (close to upper value) the cavity attracts aerofoil, at small (close cous boundary layer and decays with increasing of the distance. to lower value) circulations it repels aerofoil. It is in a qualitative agreement with the results for spherical solids [1]. Evgeny V. Semenko, Vybornaya 152-102, Novosibirsk, 630126, In the presence of intensive vibrations the light cylinder has a Russia stable position near the ceiling of the cavity and it can be held near the bottom by vibrational attractive force, too. The dependence of the lift force (attractive and repulsive ones) INFLUENCE OF TEMPERATURE AND FILLER on the distance between the solid and the boundary is investi- CONCENTRATION ON THE DYNAMIC MODULES OF gated in the wide range of dimensionless frequency of vibrations. POLYETHYLENE-BASED NANOCOMPOSITES The experimental method consists in a direct measurement of the V. V. SHADRIN L.A.KOMAR G.P.BASHIN averaged lift force by its comparison with the body weight. In overcritical domain when the heavy or light body is sus- [email protected] [email protected] [email protected] pended near the wall the threshold excitation of the body mean motion along the axis of vibration is found. The body performs a The influence of temperature and relative filler content on uniform motion along the wall and changes its direction near the the dynamic modules of polyethylene-based nanocomposites was end-walls of the cavity. It is found that oscillatory and average studied experimentally. The filler concentration was varied in dynamics of light and heavy bodies are similar. It is shown that the range from 0% to 15% with step 5%, and the temperature the average motion is caused by the breakdown of the symmetry was varied between -50 and +50 degrees. The frequency of of the body oscillations [2]. cyclic loading was 1 Hz. The analysis revealed the dependence The investigated effects are of interest for the problems of con- of the dynamic elasticity modulus and the loss modulus on the trol and manage of solid inclusions in viscous liquids and can be above-mentioned factors. In parallel with experimental studies, used in vibrational methods of liquid cleaning. the properties of polyethylene samples were investigated within The work is supported by the Russian Foundation for Basic the framework of the structural-phenomenological approach. The Research (Grant 09-01-00665a) and Academic council of PSPU results of numerical modeling indicate that the theoretical and (grant of 2010). experimental data obtained for the loss modulus are in good agreement and those for the dynamic modulus of elasticity agree satisfactorily. This allows us to conclude that the behavior of REFERENCES polyethylene is similar to that of the viscoelastic material at prescribed amplitude and frequency of vibrations. [1] A. A. Ivanova, V.G. Kozlov and A. F. Kuzaev. Vibrational hy- This study was supported by the Russian Foundation for Basic drodynamical interaction of sphere with cavity boundary // Research (grant 09-08-00339-a) and RAS Project 09-C1-1008. Fluid Dynamics. 2008. Vol. 43. No 2. pp. 194–202. [2] V.D. Schipitsyn. Behaviour of a cylindrical solid in a rectan- Gennady P. Bashin, Bukireva 15, Perm, 614990, Russia gular cavity with viscous liquid subjected to horizontal vibration // Convective flows... Vol. 4. Perm: PSPU, 2009. pp.169– 186. DYNAMICAL SYSTEMS WITH VARIOUS DISSIPATION: BACKGROUND, METHODS, APPLICATIONS

Vitaliy Schipitsyn, Sibirskaya, 24, Perm, 614990, Russia M.V.SHAMOLIN [email protected]

THE FORCE ON AEROFOIL IN STEADY This work is devoted to the development of qualitative meth- IRROTATIONAL FLOW WITH IMMOBILE POINT ods in the theory of nonconservative systems that arise, e.g., in VORTEX such fields of science as the dynamics a rigid body interacting with a resisting medium, oscillation theory, etc. This material E.SEMENKO can call the interest of specialists in the qualitative theory of or- [email protected] dinary differential equations, in rigid body dynamics, as well as in fluid and gas dynamics, since the work uses the properties of The problem of plane steady irrotational flow past aerofoil is motion of a rigid body in a medium under the streamline flow considered. To stimulate the flow with adjoining to aerofoil cav- conditions. The author obtains a full spectrum of complete in- ity we place into fluid domain the immobile point vortex con- tegrability cases for nonconservative dynamical systems having

86 nontrivial symmetries. Moreover, in almost all cases of integra- chamber. Earth’s surface deformation in the vicinity of the vol- bility, each of the first integrals is expressed through a finite lin- canic edifice has been selected as a detector of pressure variation ear combination of elementary functions and is a transcendental inside the volcanic chamber. function of its variables, simultaneously. In this case, the tran- Registration of deformation processes of the Earth’s surface scendence is meant in the complex analysis case, i.e., after the in the vicinity of the volcanic edifice is performed using geodesic continuation of the functions considered to the complex domain, and satellite measurement values. In this paper, we suggest to use they have essentially singular points. The latter fact is stipulated fiber-optic sensors (23 mm base) for surface deformation moni- by the existence of attracting and repelling limit sets in the system toring, which allows us make measurements with an accuracy to − considered (for example, attracting and repelling foci). 10 6. The process of describing the force field is a sequence of steps. The purpose of this study is to find key parameters that are We first study a preparatory model of the force field and construct most sensitive to changes in the state of the volcanic system. a family of mechanical systems whose motion has different char- The system under consideration consists of a volcanic edifice, acteristics that essentially depend on model parameters such that a magma chamber, a volcanic channel and surrounding rocks. the information about them is incomplete or does not exist at all. Rock material is assumed to be homogeneous, isotropic and elas- As a result of studying such a model, there arise questions such tic with the elastic modulus of 5000 MPa and Poisson’s ratio that the answers to them cannot be found in the framework of equal to 0.2. The volcanic system was deformed under the ac- the accepted model. Then the elaborated objects become the sub- tion of the applied pressure distributed over the surfaces of the ject of a detailed experimental study at the second step. Such chamber and the channel. The deformation field was calculated an experiment presupposes the answers to the formulated ques- using the two-dimensional axisymmetric finite element model. tions and either introduces necessary corrections to the prepara- The present approach has been used to study the influence of tory constructed model or reveals new questions, which lead to sizes, shapes and chamber location depth on the Earth’s surface the necessity of the first step repetition but in a new level of the deformation, to estimate the zone size in the vicinity of the vol- problem understanding. canic edifice where the surface deformation can be measured with In this work, we study the problem on the body motion under the given accuracy, to investigate the possibility of registration of the condition that the line of the force applied to the body does plug location in the channel, and to analyze the behavior of the not change its orientation with respect to the body and can only volcanic system in the presence of several chambers. displace parallel to itself depending on the angle of attack and, The obtained results indicate that the deformations εrr are less possibly, on other phase variables. Such conditions arise under sensitive to changes in the parameters of the volcanic system, the plate motion with the so-called “large” angles of attack in and the deformations εϕϕ can be preferably used for recording a medium under a streamline flow (in this case, the fluid is as- changes in the state of the volcanic system. sumed to be ideal in general, although all this are also true for This research was supported by the RAS Presidium Program fluids of a small viscosity, first of all, for the water) or under a (project No 09-P-1-1010) and by the Research and Educational separation flow (which is justified by an experiment completely Center “Non-equlibrium transitions in continuum media” (grant satisfactory). Therefore, the main objects of studying is a family No 19-14n-01a). of bodies such that a part of the surface of each of which has a plane part that is flowed by a medium according to the streamline Varvara A. Bogoyavlenskaya, 1, acad. Korolev, Perm, 614013, flow laws. Russia Maxim V. Shamolin, Michurinskii Ave., 1, Moscow, 119899, Russia A THREE DIMENSIONAL BRICK ELEMENT FOR NONLINEAR ORTHOTROPIC ELASTIC MATERIALS ESTIMATION OF SURFACE DEFORMATION BASED ON THE THEORY OF A COSSERAT POINT SENSITIVITY TO CHANGES IN THE CHARACTERISTIC PARAMETERS OF THE VOLCANIC L.SHARIPOVA M. JABAREEN M.B.RUBIN SYSTEM [email protected] [email protected] I.N.SHARDAKOV V.A.BOGOYAVLENSKAYA [email protected] [email protected] [email protected] In the Bubnov-Galerkin approach the response of an element At present, there are many methods for volcanic activity moni- is determined by integrals over the element region with the as- toring. The most widespread approach to this problem is based on sumption that the kinematic approximation is valid pointwise. the methods of registration and analysis of seismic waves caused This formulation leads to a robust element, but this element ex- by volcanic activity. However, seismic events can be recorded hibits unphysical locking for bending dominated structures and only at the stage of high eruptive activity. From the point of view for nearly incompressible materials. Special methods based on of prediction possibility, the quasi-static deformation processes enhanced strains, incompatible modes and reduced integration that precede dynamic processes are of primary interest. with hourglass control have been developed to overcome these One of the key factors determining the quasi-static state of problems. But these improved formulations can exhibit unphysi- the volcanic system is the hydrostatic pressure in the volcanic cal hourglassing in regions experiencing combined high compres- chamber, but it is impossible to measure pressure directly in the sion with bending. volcanic chamber. Therefore, it is necessary to develop indi- The Cosserat point element (CPE) is a new element based on rect methods for registration of pressure evolution in the volcanic the theory of a Cosserat point. Being robust user friendly element

87 for nonlinear elasticity it does not exhibit unphysical locking or this field have led to the contemporary theory of solitons [2-4]. hourglassing for thin structures or nearly incompressible materi- Following this theory, the problem of vibration energy localiza- als [1]-[5]. In contrast with standard finite element methods CPE tion for an infinite FPU chain can be asymptotically reduced to theory considers an element as a structure and introduces a strain finding a localized solution of the nonlinear Schrödinger equation energy function which characterizes the response of the struc- (NLSE), which is a completely integrable system [4] in contrast ture. The strain energy function is suitably restricted so that the with a discrete NLSE. Due to the reduction to NLSE, the math- element satisfies a nonlinear version of the patch test. The con- ematical origin of the vibration energy localization in an infinite stitutive coefficients in the strain energy function are determined chain can be clearly understood. However, it was recently [5] by comparison with exact solutions of linearized 3D theory. shown that study of not infinite but a finite (periodic) FPU chain In this paper a three dimensional brick element for nonlinear has to be chosen as a starting point of investigation to clarify the orthotropic elastic materials is derived based on the theory of a physical nature of this phenomenon. Cosserat point. The element has eight nodal director vectors that This important conclusion, obtained for symmetric FPU chain, characterize locations of the nodes of the brick element. The has allowed clarifying the conditions of the transition from com- nodal vectors are determined by balance laws of director mo- plete energy exchange between different parts of the oscillatory mentum and hyperelastic constitutive equations for intrinsic di- chain to energy localization on excited group of particles. This rector couples which specify the nodal forces. The constitutive result has been obtained on the base of LPT – concept developed constants are determined by a direct approach which compares in the papers [6, 7]. Cosserat solutions with exact solutions of the linearized equa- The symmetric model of FPU chain is valid for various me- tions of an orthotropic rectangular parallelepiped. The constants chanical problems. However, for application to the problems of determined are shown to be different from those obtained by ex- Molecular and Polymer Physics it is necessary to consider asym- act integration of the associated Bubnov-Galerkin approximation. metric (α-β) FPU problem because realistic interatomic poten- In particular, it is shown that the Cosserat approach eliminates tials of interaction are strongly asymmetric ones. unphysical locking occurring in Bubnov-Galerkin approach. The extension of our previous results [5] on the asymmetric FPU chain is the goal of given presentation. Such an extension is a complicated problem from mathematical viewpoint. Its solu- REFERENCES tion has shown that account of asymmetry in potential of interaction leads to renormalization of the coefficients of final equations [1] M. B. Rubin. On the theory of a Cosserat point and its ap- of motion. Therefore the main results obtained for symmetric plication to the numerical solution of continuum problems, model remain to be valid qualitatively. As for application to re- ASME J. Appl. Mech., 52, pp.368-372, 1985. alistic models of molecular systems, the corrections caused by [2] M. B. Rubin. Cosserat Theories: Shells, Rods and Points., asymmetry of potential of interaction can be significant. Solid Mechanics and its Applications, Vol. 79, Kluwer, The Netherlands, 2000. [3] B. Nadler and M. B. Rubin. A new 3-D finite element for non- REFERENCES linear elasticity using the theory of a Cosserat point, Interna- tional Journal of Solids and Structures, Vol. 40, pp. 4585- [1] E. Fermi, J. Pasta, S. Ulam. Los Alamos Science Laboratory 4614, 2003. Report No. LA-1940, 1955, unpublished; reprinted in Col- [4] M. Jabareen and M. B. Rubin. An improved 3-D Cosserat lected Papers of Enrico Fermi, edited by E. Segre (University brick element for irregular shaped elements, Computational of Chicago Press, Chicago, 1965), v.2, p.978 Mechanics, Vol. 40, pp. 979-1004, 2007. [2] T. Dauxois, M. Peyrard. Physics of Solitons. (Cambridge [5] M. Jabareen and M. B. Rubin. A Generalized Cosserat Point University Press, 2006) Element (CPE) for Isotropic Nonlinear Elastic Materials In- [3] R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, H. S. Morris. Soli- cluding Irregular 3-D brick and Thin Structural Elements, tons and Nonlinear Wave Equations. (Academic Press Inc., Journal of Mechanics of Materials and Structures, Vol. 3, No. London, 1982) 8, pp. 1465 - 1498, 2008. [4] A. J. Lichtenberg, R. Livi, M. Pettini, S. Ruffo. The Fermi- Pasta-Ulam problem: A status report, in Lect. Notes Phys. Leah Sharipova, Bol’shoy pr., 61, Saint-Petersburg, 199178, Rus- 728, 21, (Springer-Verlag Berlin Heidelberg, 2008) sia [5] L. I. Manevitch, V.V.Smirnov. Submitted to Phys. Rev. Lett.

TRANSITION TO ENERGY LOCALIZATION IN [6] L. I. Manevich. New approach to beating phenomenon in ASYMMETRIC PERIODIC FERMI-PASTA-ULAM coupled nonlinear oscillators chains. 8th Conference on OSCILLATORY CHAIN Dynamical Systems-Theory and Applications, Proceedings. Lodz, December 12-15, 2005, Poland, v.1, p. 119. D.S.SHEPELEV V. V. SMIRNOV L.I.MANEVICH [7] L. I. Manevitch. New approach to beating phenomenon in [email protected] coupled nonlinear oscillatory chains. Arch. Appl. Mech. v.77 (5), p.301, 2007 Energy localization in nonlinear oscillatory chains has become a problem of great interest since the moment of publication of the FPU original report [1]. A large number of investigations in Denis S. Shepelev, Kosygina str. 4, Moscow, 119991, Russia

88 dence of the WE’s mass centre movement on eccentrical mo- INFLUENCE OF SOLID COATING ON STABILITY OF ments of the unpaired VE and the paired VE. HIGHLY POROUS ELASTIC ROD SUBJECT TO The report is illustrated with a big amount of graphic materials. COMBINED LOADS

D.SHEYDAKOV Andrey Shihov, 66, Kolmogorova st, Ekaterinburg , 620034, Rus- sia [email protected]

The stability problem for three-dimensional nonlinearly elas- STUDY OF FEATURES OF DEFORMATION RESPONSE tic highly porous bodies with regard to their special properties OF INTERFACIAL ZONES IN BLOCK-STRUCTURED (in particular, a size effect) is little-studied. On the other hand MEDIUM TO LOW-AMPLITUDE DYNAMIC these properties can affect the behavior of such bodies consid- MECHANICAL ACTIONS erably, especially in the case of low-sized objects. In addition, the constructional elements made of highly porous media have, E.V.SHILKO S.V.ASTAFUROV A.S.GRIGORIEV as a rule, a composite structure (a metallic or polymeric foam, S.G.PSAKHIE covered by shell), which should be taken into account when ana- [email protected] [email protected] lyzing their stability, because according to the experimental data [email protected] [email protected] an influence of the coating on construction properties turns out to be quite substantial in some cases. The present research is dedicated to the stability analysis of An important direction in deformable solid physics and me- a nonlinearly elastic rod made of highly porous medium sub- chanics is investigation of features of mechanical response of so- ject to combined loads – an axial compression and external hy- called block-structured materials and media. This term combines drostatic pressure. The problem has been studied in the frame- materials, which mechanical behavior is defined significantly by work of a general stability theory for three-dimensional bodies. It processes of strain localization at block boundaries (interface re- was assumed that the rod has a composite structure – an interior gions between structural elements). Typical representative of this (core) made of highly porous medium and solid exterior (coat- class of materials is hierarchically organized geological medium. ing). To describe the properties of a rod core the Cosserat contin- An important feature of block-structured geomedium is that uum model was used, while the coating behavior has been studied its fragments are in complex stressed state. There is well-known in the framework of a classic continuum model. Such approach effect consisting in deformation response of prestressed interfa- allowed more accurate and detailed studying of the buckling pro- cially structured materials to dynamic actions in form of irre- cess for highly porous elastic rod with solid coating. By solv- versible relative displacement of blocks. Theoretical studies as ing the linearized equilibrium equations numerically for some well as experiments on rock samples and fragments of plane dis- specific materials the critical curves and corresponding buckling continuities in rock massifs showed strong relation between shear modes have been found, and the stability regions was constructed stress level in interfacial zone and parameters of relative block in the plane of the loading parameters. Basing on the analysis of displacement initiated by dynamic mechanical actions. This these results the main influence patterns of size effect and also leaded to the idea of possibility of estimation of shear stress level of coating properties on equilibrium bifurcation are revealed for in the fragments of active fault zones by measuring deformation a highly porous rod subject to an axial compression and external response to testing dynamic actions. Nevertheless to transform hydrostatic pressure. this idea into practical approach it is necessary to have informa- This work was supported by the Russian Foundation for Basic tion about the character of connection between shear stress level Research (grant 09-01-00459-a) and by the President of Russian in interfacial zone and deformation response to dynamic pertur- Federation (grant MK-6315.2010.1). bation of local stress state. This information could be obtained in particular on the base of computer-aided simulation of fragments Denis N. Sheydakov, Chekhova Ave., 41, Rostov-on-Don, of block interfaces. In the present paper this study was done with 344006, Russia use of the movable cellular automaton method. Two-dimensional model of block interface being under the condition of nonequaxial compression is considered. Influence STARTING DYNAMICS OF VIBRATIONAL of “impulse” dynamic actions on the value of initiated tangential TRANSPORT MACHINES WITH THREE UNBALANCED and normal relative block displacements was investigated. Sim- VIBRATION EXCITERS ulation results showed threshold character of dependence of initiated displacements on shear stress. In particular, their values A.SHIHOV S.RUMYANTSEV D.TARASOV undergo step-wise increase by order of magnitude when shear [email protected] stress value transits from quasi-elastic to quasi-plastic stage of in- [email protected] terfacial zone deformation. An important difference of boundary deformation response to low-amplitude impulse actions realized In this report we present the results of the researching of start- with use of different types of elastic waves is revealed. Specif- ing dynamics of vibrational transport machines (VTM) with three ically, on realization of dynamic mechanical actions with use of unbalanced vibration exciters (VE) mounted on the same work- elastic P-waves the character of registered dependences of initi- ing element (WE). Adding the third VE to the VTM construction ated displacements on shear stress level can significantly change principally changes the machine’s vibration. with change of normal stress value. At the same time, if energy of Using the mathematical model of VTM dynamics we have per- testing dynamic actions is transmitted mainly by elastic S-waves, formed series of numerical experiments and found the depen- the character of corresponding deformation dependences remains

89 unchanged at different values of normal stress. Simulation results of microcracks propagation near pores. Results of simulation substantiate use of “impulse” perturbation sources, which gener- have shown the Hugoniot elastic limit (HEL) of oxide nanoce- ate elastic S-waves, for obtaining the uniquely interpreted data ramics depends not only on the porosity, but also on distribution about deformation response of fragments of real fault zones to of voids and the ratio of pores size to grain sizes. Concentration testing dynamic low-amplitude actions. of nanopores in grain boundary phase causes the decreasing of Influence of the degree of nonequiaxiality of compression (it the shear strength of nanostructured and ultrafine-grained ceram- is defined as a ratio of compressive stress in the direction of inter- ics. It was shown the oxide nanoceramics behind front of a shock face strike to normal stress) on qualitative change of parameters wave with amplitudes exceeding HEL possess the spall strength. of deformation response of interface zone to testing dynamic ac- Results of numerical researches of influence of SiC or ZrB2 tions is analyzed in detail. nanoparticles in oxide nanocomposites are presented. It is shown Simulation results lay the foundation for an approach to esti- that uniform distribution of nanoparticles in ceramic nanocom- mation of proximity of stress state of plane discontinuities (faults posites can lead to essential increase of the spall strength even if and large cracks) in rock massif to critical value wherein seismo- the particles concentration is about several percent. generating unsteady slip can take place.

Evgeniya G. Skripnyak, Lenin Ave. 36, Tomsk, 634050, Russia Evgeny V. Shilko, pr. Akademicheskii, 2/4, Tomsk, 634021, Rus- sia

MECHANICAL BEHAVIOR OF ULTRA FINE-GRAINED LYAPUNOV EXPONENT OF PENDULUM MOTIONS OF ALLOYS AT HIGH STRAIN RATES RIGID BODY WITH A FIXED POINT V.A.SKRIPNYAK E.G.SKRIPNYAK A.L.SHVYGIN S.V.RAZORENOV [email protected] [email protected] [email protected] [email protected] The problem of motion of rigid body with one fixed point (Euler-Poisson problem) is under consideration. In case where During the last decade the interest in production and process- the center of gravity is located in the principal plane of the eling of ultra-fine grained (UFG) and nanostructured (NS) metal lipsoid of inertia the equations of motions admit the particular alloys have been rise sharply. It was revealed, that high strength solution which represents the pendulum motions (Mlodzeevsky of UFG alloys is combined with satisfactory plasticity. One of solution). Lyapunov exponents of this solution is investigated. the actual problems is the studying of scaling laws for prediction of physical and mechanical properties of (ns) and UFG materials. In this connection the development of models for computer mul- Andrey L. Shvygin, Moscow State University of Instrument En- tilevel simulation of mechanical behavior of NS and UFG alloys gineering and Computer Science, Russia in a wide range of strain rates and temperature assumes a great importance. We present a new model for description of inelastic deforma- MECHANICAL BEHAVIOR OF NANOCERAMICS tion of NS and UFG HCP and FCC metals in wide range of plastic UNDER SHOCK WAVES LOADING strain and strain rates. The model bases on the micro- and meso- mechanical approach to physical mechanisms of inelastic defor- E.G.SKRIPNYAK V.A.SKRIPNYAK mation and failure of NS and UFG alloys . V. V. SKRIPNYAK The model was used for the investigation of fast inelastic de- [email protected] [email protected] formation of UFG VT1-0, Ti-6Al-4V, Ti-6-22-22S, AA6063T6 [email protected] and some others alloys under dynamic compression and tension by computer simulation method. The mechanical behavior of Features of mechanical behavior of nanoceramics Al2O3 and polycrystalline (PC) and UFG titanium and aluminium alloys un- ZrO2 - 3 mol. %Y2O3 and ceramic nanocomposites under shock der quasi-static and plane shock wave loadings were simulated in loading with amplitudes up to 10 GPa were investigated by the a temperature range from 100 to 800 K. multilevel computer simulation method. The model of structured Results of simulation of the deformation behavior of pure tita- nanoceramics with porosity less 20% was proposed for investi- nium (VT1-0), and the titanium alloys Ti-6Al-4V and Ti-6-22- gation of deformation and damage nanoceramics on meso- and 22S testify the strain rate sensitivity of the yield stress under macro- levels. Results of numerical investigation of the influence quasi-static and dynamic loading is different. Alloys with grain of pores size, pore shapes, and pores distribution on the fragmen- sizes from 300 to 500 nm have the quasi-static yield strength and tation of ceramics under shock wave loading presented. Pres- the tensile strength twice higher than that of coarse grained coun- ence of mesopores and clusters of nanopores causes the initiation terparts. However the dynamic strength (the spall strength) of the of strain localization and microcracks nucleation under dynamic UFG alloys is only 10 percents above than that of coarse grained compression. The occurrence of bimodal distributions of the lo- alloys. The considerable decreasing of the strain before failure of cal particle velocities on mesolevel precedes the strain localiza- UFG alloys was predicted at high strain rates. tion on mesolevel and the nucleation of micro cracks. Pores at nano- and micro- levels prevent the creation of the failure wave in oxide nanoceramics under shock wave loadings due to stops Vladimir A. Skripnyak, Lenin Ave. 36, Tomsk, 634050, Russia

90 hardening. Metal layers are connected without possibility of rel- TRAFFIC FLOWS MODELLING AND SIMULATIONS ative displacement in normal and tangential directions till meetig a condition of stratification failure. N.N.SMIRNOV Failure of two kinds is considered: [email protected] • failure of sheet’s materials as a result of exhaustion of plasticity resource; Mathematical model for unidirectional unsteady traffic flows incorporates two non-linear partial differential equations. The • bonding failure as a result of foliating of border. model makes it possible to determine the maximum handling ca- The phenomenological damage model of Kolmogorov - Boga- pacity of a road, to follow the origination and evolution of traffic tova [1-3] is used to estimate degree of breakdown and predict jams and to perform optimization analysis of the traffic condi- failure by means of of exhaustion of plasticity resource, in accor- tions. The model is based on the findings of the predecessors, dance with it deformability of metal at cold plastic deformation and incorporates the methods for traffic flows analysis previously is characterised by plasticity dependence on stress coefficients developed. But the model also represents an essential step for- of k and µσ. The main conceptual and design parameter in the ward in traffic flows modeling. phenomenological damage model is degree of breakdown ω. Be- The mathematical model developed in the present project is fore deformation ω = 0, at the moment of macrocrack initiation not limited only by the substance conservation equation and the ω = 1, and intermediate values ω characterize accumulation level empirical relationships between density and velocity, but con- of deformative microdefects in metal. Mode of deformation at tains both the continuity (substance conservation) and momen- cold rolling of strips was estimated by means of LS-DYNA finite tum equations. It presents a closed form mathematical model for element model. Foliating failure can run by two mechanisms – it multi-lane traffic accounting for lane changes and suggests the is a separation in perpendicular direction to bonding plane, and procedure to develop the model parameters based on technical also a shift along bonding plane. [4]. characteristics of vehicles. Thus, the available experimental data In conclusion experimental data was compared with the data on the traffic flows are used only for the validation of the model. received using foregoing models. The developed model even being simplified for one-lane traffic The research was supported by Russian Academy of Sciences makes it possible to describe qualitatively and quantitatively, with program No 22. a satisfactory precision, such significant characteristics of traffic flows as the conditions of maximal handling capacity, evolution of the traffic jams on the roads, etc. REFERENCES The novelty of the suggested approach was in development of a model coupling the self-regulating traffic flows and the induced [1] V.L. Kolmogorov. Pressure, deformations, destruction. M: contaminants dispersion in turbulized air flows in big tunnels. Metallurgy, 1970. p. 252. The model could be the test site for investigating the efficiency [2] A. A. Bogatov, O. I. Mizhiritsky, S. V.Smirnov. Resource of of different strategies of traffic regulation to reduce the air pollu- metal plasticity by metal forming. M: Metallurgy, 1984. p. tion in tunnels. Thus it would be possible to perform optimization 144. of the regulatory actions so as to attain the maximal handling ca- [3] S. V.Smirnov, V.P.Shvejkin. Plasticity and deformability of pacity of a tunnel under the conditions of minimum air pollution. a carbonaceous steel. Ekaterinburg: Russian Academy of Sciences, 2008.p 256. [4] G. P.Tcherepanov. Mehanika of destruction of composite Nickolay N. Smirnov, Leninskie Gory, Moscow, 119992, Russia materials. M: The Science, 1983. p. 296.

Irina A. Golubkova, Komsomolskaya, Ekaterinburg, 620219, INVESTIGATION OF FAILURE OF MULTILAYERED Russia COMPOSITE MATERIALS RECEIVED BY EXPLOSION WELDING AND ROLLING SIMULATION OF RHEOLOGY OF PLASTICALLY S.V.SMIRNOV I.A.GOLUBKOVA DEFORMABLE 304 STEEL IN A CONDITION [email protected] DYNAMIC RECRYSTALLIZATION

A.S.SMIRNOV A.V.KONOVALOV The multilayered sheet materials consisting of several homogeneous or heterogeneous layers are the simplest in struc- [email protected] [email protected] ture. During manufacture high standards are shown to bonding strength of composites. The purpose of this study is to in- In the condition of high temperature proceed nonequilibrium vestigate failure of multilayered composite materials produced processes in metal materials connected with phase transitions and by explosion welding with the following rolling. Experimen- change of structure. Passage of these physical processes during tal research of three-layered samples after explosion welding high-temperature deformation is reﬂected on difﬁcult rheologi- and cold rolling was carried out in a following combination cal behaviour of the deformable alloys depending on strain his- “12Cr18Ni10T+St20+12Cr18Ni10T” and “Cu+St20+Cu”. Un- tory. As a result of it at the solution of a problem of deformable der theoretical consideration it is supposed (assumed) that the solid mechanics connected with the description of form change of connected metals are elastoplastic with isotropic deformation a body under external loadings, it is necessary to consider such

91 difficult rheological behaviour of environment. For this purpose METHOD OF IDENTIFICATION OF MODEL OF constitutive relation took into account rheological properties is STRAIN RESISTANCE BY RESULTS OF EXPERIMENTS created and identify by results of experiments. AT COMPRESSION SAMPLES Purpose of the given work was the complex research rheological behaviour of a 304 steel at temperatures 1223 K and 1423 K A.S.SMIRNOV A.V.KONOVALOV and also check of adequacy before created rheological models [1] on this steel. [email protected] [email protected] The nonmonotonic law of deformation of the sample in a range At compression of samples in the conditions of heats with the of ratio of deformations 0,1-4 s-1 has been used for identification large plastic deformations efficiency of applied lubrication more of the model and research rheology of a 304 steel. The relative low, than at a room temperature. It leads to increase in hetero- size of compression of samples was 60%. Experiments on com- geneity of deformation in the sample and to transition of a part pression spent on experimental installation created in Institute of of its free surface in a zone of contact with the die. Therefore at Engineering Science of the Ural Branch of the Russian Academy identification of model of strain resistance it is necessary to con- of Sciences. It is registered during experiment change of force sider heterogeneity of the stress-strain condition of the sample at of deformation of the sample, moving of captures of the instal- compression. lation and also change of temperature of a lateral surface of the In work an algorithm taking into account heterogeneity of the sample. Temperature measured by thermal imaging systems NEC stress-strain condition of the sample is offered at identification of TH-9100. After a stop of captures of installation samples quickly model of strain resistance. cooled for the purpose of fixing of structure of the alloy. This work was performed by Programs of fundamental re- It was not possible to avoid a friction between a sample and searches of Presidium of the Russian Academy of Sciences No22. die so in the sample the non-uniform stress-strain condition took place which can lead to different types of processes of softening- Alexandr S. Smirnov, Komsomolskaya st., 34, Ekaterinburg, hardening in various zones of the same sample. In a result of it at 620219, Russia identification rheological model is necessary to consider heterogeneity of the is stress-strain condition. WATER DROPLET NON-EQUILIBRIUM INTERACTION Microstructure of the sample is investigated in three zones WITH FLAME ZONE. where strain by results of a final-element simulation was most non-uniform: region close to a free surface, the centre of the sam- N.N.SMIRNOV V. F. NIKITIN V.A.NERCHENKO ple and region near contact of the sample with the die. The mi- G. MANZINI P. ANDREINI crostructure is studied by method EBSD, TEM and optical metal- lography. It was revealed that in a 304 steel goes dynamic recrys- [email protected] tallization in full volume of the sample at temperature 1423 K. At [email protected] 1223 K the main process softening in the centre of the sample – dynamic polygonization, in region near contact – dynamic recov- The presentation describes the results of investigation for non- ery, and in a zone close to a free surface of the sample – dynamic equilibrium effects in fluid droplet / flame interaction resulting in recrystallization. rapid droplet evaporation. The goals of this study are the elab- Model identification is performed with a glance of the non- orations of heat, momentum and mass transfer models of fuel uniform stress-strain condition in the sample using a final- and water spray liquid droplets (fuel spray and water mist fire element simulation by a technique published in work [2]. In a scenarios) and of fuel pool liquid (pool fire scenarios). Mainly result of identification it has been established that the model [1] the paper is focused on the heat transfer and the liquid – vapour well describes rheological behaviour of 304 steel at temperatures phase transition. Such focus is needed to clarify the free burning 1223 K and 1423 K. and the fire suppression phenomena, particularly related to water mist systems effectiveness in the control, suppression and extinc- This work was performed by Programs of fundamental re- tion of different type of fire. The problem of droplet interaction searches of Presidium of the Russian Academy of Sciences No22. with the streaming gas flow was investigated taking into account mechanical drag, atomization of droplets, non-equilibrium heat and mass transfer. In modeling droplets breakup in a gas flow the REFERENCES inertia of the process should be taken into account. Fragmenta- tion does not take place instantaneously: it needs time for small [1] A. V.Konovalov. Viscoplastic model for the resistance of droplets to separate from the initial one, i. e. it needs a definite metals to high-temperature deformation. Russian metal- time for the liquid bridges between the droplets to be established, lurgy(Metally). Vol. 2005. No 5. p. 456–459. elongated and broken. The mathematical model for non-equilibrium liquid droplet [2] A. V.Konovalov, A. S. Smirnov. Identification of model of evaporation was developed. The obtained solutions made it pos- strain resistance by results of experiments at compression sible to compare the results with that provided by the existing samples//Zavodskaya laboratoria. Diagnostika materialov. equilibrium models and to determine the applicability limits for Vol. 2010. No1. P.53–56. (In Russian). the existing equilibrium models. • The results of investigation showed that the difference between quasi-equilibrium and non-equilibrium solutions in- Alexandr S. Smirnov, Komsomolskaya st., 34, Ekaterinburg, crease on decreasing droplet radius, decreasing accommo- 620219, Russia dation coefficient and on increasing diffusion coefficient.

92 • The dimensionless criterion was developed characterizing Under modeling results it was obtained required volume of cal- the deviation of small liquid droplet evaporation process culation data for set of a and b values. It was graphed the nomo- from the equilibrium, which permits to determine applica- grams of penetration depth hind and scratch depth hscr vs coeffi- bility limits for equilibrium models. cient a when b was constant. The procedure of coefficients a and b determination was described more detailed in work [2]. Exper- • It was demonstrated that non-equilibrium models more ad- imental verification this method was carried out for two materials equately describe final stages of droplet evaporation. Those (copper, aluminum). The specimens were indented and scratched models are free from the common drawbacks of all quasi- using the atomic force microscope Nanoscan with the Berkovich equilibrium models, because non-equilibrium approach al- indenter . lows to avoid non-physical growth of the evaporation rate This work was performed together with ITAM SB RAS and on decreasing droplet radius. was supported by RFBR (project No 10-08-96050). • Investigating the behavior of individual droplets in a heated air flow allowed to distinguish two scenarios for droplet heating and evaporation. Small droplets undergo succes- REFERENCES sively heating, then cooling due to heat losses for evaporation, and then rapid heating till the end of their life time. [1] S. A. Fedosov, L. Peshek. Determining mechanical properties Larger droplets could directly be heated up to a critical of materials by microindentation: Modern foreign methods.- temperature and then evaporate rapidly. Atomization of Moscow: Faculty of Physics of MSU, 2004. P. 100. droplets interferes the heating and evaporation scenario. [2] S. V.Smirnov, E. O. Ekzemplyarova. Research methodology of strain resistance by atomic-force microscope and The present research was supported by the Russian Foundation nanohardness tester // Izvestia vyzov. Chernaya Metal- for Basic Research and EINSTEIN Consortium, Italy (Project lyrgiya, No 1. 2010. P. 68-69. (In Russian). Code 09-03-92421).

Nickolay N. Smirnov, Leninskie Gory, Moscow, 119992, Russia Evgeniya O. Smirnova, Komsomolskaya st., 34, Ekaterinburg, 620219, Russia

DETERMINATION OF STRESS-STRAIN CURVE BY WAVE FORMATION ON FREE SURFACE ON INDENTATION AND SCRATCH TEST INTERACTION WITH A BODY MOVING UNDER FREE SURFACE. E.O.SMIRNOVA S.V.SMIRNOV [email protected] M.N.SMIRNOVA K.A.KONDRAT’EV A.V.ZVYAGUIN The mechanical characteristics of materials, e.g. the true stress [email protected] - true strain relation, have been determined using the type compression or tensile test. Though in the microscale and mesoscale The two-dimensional problem of surface waves formation and it is difficult to carry out this tests. Therefore, for this purpose it interaction under the conditions of thin body motion in compress- is used different methods basing on the results of indentation. In ible fluid near free surface was regarded. The fluid is assumed the works [1] was suggested various methods for determination of occupying infinite semi-space, gravity is neglected as compared quantitative strength characteristics of materials (such as Young with fluid inertia. The solution was obtained for a problem of in- modulus, yield strength, hardness, coefficients in functional definite span plate motion in compressible fluid at a definite depth, pendence of strain resistance). constant velocity and inclination angle. The solution allows de- The object of work is to present the method for definition of termining free surface shape, drag and lift forces. In the limit- the stress-strain curve of metals from indentation, scratch test ing cases of relatively small and relatively big depth the obtained and computational simulations. The method based on the resolution permitted analytical approximation. For relatively thin sults of indenter penetration in elastoplastic material and scratch fluid layer above the moving body the resistance force depends test modeling by a finite element method. The adamantine in- on body thickness , inclination and Mach number, while for rela- denter was considered as a linear elastic isotropic material where tively thick fluid layer the force depends on body length, inclina- Young‘s modulus was 1140GPa and the Poisson‘s ratio was 0.07. tion angle and Mach number as well. Investigated material was elastic-plastic. Purely elastic deforma- The solution was obtained numerically for all the range of vari- tion occurred at the beginning of indentation and was obeyed the ation of governing parameters. Approximation formulas for nu- Hook’s law. In the plastic state material was obeyed the Mises‘s merical solution were developed, which coincide with analytical yield condition and power law σ = aεb,where σ - Mises‘s yield formulas for the limiting cases. load (strain resistance); ε - total deformation with respect to The solution was obtained for a problem of body motion in Mises; a and b - numerical empirical coefficients. Finite element compressible fluid at a depth, constant velocity and inclination computations were carried out for different combinations of co- angle. Both cases of positive and negative inclination were re- efficients for elasto-plastic properties representing common engi- garded, which means flow separation from the upper side and neering metals. The work-hardening exponent b was varied from bottom side of the wing. Both cases of infinite cavity behind the 0,1 to 0,5; the work-hardening rate a was varied from 200MPa to body and final length cavity were regarded. For the case of final 800MPa; while the Poisson‘s ratio was fixed at 0,34. The load on length cavity vapor pressure in it is less then the pressure on free the indenter was fixed at 5mH. surface. The pressure tending to that on free surface the length of

93 cavity tends to infinity, and the resistance and lift forces tend to on the material surface. Possibilities of this technique are studied that for basic solution with an infinite cavity. based on 3D modeling by movable cellular automaton method for Analysis of the obtained solution shows that it could describe ceramic coating on steel substrate. Pores position varies in depth the fluid flow induced by plates of both positive and negative cur- and in the direction of counter-body motion. It is shown that a vature, which could have singularity at the rear or at the front specific frequency at the friction force spectrum corresponds to edge. The present research was supported by RFBR (grant 09- average distance among the pores. 08-00396). Valeriy V. Sergeev, Pr. Academichesky 2/4, Tomsk, 634021, Rus- Maria N. Smirnova, Leniskie Gory, 1, Moscow, 119992, Russia sia, Tomsk, 634021, Russia

TIME-FREQUENCIES ANALYSIS OF ACOUSTIC WAVE PROPAGATION IN A VISCOUS LAYER INSIDE A VIBRATION IN FRICTION: COMPARISON OF ROTATING ELASTIC CYLINDRICAL SHELL EXPERIMENT AND SIMULATION I.N.SOLDATOV N.V.KLYUEVA A.YU.SMOLIN A.I.DMITRIEV S.A.DOBRYNIN E.A.KOLUBAEV [email protected]

[email protected] Wave processes in a viscous incompressible fluid partially filled rotating cylindrical container with elastic wall are consid- Acoustic vibrations in friction obtained from both experiment ered. We consider the case of fast rotation, when the ratio of the and simulation are analysed using wavelet transform. The exper- gravity and rotation effects is small. Dynamics of the container iments have been made on steel under sliding friction. The sim- are treated using the Donnell-Mushtari approximation for elastic ulations have been performed using movable cellular automaton thin circular cylindrical shell. The wave motion is represented method. Similarity of acoustic spectra for the experimental and as a superposition of propagating wave modes. Using analytical simulation results is revealed also as their correspondence with methods we show that each wave mode is a superposition of six wear particle size. helical harmonic: the two forms central core of flow, other forms boundary layers. The dispersion equation for harmonic waves in Aleksey Y. Smolin, Pr. Academichesky 2/4, Tomsk, 634021, the system is obtained. Elasticity of the cylindrical wall mani- Russia fests itself first of all through appearance new modes similar to surface mode. The work was supported by the RFBR (project No. 09-01-00356). NUMERICAL ANALYSIS OF RATE-DEPENDANT ROOF FALL IN COAL MINES Igor N. Soldatov, Belinskogo, 85, Nizhny Novgorod , 603024, I.YU.SMOLIN A.V.KUZNETSOVA Russia [email protected] CLASSICAL MECHANICS OF THE STRUCTURED The problem of roof fall in a breakage heading is considered. PARTICLES To describe the difference in fall spacing observed in practice for different advance rates an approach based on a visco-plastic V.M.SOMSIKOV model of media is proposed. The model of rocks takes into [email protected] account time-dependant non-associated plasticity, strain hardening and softening. Numerical modeling was carried out by In nature all bodies are the structured particles (SPs). There- TOCHNOG code based on the finite element method. Results of fore the motion and internal energy of SPs change at their inter- 2D and 3D calculations for a given geometry of the mine opening actions. The change of internal energy is usually due to friction and different sweep rates are presented. Advantages and disad- and described empirically. So a question arises, whether it is pos- vantages of the approach are discussed. sible to find rigorous mathematical description of SPs dynamics within the frames of the Newtonian mechanics, and if possible Igor Y. Smolin, Pr. Akademicheskii 2/4, Tomsk, 634021, Russia then how? We found the answer on this question by obtaining and studying the equation of SPs motion when SP is equilibrium system of potentially interacting material points (MPs) in a ther- MODELING TRIBOSPECTROSCOPY USING MOVABLE modynamic limit [1]. CELLULAR AUTOMATA It has been fined that under certain conditions dynamics of A.YU.SMOLIN V. V. SERGEEV such systems is irreversible. These conditions are formulated as S.YU.KOROSTELEV S.A.DOBRYNIN follows: [email protected] 1. the energy of an equilibrium subsystem must be presented as a sum of internal energy and the energy of SP motion as a whole. Tribospectroscopy is a non-destructive technique for detecting nanoscale pores in surface layer of a material using frequency 2. each material point in the system must be connected with a analysis of the force resisting to sliding of a small counter-body certain SP independent of its motion in space.

94 3. during all the process each SPs are considered to be equi- mechanisms: in the first case it is caused by the wave, while in librium. the second it is caused by the dependence of this amplitude on The first condition is necessary to introduce internal energy in the the depth of the bubble submergence. description of system dynamics as a new key parameter charac- tering energy variations in SP. The second condition enables not to redefine SP after mixing of material points. The last condition REFERENCES is taken from thermodynamics. It is equivalent to the condition of [1] R. H. Buchanan, G. J. Jameson. Cycling migration of bubbles weak interactions in the SP, which do not violate SP equilibrium. in vertically vibrating liquid columns. Ind. Eng. Chem. Fun- Moreover, it implies that each SP contains so many elements that damen., 1962, 1(2), 82-86. it can be described using the concept of equilibrium system. It is known that the Newton’s equation for MPs can be found [2] I. I. Blekhman, L. A. Vaisberg, L. I. Blekhman, on the basis of Noether’s theorem for energy which is defined by V.B. Vasilkov, K. S. Yakimova. “Anomalous” phenom- symmetry of space and differentiating energy on time. We have ena in fluid under the action of vibration. – Doklady Physics, shown that the SPs motion equation unlike one for MPs is de- 2008, v. 53, No 10, pp. 520–524. termined by two types of symmetry: symmetry of space and SP [3] V.S. Sorokin, I. I. Blekhman. Motion of a body with variable internal symmetry. The motion and internal energy of SP corre- volume (a gas bubble) in an oscillating fluid, Proceeding of sponds to these two types of symmetry. According to SPs motion the APM 2009, 2009, 633-642 equation the change of the motion energy of its centre of weights is caused by potential collective force. The change of internal energy is caused by nonpotential force determined by forces of Vladislav Sorokin, 29 Polytechnicheskaya st., St. Petersburg, MPs interaction. 195251, Russia We obtained the Lagrange, Hamilton, Liouville and D’Alambert equations for SP and have shown how above equations describe the equilibration process in non-equilibrium sys- SATURNS RINGS – OUR NATURAL “DYNAMICAL tems which submitted as a sets of SPs. LABS” IN SPACE

F. SPAHN REFERENCES [email protected]

[1] V.M. Somsikov. The restrictions of classical mechanics in We present examples of a successful iteration between theo- the description of dynamics of nonequilibrium systems and retical prediction and observation with the Cassini space probe in the way to get rid of them. New Advances in Physics, Vol. 2, the case of structures evolving in the dense rings of Saturn. In No 2, September 2008, pp. 125-140 this sense the latter cosmic granular gases can indeed be understood as “our cosmic dynamical labs” for understanding disk-like Vyacheslav M. Somsikov, Kamenskoe Plato, Alma-Ata, 050020, cosmic structures in general: Kazakhstan Here we report on investigations of the action of gravitational perturbers in thin cold astrophysical discs. The model includes viscous diffusion of the disc matter and gravitational scattering MOTION OF AN ELASTIC BODY IN FLUID UNDER by the perturber as counteracting processes. Two types of density VIBRATION structures are found, depending on the mass of the perturbing body and on the amount of momentum transport in the disc. A V.S.SOROKIN I.I.BLEKHMAN V.B.VASILKOV gap around the whole circumference of the disc is opened if the [email protected] perturber is more massive than a certain threshold. Alternatively, a local S-shaped density modulation is generated that we call a The effect of gas bubbles sinking in fluid under vibration was “propeller”. Beyond the perturber’s mass, the kinematic viscosity observed and described in papers [1,2]. In paper [2] the neces- of the disk comprises the second crucial parameter of the model sary condition for the arising of the effect was the presence of which describes the transport properties of the disc material. An- the standing wave in fluid, which for relatively low frequencies is alytical and numerical solutions provide the characteristic spatial possible only in gas saturated fluid. extent of the “propeller” to depend on the mass of the perturber In paper [3] it was supposed that the effect of gas bubble sink- and the disk-viscosity. ing in oscillating fluid can occur in the absence of the standing We apply these results to dense rings of Saturn perturbed by an wave, i.e. in fluid without other bubbles. In this case gas bubble embedded moonlet where our inspection of the Cassini-imaging sinking takes place due to the deformability of the bubble under data revealed 12 “propellers” in the 100-metres size range to re- the action of fast varying pressure, which occurs by oscillations side in Saturn’s A ring. Further analysis of the Cassini Imaging and is proportional to the mass of the liquid column placed above. data (ISS) has led to meanwhile about 200 “propeller” features in In the present paper this supposition is confirmed by theoretical the middle A ring. Interestingly - these moonlets jostle in three and experimental research. narrow radial bands in the A ring. From these observations we Thereby, there are two different mechanisms, two explanations conclude about a few million such few hundred metre objects - of the effect of gas bubble sinking in vertically oscillating fluid but none larger than 500 metres - to exist in the outer rings of – “wave” and “nonwave”. The presence of the gradient of the Saturn. This, in turn, has crucial consequences for the origin of force amplitude, exerted on the bubble, is common for the both planetary rings in general.

95 The second application concerns gas-dust discs around a pro- NUMERICAL INVESTIGATION OF INELASTIC tostar perturbed by a protoplanet - practically the “nursery of a DEFORMATION AND FRACTURE OF ROCKS planet”. Again either gaps or “propeller”-shaped structures can be expected to have formed within the disk. With increasing res- YU.P.STEFANOV olution of modern telescopes the chance might appear in future to be witness of a planetary growth by studying structures in circum- [email protected] stellar disks. Deformation in geological media can proceed in various modes depending on the structure, porosity, and stress state of a medium. Shear deformation often involves volumetric changes Frank Spahn, University Potsdam; institute of physics and as- and depends strongly on pressure. In this case, large cracks and tronomy, Karl-Liebknecht-Str. 24/25, Potsdam-Golm, D-14476, fractures can appear for which the preceding processes are nor- Germany mally loosening, microcrack accumulation with an increase in volume, and strain localization. In a dense medium, an increase in average pressure causes an increase in effective strength. In ATOMISTIC SIMULATION OF LASER ABLATION a highly porous medium, there is a certain pressure threshold at S.STARIKOV which the strength decreases, fracture of grains begins, and the pore space reduces. At high pressures, the transition to the dislo- [email protected] cation mechanism of plasticity is possible. In the work, a mathematic model is proposed and used in nu- Laser ablation has many technological applications in ma- merical analysis of different modes of deformation and fracturing terial microprocessing and fabrication nanostructures. At the in geomaterials. It is shown that in regions of increased porosity same time the mechanism of this process is not completely in- and cracking, shear deformation can occur with both dilatation vestigated. In this work the atomistic simulation of laser abla- and compaction depending on the state and stress of a medium. tion of gold and silicon is performed. The electronic subsystem In individual regions, shear zones and loosening and compaction was modeled as the continuous environment with the tempera- zones can be formed at a time. According to the mode of defor- ture. The electronic subsystem could inﬂuence dynamics of atom mation, the slope of localization bands about the axis of maxi- movement in two ways: the local heat exchange through the ther- mum compression depends on the stress state and porosity of a mostat and the dependence of interatomic potential on electronic medium. It is demonstrated that fracturing can be the ﬁnal stage temperature. The creation of the electron-temperature-dependent of deformation and its localization at which dilatation gives way potentials for an atomistic simulation was performed by force- to compaction and a decrease in effective strength in a narrow matching method. The method leads to new interatomic poten- material zone. tials which are obtained on the basis of synthesis of quantum and statistical mechanics. The results of atomistic simulation are dis- Yurii P. Stefanov, Akademicheskii, 2/1, Tomsk, 634021, Russia cussed. The comparison to the experimental data is performed.

Sergey Starikov, Izhorskaya 13/2, Moscow, 125412, Russia AN INTERMEDIATE ASYMPTOTIC SOLUTION OF THE COUPLED CREEP-DAMAGE CRACK PROBLEM IN SIMILARITY VARIABLES TOWARDS COMPUTING BOUNDS FOR COMPACT INVARIANT SETS OF SOME HAMILTONIAN SYSTEMS L.V.STEPANOVA WITH 2 DEGREES OF FREEDOM [email protected] K.E.STARKOV Analysis of effects of material damage on the stress and strain [email protected] fields near crack tip in non-linear materials is the very important problem for evaluation of crack behavior in elements of struc- We consider the problem of finding a domain which con- tures. The influence of damage on crack-tip fields has been the tains all compact invariant sets for the generalized Henon-Heiles subject of many papers, especially for cracks in brittle materi- Hamiltonian systems, see e.g. in [Maciejewski A. and oth., J. als, elastic-plastic-brittle cracks, creep cracks and fatigue cracks. of Dynamics and Differential Equations, vol.17, 2005, 475-488] Thus, the phenomenon of crack growth in materials undergoing and the Hamiltonian system appeared in studies of dynamics of deformations coupled with damage has been investigated exten- hydrogenic atoms, see e.g. in [Beims M.W. and oth., Physical sively over the past twenty years. Review A, vol. 62, 2000, 043410-043418]. In discussing crack growth on the basis of damage mechanics Finding such domain means its description with help of equa- it is advantageous to introduce the term “totally damaged zone”. tions and inequalities depending on parameters of a system and The totally damaged zone can be interpreted as a zone occupied is important for analysis of its long-time dynamics. This work is by microcracks oriented orthogonally to the main crack. Inside a continuation of the study realized in the paper [Starkov K.E., the totally damaged zone (TDZ) the damage involved reaches its Physics Letters A, vol. 372, 2008, 6269-6272]. critical value (for instance, the damage parameter reaches unity) and a complete fracture failure occurs. In view of material dam- Konstantin E. Starkov, Av. del Parque N 1310, Tijuana, 22510, age stresses are relaxed to vanishing. Therefore one can assume Mexico that the stress tensor components in the TDZ equal zero. Outside

96 the zone damage alters the stress distribution substantially com- WAVE PROPAGATION IN THE COMPOSITE POROUS pared to the corresponding non-damaging material. Well outside AND LIGUID LAYER GENERATED BY MOVING AND that zone the damage parameter is equal to 1. The asymptotic OSCILLATING LOAD stress, creep strain rate and damage fields near the crack tip are analyzed by solving nonlinear eigenvalue problems resulting in T. V. SUVOROVA U.V.STOLIAROV S.A.USOSHIN a new far stress distribution. The configurations of the totally H.A.USOSHINA damaged zone governed by the new far stress field are found and analyzed. suvorova [email protected] The effects of material damage on the asymptotic stress and creep strain rate fields of mode I and mode III cracks were anal- Oscillations of the composite porous and liquid layer are con- ysed on the basis of continuum damage mechanics by postulating sidered. Heterogeneous media is described by Biot-Frenkel’s power-law creep damage theory. Based on the similarity variable equations. Green matrix for the motionless oscillating load is a stress analysis is carried out for the mode I crack under plane constructed. Neutral curves which determine behaviour of a stress and plane strain conditions and for the mode III crack and wave field on a surface of media, are numerically investigated assuming the existence of a totally damaged zone near the crack and properly approximated. The analytical formulas describing tip. It is found that the Hutchinson-Rice-Rosengren solution can’t wave fields for undersonic and supersonic moving velocity are be used as the remote boundary condition and the actual far field constructed, the numerical analysis of results is given. At the far stress is obtained. The shape of the totally damaged zone is given distance from loading asimptotic representation of the decision and analysed. It is shown that the new far field stress asymp- by a method of a stationary phase is constructed. Influence of a totics can be interpreted as the intermediate asymptotic valid for water-saturation, presence of a liquid layer, various conditions of times and distances at which effects of initial and boundary con- fastening of the basis on surface waves is investigated.The phase ditions on the stress and damage distributions are lost. Higher speeds and amplitude dependences of surface waves from a direc- order fields for damaged nonlinear antiplane shear and tensile tion and speed of movement of loading, a structure and properties crack problems are analytically derived. The higher order fields of the basis are investigated. For definition of loading the data re- obtained permit the shape of the totally damaged zone modelled ceived at natural experiment at passage of a train on a railway in the vicinity of the crack tip to be determined more exactly. way are used Work is carried out at partial financial support of the Russian Larisa V. Stepanova, Akad. Pavlov, 1, Samara, 443011, Russia Fund Fundamental researches (the project 06-08-00867-a).

Tatiana V. Suvorova, prof. RGUPS, Narodnogo Opolchenia sq.,2, HYBRID ALGORITMS IN COMPUTATIONAL MODEL Rostov-on-Don, Russia, UPDATING PROCEDURES FOR HYDROMECHANICAL Iuryi V. Stoliarov , Sergei A. Usoshin, Helen A.Usoshina- posr- SYSTEMS graduates RGUPS,SFU

V.D.SULIMOV P.M.SHKAPOV [email protected] IGNITION DISTRIBUTOR

E.SVIJAZHENINOV Reliable computer modeling in hydromechanical system dynamics is very important for studies related to the safety of equip- [email protected] ment components in the nuclear power industry. Computer dynamic model updating is a powerful means to produce these re- Invention is related to the field of mechatronics, electrical en- liable models. The method uses an initial computer model and gineering and automotive industry, in particular, to the devices measured spectral data of the system to modify physical param- for intermittent current pickup, and can be used in ignition sys- eters of the initial computer model. Problems of computational tems of internal combustion engines, as well as in commuta- model validation are formulated as inverse spectral problems. In- tion, telemetry and weaponry systems, for instance, for control completeness of registered spectral data and presence of multi- of fire a volley. The ignition distributor includes stator with in- ple frequencies result in the error function being non-convex and ternal round surface, on which along circumference n contacts non-differentiable. As the function has numerous local minima, are evenly arranged with leads to spark plugs, and rotor installed it is necessary to use global optimization methods. Two novel inside the stator along its axis of symmetry and equipped with global optimization algorithms combining a Metropolis-based al- current-distributing plate, which is made in the form of regular gorithm and deterministic gradient techniques are introduced. n+1 or n-1-beam star, at that voltage is supplied to spark plugs Smoothing approximations are inserted during the local search sequentially for the time of incomplete revolution of the rotor, that makes it possible to expand the approach on the class of therefore, the required rotational frequency is reduced n+1 or n- non-differentiable problems. Practical versions of the lineariza- 1 times accordingly. The effect of application of the distributor tion method for local search subproblems on base of biparametric supplied with multi-beam rotor in comparison with the traditional smoothing approximations of criteria and the corresponding algo- single-beam rotor distributor is as follows: rithm are proposed. Implementations of the hybrid algorithms are 1. Provision of structural self-balance of the rotor. discussed. Results of computational experiments are presented to 2. Multiple reduction of nominal rotational frequency of the illustrate the efficiency of the approach. rotor. Valeriy D. Sulimov, Baumanskaya 2nd Street, Moscow, 105005, 3. Much lower vibration, mechanical and electrical heating, Russia as well as dynamical loads upon the bearing units.

97 4. Sufﬁcient reduction of wear and instability of operational EXPERIMENTAL RESEARCH OF A RUN-UP PROCESS properties. OF A VIBRATING MACHINE WITH AN UNBALANCE 5. Multiply increasing time of spark plug charge action. EXCITATION

All these signiﬁcantly increase proﬁciency and reliability of A.V.SYLYVONYUK N.P.YAROSHEVICH such a demanding device as an ignition distributor. [email protected] m [email protected] Evgeny D. Svijazheninov, 61 Bolshoy, St.Petersburg, 199178, Russia Lay out of experimental researches results of the run-up process of vibrating machine with an unbalance excitation, which is put into rotation with the help of asynchronus motor. A chart and description of the explored construction is resulted. The graphs STRUCTURAL-PHENOMENOLOGICAL MODEL OF are presented conducting of experiment got as a result. THE MECHANICAL BEHAVIOR OF ELASTOMERIC NANOCOMPOSITES BASED ON STYRENE BUTADIENE Andriy V. Sylyvonyuk, 75 Lvivska Str., Lutsk, 43018, Ukraine RUBBER

A.L.SVISTKOV A.G.PELEVIN A.A.ADAMOV SHEAR RUPTURE MECHANISM OF B.LAUKE K.W.STOCKELHUBER¨ G.HEINRICH SUPER-BRITTLENESS OF HARD ROCKS

[email protected] B.G.TARASOV [email protected] The work presents an analysis of rubber compounds by comparing the parameters of a phenomenological mechanical model The paper discusses the nature of rock brittleness variation un- of rubber for fourteen types of nanocomposites. Constitutive der the effect of confining pressure σ for intact rocks of dif- equations are constructed using a scheme that is illustrating the 3 ferent hardness failed at triaxial compression σ > σ = σ . mechanical behavior of rubber compounds. The new model rep- 1 2 3 Shear rupture is a universal form of rock failure under confined resents the superpostion of elastic, viscous, plastic and transmis- compression, both in the laboratory and in nature. It is generally sion elements. To describe the properties of each element, the assumed that shear rupture results in a substitution of the initial known equations of the nonlinear theory for elasticity, the theory cohesive material strength by frictional resistance which repre- of nonlinear viscous fluids and the theory of plastic flow under fi- sents the minimum value of the rupture resistnace. Increase in nite deformations are used. The scheme shows how the tensorial σ in this situation should lead to increased friction in the rup- nonlinear equations are combined into the system of equations 3 ture zone and, consequently, to a decrease in rock brittleness (as proposed in [1]. happens for insufficiently hard rocks). However, this assumption The model is used for simulating the mechanical behavior of is largely unexplored experimentally for hard rocks due to the different samples of rubber compounds. In all cases the rubber uncontrollable explosive-like nature of rupture. matrix consists of solution - styrene butadiene copolymers, filled Experimental results obtained on a special testing apparatus with 20 or 40 phr with pyrogenic and precipitated silicas with dif- and presented in this paper show that hard rocks exhibit extraor- ferent surface modifications by different silanes and one carbon dinary embrittlement with rising σ3. The brittleness index k = black sample as reference. Wr/We used for brittleness characterisation represents the ratio The theoretical results describe accurately the mechanical be- between the rupture energy Wr and the corresponding elastic en- havior of rubbers for all samples considered in this work. ergy We withdrawn from the specimen during this rupture pro- The work was supported by integration program performed cess. The brittleness index k characterises the capability of rock jointly by scientists SB and the Far East RAS (program RAS for self-sustaining failure due to elastic energy available from the 09-T-1-1006), Grant RFBR 09-08-00530-a, DFG-FOR 597 and specimen. The value of k = 0 corresponds to absolute material DFG-Priority Program 1369 “Polymer-Solid Contacts: Inter- brittleness when the rupture energy is equal to zero. For dolerite phases and Interfaces”. specimens tested at σ3 = 0; 30; 60; 75 MPa variation of brittleness index k is as follows: k = 1.5; 0.6; 0.02; 0.006. At σ3 = 100 and 150 MPa rupture control was not possible due to explosive-like rupture propagation. In this situation k 0.006. The conducted REFERENCES experiments show that hard rocks at high≪ confining pressure can approach conditions of absolute brittleness. [1] A. L. Svistkov, B. Lauke, G. Heinrich. Modeling of vis- On the basis of analysis of structure of shear ruptures formed in coelastic properties and softening of rubber materials // Pro- hard rocks at high confining pressures a new shear rupture model ceedings of 5th European conference “Constitutive models has been developed which can explain the extraordinary rock em- for rubbers” Paris, 2007. P. 113-118. brittlement. It is known that any shear rupture propagates due to creation of tensile cracks in front of the rupture head. The fundamental structure of shear ruptures in this case is represented by an echelon of blocks (slabs) separated by tensile cracks. These Aleksander G. Pelevin, street Academ. Koroleva, 1, Perm, blocks are subjected to rotation at shear displacement of the rup- 614013, Russia ture faces. In soft rocks at any σ3 and in hard rocks at low σ3 the

98 blocks collapse at rotation, leading to normal friction within the about the shear deformations distribution the 2D equations, de- rupture head. scribing the deformation and the free vibrations of a plate, are Increase in σ3 causes reduction in the length of the tensile delivered. Earlier the same problem was investigated in [1,2], but cracks (and, consequently, the blocks). We can suppose that in these papers the accepted hypotheses are not enough exact be- within a certain (embrittlement) pressure range the blocks because some boundary conditions at the face planes of plate are not come short enough to withstand rotation without failure. In this satisfied. In [3] the accepted here hypothesis is used for a beam, case the blocks can operate as hinges, eliminating friction. This and in this paper it is applied for a plate. can happen in hard rocks only because very strong material is The asymptotic analysis of the obtained 2D equations is ful- required to achieve this. The very high strength of the rotating filled as for static problem so for free vibrations. It is estab- blocks for these conditions is explained also by the scale effect: lished that the delivered equations are of the same exactness as the smaller the rock fragment the stronger it is. The block length the Timoshenko–Reissner equations for the orthotropic material. developed in laboratory specimens at the embrittlement pressure The paper is written with the financial support of Russian condition is a fraction of a millimetre (about 1/10 mm). At corre- Foundation of Basic Investigations, grants 09.01.92002 HHC-a, sponding rotation of the blocks-hinges this mechanism can create and 10-01.00244-a. positive forces assisting the fault displacement (effectively ‘negative friction’). This makes rocks superbrittle and the failure uncontrollable and extremely violent. The new concept advances our knowledge about hard rock REFERENCES properties at high confining pressure and great depths.

Boris Tarasov, The University of Western Australia, 35 Stirling [1] P.E. Tovstik and T. P.Tovstik. Two-dimensional models of Highway, Crawley, Perth, 6009, Australia plates made of an anisotropic material // In “Computer methods in mechanics of continuous media”. Issue 3. St. Peters- burg Univ. 2008. Pp. 4–16 (in Russian). PROPAGATION OF WAVES AT A PLANE INTERFACE BETWEEN TWO MICROPOLAR POROUS ELASTIC [2] P.E. Tovstik. Two-dimensional models of plates made of an SOLID HALF-SPACES anisotropic material // Doklady Russian Academy Sci. V. 425. No 4, Pp. 489-491.= Doklady Physics. 2009. Vol. 54. S.K.TOMAR A.KHURANA No 4, pp. 205-209.

[email protected] [3] P.E. Tovstik, and T. P.Tovstik. One-dimensional models of beam made of an anisotropic material with the inclined This paper deals with the phenomena of reflection and trans- anisotropy // Izv. RAS. Mekhanika Tverdoto Tela (to be pub- mission of plane elastic waves from a plane interface between lished, in Russian). two distinct micropolar porous elastic solid half-spaces in welded contact when (i) a set of longitudinal wave is made incident and (ii) a set of coupled transverse waves is incident. Using method of potentials, the appropriate boundary conditions at the interface are solved to obtain the amplitude ratios of various reflected Tatiana P. Tovstik, Bolshoi Prospect 61, St-Petersburg, 199178, and transmitted waves. The expressions of energy ratios corre- Russia sponding to various reflected and transmitted waves are also presented. It has been noticed that the amplitude and energy ratios are complex valued and they depend upon the angle of incidence, ROBUSTNESS OF LAYERED STRUCTURES IN frequency of the incident wave and on the elastic properties of BIOLOGY the media. Numerical computations for a peculiar model are in progress. It has been aimed to compute the variations of modu- S.VAKULENKO lus of amplitude and energy ratios against angle of incidence and [email protected] against frequency of the incident wave. The effect of presence of voids on the amplitude ratios will also be studied. We consider layered patterns emerging in many applications Sushil Tomar, Sector - 14, Chandigarh, 160 014, India (chemical, physical and biological). In biology experimental data show that these patterns are remarkably robust with respect of different perturbations (inhomogeneities, mutations, noise, external TWO-DIMENSIONAL MODELS OF PLATES MADE OF fields). AN ANISOTROPIC MATERIAL We discuss different mechanisms supporting this robustness. The proposed mathematical approach involves the geometric P.E.TOVSTIK T. P. TOVSTIK control theory, nonlinear waves, contemporary ideas of discrete [email protected] tovstik [email protected] mathemetics.

A plate made of an anisotropic material with the general anisotropy described by 21 elastic modules is studied. By us- Sergey Vakulenko, Bolshoy pr. VO 61, S Petersburg, 199178, ing the cinematic generalized Timoshenko–Reissner hypothesis Russia

99 systems with evolving and diffusing interfacial boundaries (e.g. MODELING MECHANOCHEMISTRY OF THE processes of evaporation/condensation, solidiﬁcation, polymeri- DIFFUSION CONTROLLED CHEMICAL REACTION sation, etc.). FRONT PROPAGATION IN ELASTIC SOLIDS

E.N.VILCHEVSKAYA A.B.FREIDIN Anatoliy Vorobev, University Road, Southampton, SO17 1QE, [email protected] United Kingdom

Based on the general consideration of a chemical reaction front propagation in an elastic solid developed in [1] we focus on the ANALYTICAL STUDY OF HOPF BIFURCATIONS IN statement and solution of a simplest 3D-boundary value prob- FRICTIONAL CHATTER MODELS lem. We study an oxidation of a ball under external loading. For simplicity sake we use a linearized chemical affinity tensor in a P. WAHI small strain approach. We emphasize the role of internal stresses [email protected] induced by oxide formation. We demonstrate that, depending on boundary conditions and material and chemical reaction parameters, the reaction may be blocked by the oxide growth because of In this work, I would be focusing on the primary instability of internal stresses. the origin of limit cycles from the steady cutting in some typical models for frictional chatter. Stability surfaces comprising of two distinct lobes have been drawn in the parameter space separating REFERENCES regions of stable and unstable equilibrium. Conditions have also been obtained for a codimension-2 Hopf bifurcation and also for [1] A.B.Freidin. On chemical reaction fronts in nonlinear elas- the criticality curve where the Hopf bifurcation gets degenerate. tic solids. Proceedings of XXXVI International Summer This criticality curve separates regions on the stability surface School-Conference APM’ 2009. Eds: D. A. Indeitsev, A. M. corresponding to subcritical and supercritical Hopf bifurcations. Krivtsov. St.Petersburg. Institute for problems in mechanical Engineering. 2009. P.231-237. Pankaj Wahi, Indian Institute of Technology-Kanpur Campus, Kanpur, 208016, India Elena N. Vilchevskaya, Alexander B. Freidin, IPME, Bolshoi 61, V.O., St.Petersburg, Russia COMPLEX DYNAMICS OF PENDULA SYSTEMS NUMERICAL MODELLING OF INCOMPRESSIBLE M.WIERCIGROCH MISCIBLE FLUIDS [email protected] A.VOROBEV [email protected] Complex dynamics of pendula systems is being analysed with a view of vibrational energy harvesting and vibration isolation. We study the interplay between the thermodynamic transi- The main focus of the paper is a new concept of energy con- tion and hydrodynamic flows, which would characterise a typ- version [1], where a predominantly vertical oscillatory motion is ical thermo/hydro-dynamic evolution of a binary mixture (in a being converted into rotations by means of a pendulum or pendu- dissolution/nucleation process). The Cahn-Hilliard approach is lums system. The simplest physical configuration of this idea is used to model the behaviour of diffusing interfaces. Owing to a parametric pendulum operating in the fully rotational motion. strong density gradients at liquids’ interfaces, to describe the hy- Practically this can be envisaged as a floating structure excited to drodynamic behaviour of such a system the full continuity equa- which pendulums are mounted at their pivots. These vertical os- tion should be used, even if a binary mixture consists of two in- cillations of the structure will generate the pendulums rotational compressible liquids. This makes the hydrodynamic flows be- motion. ing quasi-compressible and introduces significant complications Numerical, analytical and experimental studies have been un- to the numerical analysis. To simplify the numerical treatment, dertaken on a parametric pendulum [2-4] and a pendulum excited we, firstly, analyse the endowed quasi-compressibility of the full by a planar motion [5]. They suggest the rotational motion is per- Cahn-Hilliard-Navier-Stokes equations and, secondly, we derive sisting and occurs for a large range of frequencies and excitation the low-Mach number approximation for these equations (the amplitudes, which are the main control parameters. These inves- Boussinesq approximation for the Cahn-Hilliard-Navier-Stokes tigations reinforce the viability of this new concept of the energy equations). The obtained theoretical model is verified using the conversion. In the lecture I outline this new concept and review measurements of effective surface tension which were obtained the current theoretical and experimental developments. Specifi- by analysis of droplet transformations in a spinning capillary tube cally, I will present the physical and mathematical modelling of a (spinning drop tensiometry). We examine such a configuration single pendulum and a system comprised of two pendulums ex- numerically paying particular attention to dependences of disso- cited by a planar motion. Details of the experimental rig for this lution rate and effective surface tension on the governing parame- new two pendulums system and a correlation between the theory ters. The obtained theoretical model may be further employed as and experiments will be given. Detailed numerical and analytical a universal tool for a thermo/hydro-dynamic analysis of various results showing strongly nonlinear behavior will be discussed.

100 REFERENCES conditions (principal vector and principal moment of all forces are equal to zero): [1] Wiercigroch, M., Energy Conversion and Extraction from Oscillatory Motion Using Pendula Systems Operating in Ro- (−→n σ)dF −→P = 0, −→r (−→n σ)dF −→m = 0, (4) tational Motion, 2009 (patent pending) · − × · − ZZ ZZ [2] Xu, X., Wiercigroch, M.., Approximate Analytical Solutions where ~m - principal moment of the external forces; P~ - principal for Oscillatory and Rotational Motion of a Parametric Pen- vector of the external forces; ~n - unit normal to the tooth root dulum, Non. Dyn. 47: 311–320, 2007 surface F (x1,x2,x3) = 0; σ - stress tensor for isotropic elastic [3] Xu, X. Pavlovskaia, E.E., Wiercigroch, M., Romeo, F., medium. Lenci, S., Dynamic Interactions Between Parametric Pendu- Taking into account expressions for translation and rotation, lum and Electrodynamical Shaker: Effect of Non-ideal Exci- equations for stress tensor components are obtained. These com- tation Source, Z. Angew. Math. Mech. 87: 172–187. 2007 ponents depend on tooth root size, the place of action and direc- [4] Horton, B., Wiercigroch, M., Xu, X., Transient tumbling tion of point load and elastic properties of periodont: chaos and damping identification for parametric pendulum, (σ ~n) ~n (σ ~n) ~s (σ ~n) ~t Phil. Trans. Roy. Soc. A 366: 767 784, 2008 σn = · · ,σns = · · ,σnθ = · · , (5) ~n 2 ~n ~s ~n ~t [5] Horton, B., Nonlinear Dynamics of Pendula Systems, PhD | | | || | | || | Thesis, University of Aberdeen, 2008 where σn - stresses which acts normal to periodont, σns - stresses which acts along generatrix and tangent to periodont, σnθ - stresses which acts along guide and tangent to periodont. Ob- Marian Wiercigroch, Centre for Applied Dynamics Research, tained expressions can be used in practice by orthodontist for University of Aberdeen, Aberdeen, UK stress estimation appears in periodont in case of point load action on tooth root, particularly action from the pins of orthodontic ap- ANALYSIS OF STRESS DEFORMED CONDITION OF pliance. Information about these stresses has a great importance PERIODONTAL LAYER because they correspond to three mutually perpendicular directions and allow finding total stresses which appears in periodont K.S.YURKEVICH S.M.BOSIAKOV and predict algesia appearance. It is significant that obtained explicit functions can be used for development of computer pro- [email protected] [email protected] gram which allows to render stress field in periodont using free package ParaView. Investigation of biomechanical influence of orthodontic appli- ances on teeth roots is one of the factors of successful treatment. Kirill S. Yurkevich, Nezavisimosti avenue, 4,, Minsk, 220030, Proportion of load produced by appliance is one of the most im- Belarus portant things because incorrect application of pressure can be cause of algesia appearance and pathological process in periodont. Present work shows results of stress analysis at the contact MATHEMATICAL MODEL FOR THERMOELASTIC region of periodont and tooth root in case of point load action on DEFORMATION OF NANOSTRUCTURED FLEXIBLE fang. It is assumed that the shape of periodont is defined by two GRAPHITE O-RING SEAL PACKS complex elliptical hyperboloids in the following way: A.V.ZAITSEV N.G.ZLOBIN O.YU.ISAEV F0(x1,x2,x3)= F (x1,x2,x3)+ h0, (1) D.V.SMIRNOV A.M.KHANOV [email protected], [email protected] F (x ,x ,x )= 1 2 3 Flexible graphite is a unique nanostructured material with 2 2 (2) temperature-independent high thermo-chemical durability, low H x1 x2 2 = x3 + + p p = 0, 2 2 2 coefficient of friction and high elastic properties. Flexible − 1+ p p sak b − ! − graphite seals are easily breaking-in without erosive affection on where H - heightp of fang; p - parameter, characterize the fillet of the contact metal surfaces, fit for multifunctional usage in high- the fang top; ak = a1 when x1 < 0 and ak = a2 when x < 0, corrosive and high-reactive gas (i.e. oxygen, nitrogen, hydrogen, a1, a2, b - ellipse axles in the fang section z = H; h0 - parameter, etc.) and fluid (i.e. solutions of acids, alkalis and salts, organic characterized the thickness of periodont. solvents, petroleum and petroleum derivatives, sweet and sea- Displacement vector components of periodont points can be water, etc.) media. Flexible graphite O-ring gland seals and seal introduced as follows: packs have high reliability, they do not require additional hermetization during long-term usage and work at temperatures up to 1 ~u = (F (x1,x2,x3)+ h0)( ~u0 + ~ϕ ~r). (3) 560 ˚ C with pressures up to 40,0 MPa. h0 × At present time there is an intensive use of flexible graphite Here ~u0 - vector of translational displacements of tooth root along O-ring seals in aerospace, metallurgical, oil-and-gas and chemi- coordinate axis; ~ϕ - vector of angles of rotation along coordinate cal plants, power industry facilities, housing and communal ser- axis of elliptical hyperboloid, ~r - radius-vector drawn from the vice companies. Traditional experimental schemes (i.e. full scale corresponding strength center. Translational displacements and in-situ experiments with structures and prototypes) for the devel- angles of rotation of tooth root can be finding from equilibrium opment and optimization of seals are unreasonable due to a high

101 risk of accidents with serious environmental and economic dam- CF for the composites with fibres and particles enclosed by a ma- age. Therefore, mathematical modelling makes it possible to pre- trix layers with nonzero thicknesses and to formulate and prove dict thermophysical and mechanical properties of nanostructured theorems on a negative sign of these derivatives and on a local flexible graphite, to describe mechanical behaviour and optimum isotropy of random stress and strain fields. design of O-ring seals and seal packs corresponding to surviv- The authors acknowledge the support of the Russian Founda- ability and safety usage. tion for Basic Research (Grant RFBR–Urals No 07–01–96056). A mathematical model for thermoelastic deformation of seals is developed taking into account a cylindrical nature of anisotropy type of flexible graphite, experimental data for elastic, strength Aleksey V. Zaitsev, Mechanics of Composite Materials and Struc- and thermophysical characteristics of materials (the properties tures Dept., Perm State Technical University, Komsomolskiy differ in radial, hoop and generatrix directions). An initial op- Ave., 29, 614990, Perm, Russia eration mode of O-ring seals, produced on a large scale and used for rod seal in the gland cocks, has been considered. The influence of thermoforce loading conditions on the character of SYMBIOTIC CELLULAR AUTOMATA METHOD AND stress-strain distributions along cross-sections of seals and seal ITS APPLICATION TO SIMULATION OF THE packs has been investigated using numerical FEM solutions of MECHANICAL RESPONSE OF LIGNITE 3D stationary boundary value problems. Estimation of real dam- S.ZAVSEK A.V.DIMAKI E.V.SHILKO age mechanisms (i.e. damage from tension or compression in ra- A.I.DMITRIEV J.PEZDIC S.G.PSAKHIE dial, hoop and axial directions, and from transversal and antiplane shear) affection on initial strength, a comparison of different load- [email protected] ing modes (i.e. reciprocating motion in sealing bush or opposite direction, and torsion of the rod), height and conditions on con- Study of the behavior of complex multiphase media, which, in tact surfaces (i.e. ideal contact, friction or slip) between seals particular, include black and brown coal in natural occurrence, re- on the maximum values of radial, hoop, axial and shear stresses quires the application of effective computational methods. Such have been calculated. The locations of damaged domains ob- methods should allow description the interaction of medium com- tained from computational experiments are in a good agreement ponents occurred in various states of aggregation. There is the with the results of O-ring seals experience. That allows us to proposed method of describing of a complex multiphase medium, define an optimum hermetization pressure, to justify recommen- called the method of symbiotic cellular automata (SCA). dations to modify existing structures of O-ring seal packs and to Symbiotic cellular automata method is the combination of develop engineering techniques for refined strength analysis. classical cellular automata and movable cellular automata (MCA) approaches. In the framework of this method the investigated Mechanics of Composite Materials and Structures Dept., Perm medium is considered as a superposition of two interrelated me- State Technical University, Komsomolskiy Ave., 29, 614990, dia. One of them is described by a set of movable cellular au- Perm, Russia tomata and another – by a mesh of classical cellular automata. Novomet-Sealur Ltd., 1905 Year Str., 35, 614014, Perm, Russia The step of calculation consists of two main substeps. First of them is the step of the MCA model. At this substep the process of mass transfer of solid phase due to mechanical loading is PROBABILISTIC METHODS FOR THE ANALYSIS OF being considered. Next for the mechanical substep is performed RANDOM STRESS AND STRAIN FIELDS IN 2D AND 3D on a mesh of classical cellular automata. At this substep the pro- MATRIX-INCLUSION COMPOSITES cess of mass transfer of gas in the pores and channels is being considered, as well as the values of the forces acting on the mov- A.V.ZAITSEV V.S.KOKSHAROV able cellular automata from gas phase are being calculated. The [email protected] configuration of pores and channels through which gas propagates, is projected to the mesh of classical cellular automata from Prediction of the effective deformation properties and defini- the MCA layer. This symbiotic approach combines solutions for tion of the statistical characteristics of random stress and strain mechanical and gas dynamic problems and allows description of fields at components of fibre- (2D) and particle-reinforced (3D) multiphase heterogeneous media. matrix-inclusion composites are connected to finding the solu- We implement a simple linear model of sorption at a constant tions of stochastically nonlinear boundary-value problems using temperature, while the diffusion of gas in a solid frame and fil- various hypotheses on character of collective multiparticle inter- tering through channels in rock are not being considered. This action in the ensemble of reinforcement aggregates and approx- approximation is justified because the characteristic time of de- imations of conditional and unconditional correlation functions formation and fracture of the rock is much smaller than the char- (CF). Obtained derivatives of unconditional CF of the second and acteristic times of filtration-diffusion processes. It is possible third order, which are defined by a ration of the measures for the to implement more complex and realistic sorption models in the interphase surface (i.e. summarized perimeter or area) and for the framework of our method. fragment (i.e. volume or area), at the points corresponding to zero The proposed approach was used during the study of fracture values of the arguments, can be considered as one of a possible of lignite in atmosphere of carbon dioxide. Significant influence conditions for verification and a rejection of existing and devel- of pressure of the gas atmosphere on the fracture process was ob- oped models for random-structure composites. Usage of polydis- served. External gas atmosphere works like constrained boundary persible approximation make it possible to receive an exact rela- conditions and retards the formation of main crack, that leads to tions for derivatives of conditional and unconditional multipoint increase of effective strength of material.

102 It is clear that the proposed approach does not cover the entire PENDULUM THERMOVIBRATIONAL CONVECTION spectrum of processes in such a complex system like a multiphase IN VERTICAL ANNULUS heterogeneous medium. Nevertheless, proper selection of model parameters allows obtaining good agreement with the results of F.A.ZVEZDIN V.G.KOZLOV natural experiments. Further directions of method development are the introduction FAZ [email protected] [email protected] to the model as the state equation for non-ideal gas, which is important for realistic CO2 simulation, as the possibility of model- The thermal convection of liquid in a vertical annulus with the ing filtration processes in porous solid body. longitudinal partition subject to high-frequency torsion oscillations about its axis is studied. If the interior boundary has higher Andrey V. Dimaki, Akademicheskii av. 2/4, Tomsk, 634021, temperature the convective flow has a form of vortical convective Russia rolls. The convection is determined by thermovibrational mechanism [1] and it is characterized by vibrational number Rk [2]. According to [2] the gravitational convective flow stabilizes the THE THEORY OF CONTINUOUS MEDIUM WITH FREE thermovibrational convection appearance. ROTATION WITHOUT COUPLED STRESSES. The present research is devoted to the detailed experimental study of the thermovibrational convection of different liquids in E.ZDANCHUK V. LALIN the annulus of various aspect ratio in case of the exterior and the [email protected] interior heating. It is found that natural convection (gravitational Rayleigh New continuous medium classification has been suggested in number Ra) and the annulus aspect ratio A influence on the paper “On continuous medium classification. New models in threshold of the thermovibrational convection excitation. In the structural mechanics” by Lalin V.V. As result, new class of con- annulus of the finite height the vibrational convection is deter- tinuous medium models was found out. We called it “medium mined by vibrational number RK and parameter Ra/A. The re- without coupled stresses with free rotation”. Basic equations sults of the experiments with different liquids in the annulus for linear free rotation medium without coupled stresses are pre- of different aspect ratio are in good agreement on the plane sented. New effect in infinite linear elastic space has been found Ra/A,Rk. The increase of Ra/A (decay of relative height of the out: the existence of angles of rotation harmonic oscillations. annulus) leads to the crucial rise of the threshold Rk. The structure of vibroconvective flows is studied. It is found Elizaveta Zdanchuk, Polytechnicheskaya 29, Saint Petersburg, that the dimensionless wave number of vibroconvective rolls in- 195251, Russia creases with the decrease of vibration amplitude (at definite value of Ra/A). In the case of extremely small Ra/A (when the gravitational convection is negligible) the threshold value of Rk and the THREE-DIMENSIONAL DYNAMIC PROBLEM OF wave number of vibroconvective structures are in good agreement COUPLED THERMOELASTICITY FOR SPHERE with the theory [2]. The new type of instability is found when the exterior bound- A.ZHIGALIN ary of the liquid layer has higher temperature. In this case the [email protected] parameter Rk is negative and the thermovibrational mechanism plays stabilizing role. Nevertheless a threshold excitation of regular system of short wave vortices elongated along the vertical In this paper a closed solution of coupled problem of linear axis and located near the cavity walls could be found. The con- thermoelasticity for a solid sphere with a non-uniform tempera- vection develops in the boundary layers and does not influence on ture distribution is obtained. The solution is in the form of ex- the heat transfer, the size of the vortical structure is determined pansion in a complete system of eigenfunctions of a differential by the thickness of the Stokes layer. operator generated by the initial boundary value problem. This The research is supported by Russian Foundation for Basic operator is not self-adjoint. Corresponding eigenfunctions are not Research (grant 09-01-00665a) and Academic council of PSPU orthogonal. It is proved that the eigenfunctions of a mutually ad- (grant of 2010). joint operators pair constitute a biorthogonal system. Therefore the utilizing of eigenfunctions of the adjoint operator permit to obtain the coefficients of expansion mentioned above. As a result, REFERENCES the procedure of finding the coefficients of expansion is reduced to solving a countable sequence of Cauchy problems. [1] A. A. Ivanova, V.G. Kozlov. Thermal vibrational convection The method considered allows to formulate efficient compu- in a cavity under nontranslational oscillations // Fluid Dy- tational algorithms for solving coupled thermoelastic problems. namics. 2003. V. 38. No 3. P. 372–386. Eigenfunctions arised in the boundary value problem for solid sphere can be written in exact form. In general, numerical meth- [2] V.G. Kozlov. Vibrational thermal convection in a cavity ex- ods have to be applied to determine the eigenvalues whereas this ecuting high-frequency rocking motions // Fluid Dynamics. leads to the problem of searching complex roots of a transcen- 1988. V. 23. No 3. P. 437–442. dental equation. Feudor Zvezdin, Sibirskaya, 24, Perm, 614990, Russia Alexandr G. Zhigalin, Akademika Pavlova, 1, Samara, 443011, Russia

103 Index

Abdrashitov A. V., 16 Bulygin A. N., 17 Abid T., 16 Bushueva C. A., 29, 30 Abramyan A. K., 16, 69 Bykov N. Y., 30 Adamov A. A., 98 Adiguzel O., 16 Carini A., 31 Aero E.L., 17 Carvalho G., 32 Afanasyeva I. M., 78 Castellanos A., 35 Aifantis E. C., 70 Chaikin V., 79 Ajibose O. K., 18 Cherkaev A. V., 20 Alabuzhev A. A., 18 Cherkay M., 31 Alashova A. S., 19 Chernyaeva E., 79 Alekhin A. N., 19 Chertova N. V., 43 Alekhin V.V., 55 Chivilikhin S. A., 31, 80 Alexandrov S., 65 Chumakov Yu., 31 Alexeyeva O., 19 Chzhvai V., 32 Allag S., 19 Cresseri S., 32 Altenbach H., 19, 20, 60, 62 Andreini P., 92 de Assis J. T., 32, 33 Annin B. D., 55 Demidov V.N., 33 Antimonov M. A., 20 Denisova M. O., 27, 30, 33 Antipov K. A., 21 Dimaki A. V., 102 Antonova O. V., 21 Dmitriev A. I., 34, 54, 76, 94, 102 Antonyuk S., 21, 34 Dobroskok A., 34 Aouabdia Y., 22 Dobrynin S. A., 94 Argunova T. S., 22, 23 Dosta M., 34 Arrieta A.F., 23 Dvas N., 35 Dyatlova P., 35 Artemova A. I., 74 Dyda A. A., 35 Arutyunyan R. A., 24 Asonov A. M., 24 Einsfeld R., 32 Astafurov S. V., 24, 25, 89 Eltsov I.N., 75 Averbukh E. L., 51 Eremeyev V.A., 20, 26 Erofeeva E. S., 71 Babenkov M., 26 Espin M. J., 35 Babichev A. V., 55 Evlampieva S. E., 72 Banichuk N. V., 26 Evstigneev N. K., 36 Bashin G. P., 86 Beliakova T. A., 64 Fedina M. E., 36 Belov K. L., 26 Fedorchenko I. A., 37 Belyaev A., 26 Fedorov A. V., 36 Belyakova E. I., 23 Fedorova A. F., 37 Berinskiy I. E., 24, 27 Fedorova A. N., 37 Bezoyan E., 27 Fedorova N. N., 37 Bezrukov L. N., 70 Fiﬂimonov M. Yu., 37 Birikh R.V., 27 Filippenko G., 38 Blekhman I. I., 95 Freidin A. B., 20, 56, 60, 100 Bobylev S. V., 28 Fries L., 34 Bogdan M. M., 28 Fukalov A. A., 38 Bogdanova S. B., 28 Bogoyavlenskaya V.A., 87 Galimov E. M., 38 Borodin I. N., 28 Garishin O. K., 38, 71 Borovkov A. I., 21 Garkushin G. V., 39 Bosiakov S. M., 32, 101 Gavrilov S. N., 39, 40 Bouaﬁa Y., 45 Genna F., 32, 77 Boubertakh A., 16, 22, 29 Gerasimenko E. A., 40 Boukezzata M., 19 Gileva O.S., 71 Brandt K., 21 Ginzburg B., 81 Bratov V., 29 Gladkov S. O., 28 Breder R. L. B., 32 Glazov A. L., 40

104 Glushkov E. V., 63 Knyazeva A. G., 36, 54, 69 Glushkova N. V., 63 Koksharov V.S., 102 Goldfeld M. A., 37 Kolubaev E. A., 94 Golotina L. A., 41 Kolykhalin V., 54 Golubkova I. A., 91 Komar L. A., 54, 86 Gorkunov E. S., 42 Kondrat’ev K. A., 93 Goryacheva I. G., 42 Konovalenko Is. S., 54 Grekov I. V., 23 Konovalenko Iv. S., 54 Grekova E., 42 Konovalov A. V., 91, 92 Grekova E. F., 39 Konovalov D. A., 55 Gridasova E. A., 65 Korobeynikov S. N., 55 Grigoriev A.S., 89 Korolev I. K., 56 Grinyaev Yu. V., 43 Korostelev S. Yu., 56, 94 Gusarov V.V., 80 Korovkin V., 35 Gutkin M. Yu., 22, 23, 43, 44 Kosevich Yu. A., 56 Koshelev V., 56 Hagedorn P., 23 Kostarev K. G., 29, 30, 33 Hakem A., 45 Kostina L. S., 23 Hakobyan V., 46 Kouzov D. P., 57 Hamada S., 22, 29 Kovalev V.A., 57 Hamamda S., 16 Kovaleva M., 57 Hansson P., 68 Kovaleva N. A., 51 Heinrich G., 98 Koverga A. U., 58 Heinrich S., 21, 34 Koverga A. Yu., 60 Hockauf M., 39 Kozhevnikov E., 58 Hoover C. G., 46 Kozhevnikov E. N., 58 Hoover Wm. G., 46 Kozhevnikova L. L., 41 Hoving J. S., 46 Kozlov N. V., 59 Hwu Y., 22 Kozlov V.G., 59, 103 Krashnikov V.S., 28 Il’ichev A. T., 47 Kravets A. G., 37 Indeitsev D., 26 Krivtsov A. M., 24, 27, 60, 61, 63 Indeitsev D. A., 47 Kroener C., 60, 62 Ing J., 47 Kryzhevich D. S., 54, 82 Isaev O. Yu., 101 Kubyshkin E. P., 58, 60 Itskov M., 47 Kucherenko Ya. V., 58 Ivanov K., 48 Kucheyeva D. O., 60 Ivanova A. A., 85 Kuchmin Y., 16 Ivanova E., 48 Kudarova A. M., 61 Ivanova S. Yu., 26 Kudish I. I., 61 Ivantsov A. O., 65 Kukushkin S., 62 Jabareen M., 87 Kuleshov A. S., 61 Je J. H., 22, 23 Kumar S., 62 Kurkin A. A., 51 Kanel G. I., 39 Kutergin A. V., 38 Karyakin M., 48 Kutschke A., 60, 62 Kazakov K., 49 Kuzkin V., 35 Khanov A. M., 101 Kuzmichev S., 62 Kharlamov G. V., 49 Kuzmina I. A., 36 Khokhryakova Yu., 50 Kuznetsova A. V., 94 Khurana A., 99 Kvasha O. V., 63 Khvostova O. E., 51 Kikot I. P., 51 Lalin V., 103 Kireenkov A., 52 Lammering R., 63 Kireenkov A. A., 52 Laptev D. V., 28 Kiselev A. P., 52 Lauke B., 98 Kitavtsev G., 53 Lavrova T. B., 63 Kizilova N., 53 Le-Zakharov S. A., 24 Klimenko L. S., 53 Lebedev A. V., 29 Kloss H., 34 Lebedev S. N., 54 Klyueva N. V., 94 Lebedeva K. Yu., 18

105 Leshchev D. V., 30 Odintsova V., 76 Lima I., 32 Oesterle W., 34 Linkov A., 34 Onischuk A. A., 49 Loboda O. S., 63 Onufriev A., 51 Lomakin E. V., 64 Osipov A. V., 62 Lurie S., 64 Ovid’ko I. A., 28, 43, 70, 76 Lutov V., 35, 79 Lutsik A.I., 71 Palis F., 76 Lyamina E., 65 Palis S., 76 Lychev S. A., 67 Panpurin S. V., 44 Lyubimov D. V., 53, 65 Panteghini A., 77 Lyubimov E. V., 35 Parshakova Ya. N., 65, 77 Lyubimova O. N., 65 Parshin D. A., 77 Lyubimova T. P., 53, 65, 77 Pavlov I. S., 81 Pavlov Yu. V., 17 Majorov A., 35 Pavlovskaya E., 18, 47 Makeev E. V., 26 Pelevin A. G., 98 Mamontov A. E., 66 Perelmuter M., 77 Manevitch L. I., 51, 56, 57, 66, 88 Perunov E. N., 55 Manzhirov A. V., 67 Pessoa J. R. C., 32 Manzini G., 92 Pestov K. N., 65 Markov P.V., 67 Petinov S. V., 56, 78 Marrero Iglesias S., 33 Petrov V.E., 78 Mayer A. E., 28 Pezdic J., 102 Mazo M. A., 51 Pilishkin V.N., 78 Melentyev A., 68 Podolsky M., 79 Melin S., 68 Polyakova E., 79 Melnikov B. E., 68 Polyanskiy A., 79 Metrikine A. V., 46, 61, 69 Polyanskiy S., 80 Meyer L., 39 Polyanskiy V.A., 26, 47, 79 Michailov A. A., 21 Popov I. Yu., 80 Mikolaychuk M. A., 69 Porubov A. V., 80 Mirantsev L. V., 16, 69 Potapenko M. A., 81 Mironov V., 70 Potapov A. I., 81 Mizev A. I., 27, 71 Pozdnyakov A., 81 Mokhov E. N., 22 Prozorova E. V., 81 Molotkov L. A., 71 Psakhie S. G., 16, 24, 25, 54, 82, 89, 102 Purtov P.A., 49 Monin V., 33 Pustovalova O., 48 Morozov I. A., 71 Morozov N. F., 20, 28, 70, 72 Quintanilla M. A. S., 35 Moshev V.V., 72 Mozhaev V.G., 73 Radons G., 82 Mukhrjee A. K., 76 Razorenov S. V., 39, 90 Muratikov K. L., 40 Rega G., 83 Mushnikov A. N., 42 Rodygina O. A., 80 Myagkov N. N., 70 Roman N. V., 56, 82 Rubin M. B., 87 Nagalyuk S. S., 22 Rudoy E., 84 Nasedkin A., 73 Rumyantsev S., 89 Nasedkina A., 74 Rumyantsev S. A., 19, 84 Naumenko K., 60, 62 Ryzhak E. I., 84 Nazarov L. A., 74, 75 Nazarova L. A., 74, 75 Sabirov R. R., 59 Neganov S., 54 Sakharov A. V., 42 Neild S.A., 23 Salhi A., 45 Nerchenko V.A., 92 Salikov V., 21 Neygebauer I., 75 Sargsyan S., 85 Nikitin L.V., 75 Savin A. V., 51 Nikitin V.F., 92 Savitski A., 39 Nikitina N. Ye., 75 Sazhenkov S., 85 Nikonov A. Yu., 76 Schipitsyn V.D., 85

106 Schneider G., 21 Tomar S. K., 99 Semendyaev S., 52 Tovstik P.E., 72, 99 Semenko E., 86 Tovstik T. P., 99 Semenov A. S., 68 Tsiberkin K. B., 65 Sergeev V.V., 24, 94 Tsukanov I. A., 55 Serrar C., 16, 22, 29 Tuchkova N., 64 Shadrin V.V., 86 Shamolin M. V., 86 Usoshin S. A., 97 Shardakov I. N., 41, 87 Usoshina H. A., 97 Sharipova L., 87 Sharma I., 62 Vaganova N. A., 37 Sheinerman A. G., 70 Vakulenko S., 31, 99 Shepelev D. S., 88 Valverde J. M., 35 Shevtsova M., 73 Vasilkov V.B., 95 Sheydakov D., 89 Vasilyeva G., 19 Shihov A., 89 Vilchevskaya E. N., 100 Shilko E. V., 24, 25, 89, 102 Vjatkin A.A., 59 Shishkina E. V., 39, 40 Volkov-Bogorodskii D., 64 Shkapov P.M., 97 Vorobev A., 100 Shulgin A. V., 36 Vosel S. V., 49 Shumikhin T. A., 70 Voynov I. B., 21 Shvygin A. L., 90 Wagg D. J., 23 Skiba N. V., 76 Wagner B., 53 Skripnyak E. G., 90 Wahi P., 62, 100 Skripnyak V.A., 90 Wiercigroch M., 18, 47, 100 Skripnyak V.V., 90 Wolff M., 21 Smirnov A. S., 91, 92 Smirnov D. V., 101 Yakimova K. S., 24 Smirnov N. N., 91, 92 Yakovlev Yu., 26 Smirnov S. V., 55, 91 Yakovlev Yu. A., 47, 79 Smirnov V.V., 88 Yakushenko E. I., 42 Smirnova E. O., 93 Yakushev A., 70 Smirnova M. N., 93 Yaroshevich N. P., 98 Smirnova S. V., 93 Yurkevich K. S., 101 Smolin A. Yu., 25, 54, 56, 82, 94 Smolin I. Yu., 94 Zadvorkin S. M., 42 Soldatov I. N., 94 Zaitsev A. V., 38, 101, 102 Somsikov V.M., 94 Zakharova Yu. V., 37 Sorokin V.S., 95 Zavertan A. V., 40 Sorokova S. N., 54 Zavsek S., 102 Sosnovtsev V.V., 78 Zazin Yu. P., 64 Spahn F., 95 Zdanchuk E., 103 St¨ockelhuber K. V., 98 Zeitlin M.G., 37 Starikov K. E., 96 Zhigalin A., 103 Starikov S., 96 Zlobin N. G., 101 Stefanov Yu. P., 96 Zolnikov K. P., 16, 54, 82 Stepanova L. V., 96 Zubova E. A., 51 Stoliarov U. V., 97 Zvezdin F. A., 103 Subbotin S. V., 59 Zvyaguin A. V., 93 Sulimov V.D., 97 Suvorova T. V., 97 Svijazheninov E., 97 Svistkov A. L., 67, 71, 98 Sylyvonyuk A. V., 98

Taleb K., 16, 22, 29 Tarasov B. G., 98 Tarasov D., 89 Tarasov D. Y., 84 Tarunin E., 68 Tikhonov A. A., 21

107