Apm 2005 Book of Abstracts

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XXXIII Summer School – Conference “Advanced Problems in Mechanics”

June 28 – July 5, 2005, St. Petersburg (Repino), Russia

APM 2005

BOOK OF ABSTRACTS

http://www.ipme.ru/ipme/conf/apm2005 GENERAL INFORMATION APM 2005 is the thirty third in a series of annual summer schools held by Russian Academy of Sciences. The Summer school “Advanced Problems in Mechanics 2005” 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.

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 School was on nonlinear oscillations of mechanical systems with a finite number of degrees of 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 • V.V. Beletsky(Keldysh Institute of Applied Mathematics RAS, Moscow, Russia) • A.K. Belyaev(St. Petersburg State Polytechnical University, IPME RAS, Russia) • I.I. Blekhman (IPME RAS, Mekhanobr-tekhnika Corp., St. Petersburg, Russia) • M. Cartmell (University of Glasgow, UK) • A. Castellanos (University of Seville, Spain) • 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) • G.M.L. Gladwell (University of Waterloo, Canada) • I.G. Goryacheva (Institute for Problems in Mechanics RAS, Moscow, Russia) • E.F. Grekova (IPME RAS, St. Petersburg, Russia) • D.A. Indeitsev (IPME RAS, St. Petersburg, Russia) — Co-Chairman • E.A. Ivanova (St. Petersburg State Polytechnical University, IPME RAS, Russia) • A.M. Krivtsov (St. Petersburg State Polytechnical University, IPME RAS, Russia) — Chairman of the Organizing Committee • G.A. Leonov (IPME RAS, St. Petersburg State University, Russia) • E.V. Lomakin (Moscow State Lomonosov University, Russia) • A. Metrikine (TU Delft, The Netherlands) • N.F. Morozov (IPME RAS, St. Petersburg State University, Russia) • V.A. Palmov (St. Petersburg State Polytechnical University, Russia) — Co-Chairman • M. Wiercigroch (University of Aberdeen, UK) — Co-Chairman • P.A. Zhilin (IPME RAS, St. Petersburg State Polytechnical University, Russia) • F. Ziegler (Wien University of Technology, Austria)

1 LOCAL ORGANIZING COMMITTEE • I. Alexandrov, St.Petersburg State University of Aerospace Instrumentation • K. Aristovich, St.Petersburg State Polytechnical University, IPME RAS • I. Berinsky, St.Petersburg State Polytechnical University, IPME RAS • A. Firsova, IPME RAS — Scientiﬁc Secretary • O. Gonin, St.Petersburg State Polytechnical University • A. Krivtsov, St.Petersburg State Polytechnical University, IPME RAS — Chairman • O. Loboda, St.Petersburg State Polytechnical University, IPME RAS • A. Sergeyev, IPME RAS • E. Serogo, IPME RAS • P. Tkachev, St.Petersburg State Polytechnical University, IPME RAS • S. Vasiliev, St.Petersburg State Polytechnical University • E. Vilchevskaya, IPME RAS, St.Petersburg State Polytechnical University The conference is organized with help of our service agency “Monomax” (ﬁnd the information further).

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 generalized continua (polar and micromorphic continua, mixtures, porous media, electromagnetic continua, grains, powders etc) • solids and structures • phase transitions • nanostructures and thin films • wave motion • nonlinear dynamics, chaos and vibration • dynamics of rigid body and multibody dynamics • fluid and gas • computational mechanics • mechanical and civil engineering applications

MINISYMPOSIA AND CHAIRS [MS1] Application of Mechanics to Aerospace Systems M. Cartmell (Glasgow, UK) [MS2] Biomechanics and Mechatronics H. Irschik (Linz, Austria), A.K. Belyaev (St. Petersburg, Russia), Yu.I. Nyashin (Perm, Russia) [MS3] Nonlinear Methods and Integrable Models in Mechanics N.A. Kudryashov (Moscow, Russia), A.V. Porubov (St.Petersburg, Russia) [MS4] Wave Mechanics of Civil Engineering Structures A.V. Metrikine (Delft, The Netherlands) 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.

2 REGISTRATION FEE The conference fee is:

• for participants giving a presentation: 300 EUR • for participants not giving a presentation: 220 EUR • for students: 220 EUR The fee covers invitation costs, postage, book of abstracts, proceedings, social program (excursion, get-together party), and organizational costs.

• The fee for accompanying persons is 100 EUR and covers the invitation costs, postage, and social program. • There is a reduced conference fee of 50 EUR for participants having permanent position in NIS countries (Armenia, Azerbaijan, Belarus, Georgia, Kazakhstan, Kyrgystan, Moldova, Russia, Tajikistan, Turkmenistan, Ukraine, Uzbekistan), which partially covers the postage and book of abstracts. • There is a reduced conference fee of 25 EUR for students from NIS countries.

EUROMECH COLLOQUIUM 468 In the frames of APM 2005 is held EUROMECH Colloquium 468 Multi-scale Modelling in the Mechanics of Solids (June 29 July 1, 2005).

TIME-TABLE SCHEME

Abbreviations: O is the opening ceremony, PL are the plenary lectures, MS is a minisymposium, S is an oral session, PS is a poster session, EM is a EUROMECH Colloquium session, C is the closing ceremony.

28 29 30 1 3 4 5 DATE Tue Wed Thu Fri Son Mon Tue Room A morning O PL MS 1 PL S PL S PL MS 2 PL S PL MS 3 C evening MS 1 S S MS 2 PS 1 PS 2 MS 3 Room B morning MS 4 EM EM EM S S evening MS 4 EM EM EM S

ACKNOWLEDGEMENTS

The APM 2005 organizers gratefully acknowledge ﬁnancial support from Russian Foundation for Basic Re- search (project 05-01-10046), Branch of Power Engineering, Mechanical Engineering, Mechanics and Control of RAS, Commission on the Youths Problems of RAS and St.Petersburg’s Government (Science Committee). We are very thankful to the University of Aberdeen for the technical support.

3 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 13 years of practical experience in destination management. Our activity past 13 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 100 international and more than 700 local events held in St. Petersburg in 1997- 2005. 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: Zanevsky pr. 1, St. Petersburg, Russia Address for correspondence: PO box 168, 195112 Saint Petersburg, Russia tel. +7 (812) 444 7166, +7 (812) 445 0159, fax +7 (812) 324 7322 E-mail: [email protected] Web site: http://www.monomax.ru

4 JUNE 28, TUESDAY

Room A Room B Morning Session CHAIRMAN D.A. INDEITSEV Morning Session Minisymposium “Wave Mechanics of Civil Engineering Structures” 9:30 – 10:00 OPENING CEREMONY CHAIRMAN A.V. METRIKINE 10:00 – 10:40 A.I. Borovkov, V.A. Palmov. Basic Solutions and Regular Expansion in the Mechanics of Periodic Composites 11:30 – 12:00 G.L. Kuiper, A.V. Metrikine. Non-linear Analysis of Dynamic Stabil- 10:40 – 11:20 W. Pietraszkiewicz. Dynamically and Kinematically Exact Theory of ity of an Offshore Pipe Conveying Fluid Irregular Shell Structures and its Numerical Implementation 12:00 – 12:30 I.V. Sturova. Unsteady Behavior of an Elastic Beam Floating on the Surface of Deep Water Break Coffee break 12:50 – 13:20 A.K. Abramian. Localized Waves in Shallow Water Caused by a Soli- Minisymposium “Application of Mechanics to Aerospace Systems” ton Moving Along the Bottom 13:20 – 13:50 K.L. Muratikov, A.L. Glazov. Thermoelastic Photoacoustic Mi- CHAIRMAN M.P. CARTMELL croscopy of Vickers Indented Metals 11:30 – 12:00 A. Steindl, H. Troger. Stability Problems Related to the Space Elevator 12:00 – 12:30 R.W. Tucker, D. Burton. Twisted Ring Laser Accelerometers Evening Session Coffee break 15:30 – 16:00 L.V. Nikitin. Statics and Dynamics of Rods with Skin Friction 16:00 – 16:30 J. Mennink, F. Soetens, N.A. Kutanova. Cross-sectional Stability of 12:50 – 13:20 A.M. Krivtsov, E.M. Galimov, M.S. Legkostupov, A.V. Zabrodin. Aluminium Extrusions Simulation of Earth–Moon System Development 13:20 – 13:50 P. Duo.´ Prediction of Foreign Object Damage Coffee break 16:50 – 17:20 M. Steenbergen. On the Effect of the Interface Conditions on the Dy- Evening Session namic Response of a Beam on a Half-Space to a Moving Load 17:20 – 17:50 J.S. Hoving, F. Molenkamp. The Dynamic Response of a Discrete Lat- 15:30 – 16:00 M. Francis, D. Biamonti, A. McCall, J.P. Sanchez, G. Albaladejo, tice Containing Dry-Friction Elements to a Surface Load D. Moore, S. Graham, C. Murray, S. Shahid, P. Trivedi, D. Santhanakrishnan, 17:50 – 18:20 A.V. Metrikine. A Discontinuous-Strip Method for Calculation of the G. Radice, G. Hughes. Lunar Net-Lander: Mission Overview Dynamic Response of Periodically-Inhomogeneous 3D Structures to Moving Loads 16:00 – 16:30 M. Francis, D. Biamonti, A. McCall, J.P. Sanchez, G. Albaladejo, D. Moore, S. Graham, C. Murray, S. Shahid, P. Trivedi, D. Santhanakrishnan, G. Radice, G. Hughes. Lunar Net-Lander: Surface Components Coffee break 16:50 – 17:20 M. Francis, D. Biamonti, A. McCall, J.P. Sanchez, G. Albaladejo, D. Moore, S. Graham, C. Murray, S. Shahid, P. Trivedi, D. Santhanakrishnan, G. Radice, G. Hughes. Lunar Net-Lander: Orbital Analysis 17:20 – 17:50 M.P. Cartmell, M.C. D’Arrigo. Simultaneous Forced and Parametric Excitation of a Space Tether JUNE 29, WEDNESDAY

Room A Room B Morning Session CHAIRMAN I.G. GORYACHEVA

9:30 – 10:10 E.A. Ivanova, A.M. Krivtsov, N.F. Morozov, B.N. Semenov, A.D. Firsova. Nanomechanics: Continuum and Discrete Approaches 10:10 – 10:50 D.P. Boso, M. Leﬁk, B.A. Schreﬂer. Multiscale Modelling for Com- posites Including Continuum to Discrete Linkage 10:50 – 11:30 A. Castellanos. Correlation Between Interparticle Forces and Bulk Be- haviour in Fine Powders Break

CHAIRMAN A.M. KRIVTSOV Dynamics of rigid bodies and multibody dynamics

11:40 – 12:00 T.V.Salnikova. On the Stability of Linear Potential Gyroscopic Systems 12:00 – 12:30 I.A. Pasynkova. Bifurcations of Cylindrical Precessions of an Unbal- anced Rotor 12:30 – 12:50 V.M. Morozov, V.I. Kalenova, M.A. Salmina. Stability and Stabiliza- tion of Nonholonomic Systems of Special Class Coffee break EUROMECHCOLLOQUIUM 468 13:10 – 13:30 A.S. Kuleshov. On the Snakeboard Dynamics 13:30 – 13:50 A.A. Loktev, Yu.A. Rossikhin, M.V. Shitikova. Dynamic Stability of an Orthotropic Plate Subjected Impact Excitation “MULTI-SCALE MODELLING IN THE MECHANICS OF SOLIDS” 13:50 – 14:10 A.V. Karapetyan. Invariant Sets of Dynamical Systems with First Inte- grals: Theory and Applications

Evening Session

CHAIRMAN A.D. SERGEYEV Mechanical and civil engineering applications

15:30 – 15:50 N.V. Banichuk, S.Yu. Ivanova, E.V. Makeev, A.V. Sinitsyn. Non-local Shell Optimization with Application of Genetic Algorithm 15:50 – 16:10 A.A. Dobroskok, A.M. Linkov, A. Ghassemi. Complex Variable BEM for 2D Problems of Heat, Current or Fluid Flow within, along and across Structural Ele- ments 16:10 – 16:30 B. Wilczynski,´ Z. Mroz.´ Optimal Design of Notched Components Against Fracture and Fatigue Failure Coffee break 16:50 – 17:10 D.P. Kouzov. The Gravitation Correction for the Acoustics Equations 17:10 – 17:30 P.E. Tovstik, T.M. Tovstik, V.A. Shekhovtsov. On the Mariner Fixed Offshore Platform Dynamics Under Random Wave Forces 17:30 – 17:50 Y. Stylianos, D.G. Luchinsky, V.N. Smelyanskiy. Reconstruction of Nonlinear Dynamics from Noisy Incomplete Measurements JUNE 30, THURSDAY

Room A Room B Morning Session CHAIRMAN A.M. KRIVTSOV

9:30 – 10:10 I.G. Goryacheva. Multi-Scale Analysis in Contact Mechanics 10:10 – 10:50 P.A. Zhilin. Some Problems of the Nonlinear Rod Theory 10:50 – 11:30 K.P. Chong. Nano Science and Engineering in Mechanics Break

CHAIRMAN D.A. INDEITSEV Fluid and Gas

11:40 – 12:00 V.V. Kozlov, Yu.A. Litvinenko, L.L. Loefdahl, V.G. Chernorai, G.R. Grek. Nonlinear Sinusoidal and Varicose Instability in the Boundary Layer (Re- view) 12:00 – 12:20 A.A. Ivanova, V.G. Kozlov. Lift Force Acting the Solids in Liquid Un- der Vibrations 12:20 – 12:40 L.A. Molotkov. Velocities of Wave Propagation in Fluid Mixtures Coffee break 13:00 – 13:20 V. Shevtsova, A. Myaldun, D. Melnikov. Stability of Oscillatory Flow in Liquid Column EUROMECHCOLLOQUIUM 468 13:20 – 13:40 S.C. Paul, M.C. Paul, T. Korakianitis, W.P. Jones. Large Eddy Simu- lation and PDF for Turbulent Combustion Flows “MULTI-SCALE MODELLING IN THE MECHANICS OF SOLIDS” Evening Session

CHAIRMAN E.F. GREKOVA Solids and Structures

15:30 – 15:50 A. Scalia, M.A. Sumbatyan. Harmonic Vibration of Rigid Punch on the Free Surface of a Porous Elastic Half-plane 15:50 – 16:10 E. Scarpetta, M.A. Sumbatyan. One-mode P-wave Propagation through a Periodic Array of Interface Cracks Coffee break 16:30 – 16:50 F.A. Gilabert, A.M. Krivtsov, A. Castellanos. Computational Design and Mechanical Characterization of Microstructured Solids in Layers 16:50 – 17:10 S.A. Atroshenko, S.I. Krivosheev, Y.V. Petrov, A.A. Utkin, G.D. Fe- dorovsky. Experimental Investigation of Granite and Limestone Fracture under Impact Pulse and Static Loading 17:10 – 17:30 V.N. Kukudzhanov. Rheological Instability and Fracture of Elastoplas- tic Specimens under Tension 17:30 – 17:50 D.A. Indeitsev, N.F. Morozov, V.N. Naumov, E.I. Prokuratova. The Temperature Impact at the End of the Rod of Complicated Structure 17:50 – 18:10 A.V. Kashtanov, Y.V. Petrov. Kinetic Nature of the Incubation Time 18:10 – 18:30 R.A. Arutyunyan. Fatigue Fracture in the Gigacycle Regime JULY 1, FRIDAY

Room A Room B Morning Session CHAIRMAN V.A. EREMEYEV

9:30 – 10:10 E.V. Lomakin, H.-J. Christ. Deformation of Elastic Material with Stress-State-Dependent Properties at the Crack Tip under Longitudinal Shear Conditions 10:10 – 10:50 D.A. Indeitsev. Inﬂuence of the Kinetics of Processes on the Behavior of Thin Structures in a Continuous Medium 10:50 – 11:30 G.A. Leonov. Stability and Oscillation of Systems with Cylindrical Phase Space Break

Minisymposium “Biomechanics and Mechatronics”

CO-CHAIRMEN H. IRSCHIK, A.K. BELYAEV,YU.I. NYASHIN

11:40 – 12:10 H. Irschik. Non-Classical Terms in the Equations of Balance for a Growing Body 12:10 – 12:40 M. Stangl, N. Beliaev, A.K. Belyaev, H. Irschik. Applying Lagrange Equations and Hamilton Principle to Vibrations of Fluid Conveying Pipes Coffee break EUROMECHCOLLOQUIUM 468 13:00 – 13:30 Yu.M. Vetyukov, V.V. Eliseev. The Lagrangian Mechanics of Elastic Shells and Its Numerical Application 13:30 – 14:00 D. Okhotsimsky, O. Nikitin, Y. Gerasimenko, N. Serbenjuk, L. Mi- “MULTI-SCALE MODELLING IN THE MECHANICS OF SOLIDS” tutsova, K. Delchev, V. Vitkov, A. Platonov, V.Yaroshevsky. A Biomechanical Stimu- lator for Scientiﬁc-experimental Study of the Regeneration of Spinal Cord Locomotion Capabilities after Traumatic Break

Evening Session 15:30 – 16:00 Yu. Nyashin, E. Simanovskaya, V. Tverier. Maxillodental System as a Specialized Biomechanical System 16:00 – 16:30 V. Lokhov, Yu. Nyashin. Application of the Stress and Strain Control by Eigenstrain to Biomechanical Systems Coffee break 16:50 – 17:20 Yu.V. Akulich, R.M. Podgayets, A.I. Kholkina. Quasi-static Analysis of the Muscular Efforts and Joint Reaction Forces during the Lower Extremity Movement in the Training Machine with Weights 17:20 – 17:50 A. Anoshkin, D. Ivanov, S. Ivanov, Yu. Osorgin, A. Tashkinov, P. Ud- intsev, Yu. Zmeev. Problems of Strength Evaluation of Carbon-Carbon Hip Prostheses 17:50 – 18:20 V.V. Potekhin, A.V. Zinkovsky, L.B. Maslov, V.I. Shapin, S.E. Lvov, E.V. Bleskin. Forced Vibration Technology for Non-destructive Testing in Orthopaedy 18:20 – 18:50 A.G. Soldatos, K.G. Arvanitis, S. Papathanassiou, A. Kladas. Con- trol for Vibration Attenuation in Variable Speed Wind Turbines JULY 3, SUNDAY

Room A Room B Morning Session CHAIRMAN I.I. BLEKHMAN Morning Session CHAIRMAN E.F. GREKOVA 9:30 – 10:10 D. Harris. Granular Flow: a New Plasticity Model Containing a Length Mechanics of Generalized Continua Scale 10:10 – 10:50 C. Vallee,´ S. Charlesm, S.Ya. Stepanov, A.A. Burov. On Time Domain 11:00 – 11:20 V.A. Eremeyev, A.N. Soloviev. On Elastic Micro- and Nano-Crystals Identiﬁcation Method for Linear Oscillation Mechanical Systems Lattices Modelling Break 11:20 – 11:40 I. Smolin. Cosserat Continuum as a Model of Mesoscale 11:40 – 12:00 A. Valance, C. Pascot, R. Delannay. Shear Instabilities in Granular Mixtures under Gravity CHAIRMAN A.M. KRIVTSOV Coffee break Computational Mechanics 12:20 – 12:50 E.L. Aero, A.N. Buliguin. Nonlinear Theory of Cardinal Rearrange- 11:00 – 11:20 V.P. Fedotov, L.F. Spevak, V.B. Trukhin, V.V. Privalova. A ment of Solid Structure under Catastrophic Deformations Numerical-Analytical Technique for Solving Problems of Mathematical Physics 12:50 – 13:10 J.M. Nitsche, T.-F. Wang, G.B. Kasting. Micromechanics of Molecu- 11:20 – 11:40 A. Smolianski, O. Shipilova, H. Haario. Particle Transport Method for lar Diffusion through Multiscale “Brick-and-Mortar” Composite Materials, with Appli- convection-reaction problems cation to the Permeability of Human Skin 11:40 – 12:00 N. Loguinova. Oscillations in a Hardsphere Model under Gravity 13:10 – 13:30 T. Chen, H.-Y. Kuo. Exact Correspondences between N-dimensional Anisotropic and Isotropic Green’s Functions in Conductions Coffee break 12:20 – 12:40 V.A. Bratov, P. Stahle.˚ Viscous Cohesive Zone Model for Fracture un- Evening Session der Rapid Loading 15:15 – 16:45 Posters (PS1) 12:40 – 13:00 V.F. Koshelev, A.M. Linkov. On Numerical Implementation of Non- local Fracture Criterion to Inhomogeneous Media Break 13:00 – 13:20 A.B. Kiselev, O.V.Nekhaeva. Computational Simulation of Irreversible Dynamic Deforming and Fracture of Damageable Solids and Structures 17:15 – 18:45 Posters (PS2) 13:20 – 13:40 I.I. Kossenko. An Analysis of the Computer Model for the Unilateral Multibody Systems Dynamics 13:40 – 14:00 R. Crouch, T. Li. On the Use of Auxillary Surfaces in Computational Plasticity Return Mapping JULY 4, MONDAY

Room A Room B Morning Session CHAIRMAN P.A. ZHILIN Morning Session CHAIRMAN I.I. BLEKHMAN 9:30 – 10:10 G.M.L. Gladwell. Inverse Problems in Vibration — Past, Present and Nonlinear Dynamics and Vibration Future 10:10 – 10:50 I.I. Blekhman, L.A. Vaisberg. Adaptive Properties of Dynamic Objects 11:00 – 11:20 A. Dabrowski. Energy Flow and Synchronization in the Energy Space Break 11:20 – 11:40 I.V. Barmin, M.V. Veselov, Y.M. Epifanov. Cable Mast Dynamics of Moving with Account of Its Flexural Vibrations 11:40 – 12:00 H. Mahbadi, M.R. Eslami, M. Kamran. Analysis of Rotating Disks Minisymposium “Nonlinear Methods and Integrable Models in Mechanics” Under Cyclic Thermal and Mechanical Loads Coffee break CO-CHAIRMEN N.A. KUDRYASHOV, A.V. PORUBOV

11:00 – 11:30 M.L. Gandarias. Nonclassical Potential Symmetries for Some Diffus- Evening Session sion Equations 11:30 – 12:00 O.V. Nagornov, Yu.V. Konovalov. Modelling of Ice Flow of Glaciers CHAIRMAN S.N. GAVRILOV Subjected to Climatic Changes Solids and Structures Coffee break 12:20 – 12:50 A.V. Borisov, I.S. Mamaev. Dynamics of a Circular Cylinder Interact- 15:30 – 15:50 M.F. Ghanameh, D. Thevenet, A. Zeghloul. Effects of Joint Configu- ing with Point Vortices ration and Loading Type on Stress Concentration in Offshore Welded Tubular Joints 12:50 – 13:20 M. Hirayama, K. Ogawa. Non-linear Dynamic Behaviors of the Glider 15:50 – 16:10 S.H. Sargsyan, A.J. Farmanyan, M.N. Mutafyan, S.A. Vardanyan. Balloon; the Newly Developed Ballooning Some Problems of Statics, Thermoelasticity and Dynamics of Micropolar Bars 13:20 – 13:50 A.V. Porubov, I.V.Lavrenov. Formation of the Rogue Waves and Holes 16:10 – 16:30 A.I. Oleinikov. Anisotropic Elasto- Visco- Plasticity of Aluminum Al- in Ocean loy Coffee break Evening Session 16:50 – 17:10 E.I. Ryzhak. An Idea and Theory of Shear Device for Investigating the 15:30 – 16:00 N.A. Kudryashov. Nonlinear Heat Equations: Fifty Years Later Localization Phenomena 16:00 – 16:30 H. Tsuji, M. Oikawa. Nonlinear Internal Waves Generated by Topog- 17:10 – 17:30 A.I. Khromov, A.A. Bukhanko, S.L. Stepanov. Deformations Concen- raphy in a Stratified Fluid trators and Fracture of Plastic Bodies Coffee break 17:30 – 17:50 G. Kostin, V. Saurin. Integro-differential Approach in the Linear The- ory of Elasticity 16:50 – 17:20 E.Yu. Malyuta, M.M. Bogdan. Stability of Gap Soliton Modes in Mod- 17:50 – 18:10 M. Benabdoun, T. Dorbani, S. Hamamda, I. Zerrouk. Dilatometric ulated Nanoclusters and Calorimetric Study of an Al-Mg-Zn Alloy 17:20 – 17:40 M.A. Chmykhov, N.A. Kudryashov. Analytical and Numerical Solu- tions of Smog Propagation Problems 17:40 – 18:00 I.L. Chernyavsky, N.A. Kudryashov. System of Equations for Nonlin- ear Pressure Waves in a Viscoelastic Vessel 18:00 – 18:20 O.Yu. Efimova, N.A. Kudryashov. Two Reaction-Diffusion Models Describing the Evolution of Interrelated Populations JULY 5, TUESDAY

Room A Room B Morning Session CHAIRMAN D.A. INDEITSEV

10:00 – 12:30 ROUND TABLE.CLOSING CEREMONY List of posters July 3 Session PS1

1. K.Yu. Aristovich. Determination of elastic modulues for nanocrystal objects using molecular dynamics method 2. I.E. Berinsky. Stability analysis of graphite crystal lattice using moment interactions 3. V.A. Kuzkin. Analitical modeling of protoplanet cloud fragmentation 4. O.S. Loboda, A.M. Krivtsov. Comparison of discrete and continuum modeling for 2D nanocrystal stripe vibrations 5. P.V. Tkachev. Stability and destruction of the material with microstructure 6. S. Vasilyev, A.M. Krivtsov. Simulation of the protoplanets system growth 7. I.B. Volkovets, A.A. Le-Zakharov. Parallel implementation of Burns–Hut algorithm for simulation of planet system formation 8. E.N. Akimova, I.I. Gorbachev, V.V. Popov. Parallel matrix sweep algorithm for solving the multicomponent diffusion problems 9. A.A. Alabuzhev, D.V. Lyubimov. Resonance oscillations of a cylindrical drop 10. S. Alexandrov, E. Lyamina. A constitutive law for damage evolution near frictional interfaces at large plastic strains 11. A.I. Alexandrovich, A.A. Demidova. Solution of plane border problems of nonlinear theory of elasticity for Signorinis body 12. A.I. Alexandrovich, A.V.Gorlova. Research of the equations of the flat theory of elasticity for physically nonlinear model 13. A.I. Alexandrovich, I.A. Vorobyev, V.G. Zhovnovatyuk. The solution of some boundary-value problems for the Kortevege-de Vries equation and the sine-Gordon equation by complex analysis methods 14. A.I. Alexandrovich, V.G. Zhovnovatyuk, I.A. Vorobyev. The solution of several three-dimensional thermal- elasto-plastic problems for alloys with the shape memory by local functionals method 15. O. Alikina, E. Tarunin. Calculation of heat transfer in vortex tube 16. Yu.N. Bakhareva, Y.N. Radayev. A numerical method for solution of three-dimensional problem of the perfect plasticity 17. V.A. Arkhipov, I.M. Vasenin, V.L. Litvinov, V.F. Trofimov, U.M. Sheremetyeva. Influence of the bond number on a drop stability in a flow 18. E.B. Azarov, S.A. Rumyantsev. Mathematical model of non-stationary dynamics “the vibration machine – the electric drive” system in case of a drive from asynchronous engines with a short-circuited rotor 19. G.V. Vasilyeva, S.A. Rumyantsev. Mathematical model of dynamics two-mass vibration machine 20. A.V. Babushkin, A.A. Chekalkin. Research of stress-strain state of powdered composits at multiaxis cyclic deformation 21. G.A. Baizakova, A.K. Tomilin. Electromagnetic management of resilient members toughness 22. A.K. Tomilin. Electromagnetic method of vibrodiagnostics of composite beam internal state 23. D.V. Baldin, A.N. Zakhlevnykh. Bistable transitions in ferronematics 24. N.K. Balabaev, L.I. Manevitch, O.L. Manevitch, M.A. Mazo, G.C. Rutledge. Analysis of structure and thermomechanical properties of a nanocrystalline plate

12 25. E.K. Batsyna, V.A. Dubinina, A.A. Kurkin, O.E. Poloukhina. Nonlinear three-wave interactions of edge waves over cylindrical shelf 26. E.K. Batsyna, V.A. Dubinina, A.A. Kurkin, O.E. Poloukhina. Influence of alongshore currents on the process of edge wave focusing 27. A.A. Kurkin, O.E. Poloukhina, I.F. Nikolkina. Unsteady dynamics of edge waves above a shelf slowly varied along shore 28. A.K. Lekanov, O.E. Poloukhina, A.A. Kurkin. Anomalous continental shelf waves 29. A. Boudjada, J. Meinnel, A. Boucekkine, S. Hamamda. A combination of density functional theory calculations (DFT) and experimental studies: Molecular conformation in tri-iodomesitylene 30. A.A. Berlin, V.G. Oshmyan, M.Yu. Shamaev, S.A. Timan. Numerical modeling of the effect of nanoparticles’ bending rigidity on the mechanical properties of filled systems 31. I.I. Blekhman, G.E. Itkin. Energetic and force barriers at the transitions between the stable states of dynamic systems 32. I.I. Blekhman, V.B. Vasilkov, K.S. Yakimova. Hydrodynamic resistance of a conic shape channel undergoing vibration 33. V. Chaikin, E. Polyakova, I. Tarasenko. On the theory of constitutive equations for mediums with hereditary properties 34. E. Polyakova, V. Chaikin, O. Zuev. On the problem of driving a soft shell along a rigid body enveloped by the shell 35. E.O. Chil-Akopyan, M.V. Belubekyan. The investigation of the asymptotics for the problems of the located instability of plate with free edge 36. A.V. Chubinskiy. Wave propagation in a composite plate loaded by a viscous fluid 37. B. Cvetkov, P. Voinovich. Accuracy analysis and 2D numerical simulation of reversed wave focusing by ultrasonic wave phase conjugation 38. V.I. Dovgan, D.A. Kitaeva, Ya.I. Rudaev. About structure formation self-organization at loading of quasibrittle materials 39. O.Yu. Efimova, N.A. Kudryashov. Two Reaction-Diffusion Models Describing the Evolution of Interrelated Pop- ulations 40. V.A. Eremeyev, E.M. Kaidashev, A.V. Nasedkin. Investigations of some nano-structures made of ZnO 41. V.A. Eremeyev, A.N. Soloviev. FEM modelling of 2D ZnO nanocrystals lattice 42. E.N. Ivanov. The problems of the mathematical modelling of some tubular bones 43. D.A. Sukhov. Convective stability in horizontal layer of micropolar fluid with free surface 44. M.R. Enikeev, I.V.Penkova, O.K. Agatyan, V.A.Sergan. The biomechanics of injuries of ankle-joint ligamentous apparatus 45. D. Kubasov, V. Sikilinda, A. Polodjishnikova. Application of biomechanically proved osteosynthesis in surgery of anterior segment of foot 46. V.A. Sementzov, V.A. Sergal’, A.A. Poloishnikova, R.S. Romanov. To the biomechanics of patella fractures fixation 47. N.V. Evlampieva , A.A. Tashkinov. Study structured geometric and physical fields in problems of deformation of elastoplastic composites 48. N.Y. Evlampieva, A.B. Freidin, E.N. Vilchevskaya. Phase transformation of a cylindrical inclusion subjected to uniform external strain field

13 49. V.I. Kondaurov, A.N. Lyubicheva. Phase transitions in damaged material

50. A.L. Kazakov. Some gas ﬂows which are solutions of generalized Cauchy problem

51. N.B. Fedosenko. Construction analytical computational grids with given properties 52. N.B. Fedosenko. An approach for incompressible ﬂow calculation based on analytical solutions

53. A.V. Fedotov, S.D. Furta. On stabilization of rectilinear motion of a multilink system

54. G. Kostin, V. Saurin. Numerical analysis of linear elasticity problems based on integro-differintial approach

55. A.V. Kvanin, G.V. Kostin, V.V. Saurin. Finding optimal parameters of a beam on elastic base under various loading

56. I.V. Gnevanov. Convective motion in plane layers with internal heat sources 57. L.A. Golotina, L.L. Kozhevnikova. Modeling of the rheological behavior of ﬁlled elastomers on structural grounds

58. M. Holschneider, M.S. Diallo, M. Kulesh, K. Kurennaya. Elliptic properties of surface waves in wavelet domain

59. K. Kurennaya. Nonlinear interaction of normal elastic waves in solids

60. V.A. Il’in. Electroconvection of poorly conducting liquid in the modulated electric ﬁeld at presence of charge injection 61. A.D. Sergeyev. Buoyancy loss by a covered object due to bending of its ﬂexible shell subjected to distributed pressure

62. Y.V. Petrov, A.V. Kashtanov, G.I. Lihachev. Initial damaging in Kachanov–Rabotnov’s kinetics theory

63. A.I. Zhornik, V.A. Zhornik, A.A. Rybinskaya, P.A. Savochka. Penny-shaped crack extension in cylinders under heating

14 Session PS2

64. A.A. Ivanova, N.V. Kozlov. Inﬂuence of vibration on behaviour of light solid in rotating cavity ﬁlled with liquid

65. V.G. Kozlov, D.A. Polezhaev. Experimental study of stability of vibrational liquid ﬂow in a rotating cylinder

66. V.G. Kozlov, A.N. Sal’nikova. Behavior of two-liquid system in a rotating cavity subject to longitudinal vibrations

67. Ya. Kolpakov, P.A. Zhilin. A micro-polar theory for piezoelectric materials

68. A.A. Komarova, I.V. Shirko. The theory of limiting balance of the anisotropic loose environment

69. E.V. Shishkina, M.P. Cartmell. Vibrational stabilisation of elastic plates

70. N.V. Kurbatova. On a stretching-torsion of a naturally twisted rod 71. N.M. Kuznetsova, N.V. Kurbatova. On the FEM approach for a bending of a naturally twisted rod

72. A.V. Zaitsev, A.V. Lukin, A.A. Tashkinov, N.V. Treﬁlov. Regularities of multi-particle interactions in random structures and description of micro- and macrofailure of unidirectional ﬁbre-reinforced composites

73. I.Yu. Zoubko, P.V.Trusov. Milling balls dynamics in the mechanical alloying process and simulation of dislocation structure formation of powder particles

74. V.I. Dranovskij, V.S. Khoroshilov, A.P. Kovalenko, A.E. Zakrzhevskij. Dynamics of deployment of a multilink design in orbit

75. M.Yu. Filimonov, N.A. Vaganova. Representation of solutions of boundary-initial problems for nonlinear partial differential equations by special series 76. N.A. Vaganova, M.Yu. Filimonov. Numerical simulations of heat propagation from deepen heat sources under non-plane surface with using parallel computers

77. V.M. Fomin, E.I. Kraus, I.I. Shabalin. An equation of state for gas hydrates in high pressure region

78. Yu.G. Gordienko, E.E. Zasimchuk. Two-stage substructure transformation of Al ﬁlm as a manifestation of multi- scaling scenario

79. E.P. Kligman, T.I. Kligman, N. Sevodina. Natural vibrations of elastic shell with ﬂuid

80. E.P. Kligman, V.P. Matveyenko, N.A. Yurlova. Deﬁnition and optimization of the dissipative characteristics of electroelastic shells

81. V. Menshykov, O. Menshykov. Interfacial plane crack under time-harmonic loading

82. N. Kontchakova. The lines of deformation ﬁeld jumping in the plastic region in the Galin’s problem

83. D.P. Kouzov. The ﬂuid compressibility account of gravitation waves theory on the ﬂuid surface

84. N.A. Kovaleva, L.I. Manevitch, A.V. Savin. Nonlinear twist dynamics of DNA macromolecule

85. N.F. Kurilskaya. Vibrations of conductive string with classical and non-classical boundary conditions in magnetic ﬁeld

86. N.F. Kurilskaya. On magnetostrictive vibrations of conductive string in magnetic ﬁeld

87. N. Mestanzade, G. Yazici. Model testing of offshore structures

88. T.B. Lavrova. The properties of the plane-strain quasi-static equations for a rigid-plastic material with anisotropic hardening

89. N.F. Lebedeva-Karlsson. Analitical modeling shock waves in curved coordinates

90. V.A. Levin, N.A. Lutsenko. On inﬂuence of form of homogeneous porous fuel element on cooling process

15 91. M.S. Mkrtchyan, S.M. Mhkitaryan, S.V. Verlinsky. Stress tension of piece-wise strip with collinear system of cracks or thin insertions during antiplane and plane deformations

92. S.V. Verlinsky. Stress-deformation tension of piece-wise homogeneous elastic half-plane with system of collinear cracks or rigid insertion along with temperature impact

93. N.F. Patsegon, L.N. Popova. Autowaves in magnetic ﬂuid

94. M.N. Perelmuter. Multilevel model of strength and fracture toughness of nanocomposites

95. A.L. Levitin. The inﬂuence of notches and cuts on the specimens stability under tension

96. O.N. Lubimova. Deﬁnition of residual pressure during making three-layers of shell on base glass and metal

97. V.V. Moshev, S.E. Evlampieva. Model of a structural element in elastomeric composites ﬁlled with nanoparticles

98. N.Ye. Nikitina. Acoustoelasticity phenomenon using for in-plane stress evaluation of civil engineering structures

99. M.A. Osipenko, Yu.I. Nyashin, S.I. Brynskikh. The optimal biomechanics of the foot prosthesis with the variable thickness leaves

100. A. Pershakova, D. Luchinskii, A. Stefanovska, P.V.E. McClintock. Stochastic coupled circle maps with applications to modelling of the human cardiovascular system

101. V.E. Petrova, K.Herrmann. Thermal fracture of a bimaterial containing internal defects

102. A.N. Prokopenya. On the stability of cylindrical precession of dynamically symmetric satellite on the elliptic orbit

103. M.A. Ryakina, E.L. Tarunin. Convective ﬂow in a spherical backlash in conditions of microgravity

104. Yu.N. Sayfutdionov, S.A. Lychev. The dynamical reaction of a 3-layered viscoelastic shell

105. Yu.I. Rudnev, D.I. Borisov. Small oscillations of ideal ﬂuid in vessels with perforated bafﬂers

106. V.N. Salnikov. Nonlinear dynamics of the systems in the Nose-Hoover environment

107. D.N. Sheidakov, L.M. Zubov. Stability of a rectangular plate under biaxial tension

108. A.Sokolov. A new model of orthotropic incompressible nonlinear elastic material

109. M.I. Shmurak, V.M. Tverier, E.Y. Simanovskaya, Y.I. Nyashin. Biomechanical model of the mamma

110. M.B. Soloviev. Construction of analytical-numeric solutions of border problems of plane stationary flows of incompressible linear viscous fluid using finite analytical elements method

111. S. Soshenkov. Evolution of three-dimensional elastic bodies proﬁles in rolling contact

112. L.V. Stepanova. On creep–damage coupled boundary value problems in fracture mechanics

113. E.L. Tarunin. Asymmetrical oscillations peculiarities review

114. Yu.P. Stefanov. Numerical investigation of fracture and strain localization in geomaterials

115. A.V. Utkin, I.F. Golovnev, V.M. Fomin. Inﬂuence of the molecular characteristics of matter on detonation phenomena

116. N.P. Yaroshevich. Numerical simulation of self-synchronization phenomenon of mechanical vibro-exciters

117. E.D. Sviageninov. Resonance siren based on forward and backward running waves and its microwave analogue

118. A.L. Svistkov, L.A. Komar. Modeling of the behavior of filled elastomeric materials with regard for the influence of a filler volume fraction

119. T.P. Tovstik. On the inﬂuence of friction on the Celt rattleback motion

120. N.A. Tsikova. Transformation Matrix of vibration waves on junction of round cylindrical rods

16 121. M.M. Vetyukov, R.F. Nagaev , M.M. Utimishev. The simulation of periodic processes in vibratory jaw crusher allowing for inﬂuence of crushed material

122. I. Vulfson. Subharmonic resonance arising due to the change of effective dissipative forces with action of polyharmonic excitation

123. I.N. Wasserman. Inﬂuence of Coulomb friction on oscillations of sucker rod and tubing strings in oil wells

124. V.E. Wildemann, M.V. Polyakov, A.M. Tsiplyakov. Elements of strength analysis of deformable systems on the basis of failure kinetics description

125. A.N. Zakhlevnykh, O.R. Semyonova. Orientational and magnetic structure of a ferronematic with bistable anchoring on the boundaries of a layer

17 Defects are postulated simply in the classic crystalline theory. LOCALIZED WAVES IN SHALLOW WATER CAUSED Effects of overlapping of their corns are ignored. BY A SOLITON MOVING ALONG A BOTTOM In proposed theory may be calculated effects of short-range ANDREY K. ABRAMYAN interactions when due to modification of their corns singularities are disappear. Then defects convert to heterogeneities. Some of [email protected] them are observed as solid vortexes after passage of shock elastoplastic waves in soft metals. In the process of an earthquake the interaction of the elevation Some solutions describe wave fronts (kinks) that divide of the sea bottom with the fluid is investigated. The sea bottom is phases- regions with different crystallography symmetries. modeled as an elastic plate, subjected to compression at its edge. Method of molecular dynamics simulation confirms this result. Two problems have been solved. In the first the fluid is assumed Criteria of defects formation were formulated. The theory pre- to be noncompressible. The Bussenesqs like equations where ob- dicts in static loading an effect of raising of defect density beyond tained and solved for the case of an uneven elastic bottom.This some threshold. As a result an order of crystalline lattice may be model for the 2D case is reduced to the nonlinear equation for destroyed. an imaginary beam-strip cut out of the central part of the plate. E.L. Aero, IPME RAS, St.Petersburg, Russia The beam-strip lies on nonlinear elastic foundation and interacts with the fluid.For 3D problem the evolution of the surface waves fronts has been found. The second problem which was solved is PARALLEL MATRIX SWEEP ALGORITHM FOR the problem for a compressible fluid. For that case the assumption SOLVING THE MULTICOMPONENT DIFFUSION that the fluid velocities are such that the free surface conditions PROBLEMS may be linearized was made. Finaly,this two cases are compared with each other. ELENA N. AKIMOVA IGOR I. GORBACHEV VLADIMIR V. P OPOV A.K. Abramian, IPME RAS, V.O., Bolshoi 61, 199178, St.Petersburg, Russia [email protected], [email protected]

When modelling many processes with intensive diffusion mass NONLINEAR THEORY OF CARDINAL transfer of elements in complicated multicomponents systems it REARRANGEMENT OF SOLID STRUCTURE UNDER is necessary to know the concentration distribution of diffusing CATASTROPHIC DEFORMATIONS components at every time moment. Diffusion mass transfer in multicomponent ﬂoats is described by the system of parabolic E.L. AERO A.N. BULIGUIN differential equations. When we used the absolutely stable implicit difference scheme, this system is reduced to block- [email protected] tridiagonal system of matrix equations. One of the effective methods for solving the block-tridiagonal matrix system is the Highly nonlinear theory of elastic and inelastic microdefor- matrix sweep algorithm [1]. The classical variant of this algo- mations is developed using a model of two mutually penetrat- rithm is sequential. The parallel matrix sweep algorithm is pro- ing sublattices with strong interaction.The standard linear theory posed and proved in the paper [2]. of acoustic and optic modes of oscillations is generalized taking The purpose of this paper is the realization of the parallel ma- into account internal translational symmetry of relative shear of trix sweep algorithm on the Massively Parallel Computing Sys- the sublattices. As a result, the interaction between them is char- tem MVS-1000/16 with the MPI Fortran program library. This acterized in terms of nonlinear periodic forces described, in par- algorithm is implemented for solving the test multicomponent ticular, as a sine function of relative displacements between two diffusion problem about the plate saturation by three elements. atoms belonging to an elementary cell. The analysis of the speed up and efﬁciency of the parallel algo- The nonlinear equations(acoustic and optic ones) are con- rithm with different numbers of processors and the comparison of nected mutually due to modes interactions. Thus macroscopic the executing times is carried out. As a result we recommend this deformations and microscopic structure are linked. Thus the the- algorithm for solving the complex diffusion problems in multi- ory are valid even when relative atomic displacements exceed a components systems. lattice period. The work was supported by the RFBR (project No. 03-01- Then we can reach and overcome bifurcation points, i.e. 00099) and by the Foundation OAO MMK¬ and¬ FNO Intels¬ (project¬ thresholds of lattice stability. The universal mechanism of these No. 11-03-02). effects conclude in lowering potential barriers due to great non- REFERENCES linear displacements of atoms. [1] A.A. Samarskii, E.S. Nikolaev. The methods for solving the In the frame of this theory it is possible to calculate an incre- grid equations. Ð M.:Nauka, 1978. 593 p. ment of activation energy of structure rearrangements as a func- [2] E.N. Akimova. Parallelization of the matrix sweep algo- tion of atomic parameters and external loads. rithm. // Mathematical modelling. 1994. T. 6, N. 9, Pp. 61- Some exact solutions of nonlinear equations of acoustic and 67. optic modes describe cardinal rearrangements of a structure of initially ideal crystalline lattice, indeed. They predict catastrophic E.N. Akimova, Institute of Math. and Mech., UrB RAS, Ekaterin- deformations, phase transitions, lattice fragmentation and defect burg, Russia formation such as dislocations and disclinations of various power, V.V. Popov, Institute of Metal Physics, UrB RAS, Ekaterinburg, their rows, sliding lines, small angel boundaries etc. Russia

18 [2] Functional strain as a determinant for bone remodeling / L. QUASI-STATIC ANALYSIS OF THE MUSCULAR E. Lanyon // Calcif. Tissue Int. 1984. 36. P. 56-61. EFFORTS AND JOINT REACTION FORCES DURING THE LOWER EXTREMITY MOVEMENT IN THE [3] Kolesnikov, G.N. Diskretnye modeli mehanicheskih i TRAINING MACHINE WITH WEIGHTS biomehanicheskih sistem s odnostoronnimi sviaziami: monograﬁya / G.N. Kolesnikov. — Petrozavodsk: PetrGU, YU.V. AKULICH R.M. PODGAYETS 2004. (in Russian). A.I. KHOLKINA Yuri V. Akulich, Perm State Technical University, 29a, Komsomol- [email protected] sky Prospect, Perm, 614990, Russia

To study the processes of adaptive recovering of bone tissue of the lower extremity that proceed actively after either surgi- RESONANCE OSCILLATIONS OF A CYLINDRICAL cal intervention or trauma, it is necessary to know actual stresses DROP and strains within the bone, and consequently the loads acting on the bone. The aim of the present study is to determine muscu- A.A. ALABUZHEV D.V. LYUBIMOV lar efforts and reaction forces in the hip and knee joints, when [email protected] the patient lies on his back in the training device with additional weights, and his lower extremity is in horizontal position. The The resonance oscillations of a oscillating cylindrical drop are choice of the horizontal position of the extremity is attributed to investigated in the present work. The drop is suspended in the the fact that the total muscular loads on the femur during the ex- different fluid and confined by two parallel rigid plates, subjected tremity flexion in the training machine are the largest just in that to vibrations. The vibration axis is perpendicular to the axis of position [1]. The large strains in the bone increase the adaptive the cylinder. The contact line is free (contact angle is fixed and activity of the bone cells leading to the bone tissue remodeling is right). The both amplitude of vibrations and the thickness of [2]. The analysis of lower extremity dynamics in the given train- viscous boundary layers are small in comparison with the drop ing device [1] shows that the dynamic components of the loads radius and thickness of the layer. The boundary layer was on in- are small during all the period of the recovering motion. It al- terface drop - surrounding fluid. The viscosity is important in thin lows to a first approximation determine the muscular efforts from boundary layers only. Therefore our general problem consists of the solution of the appropriate static problem. This problem is two sub-problems: (1) inviscid problem for drop core, (2) viscous statically indeterminate, and for its solution we used the well problem for boundary layers. known in the structural mechanics method of displacements that The frequencies of natural oscillations of the drop were found. has been extended by G.N. Kolesnikov to include the biomechan- The complex shift of eigen frequency due to the small viscosity ical problems [3]. was obtained. The real part of the frequency shift is correction The lower extremity is presented as a system of three solids: to the eigen frequencies in comparison with eigen frequencies in femur, shank with a foot, and patella. Elastic rectilinear filaments inviscid case. The imaginary part of the eigen frequencies shift is present the muscles in our model, the hip and knee joints are the damping coefficient. hinges, and the articular cartilage is an elastic spherical envelope. The forced oscillations of the drop were studied for both sub- The rigidities of muscles and cartilage are assumed to be known. problems in the main order of expansion with respect to small We have considered all the 14 muscles that participate in the acts parameter - the ratio of the vibration amplitude to the drop radius. of flexion and extension in the hip and knee joints. The accu- The parametric instability of forced oscillation was investi- racy of the obtained muscular forces have been estimated by the gated. It was shown, that small viscosity leads to the appearance comparison of the total moments and reaction forces in the hip of the resonance amplitude threshold and the extension of the in- and knee joints with the corresponding values derived from the stability domain. solution of direct dynamic problem [1]. This work was made with financial support from grant CRDF The obtained muscular efforts satisfy the Lagrange equations (PE-009-0) and grant NSh-1981.2003.1. of motion within the accuracy of dynamic component, and con- Aleksey A. Alabuzhev, Institut of continuous media mechanics UB sequently they are justified. It follows from the consideration of RAS, Ak. Koroleva, 13, Perm, 614013, Russia the obtained results that the quasi-static analysis of the redundant system with unilateral constraints allows to ascertain all the active muscles of the extremity that are involved in the given act of A CONSTITUTIVE LAW FOR DAMAGE EVOLUTION movements. It should be mentioned that in the solution of the di- NEAR FRICTIONAL INTERFACES AT LARGE PLASTIC rect problem of dynamics by Akulich at al. [1] the active muscles STRAINS were chosen subjectively by the common anatomic data on the activity of the muscles of extremity, and such an approach may S. ALEXANDROV E. LYAMINA lead to qualitatively inaccurate results. REFERENCES [1] Vliyanie vida vneshney nagruzki na usiliya myshts bedra i Assuming a rigid plastic material model with conventional davlenie v bedrennom sustave v techenie dvizheniya nizh- damage evolution equations, it is shown that at the vicinity of ney konechnosti cheloveka v polozhenii lezha na spine / maximum friction surfaces the model predicts fracture occur- Yu.V. Akulich, E.V. Denisova, A.I. Kholkina // Russian rence at any however small value of strains independently of other biomechanics journal. — 2004. — Vol. 8, N. 1. — P. 68Ð75 conditions of the process of deformation. The situation is very (in Russian). similar to that in the classical mechanics of cracks where fracture

19 criteria of the strength of materials predict crack propagation at imated solutions of the equations of the theory of elasticity for any however small magnitude of the external load. In the present physically nonlinear model of an elastic body at small deforma- work, some general ideas of the mechanics of cracks are applied tions. The offered method of construction of the approximated to quite different problems: damage evolution and fracture in the solutions is based on complexification of these equations. Re- vicinity of frictional interfaces at large plastic strains. To this end, quired complex functions represent as sums on degrees of in a the linear elastic material is replaced with the rigid perfectly plas- complex connected variable with coefficients as the some holo- tic material, the concept of stress intensity factor with the concept morphic functions. of strain rate intensity factor, the sharp crack tip with the surface Substitution of the given representations leads to to system of of maximum friction, and crack propagation with damage evolu- the recurring differential equations concerning required holomor- tion. As a result, a new constitutive law for damage evolution is phic functions. Research of the considered equations leads to to obtained. The law can be combined with the rigid plastic material construction of the iterative process, allowing to construct a re- model without any contradiction. An example of applications of quired field of movings depending on two any way set holomor- the new law is given. phic functions. Sergei E. Alexandrov, Institute for Problems in Mechanics of Rus- For construction of the approached decision of a regional prob- sian Academy of Sciences, 101-1 Prospect Vernadskogo, Moscow, lem the method of minimization of root-mean-square evasion 119526, Russia from the set boundary conditions is offered, allowing to restore two generating holomorphic functions. The given method is SOLUTION OF PLANE BORDER PROBLEMS OF tested on problems with set on border of considered area a vec- NONLINEAR THEORY OF ELASTICITY FOR tor of speed and a vector of a pressure. Results of numerical SIGNORINIS BODY experiment on construction of the approached decisions of the considered equations are resulted. ALEXANDER I. ALEXANDROVICH A.V. Gorlova, Dorodnycyn Calculation Center, Vavilova street 40, ANASTASIA A. DEMIDOVA Moscou, 117967, Russia [email protected], [email protected] THE SOLUTION OF SOME BOUNDARY-VALUE A new method of solution of basic classical border problems PROBLEMS FOR THE KORTEVEGE-DE VRIES of nonlinear theory of elasticity for Signorinis elastic and hyper EQUATION AND THE SINE-GORDON EQUATION BY elastic body for plane state of strain is considered in this work. COMPLEX ANALYSIS METHODS For target setting Almansies tensor of finite strains and Cauchys tensor of true stresses are used. Complexification of equations of ALEXANDER I. ALEXANDROVICH IVA N A. VOROBYEV balance is carried out. With use special decomposition of com- VLADIMIR G. ZHOVNOVATYUK plexified fields of movements by systems of holomorphic and an- tiholomorphic functions initial equations are solved in form of [email protected] recurrence relations, which show that definition of two holomorphic functions gives possibility to obtain all remaining holomor- In the report two equations playing the central role in the the- phic functions, and so, obtain approximate field of movements. ory of solitons, the sine-Gordon equation and the Kortevege-de Two of the holomorphic functions, determining the solution Vries equation are considered. They have the wave solutions in- of the task, are being searched realizing an iterative method. It vestigated by various methods. The Kortevege-de Vries equation is based on the minimization of a root-mean-square deviation of is used in the theory of waves on fine water, the theory of lattices, solution from the boundary solutions are given. physics of plasma and magnetohydrodynamics. Appendices of Signorinis model describing plane mode of deformation of sine-Gordon equation cover various areas, such as location in elastic body with finite strains leads to nonlinear equations of crystals, superconducting contacts of Josephson, waves of charg- fourth order concerning movements and their gradients. ing density in one-dimensional organic conductors and models of Results of numeric experiments of construction of sought field the theory of a field. In the report the new method of obtaining the of movements and field of stresses for simple border problems are analytic-numerical approximated solutions of some boundary- shown in this work. Differences of obtained mechanical fields for value problems for nonlinear equations in the partial derivatives elastic and hyper elastic models of bodies with same boundary such as sine-Gordone equation and Kortevege-de Vries equation conditions are analyzed. is presented. The solutions construct in the class of the func- A.A. Demidova, Dorodnicyn Calculation Centre, Vavilova street tions, which have the special presentation through the golomorfic 40, Moscow, 117967, Russia functions of complex variable. The offered method is based on recording the initial equations in the complex form. On the first stage required complex-valued functions are represented as se- RESEARCH OF THE EQUATIONS OF THE FLAT ries on system of power antigolomorfic functions with factors as THEORY OF ELASTICITY FOR PHYSICALLY some golomorfic functions in the limited areas. Substitution of NONLINEAR MODEL the specified representations in the investigated equations leads to system of the hooked differential equations in golomorfic func- ALEXANDER I. ALEXANDROVICH tions. These functions are included into representations required ANASTASIYA V. G ORLOVA complex-valued functions and their complex conjugation. The [email protected] system of the equations received this way is transformed into the form of recurrence relations, allowing to allocate some ini- Let‘s consider the new method of construction of the approx- tial independent golomorfic functions that define the required so-

20 lution. On the second stage for obtaining the approximated so- A.I. Alexandrovich, Vavilova st., 40, room 220, Moscow, 117967, lution of boundary-value problem for the examined equations in Russia the simplest simply connected region is proposed the method of minimization of root-mean-square deviation from the assigned boundary conditions, which allows to select two (the sine-Gordon CALCULATION OF HEAT TRANSFER IN VORTEX equation) or three (the Kortevege-de Vries equation) initial holo- TUBE morphic functions. Some numerical experiments of the analytic- OLGA ALIKINA EUGENY TARUNIN numerical approximated solutions of some boundary-value problems are given in the report. [email protected] I.A. Vorobyev, CC RAS, Vavilova st., 40, room 220, Moscow, 117967, Russia Vortex effect or Ranque’s [1, 2] effect is very well-known phenomenon. It consists in the separation of a rotated gas flux into two parts. One of the fluxes has a temperature that is higher THE SOLUTION OF SEVERAL THREE-DIMENSIONAL than the entering flux temperature, the other has a lower one. Al- THERMAL-ELASTO-PLASTIC PROBLEMS FOR most all research done in the area is experimental [5, 10]. At ALLOYS WITH THE SHAPE MEMORY BY LOCAL the moment we know several numerical works, devoted to the FUNCTIONALS METHOD effect, but amount of dependencies obtained in [12, 13] is not large. The researchers in laboratories tune the tube using empir- ALEXANDER I. ALEXANDROVICH ical data and intuition [1, 2, 4, 5, 8, 10]. By means of mathe- VLADIMIR G. ZHOVNOVATYUK IVA N A. VOROBYEV matical calculation the effect was calculated in [11]. In that work [email protected] compressibility of the media was taken into account only in the equation of heat transfer. This article is continuation of compu- Materials with the shape memory effect drew in the recent tational experiments of [11]. Solutions were obtained from full decades the considerable attention of researchers in connection Navies-Stocks’ equations with viscosity and temperature conduc- with the opened prospect for their practical use in many regions tivity dependence on temperature taken into account. Calcula- of science, technology, medicine and industry. In the long pe- tions were carried out in natural variables on a non-uniform mesh riod this materials were studied by material engineers and met- until stationary solutions were obtained. Dependencies of the allographers. As a result the alloys with the shape memory ef- temperature separation effect on different parameters of the vor- fect, which had become classical: CuAlNi, CuZnAl, AuCd, NiTi tex tube were obtained and, thus, any doubts about the possibility and other, were determined. In connection with the unique defor- to describe the effect by gas dynamics were removed. mation effects, they became to attract attention of physicists and REFERENCES mechanics. For calculating the elements of constructions from [1] Ranque G.L. J. Phys. Radium. Paris,1931,V.4. 112Ð115. such materials it is proposed to use the method of the approx- [2] Hilcsh R. Die Expansion von Gasen in Zentrifugalfeldes als imated numerical solution of three-dimensional thermal-elasto- Kalte prozes Z. Natroforsch.1946. 1. 208Ð214. plastic problems, based on the modification of the method of finite elements in the form of local functionals. Calculations are [3] Landau L.D., Lifshits E.M. Theoretical physics V.6. Hydro- carried out for the preset process of changing the temperature dynamics. Moscow, Nauka, 1986. field with the complex process of mechanical loading. In the [4] Merculov A.P. Vortex effect and its application in technics view of the complexity of the process of the deformations which Moscow, Engineering, 1969. appear in the particles of material, as the basis the concept of [5] Gutsol A.F. Vortex effect V.167. N 6. J. Physical successes, the theory of the elasto-plastic processes of A.A.Ilyushin and in 1997. 665Ð687. particular the model of the translationally isotropically strengthened medium is accepted. This model is modified for using fi- [6] Tarunin E.L. Computational experiments in problems of free nite deformations of Almancy. This approach corresponds to the convection Irkutsk University Publishers, 1990. 226. Setkhs model in the theory of elasticity, proposed in the 30’s [7] Rachmatulin H.A. Gas dynamics Moscow High School which, as indicates A.I.Lur’e in its book ”The nonlinear theory Publishers, 1965. 723 p. of elasticity”, sufficiently well described some special features [8] Arbusov V.A., et al. Observation of large-scale hydrody- of large deformations. A postulate of plastic flow in the form namics structures in vortex tube and Ranque’s effect Letters of A.A.Ilyushin instead of the analogous postulate of Drucker is to J. of Thermal Physics. V.23. No 23. 1997. 84Ð90. used. It allows to inscribe naturally the relationships between the stresses, the processes of deformations and temperature for the [9] Rouch P. Computational hydrodynamics Moscow Mir Pub- examined model of body into the computational process of the lishers, 1980. local functionals. In the report this process is constructed for the [10] Piralishvili Sh.A. et al. Vortex effect. Experiments, theory three-dimensional octagonal elements. On the basis of this con- and technical solutions Moscow, 2000. 412. cept some of numerical experiments for the simple forms of the elements of constructions with the assigning process of chang- [11] Lubimov D.V. et al. The theroretical model of Ranque’s ef- ing of the temperature field and with the complex processes of fect Perm University Publishers // Scintific J. ”Mathemat- the mechanical loading of these elements are carried out. In the ica”, 1994. pp. 162-177. report some results of numerical experiments are represented. [12] Bezprozvannykh V., et al. The RanqueÐHilsch effect. CFD The work was performed with a partial support of the Russian modelling Proceedings of International Conference ”Ad- Foundation of Basic Research (grant N 04-01-00473) vanced problems in Thermal Convection” Perm, 2003.

21 [13] Skovorodko P.A. Simulation of the flow in Ranque-Hilsch The attempts to use maximal tensile stresse defined from the tube Nauchnye ItogiÐ98, 1999. Novosibirsk. 11-12. tests on three-point bending as a strength limit show underesti- mated values of limit load compared to that of experiment. It’s O. Alikina, Perm State University, Bukireva, 15, Perm, 614099, necessary to define ”strength material constant” of the material Russia (tensile strength limit in warp direction) out of the eccentrical compression test in the embedment for one type and size of hip prosthesis specimens. PROBLEMS OF STRENGTH EVALUATION OF The work of PSTU authors is supported by Russian Fund of CARBON-CARBON HIP PROSTHESES Basic Rerearch grant 03-01-00394. Sergey G. Ivanov, Perm State Technical University, Koroleva, A. ANOSHKIN D. IVANOV S. IVANOV Perm, 614013, Russia YU.OSORGIN A. TASHKINOV P. U DINTSEV YU.ZMEEV [email protected] DETERMINATION OF ELASTIC MODULUES FOR NANOCRYSTAL OBJECTS USING MOLECULAR Carbon-carbon composite materials are characterized by bio- DYNAMICS METHOD compatibility with human tissues. Hip joint carbon-carbon pros- KIRILL YU.ARISTOVICH theses from CarBULAT have been developed in Institute of Ther- mochemistry, Perm. The composite material is produced on the [email protected] base of 4-layered fabric by chemical vapour deposition of pyro- carbon matrix on fibrous framework. The presented work is based on the determination of the first The promotion of hip joint prostheses to European market de- natural frequency for longitudinal oscillations of thin nanorod. mands the realization of tests in certified laboratories, including Using the molecular dynamic method the gain-frequency char- eccentrical compression in embedment static failure tests. High acteristics of 2D and 3D nanorods with different thicknesses is manufacturing cost of carbon-carbon materials makes worth- computed. Then the first natural frequency (FNF)is calculated. while the preliminary mathematical modelling of stress strain Young’s modulus is computed from the FNF by simple equation: = ( ωl )2 state and limit load prediction of hip prostheses. E ρ 2π ; where ρ is density, l- length, ω - FNF . The results In ANSYS 5.5 software the calculations of hip prosthesis are: stress strain state with regard for material anisotropy have been - FNF changes according the thickness of the rod (it depends made. The influence of the loading and fastening conditions on on the number of molecular layers). Hence, the Young’s modulus the stress strain state and static strength of prostheses has been is not constant and depends on the size of nanostructure. studied. Numerical solution of homogeneous, locally orthotropic - The obtained Young’s modulus is not equal to the modulus, elasticity boundary value problem with ideal contact conditions calculated by continual theory of elasticity, for the small number shows significant concentration of stresses in local zones and of molecular layers (2-7). When the number of molecular lay- seems to give stress singularity in vicinity of embedment and ers is more than 10 then Young’s modulus tends to its continual neck region. Along with diminishing of finite elements the max- value. Then we compute the Young’s modulus for another dif- imum values of axial stresses in the direction of fabric warp in- ferent nanocrystal objects (nanotube, nanocube) and compare it crease, however the character of axial stress axial coordinate de- with the results, calculated before. pendence doesn’t change. Other stress tensor components appear To calculate the Poisson’s ratio the gain-frequency character- to be essentially less in value. The results of various experiments istics for lateral oscillations is used. All results of this work are confirm the presence of the most probable failure in two zones of compared with the works [1], where the Young’s modulus and stress concentration: in the region of prosthesis steel head and in Poisson’s ratio was computed using the static method. the region of embedment prosthesis stem contact. REFERENCES Non-linear contact conditions with friction are more realistic [1] Krivtsov A.M., Morozov N.F.On Mechanical Characteristics but they demand special experiments to define parameters of fric- of Nanocrystals. // Physics of the Solid State. 2002. 44(12). tion contact. In order to use FE solution of elasticity problem 2260-2265. with ideal contact conditions one need a non-local strength criterion to predict failure of structure. Experimental and theoretical K.Yu. Aristovich, St. Petersburg State Polytechnical University, development of new limit stress state criteria of woven carbon- IPME RAS, St. Petersburg, Russia carbon composites is desirable for meaningful refined strength estimates. However there exists simple alternative in considered case: the application of simplified technique based on strength of INFLUENCE OF THE BOND NUMBER ON A DROP materials methods. With the help of this technique the evalua- STABILITY IN A FLOW tion of failure load under the test of different types and sizes of VLADIMIR A. ARKHIPOV IGOR M. VASENIN prostheses has been made. VALENTIN L. LITVINOV VYACHESLAV F. TROFIMOV Under the tests of some samples the failure takes place in the ULYANA M. SHEREMETYEVA area of sphere head prosthesis neck contact with the circumferential direction crack on the stretched side of specimen. Prosthesis [email protected] neck is grinded to coincide with the tapered opening of the head. Calculations have shown that the strength of the neck material af- Processes of drops deformation and crushing in a gas or liq- ter grinding is essentially less than that of the untreated material. uid flow are the classical problems of hydrodynamics which has

22 attracted the attention of researches for many years, and also are limiting processes are considered. In the ductile region it coin- of interest for the solution of the applied problems in the theory cides with the Coffin empirical frequency dependence in suppo- of two-phase flows. In the majority theoretical and experimental sition on the material incompressibility. Another limiting case researches key parameters determining deformation and stability (for compressible materials) corresponds to the criterion of brit- of a drop in a flow, considers the Weber number. Under cer- tle failure. Theoretical fatigue curves with two limiting strengths tain conditions the drop may crush in an accelerated flow due to are plotted. According to the experiments, the fatigue curves shift Rayleigh-Taylor instability which comes at achievement of criti- into the domain of great longevities with the increase of loading cal value of the Bond number. frequency. In the present report results of the theoretical analysis and an REFERENCES experimental research of drops stability in a viscous liquid flow [1] 1. Sakai T., Takeda M., Shiozawa K., Ohi Y., Nakajiama M., are submitted at values of Reynolds number Re < 120. The given Nakamura T., Oguma N. Experimental evidence of duplex range is of interest, in particular, for the modeling of two-phase characteristics in wide life region for high strength steels // flows in the nozzles of solid propellant rockets. Proc. 7th Intern. Fatigue Congress “Fatigue 99”, 8Ð12 June From Adamar’s solution for a drop falling in a viscous liquid 1999, Beijing, P.R. China. Beijing: Higher Education Press. by gravity at small value of Reynolds number, follows, that the 1999. V.1. P. 573-578. difference of normal pressure on all surface of a spherical drop [2] Bathias C., Baudry G. Application de la resistance a la fa- is constant and to not aspire to a drop deform. Proceeding from tigue gigacyclique au calcul de structure // Bulletin S.F.M. this, Batchelor judges independence of the drop form (at small Revue Francaise de Mecanique. 2000. N 2000 1. P. 13-18. values of Reynolds number) from values of the Weber and Bond numbers [1]. [3] The fatigue tests in high frequency loading / V.A. Kuz- With the purpose of check of this conclusion the linear analysis menko, L.E. Machnotyuk, G.S. Pisarenko et al. Kiev: of problem about a finding of critical parameters set is carried out Naukova dumka. 1981. 241p. (in Russian). in view of influence of internal flows on the drop form. The drop [4] Arutyunyan R.A. On the frequency dependence of the fa- form was approximated by a spheroid; the acceleration which is tigue fracture criterion // Problemy prochnosty. 1985. 12. included in the Bond number; was obtained by solving the equa- P.63-65. (in Russian). tion of gravitational deposition of the drop. It is shown, that for R.A. Arutyunyan, St.-Petersburg State University, Institute for any disturbances probably occurrence of drop instability at values Mathematics and Mechanics, Saint Petersburg, Petrodvoretz, of the Bond number Bo ≈ 20. Universytetskii prospekt, 28, 198504, Russia The experimental research was carried out by filming process of gravitational sedimentation of mercury drops in a water- glycerin solutions and emersion and sedimentation of olive oil EXPERIMENTAL INVESTIGATION OF GRANITE AND drops in hydro-alcohol solutions. Dependences of a drop de- LIMESTONE FRACTURE UNDER IMPACT PULSE AND formation degree on Weber number are received and occurrence STATIC LOADING of drop instability at values of the Bond number Bo > 10 was founded. S.A. ATROSHENKO S.I. KRIVOSHEEV Results of researches are analyzed together with the published Y.V. P ETROV A.A. UTKIN G.D. FEDOROVSKY data on a considered problem. REFERENCES [1] Batchelor J. Introduction to the hydrodynamics (in Rus- The work is devoted to the comparative investigation of dy- sian). Moscow: Mir.1973.- 760 p. namic and static fracture of two rocks with different structure. For study of dynamic crack formation, shock loading in microsecond V.A. Arkhipov, Research Institute of Applied Mathematics and range using magnetic pulse installation was carried out. For this Mechanics, ave. Lenin, 36, Tomsk, Russia testing specimens in the form of plate with the middle through I.M. Vasenin, V.L. Litvinov, V.F. Trofimov, U.M. Sheremetyeva, notch was used. The given load was applied to the faces of notch. Tomsk State University, ave. Lenin, 36, Tomsk, Russia The length of growing crack was established in testing. Spall fracture was investigated under shock on the end of rod specimens with determination of threshold fracture load. Propagation FATIGUE FRACTURE IN THE GIGACYCLE REGIME velocities of longitudinal and transverse waves in the mediums were measured using laser interferometer. Initial density of mate- ROBERT A. ARUTYUNYAN rials is measured. Static testing have been done for identification [email protected] of the main characteristics of dynamic crack resistance and static characteristics of strength and crack resistance of rocks. Static Some recent fatigue tests [1, 2] indicate that the fatigue curves testing was realized using rupture machine under deformation ve- have two fatigue limits (horizontal asymptotes). A review of the locity 10 mm/min of rod and beam specimens without notch, and available experimental data [3, 4] also indicates the strong fre- also long plates and beam with notch, under uniaxial tension and quency dependence of the fatigue fracture process. These two three points bending. During this static testing fracture forces effects are the main topics of our presentation. A mechanical were established. Investigation of surface and profile fracture was model describing the frequency dependent damage evolution in carried out using microscopy. On data of specimens and static metallic materials is developed and the fatigue fracture criterion testing, static strength and stress intensity coefficient character- is formulated. It is shown that this criterion is capable to describe izing crack resistance were determined. The main parameters the above mentioned fatigue behavior of metallic materials. Two for dynamic range were determined using numerical solution of

23 corresponding edge problems of mechanics for plate with notch granules porous macroisotropic composite on a basis of techni- and for rod under threshold load values and the application of in- cally pure iron PGR 3.200.28. For experiments axisymmetric cubation time criterion for evaluation of dynamic strength. The samples with central fluting plane were in-process used. Samples main parameters of this criterion are incubation time and afore- had the following sizes: length of samples L = 125 mm, diameter mentioned static parameters. The parameters of static strength of samplesD=12mm,radius generatrix fluting planes R = 50 and crack resistance of granite as a whole is essentially higher mm, diameter of a neck d = 8 mm. A waste of samples have been of limestone parameters and under bending they are essentially treated for strengthening in captures of the test machine. Trials higher than under tension. Incubation time is noticeable lower are lead on the fatigue machine - bellcrank. The machine ensures than under spallation in rods. a back - rotary motion of a spindle of mobile capture. The am- This work is a part of a research project supported by RFBR plitude of twirl of a spindle is rigidly set with the help eccentric grant 02-01-0135, RFBR- SFNS grant 03-0139010 and ministry node. Special accommodating was applied to trials. Accommo- of education E 02-4.0-90. dating looks like the cut circular table. Samples can be fixed in Svetlana A. Atroshenko, IPME RAS, V.O. Bolshoy, 61, St. Peters- accommodating under any angle to a spinning ax of a spindle. burg, 199178, Russia Thus in extreme positions the sample is loaded either only with curving, or only a torsion. And in intermediate positions - curving and a torsion jointly. Depending on an angle of an arrangement MATHEMATICAL MODEL OF NON-STATIONARY of a sample concerning a spinning ax of a spindle the share of a DYNAMICS “THE VIBRATION MACHINE – THE bending strain and a torsional strain proportionally varies. Test ELECTRIC DRIVE” SYSTEM IN CASE OF A DRIVE datas of powdered iron are in-process reduced at magnitude of FROM ASYNCHRONOUS ENGINES WITH A an angle of an arrangement of samples concerning an axis of a 90◦ SHORT-CIRCUITED ROTOR spindle of active capture in 0, 15, 30, 45 and . Mathematical and the numerical modeling of deformation and EUGENY B. AZAROV SERGEY A. RUMYANTSEV destruction of powdered composites at a cyclic loading are carried out at the following basic conjectures. The microstructure of [email protected], [email protected] a powdered material is formed by countable number of components. There is a representative volume of a heterogeneous mate- In the message the mathematical model of the “vibrational rial which defines mechanical behaviour of a powdered compos- transport machine with self-synchronized vibration exciters Ð ite. Mechanical properties in limits of structural components are asynchronous engines with a short-circuited rotor” system, al- homogeneous and phenomenologically definable. Components lowing to describe transient dynamic processes, which are ac- of structure in representative volume are distributed statistically company the start of machine from the rest and impact loads is and completely fill in him. The shape and a positional relation- represented. ship of components of structure of a material is defined statistical The differential equations of movement of “the vibration ma- moments by functions of the first and second order. It is sup- chine Ð the electric drive” system in case of a drive from asyn- posed, that for representative volume of a powdered composite chronous engines with a short-circuited rotor with random quan- and all components of structure in everyone micro- and macro- tity of pairs of poles (including different quantity in various pairs volume the quasistatic equations of mechanics of a deformable of engines on one machine) are written. solid are executed. This case allows to model transient dynamic processes both in Thus, comparison of results of physical and mathematical ex- the machine, and in the engine, taking into account their mutual periments is in-process carried out. Also, according to a princi- action against each other. Results of numerical modelling of tran- ple of superposition, the multiaxis loading has been divided into sient and steady-state dynamics are represented. The information components of curving and a torsion, and calculation of stress- is submitted in graphic form which allows to analyze transient strain state of separate components of a multiaxis loading is car- dynamic processes both in the machine, and in the engine. ried out. It has enabled to compare with results of the physical E.B. Azarov, Ural State University of Railway Transport, Kol- and calculated experiments which have been carried out earlier mogorov st., 66, Ekaterinburg, 620034, Russia at “pure” curving and a torsion of samples of powdered iron, to results of the given experiment. Operation is executed at a financial support of the Russian fund RESEARCH OF STRESS-STRAIN STATE OF of basic researches (the grant of RFBR “Ural - 2004” N 04-01- POWDERED COMPOSITS AT MULTIAXIS CYCLIC 96052). DEFORMATION Andrey V. Babushkin, Perm State Technical University, Komso- molskiy pr.,29a, Perm, 614600, Russia A.V. BABUSHKIN A.A. CHEKALKIN [email protected], [email protected] ELECTROMAGNETIC MANAGEMENT OF RESILIENT In the majority of constructions real machine components MEMBERS TOUGHNESS work in conditions of the complex stress-strain state. Generally G.A. BAIZAKOVA A.K. TOMILIN the loading occurs cyclically. However simulation and an experimental research of similar processes for composites is hampered. [email protected] In the given operation results of simulation and an experimental research of multiaxis fatigue of samples powdered structurally The first works [1,2] are regarded to the problems of electrical a heterogeneous material presented. The material represents the hanger oscillations in magnetic field involving reactive electrical

24 operation factors (condenser, coil) in external make circuit. It is example, the problem of localization of plastic strains within a shown that the presence of electro-capacity leads to increasing neck in a cylinder specimen is considered. Limiting stretching of accelerative properties, while inductance changes resiliency of force is estimated. The results are compared with the values ob- the system. Thus there is a possibility to influence level of nat- tained earlier by P.W. Bridgman (see [2], [3]) and are in a good ural frequency of system oscillations due to reactive operation agreements with them. factors. Obviously this effect takes place in more complicated REFERENCES electromechanical systems. We consider the issue concerning [1] Radayev Y.N. On the Poincare canonical transformations natural vibrations of coil spring made of nonmagnetic substance and plastic equilibrium equations invariants// Izv. AN in external stationary radial magnetic field. In the first case only SSSR. Mech. Solids. No. 1. 1990. P. 86-94. (in Russian) pure resistance of the spring is considered. Electromagnetic ef- [2] Hill R. The mathematical theory of plasticity. Oxford. fect here produces dissipative force. In the second case electric Claredon Press. 1950. circuit contains condenser and it leads to quasi-inertial coefficients of differential motion equations. In the third case increase [3] Radayev Y.N., Bakhareva Y.N. A numerical method for so- of quasi-elastic coefficients is due to the inductance. Partial os- lution of three-dimensional problem of the theory of plas- cillations frequency and longitudinal toughness of the spring are ticity// Vestnik Samara State University. The second special calculated in all three cases. Electromagnetic effect is character- issue. 2004. (in Russian) ized by nondimensional parameter - Stewart number. Numerical Yuliya N. Bakhareva, Samara State University, Acad. Pavlov computations of partial frequency and longitudinal toughness re- street, Samara, 443011, Russia lation were carried out due to Stewart number. REFERENCES [1] Tomilin A.K., Baizakova G.A. About dynamic property of ANALYSIS OF STRUCTURE AND electromechanical vibratory system // International scien- THERMOMECHANICAL PROPERTIES OF A tific practical conference “Tretyi Okunyovskie chteniya”. NANOCRYSTALLINE PLATE Report materials in 2 volumes. V.-2 Theoretical and applied mechanics.- St. Petersburg . BSTU, 2002. 157-158 p. N.K. BALABAEV L.I. MANEVITCH O.L. MANEVITCH M.A. MAZO G.C. RUTLEDGE [2] Tomilin A.K., Baizakova G.A. Parametric oscillation of systems in nonstationary magnetic field involving external [email protected], [email protected], circuit of electro-capacity// Vestnick EKSTU 2, 2003. Ust- [email protected], Kamenogorsk.- P. 39 44 [email protected], [email protected] [3] Hvingiya M.V. “Springs vibration” Mechanical Engineer- ing, Moscow, 1969. We present the results of molecular dynamics simulation for single and coupled nanoplates of pyro-phyllite 2:1 dioctahedral Alexander K. Tomilin, Eastern Kazakhstan State Technical Uni- clay mineral consisting of two tetrahedral sheets SiO4 and the in- versity named after D. Serikbaev, 12 Burov Street - ap. 45, Ust- tervening octahedral sheet AlO6. Because of phyllosilicates ex- Kamenogorsk, Kazakhstan, 492000 tensively used as constituents of polymer-clay nanocomposites, estimation of structural characteristics and thermo-mechanical properties of nano-plates is one of key problems in this field. A NUMERICAL METHOD FOR SOLUTION OF Considered model consists of one crystallographic cell in Z- THREE-DIMENSIONAL PROBLEM OF THE PERFECT direction under periodic boundary conditions in the XY-plane. PLASTICITY Corresponding chemical formula is [Al2Si4O10(OH)2]2. There were 600 atoms in calculated cell. Periodic boundary condi- YU.N. BAKHAREVA Y.N. RADAYEV tions in Z-direction was introduced for simulating of interaction [email protected], nanoplate. In this case there were 2400 atoms in calculated cell. [email protected] Calculations were performed in temperature diapason 100-500 K. We have performed a detail structural analysis of pyrofyllite A numerical method for three-dimensional analysis of equa- crystal using different potentials of interaction and provided a tions of the theory of perfect plasticity in the case when a stress good correspondence with X-ray data. It was shown that coupling state corresponds to an edge of the Tresca prism is proposed. The of nanoplates does not lead to noticeable deviations for structure static equilibrium equations are represented in the stress princi- of single plate. pal lines co-ordinate net (isostatic net) (see [1] for further details). Optimization of the potential of interaction for numerical mod- Numerical implementation is based on the solution of a system of eling of layered silicates is labored because of being of aluminum three nonlinear equations formulated in the canonical co-ordinate and oxygen in ionized form. There are several proposed versions net by finite difference technique. This method is efficient for of force fields that are essentially different ones and adopted for problems with a free boundary. Stress boundary conditions allow special goals. We performed the numeri-cal simulation that pro- to formulate the initial data for the resolving system. vided a choice of the force field appropriate to quantum mechan- This method is also applicable to axially-symmetric problems. ics calculation for similar molecules and clusters as well as to ex- Under conditions of axial symmetry it is possible to separate the perimental structural data. The main attention was paid to param- angular isostatic co-ordinate and then to reduce a problem to the eters for silicon and oxygen atoms forming a surface of nanocrys- integration of system of two nonlinear equations and separate an tal. The reason is that potentials of interaction of these atoms additional isostatic co-ordinate. It is found, that on a free bound- determine not only a crystalline structure but also interaction of ary one of isostatic co-ordinates is the natural parameter. As an nanoparticles with surrounding molecules.

25 We analysed the contributions of different types of interaction NON-LOCAL SHELL OPTIMIZATION WITH into general potential energy. Then the temperature dependences APPLICATION OF GENETIC ALGORITHM of structural characteristics were calculated. On this basis we esti-mated coefficients of thermal expansion in wide temperature N.V. BANICHUK S.YU.IVANOVA E.V. MAKEEV diapason. A specificity of the molecular mobility mechanisms A.V. SINITSYN determining these coefficients for nanoplate relative to common crystals is dis-cussed. The mechanical properties of nanoplate [email protected], [email protected], were calculated from curves of force versus displace-ment ob- [email protected] tained at slow rates of deformation. The atomistic results are re- formulated in terms of contin-uum model for thin plates. In this research the questions of design the minimal weight shells are made from quasi-brittle materials are investigated. The The study was sponsored by the CRDF (project no. 13807). problem of constructing quasi-symmetric shapes under cyclic forces which generates fractures are formulated and solved. The Mikhail A. Mazo, Semenov Institute of Chemical Physics Russian given problem is characterized by incomplete information con- Academy of Science, Kosygin str. 4, Moscow, 119991, Russia cerning initial crack size, position and orientation. Presented problem formulation and solutions are based on the guaranteed (minimax) approach. The modified algorithm of genetical (evolutionary) method is applied for searching the global optimum solutions. The outcomes of shells optimization and some data, BISTABLE TRANSITIONS IN FERRONEMATICS describing convergence of applied algorithm are reduced. N.V. Banichuk, S.Yu. Ivanova, E.V. Makeev, A.V. Sinittsyn, Rus- D.V. BALDIN A.N. ZAKHLEVNYKH sian Academy of Sciences, Lab of mechanic and optimization of structures, pr. Vernadskogo 101, Moscow, 117526, Russia [email protected]

Within the framework of continuum theory the equilibrium CABLE MAST MOVING DYNAMIC WITH THE distribution of director and magnetization in a magnetic field for a ACCOUNT OF FLEXURAL VIBRATIONS ferronematic (FN), i.e. dilute suspension of needle-like magnetic I.V. BARMIN M.V. VESELOV Y.M. EPIFANOV particles in nematic liquid crystal, with soft particle anchoring with a liquid-crystalline matrix is considered. We propose new form of energy of coupling of magnetic particles with a liquid- crystalline matrix, which include the additional term of expan- Cable mast (CM) is a part of the launch site technological sion, and give good description of orientational coupling in rather equipment. It provides a communications laying to a launch ve- strong magnetic fields, and for some critical value of additional hicle (LV) during launch preparation. Directly before LV launch, anchoring parameter leads to bistable behavior on the surface of the CM is moving away into a safe position under action of coun- magnetic particles. terbalance weight. The saved kinetic energy is damped by the hydrobuffer on the final stage of CM’s movement, thus dynamic It is shown that for low fields the FN state I with homeotropic loads on CM take place. In the paper, the methods of decision conditions of coupling of magnetic particles with a matrix is ther- of CM moving dynamic task are discussed for a case of consid- modynamically stable. In this phase the ordering has the dipolar eration of its structure as a rigid body, and with the account of nature. When the field strength achieves the threshold value, this its flexural vibrations. Comparison of calculated parameters with state is replaced by the angular FN phase II, in which the vec- the results of natural measurements is carried out. tors of field strength, the director and magnetization are coplanar, I.V. Barmin, M.V. Veselov, Y.M. Epifanov, Barmin Design Bureau and the angles of orientation of a magnetization and director con- of General Machinery, Moscow cerning a field are functions of material parameters of a FN and field strength. The FN state II grows out from the competition between the dipolar and quadrupolar ordering. In FN phase II NONLINEAR THREE-WAVE INTERACTIONS OF EDGE there is a change of conditions of coupling of magnetic particles WAVES OVER CYLINDRICAL SHELF with a matrix from homeotropic to planar. With a field increasing the FN state II is replaced by a FN state III, in which the direc- E.K. BATSYNA V.A. DUBININA A.A. KURKIN tor, the vectors of magnetization and field strength are parallel, O.E. POLOUKHINA i.e. in phase III the quadrupolar mechanism exhibits itself to the [email protected] full, together with the dipolar one. The dependence of threshold fields from material parameters of a FN is found. It is shown Edge waves of large amplitude are the essential part of wave that for critical value of additional anchoring parameter the first conditions of ocean coastal zone, and their existence was proved order transition between homeotropic and planar orientations of to be true by the numerous observational data [1]. They are director and magnetization axes takes place. thought to produce the periodic forms of a coastal line often ob- This work was supported in part by grants PE-009-0 from served in a nature [2]. The resonant nonlinear interactions of edge CRDF. waves are considered as one of the basic mechanisms of redistribution of wave energy in a coastal zone. The investigations in this D.V. Baldin, Perm State University, Perm 614990, Russia direction are carried out already during more than 30 years. For

26 example, the existence of resonant triads of Stokes edge waves theory. Numerical calculations of dispersion relations and wave above a constant sloped shelf was proved even in paper [3], and patterns for different modes are presented for different bottom to- then was confirmed in the following papers [4, 5]. Three-wave pography and flow profiles. The influence of alongshore current resonances arise for edge waves also above a cylindrical shelf of on the process of edge wave dispersive focusing is studied. The arbitrary profile [6, 7]. More difficult problem is the evaluation possibility of anomalous large-amplitude edge wave formation is of coefficients of nonlinear interaction of edge waves. Existing discussed. attempts of their evaluation are bounded usually by a choice of Ekaterina K. Batsyna, Nizhny Novgorod State Technical Univer- fixed triads, and some of these interaction coefficients have ap- sity, Minin St. 24, Nizhny Novgorod, 603600, Russia peared to be zero [5]; thus the role of three-wave nonlinear interactions in dynamics of edge wave field remains unclear. The purpose of the present work is the analysis of possible DILATOMETRIC AND CALORIMETRIC STUDY OF AN nonlinear effects in interactions of edge wave triads in a coastal AL-MG-ZN ALLOY zone. Here the interaction processes for three edge waves (propagating in the same, or in opposite directions) for three different M. BENABDOUN T. DORBANI S. HAMAMDA shelf profiles (linearly sloped shelf, concave exponential shelf, I. ZERROUK and step shelf) are considered. The conditions of synchronism of [email protected] interacting triads are defined, and the coefficients of interaction for described cases are computed for the possible triads consist- The aim of this study is the determination of the thermal ex- ing of waves belonging to four lowest modes. pansion coefficient in relation to the temperature of the alloy Al- REFERENCES 6%Zn-3%Mg having undergone various thermal and mechanical [1] A.B. Rabinovich, Long gravity waves in the ocean: trap- treatments. After the solution treatment, the α(T) curves along ping, resonance, propagation. (in Russian), S.Pb: Gidrome- three (chosen) directions remained practically the same as those teoizdat, 1993. of the material in its raw state. [2] G. Masselink, Alongshore variation in beach cusp morphol- The Al-6%Zn-3%Mg alloy, obtained through conventional ogy in a coastal embayment. Earth Surface Processes and casting, exhibits a very weak anisotropy. The shape of the curve Landforms, 1999, V. 24, P. 335 ? 347. representing the variation of the thermal coefficient of linear expansion with temperature, α(T), does not depend on the direction [3] K.E. Kenyon, A note on conservative edge wave interaction. along which it is measured. The first two phase transformations, Deep Sea Res, 1970, V. 17, P. 197 ?201. namely GP zone dissolution and germination of the η’ phase, oc- [4] I.E. Kochergin, E.N. Pelinovsky, Nonlinear interaction of cur at the same temperature and have similar intensities in the edge waves triad. (in Russian), Oceanology, 1989, V. 29, N three directions studied. On the contrary, we observe that, for 6, P. 899 ? 903. the two last transformations, the precipitation and dissolution of the equilibrium η phase (MgZn2), while they do occur at the same [5] J.T. Kirby, M. Asce, U. Putrevu, A.M. Asce, H.T. Ozkan- temperature in the three directions, their intensities vary from one Haller, Evolution equations for edge waves and shear waves direction to another. on longshore uniform beaches. Proc. 26th Int. Conf. Coastal The effect of different thermo-mechanical treatments on α(T ) Engineering (Copenhagen. June 1998), 1998, P. 203 ? 216. along each of the directions studied brings us to the conclusion [6] I.E. Kochergin, Triad of nonlinear interacting edge waves that the first three transformations, occurring in the tempera- for different type of coastal topography. Intercollegiate col- ture range [25-425C] are more or less influenced by the solu- lection of scientific proceedings ?Oscillation and waves in tion treatment, while the fourth, corresponding to the dissolution continuum mechanics (analytically and numerically meth- of the equilibrium η phase (MgZn2), remains practically unaf- ods)?, Gorki, GPI, 1989, P. 79 ? 83. fected. From another point of view, plastic deformation favours [7] Galletta V., Vittori G, Nonlinear effects on edge wave de- to a higher degree the phase transformations that could occur dur- velopment. European Journal of Mechanics B Fluids, 2004, ing the (ramp) heating of the sample. This explains the reasons V. 23, P. 861 ? 878. why the phenomena linked to the dissolution of the zones, and to the precipitation of the two phases (η and η), occur earlier, at Valentina A. Dubinina, Nizhny Novgorod State Technical Univer- lower temperatures, than in the case of raw and/or non-solution sity, Minin St. 24, Nizhny Novgorod, 603600, Russia treated, material. Smail Hamamda, Departement« de Physique, Universite« de Con- stantine, Constantine 25000, Algeria INFLUENCE OF ALONGSHORE CURRENTS ON THE PROCESS OF EDGE WAVE FOCUSING STABILITY ANALYSIS OF GRAPHITE CRYSTAL E.K. BATSYNA V.A. DUBININA A.A. KURKIN LATTICE USING MOMENT INTERACTIONS O.E. POLOUKHINA [email protected], [email protected], IGOR E. BERINSKY [email protected]

An analysis of the effect of an alongshore shear ﬂow on dis- This work continues math modelling of low-packed crystal lat- persive characteristics of edge waves propagating above a cylin- tices. Stability of such lattices can be provided by different ap- drical shelf is performed in the framework of linear shallow water proaches. This work suggests to obtain stability using complex

27 Lennard-Jones potential which has simple physical enterpreta- higher the bend of nanoparticles. The distribution of stress and tion. To solve this problem additionally to the force interactions strain fields was also obtained and analyzed. moment interactions are used [1]. Acknowledgements The main goal of this work is to create and study stability of This work is supported by the Russian Foundation for Ba- the graphite lattice using moment interactions. Plane layer of the sic Researches (project Nos. 02-03-32761, 03-03-32018, 04-03- lattice is considered. Carbon atom is modelled by three rigidly 32052). connected particles forming an equilateral triangle. Second-rank Mikhail Y. Shamaev, N.N. Semenov Institute of Chemical Physics stiffness tensors characterizing elastic bonding of the triangles are of Russian Academy of Sciences Russia, 119991 Moscow, 4 Kosy- determined. The configuration containing two triangles is con- gin str. sidered. Stability of this configuration depending on the distance between triangles and their geometry is investigated. Fourth-rank tensors characterizing elastic properties of the lattice are found. ENERGETIC AND FORCE BARRIERS AT THE Stability of the lattice is investigated. TRANSITIONS BETWEEN THE STABLE STATES OF REFERENCES DYNAMIC SYSTEMS [1] Ivanova E.A., Kirvtsov A.M., Morozov N.F., Firsova A.D. Decsription of crystal particle packing considering moment ILIJA I. BLEKHMAN GERMAN E. ITKIN interactions. // Izvestiya RAN, Mekhanika tverdogo tela, [email protected] 2003, 4, 110-127.(In Russian,to be translated in Mechanics of Solids). It can be said about many dynamic objects that there are po- I.E. Berinsky, St. Petersburg State Polytechnical University, St. tential and force barriers between their stable states. The transfer Petersburg, Russia of the object from one stable state to another is the essence of many technological processes. The “overfall” of energy between the initial and final states of equilibrium (∆E)N can in this case NUMERICAL MODELING OF THE EFFECT OF be either positive or negative, i.e. the transition can be accompa- NANOPARTICLES’ BENDING RIGIDITY ON THE nied either by absorption or by release of energy (in chemistry it MECHANICAL PROPERTIES OF FILLED SYSTEMS corresponds to endothermic or to exothermic reactions). This presentation pays special attention to the fact that in tran- A.A. BERLIN V.G. OSHMYAN M.YU.SHAMAEV sitions in case (∆E)N > 0 it is usually necessary to spend S.A. TIMAN much more energy than the value (∆E)N and in the case when [email protected], (∆E)N < 0, less energy than |(∆E)N | is obtained. Special at- [email protected] tention is paid to the fact that in this case the corresponding loss of the energy (∆E)Add can greatly exceed the value |(∆E)N |. Polymer/clay nanocomposites exhibit enhanced elastic prop- Often the above mentioned loss is irreversible and is like a fine erties at low volume fractions of clay. Effective elastic moduli which it is necessary to pay for the “the fight against nature”, of the nanocomposite are strongly dependent, in particular, on which has made the corresponding states stable. So for the pro- the clay particles aspect ratio, on their orientation distribution, cess of breaking the solid bodies (∆E)Add are, on estimated data, and on the ratio between particle and matrix mechanical proper- four orders of magnitude larger than the value |(∆E)N |. The ties. The clay particle’s structure is either exfoliated thin plates task of creating energyÐsaving technologies is in many cases re- or stacks of parallel clay plates separated from one another by duced to decreasing the value (∆E)Add. One of the purposes matrix. In the present work we propose the model for description of the presentation is to draw the attention of the specialists — of mechanical properties of polymer composite filled by highly mathematicians, physicists, and engineers — to the problem of asymmetrical particles. In the model, on the mesoscale these par- transitions. ticles are considered as thin plates or long rods. The difficulties The work has been carried out with the support of Russian both in analytical and numerical calculation of effective proper- Foundation for Basic Research (project 04Ð01Ð00053). ties of such a materials are connected with the necessity of taking IPME RAS, St. Petersburg, Russia into account the bending of thin inclusions. In numerical simulation a strong discretization of continual equations is needed because of the large aspect ratio (the ratio of characteristic di- ADAPTIVE PROPERTIES OF DYNAMIC OBJECTS mensions) of filler. Suggested computational model is based on finite element method and doesn’t require fine partition of inte- ILIJA I. BLEKHMAN LEONID A. VAISBERG gration domain on finite elements. The key idea is an additive calculation of the energy of deformation both finite element and By adaptability we understand a property of objects to retain fragment of inclusion(s) (if any) located within. Both lower di- their individuality and main functional qualities, adapting to the mension of fragment and compatibility of affine deformation of change in outer conditions. And the more complicated and highly fragment(s) and given finite element are supposed. Different mor- organized the object, the more adaptive, as a rule, it is. phology (intercalated or exfoliated) of nanoparticles may be took The adaptability of living organisms is well known. Human into account. Model allows to consider various variants of dis- beings are highly adaptive. A glowing example of it is the change position and orientation of nanoparticles from unidirectional to in the frequency of heart contractions with the change in emo- chaotic. Numerical results obtained showed the important role tional or physical load. The outstanding physicist Roger Penrose of bending deformation of asymmetrical inclusions. In particu- considered human beings to be the most stable structures of the lar, the elastic moduli of composites are decreased the greatly the physical world.

28 A great number of investigations has been devoted to the study sating in the pipe-line, while the conic insert was not affected by of the adaptive properties of a human organism. Works by Hans vibration at all. Dependencies were obtained of the ratio of hy- Selye, awarded the Nobel prize, take a prominent place among drodynamic resistances in the conditions of vibration and of the them. They are presented in his brilliant book “Tale of an adap- free outflow of the fluid to the value of hydrostatic pressure. tation syndrome”. The work has been carried out with the support of Grants Adaptability of living organisms can be explained by a natural Board by the President of RF (Program ”Leading Scientific selection: only those organisms, which were able to adapt to the Schools” 1521.2003.8) and of RFBR (grant 04-01-00053). changes of the outer conditions, have survived in the process of I.I. Blekhman, V.B. Vasilkov, K.S. Yakimova, IPME RAS & evolution. “Mekhanobr-tekhnika” Corp., St. Petersburg, Russia It is surprising, however, that some technical systems also possess a certain adaptability. This presentation is devoted to the de- DYNAMICS OF A CIRCULAR CYLINDER scription of some examples of adaptability of mechanical objects INTERACTING WITH POINT VORTICES that the authors came across in their scientific and engineering practice. Perhaps specialists in other branches can give other, no A.V. BORISOV I.S. MAMAEV less dramatic examples. At the beginning of this presentation we give a “working” We consider the system of a rigid body interacting dynami- definition of some main notions, connected with adaptability — cally with point vortices in a perfect fluid. For arbitrary value of selfÐorganization, stability, steadiness of stationary states and vortex strengths and circulation around the cylinder the system is motions. Then we consider adaption effects during the self- shown to be Hamiltonian. We also reduce the number of degrees synchronization of unbalanced rotors, in the conical inertial of freedom of the system by two using the reduction by simme- crushers of the “Mekhanobr Institute”, adaption of the profiles try technique and perform qualitative analysis of the integrable of the working chambers of crushers when they are warn out, ef- system of a cylinder interacting with a vortex. fects of self-adjustment of vibro-absorbers and balancing devices, A.V. Borisov, Institute of Computer Science, 1, Universitetskaya adaption and reservation when oscillations are excited by a chain str., 426034 Izhevsk, Russia of generators, etc. It should be noted that the adaption properties of some machines are not accidental, but are the result of purposeful work of BASIC SOLUTIONS AND REGULAR EXPANSION IN investigators, designers, and inventors, though based mainly on THE MECHANICS OF PERIODIC COMPOSITES intuition. In fact, any well designed machine possesses, in some way or another, adaptive properties. ALEXEY I. BOROVKOV VLADIMIR A. PALMOV This report discusses the way of providing or improving the [email protected] adaptive properties of a device by means of increasing the number of the degrees of freedom, say, by a transition form a cinematic Elastic periodic composite is under approach consideration. drive to a dynamic one. It is noted that it is also possible to use We propose new approach for the analysis of mechanical be- the approaches of mechatronics. havior of this composite: we use the so called “basic solutions” Some general principles are considered which underlie the which represent exact or approximate numerical solutions of six adaption effects in mechanical systems. In particular, a suppo- boundary value problems, formulated for unit cell of composite. sition is discussed here that in a number of cases adaptability is a We suppose for the simplicity that any cell has three planes of result of extreme properties of the corresponding processes. symmetry, which are crossing in the centre of the cell and which The work has been carried out with the support of Grants are parallel to the coordinate planes of the orthogonal coordinate Board by the President of RF (Program ”Leading Scientific system. Schools” 1521.2003.8) and of Presidium of Russian Academy of Linear combination of “basic solutions” represents the so Sciences. called “regular expansion” of stresses, strains and displacements. IPME RAS & “Mekhanobr-tekhnika” Corp., St. Petersburg What can be done using new approach? Firstly we propose new method of homogenization. In particular, we present new formulae for effective moduli of elasticity. HYDRODYNAMIC RESISTANCE OF A CONIC SHAPE They represent exact values of effective moduli for homogeneous CHANNEL UNDERGOING VIBRATION in space averaged stresses and strains of composite. Secondly we present new motivation which leads to the so ILIJA I. BLEKHMAN VLADISLAV B. VASILKOV called homogenized problem. It is shown that “regular ex- KIRA S. YAKIMOVA pansion” represents effective solution of exact elasticity equa- [email protected] tion for initial nonhomogenized composite if averaged stresses, strains and displacements satisfy to the equations of homogenized The work contains the results of the experimental investi- boundary value problem. gation, allowing the hydrodynamic wastes to estimate when a At least, thirdly, the boundary layer concept was applied for stream of fluid flows inside the widening or narrowing sections the formulation of boundary conditions for homogenized prob- of a pipe-line under the action of vibration. The outflow of the lem, which leads to the satisfaction of initial nonhomogenized fluid from a hole of a vibrating vessel was investigated for the boundary conditions of composite. case when a conic insert was placed either right in the hole on Numerical examples are presented and results are discussed. the bottom of the vessel, or in the pipe-line at the certain distance A.I. Borovkov, V.A. Palmov, St.Petersburg State Polytechnical from the bottom. In the latter case a stream of the fluid was pul- University, Russia

29 M. Leﬁk, Geotechnical Engineering and Engineering Structures MULTISCALE MODELLING FOR COMPOSITES Technical University of Lodz, Al.Politechniki 6, 93-590 Lodz, INCLUDING CONTINUUM TO DISCRETE LINKAGE Poland D.P. BOSO M. LEFIK B.A. SCHREFLER

[email protected], [email protected] A COMBINATION OF DENSITY FUNCTIONAL THEORY CALCULATIONS (DFT) AND Many natural and man-made materials exhibit an internal EXPERIMENTAL STUDIES: MOLECULAR structure at more than one length scale. These internal structures CONFORMATION IN TRI-IODOMESITYLENE may be of translative nature, where the structure is more or less invariant with respect to a translation of a vector corresponding ALI BOUDJADA JEAN MEINNEL to the smallest length scale. ABDOU BOUCEKKINE SMA¬IL HAMAMDA Superconducting (SC) cables can be regarded as very good ex- [email protected] amples of hierarchical structures where lower levels take part in the global behaviour. According to the current design, the SC The rotational motion of methyl groups in solids has been ex- alloy is formed into fine filaments, which are embedded in a low- tensively studied. The methyl group provides perhaps the sim- resistivity matrix of normal metal to make the elementary strand. plest example of large amplitude motion of light atoms governed After that more than one thousand strands are twisted together ac- by no covalent forces, thus it is interesting to establish the inter- cording to a multi-level twisting scheme to form the final cable. action between this quantum rotor and the frame to which it is We can distinguish three scales: linked. • the filament (micro scale); The main purpose of this work is to study the vibration spectra • the strand (meso scale); of the methyl groups, especially the torsion modes in the case of weakly hindered rotors. A first problem is that the correspond- • the cable (macro scale). ing frequencies fall in the same frequency range as lattice vibra- One of the main problems for the analysis of such a structure tions and some internal modes for aromatic molecules with heavy is the estimation and the evolution of the global constitutive char- atoms substituents. Quantum mechanical calculations (DFT) are acteristics. Because of the scale separation between structural now able to find an accurate molecular conformation of isolated levels a spatial discretization — e.g. that of a finite element mesh molecules [1] and then to estimate the frequencies (and their - fine enough for the micro level would result in a huge number spectroscopic intensities) of all the molecular normal modes of of elements and unknowns at the macro level. This would be vibration. numerically difficult to manage and, first of all, not necessary. In trihalogenomesitylenes there are no adjacent methyl groups We start the procedure at the micro level with estimation of in the molecule, and thus intermolecular effects are more easily the effective mechanical and thermal coefficients of the compos- distinguishable, from intramolecular effects, than in hexamethyl- ite [1]. This is done by an analysis of the micro-cell of periodicity, benzene [2]. In similar compounds [3], molecules with a three- according to the classical theory of homogenisation. At the meso fold axis of symmetry show, from a crystallographic viewpoint, scale a fibrous beam model is used [2], which inherits the prop- a specific behaviour. The methyl groups are symmetrically sur- erties of the composite calculated at the micro level. We choose rounded by halogens and the potentials hindering methyl rotation as unit cell a triplet of strands, which is the first cabling stage. At are unexpectedly low in isolated molecules and give rather large this level the problem is different: the mechanics is no more con- tunnelling excitations in the crystalline state [4].We justify the tinuous, but becomes discrete. As a consequence in the definition study of the 1,3,5-triiodo-2,4,6-trimethylbenzene or triiodome- of the effective properties at this level we take into consideration sitylene (TIM) by several points: the contact phenomena occurring between the strands. Going to the macro level, a sub-cable stage is analysed using the thermo- • the iodine is the halogen with the biggest van der Waals mechanical properties calculated at meso level. In this way we radius, it is in TIM that the methyl groups are the less hin- can analyse a bundle of strands via a continuum model. dered [4]. We discuss the advantages and disadvantages of the method • the crystal structure has already been studied by our group with the example of a finite element solution obtained for an ide- and has revealed unexpected conformation for the methyl alised model of a superconducting sub-cable in its operating con- groups at 14 K . In the range 300-15 K the crystal struc- ditions. ture (TIM) remains triclinic with a space group P-1. X- REFERENCES ray measurements indicating small conformational changes [1] D.P. Boso, M. Lefik and B.A. Schrefler “A multilevel ho- [5] on cooling, we have studied the structure of TIM at 15 mogenised model for superconducting strands thermome- K by neutron diffraction [6]. TIM molecules have not the chanics”, in print on Cryogenics. expected symmetry, but rather a Cs one: one CH3 has an [2] M. Lefik, B.A., Schrefler, 3D finite element analysis of eclipsed conformation (one C-H in the ring plane), a sec- composite beams with parallel fibres based on the homog- ond is rotated by 180 while the third undergoes a rotation of enization theory, Computational Mechanics, 14, 1, 2Ð15, about 80. 1994. • the final goal being the comprehension of the tunnelling for D.P. Boso, B.A. Schrefler, Department of Structural and Trans- the three methyl groups, it is necessary to test theoretical portation Engineering, University of Padua, Via Marzolo 9, programs computing molecular geometry in order to locate 35131 Padua, Italy more precisely the position of the hydrogens in the isolated

30 molecule and, consequently, the height of the potential bar- distance from the crack tip. In the other case constant distributed rier hindering the methyl group rotation. We used the Gaus- load is suddenly applied at crack faces. sian programs and the DFT calculation using MPW1PW91 Influence of coefficient ζ controlling viscosity behavior on co- functional with the Lanl2DZ** basis set have shown that if hesive zone development is studied. Obtained dependencies of the conformation C3h is stable for TIM, a 180 rotation for “time-to-fracture” of load rate (concentrated force case) or load one CH3 does not affect the molecular energy of formation, amplitude (distributed pressure case) for different ζ values are while for a 90 rotation the conformation is unstable in the compared with estimations obtained using Morozov-Petrov crite- gas phase. This gives another stable conformation for TIM rion for dynamic crack initiation (see for ex. [1]). molecule: a Cs symmetry. REFERENCES REFERENCES [1] Morozov, N., Petrov, Y., (2000), Dynamics of Fracture, [1] Zhou Z., Fu A. and Du D. (2000) Int. J., Quant. Chem., 78, Springer-Verlag, Berlin-Heidelberg-New York 58-60. V.A. Bratov, Malmoh¬ ogskola,¬ Ostra¬ Varvsgatan 11H, Malmo,¬ [2] Babushkina, T. A. (1966). Zh. Strutkt. Khim. 7, 631-633. 20506, Sweden [3] Babushkina, T. A., Kozhin, V. M., Robas,V. I., Sfin, I. A., Semin, G. K. & Khostsyanova, T. L. (1967), Kristallogrfiya, SIMULTANEOUS FORCED AND PARAMETRIC 12(1), 143-144. EXCITATION OF A SPACE TETHER [4] Meinnel, J., Hausler,¬ W. & al.(1992). Physica B 181/181, 711-713. M.P. CARTMELL M.C. D’ARRIGO [5] Boudjada, A., Hernandez, O., Meinnel, J., Paulus,¬ W. [email protected] (2001), Acta. Cryst. A58, 1106-1108. [6] Boudjada, A., Meinnel, J., Boucekkine, A., Hernandez, O. It has been successfully shown that forced excitation of a sym- et al. (2002), J. Chem Phys. 117, 10173-10183. metrical momentum exchange tether by means of a continuous torque can accelerate the tether and increase the relative velocity of payloads fitted to each end. When advantageously coupled Ali Boudjada, Laboratoire de cristallographie, Departement« de with orbital mechanics the payloads can be released at a posi- physique, Universite« de Constantine, Constantine, Algeria tion approximately normal to the tangent to the orbit so that the Jean Meinnel, Abdou Boucekkine, Institut de Chimie, Univeriste« outer payload is boosted and the inner payload is de-boosted. A de Rennes1, LCSIM UMR 6511, 35042 Rennes, France methodology has been described by the author and others, in pre- Sma¬õl Hamamda, Laboratoire des propriet« es« thermodynamiques, vious publications, for exploiting symmetrical loadings and un- Departement« de physique, Universite« de Constantine, Constan- loadings. Whilst this is undoubtedly challenging in terms of the tine, Algeria logistics, it offers a convenient and reusable facilty for both payload exchange between two planets. However, the motor torque required to drive a large momentum exchange tether system into VISCOUS COHESIVE ZONE MODEL FOR FRACTURE continuous rotation is both substantial and dependent on the use UNDER RAPID LOADING of a contra-rotating counter inertia of sufficient size. One of several issues yet to be fully resolved is how to minimse the mo- û VLADIMIR A. BRATOV PER STAHLE tor torque necessary for guaranteed monotonic spin-up, and this [email protected], paper offers a potentially interesting way forward by means of [email protected] inertial parametric excitation. This could be achieved in practice by a harmonic modulation of a portion of the payload mass, Dynamic fracture has a lack of models with the ability to de- at each end, by manipulation of the radius of gyration of addi- scribe recent experimental results in rupture caused by loads ap- tional lumped masses. Preliminary calculations suggest that this plied or changing at high rates. First of all this concerns crack unusual form of parametric excitation can modify the spin-up re- initiation. sponse of the tether to a constant motor torque, with interesting The work presented is an attempt to describe crack initiation implications for the eradication of tumbling and chaotic motions based on the material microstructure response on suddenly ap- at threshold motor torques. plied or rapidly changing load. To make this a cohesive zone Matthew P. Cartmell, Glasgow University, James Watt Building, with viscous behavior is introduced at the tip of a crack. The University Avenue, Glasgow, G12 8QQ, United Kingdom constitutive equation giving the distribution of stresses in the cohesive zone is given by: σs(x)=σ0 + ζ ∗ u˙(x), where σ0 and ζ are material constants representing cohesive stress and viscosity, CORRELATION BETWEEN INTERPARTICLE FORCES u(x) is half the displacement across the cohesive zone and dot AND BULK BEHAVIOUR IN FINE POWDERS denoting differentiation with respect to time. Crack tip opening ANTONIO CASTELLANOS is used to predict fracture. Finite element method simulations are performed for a large [email protected] symmetrically impact loaded crack in a large plane. The model is repeating analytical problem for a semi infinite crack in an in- The aim of this presentation is to examine the relationship finite plane. Two types of loading are studied. In the first case a between attractive interparticle forces at grain level, with solid concentrated load linearly growing with time is applied at a large bulk properties at low consolidations (solid fraction, stresses),

31 fluidization (aggregation, settling) and flow regimes boundaries ON THE THEORY OF CONSTITUTIVE EQUATIONS (plastic flow, inertial flow, fluidization and suspension). In a pre- FOR MEDIUMS WITH HEREDITARY PROPERTIES vious lecture (Proceedings APM 2004, pp. 82-91) it was presented the method and results for measuring the interparticle VICTOR CHAIKIN EKATERINA POLYAKOVA forces by means of the Atomic Force Microscope (AFM) as a IVA N TARASENKO function of load, as well as the measurement of axial tensile stress by means of the Sevilla Powder Tester (SPT). Using simple con- v [email protected], [email protected] cepts from micromechanics, we estimated from SPT measurements the interparticle forces, and correlated them to the direct As the most general form of the linear constitutive equations AFM measurements. They differed by a factor between 5 and 10, for mediums with hereditary properties the form initially pro- though both set of measurements agreed qualitatively in accor- posed by V.Volterra and L. Boltzman usually is considered. This dance with theories from contact mechanics. form represents the stresses in a medium as a convolution of its strains with the kernel that characterize the medium’s relaxation properties. This form is indeed the most general one if as the Here we refine our estimation of contact forces in the bulk tak- space of strains all continuous with respect to time functions or ing into account the state of the powders in the fluidized bed . We all distributions are chosen. However no real medium can sustain show that particles are strongly agglomerated and we determine any continuous strains and keep from destruction. It is assuredly through collapse experiments the average size and average num- true with reference to the space of distributions. The real medi- ber of particles of aggregates as a function of the cohesive gran- ums can exhibit linear properties only on rather narrow classes of ular Bond number. Then we investigate the compaction of the strains. Therefore convolutions of some kernels with derivatives settled layer as we increase the confining axial pressure. This al- from strains with respect to time must be added in constitutive low us to determine the evolution of aggregates as compaction is equations for real mediums. The mechanical models of mediums increased from zero to very low values (jamming transition) and that have constitutive equations with such additional convolutions then to higher values. We show that there is a point at which the are presented. aggregates begin to be progressively disrupted. This is marked by V. Chaikin, Saint-Petersburg State University of Technology and a transition from a potential law to the classical logarithmic law Design, 18, Bolshaya Morskaya str., Saint-Petersburg, 191186, of soil mechanics for the solid void fraction as function of load. Russia Including the existence of aggregates in the estimation of contact forces in the bulk makes the agreement between AFM and SPT EXACT CORRESPONDENCES BETWEEN measurements to agree within a factor of 2. N-DIMENSIONAL ANISOTROPIC AND ISOTROPIC GREEN’S FUNCTIONS IN CONDUCTIONS

Next we discuss the implications of these interparticle forces TUNGYANG CHEN HSIN-YI KUO in the flow regimes of powders. Granular materials may have [email protected], plastic flow, inertial flow more or less rapid, fluidization and sus- [email protected] pension as a limit case of the latter. In each one of these regimes transfer of momentum between particles is governed by differ- We establish exact linkages between the N-dimensional ent physical phenomena. By friction and cohesive forces (if the anisotropic and isotropic Green’s functions in conduction for an latter are present) in the plastic flow, through collisions in the in- infinite space, a half-space, and a bimaterial space. The purpose ertial flow, and by drag forces and collisions in the fluidization of this work is not to attempt to present a solution procedure for and suspension regimes. Using scaling laws we will show that these boundary valued problem, but to focus on the general condi- fine powders never present the inertial regime, on the contrary tions and situations in which the physical problem can be directly they go from the plastic flow to a state of gas-fluidization. We de- linked with the corresponding Green’s functions of a similar con- termine the boundary between both regimes as a function of the figuration but with isotropic constituents. In the formulation, we Cohesive Bond number and the imposed interstitial gas velocity. make use of an affine coordinate transformation, incorporating with a specific transformation matrix relevant to the Cholesky decomposition, for all configurations. For each constituent region After a summary we finish the presentation with a discussion in a bimaterial space, we introduce a single transformation ma- of the open problems. Among them we may cite the problem trix relevant to its conductivity tensor. The field in each region of developing a continuum theory describing the complex pow- is then governed by a standard Laplace one, with a certain mod- der behaviour, the problem of relating more precisely the mi- ification on the boundary term and/or the interfacial continuity croscopic and the macroscopic forces and stresses, the transition conditions. For Green’s functions of a half-space, three different from a discrete to a continuous description of the powders and kinds of homogeneous boundary conditions in conductions are the problem of the relation between the average stresses, motions considered. Perfect bonding conditions are assumed to prevail at and their fluctuations, both in the ”solid” state, in plastic flows interfaces between any two adjacent regions for a bimaterial. We and in fluidization. show that, for Green’s functions of an infinite and a half-space and for all two-dimensional configurations, the exact correspondence between the anisotropic and isotropic ones can always be A. Castellanos, Departamento de Electronica« y Electromag- established without any regard to the constituent properties. And netismo, Facultad de F«õsica, Avenida Reina Mercedes s/n. 41012 thus knowing the isotropic Green’s functions will readily provide Sevilla, Spain. explicit expressions for anisotropic Green’s functions upon back

32 transformation. For three- and higher-dimensional bimaterials, ers equation is determining. On the second stage the elasticity of the correspondence can also be proven but the constituent con- wall plays the main role and KdV equation is describing the pro- ductivities need to satisfy further algebraic constraints. When cess. Eventually on the third stage (of long wave approximation) these constraints are fully complied, then the anisotropic Green’s the viscosity of wall begins its leading and rKS equation is most functions can also be obtained from those of the isotropic ones. inﬂuence on the waves propagation. H.-Y. Kuo, National Cheng Kung University, 1, University Road, REFERENCES Tainan, 701, Taiwan [1] O’Rourke M., Mechanical principles in arterial disease, Hy- pertension. 1995 Jul;26(1):2-9.

SYSTEM OF EQUATIONS FOR NONLINEAR [2] L.Formaggia, D.Lamponi and A.Quarteroni, One dimen- PRESSURE WAVES IN A VISCOELASTIC VESSEL sional models for blood flow in arteries, Journal of Engi- neering Mathematics, No.47, pp.251-276, 2003 I.L. CHERNYAVSKY N.A. KUDRYASHOV [3] Kudryashov N.A., Analytical theory of nonlinear differen- [email protected], [email protected] tial equation, Moscow-Izhevsk, IKI, 2004 Igor L. Chernyavsky, Moscow Engineering Physics Institute The investigation of nonlinear wave processes in a liquid flow (State University), Kashirskoe shosse, 31, Moscow, 115409, Rus- through a viscoelastic tube is the task of a great importance as it sia is similar to the blood flow in vessels of cardio-vascular system. The modeling of flow in a viscoelastic tube should be aid for understanding and prediction of various diseases. THE INVESTIGATION OF THE ASYMPTOTICS FOR It is well known THE PROBLEMS OF THE LOCATED INSTABILITY OF citeRourke that vessels (arteries, arterioles) have two main func- PLATE WITH FREE EDGE tions: conductivity and damping. The conductivity function is responsible for delivering full of oxygen blood to the organs and EDUARD O. CHIL-AKOPYAN MELS V. B ELUBEKYAN tissues. The damping function implies the smoothing of pressure [email protected], impulses outgoing from the heart, thus the flow sufficiently far [email protected] from aorta is close to stationary flow. Atherosclerosis disturbs the first function via the plaques on Everywhere in computational mechanics are considered the internal layer of an arterial wall (intima). Arteriosclerosis boundary value problems for unlimited areas. Meanwhile it is damages the damping function due to the degeneration and struc- clear, that in a reality any continuous medium is limited. Less tural changes in the medium elastic layer of an arterial wall (me- often, than followed, it is underlined, that the problems for un- dia). It also calls remodeling of arterial wall. Arteriosclerosis im- limited mediums can have physical sense only as limiting - when plies to the arterial hypertension and additional vessels destruc- all exterior boundary or its part leaves on infinity. tions. Naturally there is a problem; under what conditions the model We consider the model of fluid-wall interaction taking the as- of an unlimited medium describes a reality well enough [1]. The sumptions of axial-symmetric tube and incompressible fluid into investigation of the given problems in particular, will show us, as account. Some related models were established in the work far as external boundary of the medium must be situated in or- citeQuarteroni. der to ignore its influence, and consider that it “already left on Taking more detailed estimates under consideration the dif- infinity”. The problems with similar boundary conditions were ferential link between pressure in liquid flow and tube radius is already considered including in [2, 3], but we are especially inter- found via the variational principle. The system of nonlinear PDEs ested in a question on a limit of removal of the external boundary for describing velocity, pressure and tube radius perturbations in (a- infinity). a liquid flow through a viscoelastic tube is derived. Limiting problems for infinite or semi-infinite plates and shells Using the rescaling approach the family of nonlinear evolution are usually much easier, than the problems for finite plates and PDEs for describing pressure, velocity and radius disturbances shells and in many cases they suppose explicit solutions. There- is obtained for the different spatio-temporal scale. This family fore it is very interesting to know the conditions (as far as long consists Burgers, KdV and reduced Kuramoto-Sivashinsky (rKS) should be a or b), allowing to be limited to cases a- infinity or b- equations. infinity . The first equation can be linearized by the Coul-Hopf substi- Let us consider the rectangular plate [0 ≤ x ≤ a, 0 ≤ y ≤ b, tution. The second equation possesses the Painleve property and −h ≤ z ≤ h], uniformly compressed towards OY by the load can be integrated by the inverse scattering method. The third, of intensity P ∗. The equations of instability problem are selected rKS equation is not an integrable one. We had it does not pass the according to the theory of thin plates, based on the Kirchhoff Painleve test. Applying the Simplest Equations method hypothesis. citeKudryashovbook some exact solutions expressed via rational Two variants of boundary conditions are considered. Its ac- and Jacobi elliptic functions are presented. cepted that the loaded edges at y =0,bare simply supported, the The numerical studying of nonlinear pressure waves described edge at x =0is free, and then: by Burgers and rKS equations with the finite-difference implicit 1. the edge x = a is simply supported scheme is conducted. The difference between dissipative pro- 2. the edge x = a is fixed cesses for regimes related to these equations is highlighted. If the solution satisfies this condition is exist, we can say that There are three main stages for flow through a viscoelastic the located instability of plate takes place. The conditions of ex- tube. On the first stage the viscosity of fluid dominates and Burg- istence of the located instability are received. On the base of

33 these conditions the tables describing the dependence of stability Exact solutions of some problems were found. Comparisons parameters from the Poissons ratios and the correlation of plate of different models were given. The speed smog propagation was sides are brought. When a/b −→ ∞ the equations of the located obtained. As result of analytical solution the law of motion pol- instability problem of semi-infinite plate-strip with free edge are lution boundary is presented. We need to have these expression received. The conditions describing, as far as the external bound- to predict pollution movement. ary of the medium must be situated in order to ignore its influence M.A. Chmykhov, Moscow Engineering Physics Institute (State are received. The tables describing the asymptotics, when are re- University), 31 Kashirskoe shosse, Moscow, 115409, Russia sulted. REFERENCES NANO SCIENCE AND ENGINEERING IN MECHANICS [1] M.V. Belubekyan, “The problems of the located instability of plate”, The problems of optimum control, stability KEN P. C HONG and durability of mechanical systems, Yerevan: Publishing House of the Yerevan University, p. 95-99, 1997, (in Rus- sian). The transcendent technologies include nanotechnology, micro- [2] S.P. Timoshenko, “Stability of rods plates and shells”, M. electronics, information technology and biotechnology as well Nauka, 808 p. 1971, (in Russian). as the enabling and supporting mechanical and civil infrastruc- [3] A.S. Volmir, “The stability of elastic systems”, M: Gos. Iz- ture systems and materials. These technologies are the primary dat. fiz-mat. literaturi, 880 p. 1963, (in russian). drivers of the twenty first century and the new economy. Me- [4] N. A. Alfutov, “The bases of calculation on stability of elas- chanics and materials are essential elements in all of the tran- tic systems”, M.: Mashinostroenie, 336p. 1991, (in Rus- scendent technologies. Nanotechnology is the creation of new sian). materials, devices and systems at the molecular level - phenomena associated with atomic and molecular interactions strongly E.O. Chil-Akopyan, Institute of Mechanics of National Academy influence macroscopic material properties [according to I. Aksay of Sciences of Armenia 24-B, Marshal Bagramyan Avenue, Yere- of Princeton]; with significantly improved mechanical, optical, van, Armenia 375019 chemical, electrical... properties. Research opportunities, education and challenges in mechanics and materials, including multi- ANALYTICAL AND NUMERICAL SOLUTIONS OF scale modeling, nanomechanics, carbon nano-tubes, bio-inspired SMOG PROPAGATION PROBLEMS materials, coatings, fire-resistant materials as well as improved engineering and design of materials are presented and discussed MIKHAIL A. CHMYKHOV NIKOLAI A. KUDRYASHOV in this paper. K.P. Chong, Mechanics and Materials Directorate for Engineer- [email protected], [email protected] ing National Science Foundation Arlington, VA 22230, U.S.A.

The problem of the smog propagation is very important for cities with the advanced chemical industry. Formation of pollu- WAVE PROPAGATION IN A COMPOSITE PLATE tion level of cities is determined by the emission intensity, and LOADED BY A VISCOUS FLUID meteorological conditions. We call smog as gas mixture with ALEXANDER V. C HUBINSKIY firm and liquid particles. We assume that the smog density is more than the air density. alex [email protected] The present work is devoted to the study of smog propagation problems. Analysis of wave propagation in a wave-guide, which consists First of all we use the theoretical model similar to the shallow of two parallel infinitely long plates with a fluid between them, is water theory. In this case a hyperbolic system of equations for de- a classic problem in stationary structural acoustics. In the most scription of smog propagation is applied. We suppose that speed publications on this subject, a classic Kirchhoff plate model and of smog movement is determined by the smog height. These a standard model of a compressible fluid (an acoustic medium) equations are written taking cylindrical coordinates into account. are used. However, in many technical applications, for example We suppose that smog is not mixed with air. We insert force in marine engineering, fluid-loaded structures are composed of of environment resistance in equation of impulse conservation. sandwich plates. Moreover, these plates are often assembled so Here the linear law of resistance is used. We have obtained an- that the layer of a fluid between them is fairly thin and the viscous other model without inertial effects in equation of impulse con- effects in a fluid may be pronounced. The research reported in servations. These equations are met at problems of underground this paper is aimed to study an influence of viscosity on the wave hydrodynamics. propagation in the duct bounded by a sandwich plate from one We considered two problems. One of them was the smog prop- side and by a rigid wall from the other side. agation problem taking instantaneous point source into consider- The theory of sandwich plates suggested in reference [1] and ation. Another problem was the smog propagation in the case of [2] is used for the analysis of relatively low-frequency vibrations. the smog flow. These problems were solved by the numerical and It deals with coupled flexural and shear waves, which could be ei- analytical approach. For numerical solution of the first problem ther of an evanescent or of a travelling type (depending on the fre- we have used the well known Flux Correction Transport method. quency excitation). The theory proposed by A.N. Guz (ref. [3]) We also have found some analytical solutions of the first and sec- is used for description of behaviour of a viscous fluid. Unlike the ond problems. Some of these problems were solved by using self standard model of an acoustic medium, which dynamics is de- similar variables. scribed by the scalar potential, the vector potential is introduced

34 in this theory. Therefore, dynamics of a viscous fluid is described ACCURACY ANALYSIS AND 2D NUMERICAL by equations for a scalar and a vector potential. The transcenden- SIMULATION OF REVERSED WAVE FOCUSING BY tal dispersion equation is derived from two equations of dynamics ULTRASONIC WAVE PHASE CONJUGATION of a viscous fluid, two equations of vibrations of a sandwich plate and continuity equations formulated at the interface between a B. CVETKOV P. VOINOVICH fluid and a plate, which involve the amplitudes tangential and normal stresses and velocities. The efficient numerical technique of [email protected], [email protected] solving this equation is developed, in which viscosity parameter is used as a small perturbation parameter. Dispersion curves are The effect of parametric wave phase conjugation (WPS) in ap- thoroughly analyzed and compared with those obtained for a duct plication to ultrasound acoustic waves in magnetostrictive solids with two rigid walls and for a duct with a Kirchhoff plate wall and has been addressed numerically by Ben Khelil et al (2001) in 1D a rigid one. Several similarities and differences between location and by Voinovich and Merlen (2004) in 2D unsteady formula- of dispersion curves in various cases are identified and explained tion. Here we elaborate earlier results by Voinovich and Merlen in the course of analysis. The detailed parametric study of the (2004). A comprehensive accuracy analysis of the method has influence of several parameters, e.g., fluids viscosity and the pa- been performed confirming theoretical second-order accuracy of rameters of sandwich plate composition, on location of dispersion the numerical scheme. Focusing of the amplified conjugate wave curves is performed. As a reference case, a limit situation, when at the location of the incident wave source has been analysed de- a sandwich plate is loaded by an unbounded volume of a fluid at pending on the coupling modes between the elastic properties of one side is also studied. the magnetostrictive solid and the external magnetic field, and on REFERENCES the boundary conditions at the solid walls. [1] Skvortsov, V.R. Transactions of the Academy of Sciences of Ben Khelil S. et al (2001) J. Acoust. Soc. Am. 109 (1) Russia, Mechanics of a Rigid Body 1, 162-168, Symmetri- Voinovich P. and Merlen A. (2004) APM-2004 cally inhomogeneous through thickness plate as a sandwich P.A. Voinovich, St.Petersburg Branch of Joint Supercomputer plate having a soft core, 1993 (in Russian). Center, RAS, Politekhnicheskaya 26, St.Petersburg, 194021, Rus- [2] Sorokin, S.V., Peake N. Vibrations of Sandwich Plates With sia Concentrated Masses and Spring-Like Inclusions, Journal of Sound and Vibration, 237(2), 203-222, 2000. ENERGY FLOW AND SYNCHRONIZATION IN THE [3] Guz, A.N. Mechanics of the Brittle Fracture of Materials ENERGY SPACE with Initial Stresses, Naukova Dumka, Kiev, 1983 (in Rus- sian). ARTUR DABROWSKI A.V. Chubinskiy, St. Petersburg State Marine Technical Univer- [email protected] sity, St. Petersburg, Russia During the motion of vibrating systems, continuous energy changes from the potential energy into the kinetic one and vice ON THE USE OF AUXILLARY SURFACES IN versa occur. There can also exist an energy flow between some COMPUTATIONAL PLASTICITY RETURN MAPPING parts of the system, an energy flow through the system, energy flow synchronization, an energy dissipation, accumulation of en- ROGER CROUCH TIANBAI LI ergy in the system or some parts of it. An energy flow modelling [email protected] still arouses interests in the scientific world. Different methods are applied to solve problems connected with energy flow: Sta- At the core of elasto-plastic structural analyses lies the task tistical Energy Analysis [1, 2, 3], Finite Element Method [4, 5]. of returning stress increments (and the up-dated internal history But these methods do not allow for a special kind of a geometri- variables) given strain increments at each sampling point within cal view on energy changes, which could develop our intuitional a dicretised continuum. The efficiency and stability of this inner- knowledge on energy flow phenomenon. This intuition is very most loop of the global algorithm has a direct influence on the important especially in modelling the systems, where we still can accuracy and robustness of the structural analysis. not measure energy flow, such as telepathy - the problem of en- This paper presents recent findings that illustrate the benefits ergy flow between two electromagnetic systems, bioenergy (ob- of operating in a ’work-mapped’ stress space rather than conven- served thanks to Kirlian effect [6,7]), and so on. Nowadays one tional stress space. It is shown that there is strong graphical in- of the most important tools used in dynamical system investi- terpretation associated with the return path and in certain useful gations is the phase space. In the case of vibrating mechanical cases closed-form solutions exist, removing the necessity for iter- systems, one can observe the motion of the system, changes of ative schemes. For those cases where an analytical solution is not velocities and displacements of some parts of the system, using possible, the paper offers a strategy for arriving at the correct re- the phase space. One can conclude about the character of the turn state (via an augmented Newton-Raphson scheme) through motion, observe the trajectory of the motion, and it allows for the use of auxillary surfaces and ’predictive hardening’. the intuitional, geometrical view on systems dynamics. In this Examples of iso-error maps and iteration counts for trial stress presentation the conception of energy space is proposed, and dif- states far removed for the yield surface are given. This work leads ferent types of the energy flow and synchronization are shown. to significant benefits in large-scale non-linear structural analysis. Presented energy space allows for a geometrical view on energy R. Crouch, University of Sheffield, Mappin Street, Sheffield, changes in mechanical vibrating systems. This space posses all S13JD, United Kingdom the advantages of the phase space but it shows also an amount of

35 the energy accumulated in the system, the energy changes, flow, A. Dabrowski, Technical University of Lodz, Stefanowskiego1/15, dissipation, synchronization, energy attractors. The construction Lodz, 90-924, Poland of the energy space is based on the phase space. It would be better to understand the idea of this construction if we say, that the energy space is almost squeezed and stretched phase space. COMPLEX VARIABLE BEM FOR 2D PROBLEMS OF It is made in the way that if the system has the constant energy, HEAT, CURRENT OR FLUID FLOW WITHIN, ALONG then the trajectory of the motion lies on the multi-dimensional AND ACROSS STRUCTURAL ELEMENTS sphere. The radius of this sphere in the proposed energy space is equal to the square root of the total energy accumulated in the ANASTASIA A. DOBROSKOK ALEXANDER M. LINKOV system. Thanks to that one can conclude about the energy flow, AHMAD GHASSEMI accumulation and dissipation from the change of the radius of the [email protected], trajectory. Some special advantages brings the consideration of [email protected] the energy planes which are the equivalents to the phase planes. They are the projections of the trajectory onto the energy planes. The work presents the complex variable (CV) BIE and the CV- One can consider the plane showing the energy accumulated in BEM for a piece-wise homogeneous medium when 2D flow of some part of the system. More often it would be the potential heat, liquid or current occurs both in structural blocks (grains) energy on the horizontal axis and the kinetic one on the vertical and along and/or across their boundaries. Cracks, thin conductive axis. From the image of the trajectory projection onto that plane, channels or thin insulators are particular cases of the boundaries. one can see energy changes from kinetic into potential, whereas In general, we impose contact conditions, which include (i) the radius of this projection shows the total energy accumulated the discontinuity and the average value of a potential (tempera- in that part of the system. Moreover, one can conclude about all ture, head, voltage) and also (ii) the discontinuity and the average aspects of the motion, like from the phase plane. It is possible value of the flux. Consequently, in general, CV-BIE, singular and thanks to the fact that the kinetic energy is the function of ve- hypersingular, are used. Their solution is obtained by applying locity and the potential one the function of displacement. From the CV-BEM with complex variable basis functions. The latter that point of view the energy plane is almost the phase plane with are taken of the Lagrange type what allows us to have coefficients displacement on the horizontal axis and velocity on the vertical of the approximation. This simplifies computational procedures. one. In that, simple way one can obtain the energy space, which All the integrals over an arbitrary curvilinear element are evalu- allows for a new, geometrical view on energy changes in vibrat- ated by using efficient recurrence formulae. Thus, we obtain a ing systems. Proposed energy space can be applied to the linear tool for solving the problems considered. and nonlinear vibrating systems. In the presentation the appli- Heat extraction from a fracture in hot dry rock is considered cation of the energy space to the mechanical vibrating systems, as an example. In this case, the temperature is constant across a with no restrictions on the nonlinearities and discontinuities of fracture, while the flux discontinuity is proportional to the tangent the other types, such as nonlinear springs, nonlinear damping, derivative of the water temperature. The only (singular) equa- impacts, friction etc is discussed. tion is sufficient to solve the problem. Numerical results on the REFERENCES temperature distribution in a geothermal reservoir illustrate the [1] G.Maidanik, K.J.Becker, Dependence of the induced loss approach. We emphasize that the approach may also serve as a factor on the coupling forms and coupling strengths: linear basis for unsteady flow problems. analysis. Journal of Sound and Vibration (2003) 266, 15-32, A.A. Dobroskok, IPME RAS, 61 Bol’shoi pr., V.O., Saint Peters- [2] Y.Kishimoto, D.S.Bernstein, S.R.Hall, Energy flow model- burg, 199178, Russia ing of interconnected structures: a deterministic foundation for statistical energy analysis. Journal of Sound and Vibra- tion (1995) 183(3), 407-445, ABOUT STRUCTURE FORMATION SELF-ORGANIZATION AT LOADING OF [3] B.R.Mace, Power flow between two continuous one- QUASIBRITTLE MATERIALS dimensional subsystems: a wave solution. Journal of Sound and Vibration (1992) 154, 289-320, V.I. DOVGAN D.A. KITAEVA YA.I. RUDAEV [4] H.D.Xu, H.P.Lee, C.Lu, Numerical study on energy trans- [email protected], [email protected] mission for rotating hard disk systems by structural intensity technique. International Journal of Technical Sciences (2004), 46, 639-652, The problem of the model formulation suitable for the description at axial compression full diagram of deforming materials, in [5] H.H.Jang, S.H.Lee, Free vibration analysis of a spinning which loosening component to elastic and plastic deformation is flexible disk-spindle system supported by ball bearing and added, is considered. In particular, among the stated materials are flexible shaft using the FEM and substructure synthesis. initial heterogeneous mediums as the rocks and artificial building Journal of Sound and Vibration (2002) 251, 59-78, materials, for example concrete. Such materials are in station- [6] J. Opalinski, Kirlian type images and the transport of thin- ary state, which stable for small disturbances, after the limit of film materials in high voltage corona discharges, Journal of elasticity can be interpreted as dissipative structures. Their the Applied Physics (1979) vol.50 no.1, 498-504, deformation and destructions processes are analysed as instabil- [7] N.G.Bankovskiv, K.G.Korotkov, N.N.Petrov, Physical ity hierarchy, resulting from self-organization. Methods of math- imag-forming processes in gas-discharge visualisation (th ematical theory of the catastrophes are attracted for the model Kirlian effect)(reviev), Soviet Journal of Communications construction. The energy state function is presented as the sum and Technology (1986) v31, 29-45. of the potential function, which responsible for reversible defor-

36 mation, and disturbance. In the last parameter (the control param- of LS-DYNA as a tool to predict FOD damage. Different im- eter), connected with damaging, is introduce. Responsibility for pact cases will be presented, with particular enphasis on the in- structurization process was entrusted on imperfection parameter. fluence of damage size on the residual stress distribution. X-Ray The state equation is received by minimization to energy function and synchotron methods for residual stress measurements will on the order parameter and is supplemented with kinetic equation be discussed and some preliminary results presented and com- for imperfection parameter. Availability attraction of the syner- pared with the finite element predictions. Further, more recent getic methods for problems of the formulation physically well- experimental results will be presented in which FOD damaged grounded nonlinear defining equations is shown. specimens are tested in as-received and stress relieved condi- Yakov I. Rudaev, Kievskaya str., 44, Bishkek, 720000, Kyrgyzstan tions. These results underline the key role of residual stress in controlling post-FOD fatigue performance. Methods of incorporating residual stresses into a short crack arrest analysis will be discussed. DYNAMICS OF DEPLOYMENT OF A MULTILINK P. D u o,« Department of Engineering Science, Oxford, OX1 3PJ, DESIGN IN ORBIT United Kingdom

V.I. DRANOVSKIJ V.S. KHOROSHILOV A.P. KOVALENKO A.E. ZAKRZHEVSKIJ TWO REACTION-DIFFUSION MODELS DESCRIBING [email protected], [email protected], THE EVOLUTION OF INTERRELATED POPULATIONS [email protected], OLGA YU.EFIMOVA NIKOLAI A. KUDRYASHOV [email protected] [email protected], [email protected] Transformable designs in orbit are one of basic components of modern space vehicles. Such designs may be not always inves- The development of population with allowance for the diffu- tigated within the framework of the classical theory of systems sion migrations, mortality and under limited food conditions can of bodies, especially at taking into account their elasticity. Here be described by the equation the generalized mathematical model is constructed and computer = + 2 − modeling is carried out of the dynamics of the space vehicle dent D n αmn kn (1) ploying in orbit multilink pantograph design as the gravitational where n is the concentration of animals, D is a diffusion constant, stabilizer carrying solar arrays. k is a death-rate and m is the quantity of available food. If m The objective of this research is generalization of the theory of linearly depends on the concentration of animals, than equation multibody systems, containing multilink elastic designs. (1) is the Burgers-Husley equation. The case of program changing of the pantograph configuration If we describe the evolution of populations of predators and is considered. The behavior of a system is investigated at deploy- their preys, a death-rate of preys depends on the concentration ment and retrieval depending on design features of a pantograph, of predators, and the quantity of food, available for predators, its elasticity and character of friction in joints. Illustrative exam- defines by the concentration of preys. ple is given. The development of two populations competing for food can Alexandr E. Zakrzhevskij, Institute of mechanics NAS of Ukraine, also be described by the equation (1), where the quantity of avail- 3, Petra Nesterova Str., Kiev, 03057, Ukraine able food depends on concentrations of animals of both populations. The present work is devoted to the study of these two sys- PREDICTION OF FOREIGN OBJECT DAMAGE tems, describing the evolution of interrelated populations. Under PIERANGELO DUO« some conditions the exact partial solutions of these systems were found. Also several problems were solved numerically. [email protected] O.Yu. Efimova, Moscow Engineering Physics Institute (State Uni- versity), Kashirskoe shosse, 31, Moscow, 115409, Russia FOD is a major cause of engine removal and repair, particularly from military aircraft. Manufacturers are then seeking to improve the FOD tolerance of engines at the design stage and THE BIOMECHANICS OF INJURIES OF ANKLE-JOINT thereby reduce the (significant) costs of ownership. A design LIGAMENTOUS APPARATUS methodology is, therefore, required with which to predict the loss of fatigue life resulting from FOD on blades. M.R. ENIKEEV I.V. PENKOVA O.K. AGATYAN This paper describes an overview of the programme of work, V.A. SERGAN pursued at the University of Oxford, in collaboration with an industrial partner. The intention is to discuss a full range of experimental results and demonstrate the importance of incorporat- While carrying out a standard roentgenological examination ing the residual stresses in the predicition of experimental results. of a distant part of a shin and an ankle-joint a mistake is of- The overall philosophy of the approach includes using represen- ten made: the area of an injury is not situated in the center of tative blade-like specimens and ballistic impact damage in order a roentgenogram; in doubtful cases a comparative roentgenolog- to closely reproduce service conditions. ical examination of the opposite joint is not carried out (Ho K., Recent work has concentrated on the modelling side of the et al 1996; Loredo R., Sanders T.G., 2001). According to some problem. The paper will present several results showing the use authors, an examination of symmetrical joints allows to reveal

37 individual differences and abnormalities (Lesnyak O.M., 2004). 1985; Rockwood C.A., Green D.P., 1996). So the refusal of additional examinations of a joint in the oblique The breadth of the “nest” is the sum of external and internal projection and of carrying-out a roentgenological reexamination astragaloid surfaces. It is determined by the difference of the dis- in case of a degraded image is incompetent. tance between articular surfaces of ankle-bones and the breadth Three projections are used: anteroposterior projection, antero- of an astragaloid bone (Schatzker J., Tile M., 1987). Its value is posterior projection with internal rotation of a shin for 15-30 normally from 2 up to 6 mm. degrees (the so-called projection “nest”) and lateral projection 3) Side view. The exposure of the back and front displacement (Brusenskaya E.I., 2002; Grigory J., Patzakis M.J., Harvey J.P., of an astragaloid bone is possible. Normally longitudinal axis 1976; Wheelhouse W.W., Rosenthal R.E., 1980; Phillips W.A., of a shin-bone halves an astragaloid bone, and the breadth of a Schwarz Y.S., Keller C.S., 1985; Muller M.E., Allgower M., joint space in front and back parts is equal. Subluxation and front 1992; Rockwood C.A., Green D.P., 1996; Haraguchi N., Kato and back displacements of an astragaloid bone are distinguished F., Hayashi H., 1998). The roentgenography, made during the (Maikova-Stroganova V.S., Rochlin D.G., 1957). When we have weight bearing to ankle-joint ligaments, is used (Muller M.E., front displacement, the peak of the wedging of a joint space is Allgower M., Schneider R., 1992). In some cases, taking into directed onward, in case of back displacement - backward. The account the possibility of an individual reciprocal disposition of back displacement of a foot can combine with fractures of the bones, constituting an ankle-joint, the joint’s roentgenography of back edge of a shin-bone. an unaffected limb is required for comparison. 4) The roentgenography with the weight bearing to ankle-joint On the roentgenogram the following structures are examined: ligaments. It is used to diagnosticate fresh and chronic displace- 1) Tibiofibular syndesmosis. A normal tibiofibular diameter ments of ligaments of a joint (Shabus R., Orlanski V., 1997). In should not exceed 5 mm (the diameter between the inner cir- moot cases an analogous roentgenogram of an unaffected ankle- cuit of a splint-bone and the back edge of a fibular incisure of joint is carried out. The condition of splint-bone ligaments is a shin-bone) (Inman V.T., 1976; Phillips W.A., Schwarz Y.S., evaluated by a roentgenogram, carried out in the anteroposterior Keller C.S., 1985; Sclafani S.J., 1985; Harper M.C., Hardin G., projection with internal rotation of a shin for 15-20 degrees with 1988). “Superposition” of shadows of cannon-bones forms an the flexion of a foot for 10 degrees (Martin D.E., Kaplan P.A., overlapping triangle or a tibiofibular overlapping between the in- Kahler D.M., 1996). The feet are with the 8-10 kg weight bearing ner circuit of a splint-bone and the front edge of a fibular in- in the position of the supination or adduction. If the tilt angle of cisure of a shin-bone - 10 mm. Enlargement of a tibiofibular an astragaloid bone is appearing with the weight bearing, and it diameter indicates a complete or partial laceration of ligaments is more than 1 degree or twice more than that of an unaffected of the syndesmosis or a variant of the standard (comparison with ankle-joint, we can speak about the front astragaloid-peroneal an unaffected limb is required). (Inman V.T., 1976; Wheelhouse ligamentous disruption (Cox J.S., Hewes T.F., 1979; Glasgow W.W., Rosenthal R.E., 1980; Pettrone F.A., Mitchell G., David P., M., Jackson A., Jamieson A.M., 1980; Ahovuo J., Kaartinen E., Thomas F., 1983; Phillips W.A., Schwarz Y.S., Keller C.S., 1985; Slatis P., 1988; Rockwood C.A., Green D.P., 1996; Boardman Rockwood C.A., Green D.P., 1996). Joint space of an ankle-joint D.L., Lui S.H., 1997). For the evaluation of the syndesmosis should be equal: the difference in the breadth of external and in- injury the following projections are offered (Sheichtman A.G., ternal parts of a joint space of a joint should not exceed 2 mm, 1990): anteroposterior projection, lateral projection, anteroposte- and in the radial value - 1.5 degrees (Wheelhouse W.W., Rosen- rior projection with internal rotation of a shin for 10, 20, 30, 40, thal R.E., 1980; Rockwood C.A., Green D.P., 1996). 50 degrees, anteroposterior projection with internal rotation of a 2) The roentgenography with internal rotation of a shin (the shin for 30-40 degrees in upright position (for chronic injuries). projection “nest”). On the roentgenogram, carried out in this pro- Comparative roentgenogrammetry, roentgenography in two projection, the angle, formed by a line and a perpendicular, drawn jections with image magnification and image spatial filtering of to the articular surface of a shin-bone, is called a talocrural an- segment roentgenograms are recommended. Standard settings gle and it is 75-87 degrees, on average 83+/-4 degrees (Phillips such as lateral external projection, posterior and posterior projec- W.A., Schwarz Y.S., Keller C.S., 1985; Rockwood C.A., Green tion with internal rotation (for 10-15 degrees), used to examine D.P., 1996). an ankle-joint, cannot always present all necessary information This angle can be measured in a different way. This case, it for traumatologist. That’s why a patient with an ankle-shin injury is measured between the intermalleolar line and the line, drawn is recommended to have roentgen computer tomography (CT). along the articular surface of a shin-bone. In the case of this Michael Enikeev, Rostov State Medical University, Nachichevan- measurement the angle is 8-15 degrees (Rockwood C.A., Green ski, 29, Rostov-on-Don, 344022, Russia D.P., 1996). The tilt angle of an astragaloid bone can be measured in linear or radial values. The angle is formed by lines, drawn through the articular surface of a shin-bone and astragaloid bones. INVESTIGATIONS OF SOME NANO-STRUCTURES Its average value is 0 degree (from -1.5 up to +1.5 degrees) and it MADE OF ZNO is determined by the difference of the angles a and b. VICTOR A. EREMEYEV EVGENI M. KAIDASHEV The internal astragaloid surface is the distance between artic- ANDREW V. NASEDKIN ular surfaces of a medial malleolus and an astragaloid bone. Nor- mally it is 2 mm and it should not excced 4 mm. Enlargement [email protected], of an internal astragaloid surface for more than 4 mm testifies [email protected], [email protected] the laceration of a deltoid ligament (not necessarily complete) and the outer displacement (subluxation) of an astragaloid bone The elastic and piezoelastic properties of multilayered ZnO (Harper M.C., Hardin G., 1988; Pettrone F.A., Mitchell G., David thin films and nano-composites made on the base of ZnO nano- P., Thomas F., 1983; Phillips W.A., Schwarz Y.S., Keller C.S., crystals is investigated.

38 For the layer by layer growing multilayered epitaxial ZnO thin materials and investigations of their properties [1]Ð[4]. Semi- films on c-plane sapphire grown by multi-step pulsed laser de- conductor nano-crystals and lattices of these crystals are very per- position, the mathematical model is proposed which possess to spective materials for nanophotonics and nanomechanics. Char- optimize the stress filed in the upper layer of film. The finite ele- acteristic examples of such one-dimensional nanostructures are a ment simulation is applied for definition of thermal and piroelec- system of ZnO crystals on c-plane sapphire grown by multi-step trical stresses in multilayered films. The elastic and piezoelectric pulsed laser deposition or array of multi- or single-walled carbon behavior of ZnO films are investigated. The proposed model is nanotubes grown on metal substrate. based on the constitutive equations of pre-stressed piezoelectric In this work the mathematical model of system of micro- or bodies under small strains which take into account different phys- nano-crystals lattices Ð substrate is proposed. The model based ical properties of films layers. on the continuum mechanics approach. Each crystal modelled as The static and dynamic problems of 1-3 nanocomposites, an elastic beam and substrate modelled as an elastic half-space. which are based on ZnO nano-crystals lattice, are investigated. The longitudinal and bending oscillations of beam are consid- The composites with chaotic and regular distribution of nano- ered. For elastic half-space with the discrete system of beams, crystals are considered. The effective elastic, piezoelectric and surface waves of Rayleigh type are investigated with respect to dielectric constants are obtained. Some models of nanopiezo- material properties of the beam. As a special case of large den- composites of 1-3 connectivity are created using the effective sity of nano-crystals, the continuous distribution of elastic beams modules method. These models have heightened piezosensi- is considered and surface waves in this system are explored. From bility and low acoustic impedance in comparison with pure the physical point of view this system is equivalent to the elastic piezoceramic. Static, modal and harmonic analyses of 1-3 half-space with special coating. Mathematical statement of this nanopiezocomposite transducers are realized by using finite el- problem consists of motion equations and the boundary condi- ement method and computer packages. FEM packages ANSYS tions of third type describing crystalÐsubstrate interaction. The and ACELAN are used. model with surface coating is similar to proposed model of de- Some applications of multilayered films and nano-composites formed body with distributed oscillators [5]. The dispersion rela- for micro- and nano-sensors and other MEMS are considered. tion is obtained. The dynamic response to the changes in elastic This research was partially supported by RFBR. and geometrical properties of nano-crystal lattice or external in- V.A. Eremeyev, Southern Scientific Center of RAS & Rostov State fluences is discussed. University, Rostov-on-Don, Russia For ZnO nanocrystals on sapphire substrate, the results of dy- E.M. Kaidashev, Institute of Applied Mathematics and Mechanics namic processes modelling are obtained by using finite elements of Rostov State University, Rostov-on-Don, Russia method (FEM). The FEM package ACELAN [6] is used. A.V. Nasedkin, Rostov State University, Rostov-on-Don, Russia This research was partially supported by RFBR. REFERENCES [1] Springer Handbook of Nanotechnology. B. Bhushan (Ed.) FEM MODELLING OF 2D ZNO NANOCRYSTALS Springer. 2004. XXXVI, 1222 p. LATTICE [2] Handbook of Nanoscience, Engineering, and Technology. VICTOR A. EREMEYEV ARKADIY N. SOLOVIEV W. A. Goddard et al. (Eds.) CRC Press LLC. 2003. [3] C. P. Poole, Fr. J. Owens. Intoduction to Nanotechnology. [email protected], [email protected] John Wiley. 2003. (Russian translation Ð Technosfera. 2004. 328 p.). Within framework of plane linear theory of piezoelasticity the dynamic and stationary problems for ZnO nano-crystals lat- [4] E. Rosencher, B. Vinter. Optoelecronique.« Dunod. Paris. tice based on the sapphire substrate is investigated. Finite ele- 2002. (Russian translation Ð Technosfera. 2004. 592 p.). ment package ACELAN is used. The natural frequencies and [5] V. A. Palmov. Oscillations of elastoplastic bodies (in Rus- amplitude-frequency curves of system are obtained. Sensitiv- sian). Moscow: Nauka, 1976. 326 p. ity of dynamic properties of system nano-crystal latticeÐsubstrate [6] A. B. Belokon, V. A. Eremeyev, A. V. Nasedkin, A. N. with respect to external influence is researched. Soloviev//PMM. 2000. Vol. 64. N 3. P. 381Ð393. An application of nano-crystals lattice system to nanosensors V.A. Eremeyev, Southern Scientific Center of RAS & Rostov State producing is discussed. University This research was partially supported by RFBR. A.N. Soloviev, Don State Technical University & Rostov State V.A. Eremeyev, Southern Scientific Center of RAS & Rostov State University University, Rostov-on-Don, Russia A.N. Soloviev, Don State Technical University & Rostov State University, Rostov-on-Don, Russia PHASE TRANSFORMATION OF A CYLINDRICAL INCLUSION SUBJECTED TO UNIFORM EXTERNAL STRAIN FIELD ON ELASTIC MICRO- AND NANO-CRYSTALS LATTICES MODELLING NADEZHDA YU.EVLAMPIEVA ALEXANDER B. FREIDIN ELENA N. VILCHEVSKAYA VICTOR A. EREMEYEV ARKADIY N. SOLOVIEV [email protected] [email protected], [email protected] The presented work is investigation of phase transformations Development of nanotechnology depends on producing new in a cylindrical inclusion under the homogeneous external field

39 transmitted by a linear elastic matrix. If phase transformation stresses and deformations of porous material for rarefied struc- takes place in some part of the inclusion, phase boundary can be tures under given conditions loading (clean shift, uniaxal sprain). considered as surface across which the deformation gradient is Diagrams of deformation for structures with spherical and el- discontinuous. On the equilibrium interface an additional ther- lipsoidal inclusions, dependences of the conditional and un- modynamic condition is put on the boundary in addition to the conditional moments of structural stresses and deformations on conventional kinematic and force continuity conditions. This is a macrodeformations are constructed. Research is carried out de- problem with unknown boundary and the thermodynamic condi- pending on a volume fraction and depending on a kind of lin- tion acts as the restriction on the geometry of the interface. Equi- ear hardening. Comparison of behaviour elastoplastic composites librium two-phase states of the inclusion on the various loading with spherical and ellipsoidal pinhole is carried out. paths are studied. Researches are carried out with the use of finite Work is executed at financial support of the Russian Founda- elements program ANSYS. Shape and orientation of new phase tion of basic researches (the grant No. 03-01-00394). nucleus depend on the external field due to potential energy min- Natalia V. Evlampieva, Perm State Technical University, Komso- imization requirements. The results are compared with cylindri- molskiy prospekt 29A, Perm, 614000, Russia cally symmetric two-phase states which were considered earlier [1]. This work is supported by Fundamental Research Program of CONSTRUCTION ANALYTICAL COMPUTATIONAL RAS. GRIDS WITH GIVEN PROPERTIES REFERENCES NADEGDA B. FEDOSENKO [1] E.Vilchevskaya, A.Freidin. On phase transformations of an inclusion in an external strain field, Proc. XXXII Summer [email protected] School APM-2004, IPME RAS, St. Petersburg, 447-454. N.Yu. Evlampieva, SPbSU, Bibliotechnaya pl. 2, Petrodvorets, The method of construction of grids is suggested, in which St.Petersburg, Russia the function mapping computing space on physical one is a linear combination of the before obtained final analytical solutions of boundary value problems for some elliptical equations [1-4]. STUDY STRUCTURED GEOMETRIC AND PHYSICAL Thereby, this function is guaranteed analytical, but the formula- FIELDS IN PROBLEMS OF DEFORMATION OF tion of an equation to which it corresponds is not required. The ELASTOPLASTIC COMPOSITES method has no limitation on dimension of space. The received grids have all advantages of elliptical grids, but N.V. EVLAMPIEVA A.A. TASHKINOV are under construction almost instantaneously, as the construction [email protected] do not required the solution of some equation. For example, the three-dimensional grid of 6 million cells is Studied casual fields by the stresses and deformations in prob- written into the file in 3 sec. on the computer 200 MHz Pentium lems on elastoplastic deforming matrix composites with spher- This property is especially expedient at calculations of problems ical and ellipsoidal cut-ins. Considered elastoplastic marginal with moving boundaries. The method of control of grid proper- problem for structured inhomogeneous medium with regard for ties is suggested and three kinds of adapting of are realized: 1) moment functions by the module elasticity. Decision of prob- control of general character of behavior of grid lines 2) control of lem is found in integro-differential equations by the method the behavior of grid lines with going far from boundary 3) control of functions Grina in full correlation approach. The method of sta- behavior of grid lines in a given points The method is illustrated tistical averaging of system of the equations of balance elasto- by examples of 2D and 3D grids. plastic isotropic composite material is applied, within the limits REFERENCES of volume of each component of a composite the pulsations of [1] N.Fedosenko and E.Sokolov, Application of exact solu- deformations worth in an atmosphere of material functions are tions of some elliptic equations for generation of two- and neglected. The elastoplastic regional problem is solved with the multidimentional analytical grids. Proceedings of ALGO- help of numerical integration. At the calculation of moments of RITMY 2002 Conference of Scientific Computing, pp. 253- fields of stresses and deformations are used approximations of 259. moment functions by the structured module of elasticity of sec- [2] N.B. Fedosenko. Analiticheskoe reshenie zadachi stat- ond and third orders. sionarnogo raspredeleniya temperatury v N-mernom For this purpose conducted geometry study of statistical uni- pryamougol’nike s proizvol’nymi granichnymi usloviyami form isotropic fields by the structured module of elasticity. Con- i nalichiem istochnikov ili stokov tepla. P. 104 // II sidered different types of synthesized structures, for which in nu- Russian Conference. Irreversible processes in nature and merical experiment are built moment functions by the module technology. Abstracts. January 22Ð24, 2003. Moscow. (in elasticity of second and third orders. Installed differences in kind Russian). in the nature of fading the moment functions, built for structures with spherical and ellipsoidal cut-ins. Offered general for struc- [3] N.B. Fedosenko, Analiticheskoe reshenie zadachi o tures with the different volume contents an expression for ap- medlennom techenii vyazkoy zhidkosti v oblasti s proximations of moment functions by the structured module of proizvol’nymi granichnymi usloviiami. P. 167Ð168 // Inter- elasticity. national Conference on Mechanics. III Polyakhov’s read- As an example of the decision of a regional problem porous ings. February 4Ð6, 2003. St. Petersburg. (in Russian). materials with physically nonlinear matrix with linear harden- [4] Sokolov E.I., Fedosenko N.B. Ob odnom metode ing are considered. Received new numerical results on fields of postroeniya ellipticheskih setok, zadavaemyh yavnymi

40 analiticheskimi vyrazheniyami // Matematicheskoe mod- REFERENCES elirovanie. Vol. 15, N 6, 2003. P. 101Ð106 (in Russian). [1] A.V. Fedotov and S.D. Furta. On stability of motion of a chain of n driven bodies. Reg. and Chaot. dyn., 2002, vol. 7, N.B. Fedosenko, Center for Advanced Studies, Saint-Petersburg No. 3, p. 249Ð268. State Technical University, Politechnicheskaya,29, Saint- Petersburg, 194251, Russia Alexander V. Fedotov, Moscow State Institute of electronic technology, Solnechnaya alleya, 5, Moscow, 124498, Russia

AN APPROACH FOR INCOMPRESSIBLE FLOW A NUMERICAL-ANALYTICAL TECHNIQUE FOR CALCULATION BASED ON ANALYTICAL SOLUTIONS SOLVING PROBLEMS OF MATHEMATICAL PHYSICS

NADEGDA B. FEDOSENKO V.P. F EDOTOV L.F. SPEVAK V.B. TRUKHIN [email protected] V.V. P RIVALOVA [email protected], [email protected], It is known, that the two dimensional non-steady equations of [email protected] viscous incompressible flows are reduced to one equation of the fourth order concerning a stream function. The outcomes of the Practical problems of mathematical physics (elasticity, plastic- given work are obtained at direct usage of this equation. ity, heat conduction, diffusion etc.) require both a high accuracy The analytical solution of a boundary value problem for the of the solution and a considerable computation speed, often on Stokes is esteemed. As against the known solutions of infinite line. The paper presents developed parallel algorithms for solv- series, is suggested the solutions in a final form. ing problems of the kind by a technique enabling a solution to The main attention is given to statement of boundary condi- be obtained at a speed many times exceeding that of numerical tions. The main difficulties arise in a three-dimensional case. By finite-element computations of equivalent dimensionality, the ac- analogy with the introducing of a stream function, is approach curacy being the same or even better [1]. with the introducing of a vectorial potential. The calculation technique is based on the boundary element In this approach the initial system will be converted to three method [2, 3]. The required functions (velocity, strain, stress, equations at the component of a vectorial potential. Equations temperature, flow, concentration etc.) are represented analyti- of a system it appears by bound only through boundary condi- cally inside the body in terms of the values on the boundary. The tions (each speed depends on derivative a component of a vecto- boundary is partitioned into elements straight line segments or rial potential on two directions.) It creates the main difficulties at circular arcs for plane problems and planar triangular or rectangu- closing of a boundary value problem. lar elements for space problems. Inside an element the quantities In the given work the path of solving these difficulties is sug- are considered to be constant. This only numerical approximation gested. The solutions are obtained not only in two dimensional, exerts little effect on the solution accuracy, as all the subsequent but also in a three-dimensional case are obtained. operations are performed analytically; besides, high computation The computational method of equations with the convective speed enables a fairly fine boundary partitioning to be used. terms is offered. Thus the results obtained for the solution of Thus, to obtain the analytical representation of required quan- equations of the Stokes will directly be used. tities inside the body, one needs to determine their values on the All results are widespread on cases with arbitrary domains. boundary elements. Some of them are specified by the bound- For this purpose the special analytical transformation of coordi- ary conditions of the problem. The missing values are found nates is obtained. from a system of linear algebraic equations whose coefficients are N.B. Fedosenko, Center for Advanced Studies, Saint-Petersburg the integrals of the components of influence functions and their State Technical University, Politechnicheskaya,29, Saint- derivatives taken over the boundary elements. Analytical formu- Petersburg, 194251, Russia lae have been derived to compute the necessary integrals for the above-mentioned types of elements. As parameters, the formulae involve elastic moduli (heat conductivity and diffusion factors) and geometrical characteristics of the elements. This has enabled ON STABILIZATION OF RECTILINEAR MOTION OF A the formulae derived to be applied to an area of any geometry MULTILINK SYSTEM and with any elastic (heat or diffusion) properties. The formulae A.V. FEDOTOV S.D. FURTA obtained offer higher computation accuracy in comparison with numerical integration and significantly accelerate the process of [email protected] system matrix filling. For instance, when the boundary of the plane area is partitioned into 6000 elements, the process becomes The work is devoted to the problem of rectilinear motion sta- 2500 times as fast, the accuracy being higher. Note that the algo- bility of multi-link chains of rigid bodies moving in resisting rithms for system matrix filling and system solution can be easily medium. As a mechanical model, we study a system which con- parallelized, and this, along with the advantages of analytical in- sists of a material particle moving at a constant velocity and a tegration over numerical one, offers a considerable acceleration chain of n links attached to that particle by means of a weightless of computations. rod. We assume that the motion of the rigid link is restricted by a The values obtained on the boundary and the analytical inte- nonholonomic constraint. In [1] the authors give possible expla- gration formulae enable one to determine the process character- nation of flutter-type instability of rectilinear motion based on the istics inside the area analytically. In this case, even quantities re- influence of a nonintegrable constraint. We investigate the action quiring differentiation, like stress or flow, can be determined to a of additional spring attachment and dampers in the hinges on the stabilization of the chain motion. 41 high accuracy, which cannot beattained by classical computation AN EQUATION OF STATE FOR GAS HYDRATES IN techtiques. The characteristics can be computed independently HIGH PRESSURE REGION at a great number of internal points, and this enables a limitless level of parallelizing to be used here too. On the whole, with suf- VASILII M. FOMIN EVGENII I. KRAUS ficiently powerful actual computers, the algorithm obtained can IVA N I. SHABALIN offer on-line monitoring of structures at varying boundary conditions. A number of plane and space problems of deformation and Gas hydrates are crystalline molecular complexes formed from heat conduction have been solved by the technique proposed, mixtures of water and suitably sized gas molecules. The wa- among them problems for bodies with defects of various geome- ter (host) molecules, upon hydrogen bonding, form unstable lat- tries. tice structures with several interstitial cavities. The gas (guest) molecules can occupy the lattice cavities and when a minimum number of cavities are occupied the crystalline structure will become stable and solid gas hydrates will form, even at temperatures well above the ice point. In the present paper, we propose REFERENCES model equations for thermodynamic functions of ice and water [1] V.P.Fedotov, L.F.Spevak, T.D.Dumsheva, E.S.Zenkova, based on the dependence of the Gruneisen coefficient Γ(V,T) V.V.Privalova. Numerical-analytical method for solving on volume and temperature. The difference between the elastic problems of elasticity and heat conductivity // XXXII Sum- (“cold”) components of energy and pressure for water and ice is mer School Conference Advanced Problems in Mechanics. taken into account. Configurational entropy of a water providing Book of abstracts. St. Petersburg, 2004. Pp. 43-44. a measure of its disorder and resulting in the finite values of the total entropy in the zero-temperature limit is introduced. [2] C.A.Brebbia, J.C.F.Telles, L.C.Wrobel. Boundary Element Gas hydrate is modelled by mechanical mixter of hexagonal Techniques. Springer-Verlag Berlin Neidelberg New-York ice and gas. Tokyo, 1984, 524 p. E.I. Kraus, Institute of Theoretical and Applied Mechanics, 4/1 Institutskaya str., Novosibirsk, 630090, Russia [3] P.K.Banerjee, R.Batterfield. Boundary Element Methods in Engineering Science/McGraw-Hill Book Company (UK) LUNAR NET-LANDER: MISSION OVERVIEW Ltd, 1981, 494 p. M. FRANCIS D. BIAMONTI A. MCCALL Vladimir P. Fedotov, Institute of Engineering Science, Urals J.P. SANCHEZ G. ALBALADEJO D. MOORE Branch of RAS, 34, Komsomolskaya St., Ekaterinburg, 620219, S. GRAHAM C. MURRAY S. SHAHID Russia P. T RIVEDI D. SANTHANAKRISHNAN G. RADICE G. HUGHES

REPRESENTATION OF SOLUTIONS OF Previous Lunar missions performed by the Russians and BOUNDARY-INITIAL PROBLEMS FOR NONLINEAR Americans in the mid 20th Century developed a platform for sci- PARTIAL DIFFERENTIAL EQUATIONS BY SPECIAL entific exploration within our solar system. Even with the wealth SERIES of missions sent to the lunar surface, very little is known about this specific aspect of the closest neighbour to Earth. The mission MIKHAIL YU.FILIMONOV NATALIA A. VAGANOVA presented in this paper is a Flexi-class Lunar Net-Lander Mission [email protected] to the far side of the Moon for a sequence of experiments, with launch in 2015. The scientific objectives consist of geological studies of the surface and interior of the Moon, as well as inves- A method of special series with recurrently calculated coef- tigation of astrodynamic areas of interest and phenomenon in the ficients is presented. This method is an expansion of solutions close vicinity, such as the trans-lunar Lagrange point, L2. The into a series by the power of one or several special functions pro- success criteria were to achieve retrieval of data from the scien- vided the coefficients of the series to be computed recurrently. tific surveys as well as achieving payload effectiveness. Three This approach allows to obtain constructive solutions in the form spacecraft have been designed to meet the mission objectives; L2 of convergent series also for boundary-initial problems for a cer- orbiter that would separate from the main space bus and perform tain class of nonlinear equations. New theorems of existence of a stable manifold insertion to an L2 halo orbit, where it will carry classic solution for evolutionary Korteweg-de Vries equations are out experiments and act as a communications relay with Earth presented. Results of numerical equations presented for nonlin- for the Landers. A Micro satellite is required to perform map- ear wave equation. ping of the Moon to determine suitable landing sites for the lan- The work is supported by RFBR and interdisciplinary project ders. Three landing sites have been targeted and form a triangular of UrB and SB RAS. shape with each lander being over 1000km apart. The Landers M.Yu. Filimonov, N.A. Vaganova, Institute of Mathematics and will be deployed from the space bus after mapping is complete Mechanics UrB RAS, 16, S.Kovalevskaya str., Ekaterinburg, and land at their respective sites, after which time the rovers will 620219, Russia be deployed and mission operations continue for three-months.

42 Mark Francis, Department of Aerospace Engineering, Univer- guided rovers with a 10 meter roaming capability. A number of sity of Glasgow, University Avenue, Glasgow, G12 8QQ, Great theoretical design proposals, such as flat packed rocker-boogie Britain suspension, laser line-of-sight navigation, height adjustable lunar surface-contact payload structure and umbilical life support are also presented. End of life problems are also accounted for in LUNAR NET-LANDER: ORBITAL ANALYSIS the design. Maximum reliability and cost efficiency are achieved M. FRANCIS D. BIAMONTI A. MCCALL using a series of off the shelf, space proven parts in a variety of J.P. SANCHEZ G. ALBALADEJO D. MOORE configurations. S. GRAHAM C. MURRAY S. SHAHID Andrew McCall, Department of Aerospace Engineering, Univer- P. T RIVEDI D. SANTHANAKRISHNAN G. RADICE sity of Glasgow, University Avenue, Glasgow, G12 8QQ, Great G. HUGHES Britain

NONCLASSICAL POTENTIAL SYMMETRIES FOR Nested in the general practice of Mission Analysis and De- SOME DIFFUSION EQUATIONS sign, the actual considerations for a preliminary design of orbital related mission segments lead to a highly iterative process. MARIA L. GANDARIAS Trade-offs at system and subsystem level are performed to de- [email protected] sign a space system that meets requirements and constraints and achieves mission objectives. In this paper, a dedicated space bus In this work we consider a class of diffusion equations which and relay satellite are designed based on the requirements for a are of interest in mathematical physics. For some of these equa- Lunar-Net-Lander mission to the Moon, illustrating system bud- tions nonlocal potential symmetries are derived. These nonclas- gets, explaining subsystem designs and presenting a preliminary sical potential symmetries allow us to increase the number of conﬁguration for the whole space system. The main purpose of solutions. These solutions are neither solutions arising from non- the space bus is to carry and support 3 landers, a relay satellite classical symmetries nor solutions arising from classical potential and a microsatellite. Several interesting aspects of the Restricted symmetries. Circular 3-Body Problem for Earth-Moon system are exploited M.L. Gandarias, University of Cadiz,« Rio San Pedro, Puerto to achieve the orbiter relay satellite operational Halo orbit as well Real, 11510, Spain as to provide the correct insertion of the space bus into the lunar parking orbit for lander deployment. Careful considerations on ∆ planning V bursts are proven to successfully achieve a stable EFFECTS OF JOINT CONFIGURATION AND LOADING manifold insertion. Active and Passive stabilisation methods are TYPE ON STRESS CONCENTRATION IN OFFSHORE used during each phase of the mission, such as spinning motion WELDED TUBULAR JOINTS on the elliptical transfer orbit and gravity gradient on the lunar parking orbit. Finally, an appealing solution addresses the end of M. F. GHANAMEH D. THEVENET A. ZEGHLOUL life issues for the space system. Davide Biamonti, Department of Aerospace Engineering, Uni- [email protected], versity of Glasgow, University Avenue, Glasgow, G12 8QQ, [email protected], Great Britain [email protected]

In building offshore platforms, tubular members are used be- LUNAR NET-LANDER: SURFACE COMPONENTS cause of their good structural and mechanical properties. Tubu- lar members are connected to one another by special welded RANCIS D. BIAMONTI A. MCCALL M. F joints which represent structural discontinuities causing signifi- J.P. SANCHEZ G. ALBALADEJO D. MOORE cant stress concentrations in the vicinity of the weld. For some S. GRAHAM C. MURRAY S. SHAHID joints, the stress concentration can produce a maximum stress at P. T RIVEDI D. SANTHANAKRISHNAN G. RADICE the intersection as high as thirty times the nominal stress acting in G. HUGHES the members. Under repeated loading, fatigue failures always ini- tiate at the areas which referred to as “hot-spot” where there are high-level stress concentrations which called “hot-spot stress”. The proposed mission encompasses the design of three iden- For this reason, accurate computation of stress concentration is tical landers and rovers. The landers have two main purposes: very important in tubular joints design. surface exploration and experimentation. Lunar soft landing is The stress concentration depends on joint configuration and achieved using a bi-propellant four thruster-cluster and sixteen loading type. Because of this fact, a lot of effort has been con- monopropellant thrust vector (attitude control) thrusters. The centrated by many researchers on studying simple tubular joint Guidance, Navigation and Control system provides a fully auto- configuration subjected to different loading conditions. This pa- mated landing. A range of scientific test equipment is utilized to per presents a new study of different types of tubular joints. They provide spectrometer analysis, digital imaging and ground den- were subjected to Combined loading (Com): Axial (Ax) and ro- sity experiments. The proposed design encapsulates a number of tational bending “Combined Bending (CB)” loading since a lot novel technologies to overcome several mission constraints. The- of studies already done for the joints subjected to one or two of oretical designs for folding landing gear and other space saving the three types of loading applied to the tubular joints: “In Plane devices are presented. The design includes three identical cable Bending (IPB), Out of Plane Bending (OPB) and Ax loading”.

43 A tubular joint can be described by the use of five non- tem that exhibits certain vibratory behaviour. The earliest dis- dimensional parameters α, β, γ, τ and θ. Initially, a single crete inverse problems studied by Gantmakher and Krein, and the set of joint parameters was studied using various different joint analogous continuous problems studied by Gelfand and Levitan, types subjected to the four cases of loading Ax, IPB, OPB and fall into this category. The former studied sets of concentrated Combined loading. For numerical simulations, the finite element masses on a taut string, the latter the Sturm-Liouville differential package ABAQUS was used. This decision was based primarily equation. Both identified the data necessary to specify a unique on the degree of automation that it was possible to introduce to system, and provided rudimentary algorithms for constructing the the mesh generation, model solution and results extraction. system from the data. There are now efficient algorithms for solv- Mohamad F. Ghanameh, LPMM, Ile du Saulcy, Metz Cedex 01, ing such problems. 57045, France Now, inverse problems are interpreted in a looser sense: as the construction (not reconstruction) of a system, or a family of systems that exhibits certain vibratory behaviour — principally a COMPUTATIONAL DESIGN AND MECHANICAL set of natural frequencies — a so-called spectrum. This leads to CHARACTERIZATION OF MICROSTRUCTURED the important concept of an isospectral family- a set of systems of SOLIDS IN LAYERS a certain type that all have a common spectrum. Procedures for finding such isospectral families can be algebraic or differential. FRANCISCO A. GILABERT ANTON M. KRIVTSOV ANTONIO CASTELLANOS Discrete vibrating systems can be classified according to the connections between the parts; this leads to the idea of the graph [email protected] associated with a system. There has been steady progress in solving inverse problems, and constructing isospectral families of Computational designs of solids built from simplified mi- systems, on specific simple graphs, but there are still many open croscopic laws of interaction have become an extremely useful problems: related to more general graphs, non-spectral data, in- tool to model materials with prescribed macroscopic properties. variance of sign properties,, etc. Many of them are devoted to understand a multitude of not well known phenomena such as the plasticity, phase transitions, creeps G.M.L. Gladwell, University of Waterloo, 200 University Avenue propagation, etc. Nevertheless, these realizations still need high West, Waterloo, N2L 3G1, Canada performance calculations in view of the large number of particles. The present work reports a computational procedure to design multilayered bulk materials in the framework of the Molecular CONVECTIVE MOTION IN PLANE LAYERS WITH Dynamics simulations. The main goal is to model a computer INTERNAL HEAT SOURCES material with well defined mechanical properties considering a simple interatomic law of Lennard-Jones type. The novelty is IVA N V. G NEVANOV the internal structure, which is arranged in layers with different atom sizes. Given a fixed space volume to be occupied by the [email protected] solid, this structural arrangement allows to decrease drastically (∼ 60 − 80%) the necessary number of atoms compared with the Inclined plane viscous fluid layer is considered. The bound- case of a homogeneous solid. The result is a considerable speed- aries are at the same temperature, internal heat sources are up of the calculations. This work deals with two main aspects of present. Small perturbations related stability of motion and equi- this realization: first, the programming and numerical details nec- librium is investigated. The influence of uniformly distributed essary to make the calculations efficient, and second the physical heat sources on horizontal layer equilibrium stability and inclined considerations of the interaction law needed to obtain the desired layer motion stability is discussed in [1,2]. mechanical behaviour of the bulk material. We show that impos- This work examined the case of heat sources, concentrated on ing an appropriate scaling law between the interatomic potentials the layer axis. This situation may be implemented by transmitting from different layers, we obtain the same elastic moduli as if the electric current through the conductor in the fluid. material was homogeneous. F. A. Gilabert, A. Castellanos, Departamento de Electronica« y Extreme cases were investigated — horizontal and vertical lay- Electromagnetismo, Facultad de F«õsica, Avenida Reina Mercedes ers. Netral curves, describing the instability region, and overcrit- s/n. 41012 Sevilla, Spain ical motions pictures were found. Comparison were made with A.M. Krivtsov, St. Petersburg State Technical University, Depart- the uniform heat sources case. Linear stability theory and numer- ment of Theoretical Mechanics, Politechnicheskaya Street 29, ical calculations were used for this research. 195251 St.Petersburg, Russia REFERENCES

INVERSE PROBLEMS IN VIBRATION — PAST, [1] Gershuni G.Z., Zhuhovitsky E.M. Convective stability of PRESENT AND FUTURE incompressible ﬂuid // Moscow, Nauka, 1972, 392p.

GRAHAM M.L. GLADWELL [2] Gershuni G.Z., Zhuhovitsky E.M., Nepomnyaschy A.A. Stability of convective ﬂows // Moscow, Nauka, 1989, 320p. [email protected]

When interpreted strictly, inverse problems in vibration are I.V. Gnevanov, Perm State University, Bukireva, 15, Perm, concerned with the reconstruction of the unique vibrating sys- 614099, Russia

44 as 2D surface substructure transformation from the irregular in- MODELING OF THE RHEOLOGICAL BEHAVIOR OF termediate polycrystalline state to the regular surface state (in the FILLED ELASTOMERS ON STRUCTURAL GROUNDS shape of the initial grid-like arrangement of microscopic extrusions and intrusions). L.A. GOLOTINA L.L. KOZHEVNIKOVA After subsequent cyclic tension it is continued by the 2nd qual- [email protected], [email protected] itatively different stage of evolution with appearance of the qualitatively new tweed-like polycrystalline state in the bulk with the This paper is concerned with the study of grain composite elas- regular grid-like arrangement of microscopic pores ( 20-30 mi- tomeric materials filled with particles of size 10 micron at most. crons). They stimulate the whole disappearance of the Laue- There are different approaches to model the mechanical be- picture of initial single crystal and appearance of split Debay havior of these materials, of which one implies the use of repre- rings. Such X-ray picture conforms to polycrystalline structure sentative structural cells as a basis for constructing macroscopic with dimensions of crystals 1 micron. One can describe it continuum relationships. In our research we use a structural cell as 3D bulk substructure transformation from the regular surface in the form of elastomeric cylinder of height equal to its diameter state on the lower scale to the regular bulk state on the higher and with a rigid inclusion in the central part of the cylinder. The scale (the tweed-like polycrystalline state with the regular grid- cells subjected to elastic deformation are studied taking account like arrangement of microscopic pores)[Yu. G. Gordienko, M. V. of boundary conditions that keep their packing density under de- Karuskevich and E. E. Zasimchuk, Single Crystal ”Smart” Sen- formation. The mechanical evolution of structural cells caused by sors of Fatigue Damage (summary on the results of the scientific damage generation and propagation is investigated. To this end, a project INTAS-AIRBUS-1547-99), Proc. of Joint meeting NIS- series of boundary- value problems for structural cells are solved. participants INTAS, September 2002, Katsiveli]. Based on the obtained results, some peculiarities of the macro- The transformations are followed by appearance of the ”appar- scopic behavior of composites under extension and compression ent fractal behavior” [D.A. Hamburger, O. Biham, and D. Avnir, are explained. Phys. Rev. E 53, 3342 (1996)] in a range between physically A method is offered which allows the material structure to be relevant cutoffs. The equidimensional maps and pole plots (that described in terms of continuum relationships. With the method characterize anisotropic nature of surface relief) for ”the apparent it is possible to treat the obtained results by continuum mechanics fractal dimension” of a film surface were plotted. They allow not techniques.Calculated tensile curves agree well with the available only to find strain localization places, but also to envisage their data obtained from constant rate extension tests. However, it has dynamics in the structurally heterogeneous materials with a priori been found experimentally that the materials examined demon- known zones with different structure and hardness. The possible strate marked rheological properties depending on the structural non-linear jamming scenarios are considered for the transforma- material parameters such as filler sizes, volume concentration etc. tions and their relations with the observed scaling relationships Consideration of the viscoelastic properties of the binder (elas- are discussed. tomer) with the aid of the structural cell allows modeling of the Yuri Gordienko, Institute of Metal Physics, 36 Vernadsky Street, creep process, accompanying with vacuole generation and evolu- Kiev, 03142, Ukraine tion, in tensile tests. Calculations provide qualitative description of experimentally observed mechanisms. Creep and failure processes slow down as the particle size decreases. On the other MULTI-SCALE ANALYSIS IN CONTACT MECHANICS hand, the manifestation of creep becomes more pronounced with increase in the filler content and at higher stress level. IRINA G. GORYACHEVA Acknowledgments. This work was supported by the RFBR [email protected] grant 04-01-96038 of the regional contest ”Urals”. Lyudmila A. Golotina, Institute of Continuous Media Mechanics, Analysis of the processes occurring in contact interaction of 1, Ak. Korolev str., Perm, 614013, Russia solids is carried out at least at two scales. One scale is related to the characteristic wear particle dimension, and to the contact spot diameter which is typically within TWO-STAGE SUBSTRUCTURE TRANSFORMATION OF the interval from 1 to 10 µm. Modeling of the processes of con- AL FILM AS A MANIFESTATION OF MULTI-SCALING tact interaction at this scale let us study the dependence of the real SCENARIO contact area (the total area of real contact spots of two contacting rough bodies) and the real contact pressure on the loading condi- YU.G. GORDIENKO E.E. ZASIMCHUK tions, and on mechanical and geometrical characteristics of bod- [email protected] ies. The analysis of internal stresses and the temperature distribution near the contact spots, and their redistribution in the friction The plastically deformed Al pure single crystal foils rigidly at- process makes it possible to predict the type of surface fracture tached to aluminum alloy specimens under cyclic tension were and the size of the wear particles. The approaches to calculate investigated. The 1st stage of surface evolution is started by the real contact area and the real contact pressure, as well as the appearance of a polycrystalline state with the regular grid-like internal stresses, and to study the contact fatigue fracture due to arrangement of microscopic extrusions and intrusions ( 1-2 mi- damage accumulation process in cycling loading in friction of crons) only at the surface, which was known before for other rough surfaces are discussed. objects [M.Videm, N.Ryum, Mater.Sci.Eng., A219, 1996, 1-20]. The other scale is related to the characteristic size of the nomi- This initial stage of relief formation is accompanied by growth nal contact region which is the minimal connected region enclos- of asterism of Laue spots on X-ray picture. One can describe it ing all the contact spots. Solution of the contact problems for

45 bodies with various macroshapes and contact conditions enables this paper we investigate some of these extensions to the model us to study the dependence of the nominal contact area and the and consider some representative problems. nominal (averaged) contact pressure, approach of the contacting bodies on the load applied to them, material properties and the D. Harris, University of Manchester, Sackville Street, Manch- contact conditions (normal, sliding, or rolling contacts, etc.), and ester, M60 1QD, United Kingdom to calculate the averaged internal stresses arising from the action of the averaged normal and shear contact stresses. In addition, solutions of the wear contact problems let us analyze the evolution of all the contact characteristics (stresses, contact area, approach of contacting bodies) in friction process taking into account the macroshape variation due to wear of contacting bodies. The solution of the contact problem at this scale makes it possible to predict the durability of moving parts of machines during opera- NON-LINEAR DYNAMIC BEHAVIORS OF THE GLIDER tion. BALLOON; THE NEWLY DEVELOPED BALLOONING The processes taking place at different scales are interdepen- dent. The approaches based on the principle of localization and MASASHI HIRAYAMA KINYA OGAWA on the method of averaging are developed to calculate all the contact characteristics taking into account both macro- and microge- masashi [email protected], ometry of contacting bodies. The wear process is also consid- [email protected] ered at two scales related with surface microgeometry evolution due to wear particle detachment, and with the macroshape variation. The investigation of the macroshape variation is based on Ballooning is one of the most exciting sky sports. A hot air the wear equation obtained from a mathematical simulation of the balloon is usually adopted, but it is not easy to control the motion processes which occur at rubbing surfaces at microscale. The two because of its huge size. On the other hand, a gas balloon is much scale analysis developed is also used to study the contact interac- more controllable in virtue of its small apparent mass and will tion of inhomogeneous bodies. provide quickness in motions also due to its compactness. About The research was carried out under the financial support of the fifty years ago, there was invented a jumping balloon which is Russian Foundation for Basic Research (grant N 04-01-00766) described as follows; The pilot is suspended from a small gas and INTAS (grant N 03-51-6046). balloon and he stands on the ground in a slightly over weighted I.G. Goryacheva, Institute for Problems in Mechanics of Russian against buoyancy. He kicks the ground to jump up and he can en- Academy of Sciences, Moscow, Russia joy a short time floating in the air. It is, however, not easy to make a fine adjustment of balancing system. Recently, we have developed an alternative system with a balancing rope that he hold, he GRANULAR FLOW: A NEW PLASTICITY MODEL will go up, but the balancing rope is also pulled up with him, and CONTAINING A LENGTH SCALE finally an equilibrium state will come out. Thus, the pilot can en- joy the up and down movements in the air as far as he likes, since DAVID HARRIS the system has the self-equilibrium at his optional vertical positions. When a non-spherical balloon, such as an ellipsoid balloon, is provided, aerodynamic lift and drag can be expected in addi- The continuum modelling of the deformation and flow of gran- tion to the buoyancy and will be served to glide the system. In ular materials has proven to be a very difficult problem and there the present, we have investigated dynamic behavior of the newly is no consensus in the literature as to the correct way to proceed. developed system named The Glider Balloon which consists of Standard continuum models involving sets of partial differential a prolate ellipsoid balloon and the balancing rope and can move equations are beset with a number of theoretical difficulties, for in horizontal direction like a glider as well as vertical direction. example, ostensibly evolutionary problems turn out to have no Two-dimensional nonlinear differential equations for motion of solution to problems with initial Cauchy data. On the other hand, the glider balloon were derived with and without taking account non-standard continuum models based on a full Cosserat theory of flexibility and damping of the rope and were numerically an- are very complicated and there appears to be no experimental ba- alyzed for several situations to simulate the flight of the system. sis for the use of couple stress. We present here a model based Changing the attitude of the balloon, the motions of the system upon a reduced Cosserat continuum, namely one which containts were simulated and conditions for acquiring a long distant flight an intrinsic spin and associated inertia density, but in which cou- were clarified. The motion of rope was found to have a signifi- ple stress is absent. The model combines some aspects of three cant effect on the GB movement, and the maximum height and standard continuum models, namely the plastic potential, double distance were decreased when the flexibility and damping of the shearing and the double sliding free rotating models. The theory rope was considered. This is well understood in connection with is well-posed, thus enabling the solution of initial value problems the dissipation of energy in the balancing rope. In the case of and appears able to provide an explanation of a number of prop- small damping, the rope oscillates violently and the system be- erties of granular materials, for example, non-coaxiality, dilata- comes unstable. tion and rotation associated with shear flows. By using an invariant formulation, extending the model to three dimensional prob- M. Hirayama, Department of aeronautics and astronautics, lems, the inclusion of elastic effects, and strain or density hard- Graduate school of engineering, Kyoto University, Yoshida Hon- ening/softening a wide variety of problems may be considred. In nmachi Sakyo-ku, Kyoto, 606-8501, Japan

46 ELLIPTIC PROPERTIES OF SURFACE WAVES IN THE DYNAMIC RESPONSE OF A DISCRETE LATTICE WAVELET DOMAIN CONTAINING DRY-FRICTION ELEMENTS TO A SURFACE LOAD M. HOLSCHNEIDER M.S. DIALLO K. KURENNAYA JEROEN S. HOVING [email protected], [email protected], [email protected] [email protected] At present, the development of railway transportation focuses on the reduction of travelling time and the increase of comfort to passengers. Speed increase has brought new problems to railway This contribution is concerned with the elastic surface engineering, especially the increase of train, track and soil vibra- Rayleigh waves. The fluctuation trajectory of the volume ele- tion amplitudes and rapid dynamically-induced soil settlement. ments while the Rayleigh waves propagation process has an el- To predict the dynamic soil settlement, a non-linear model is to liptic form and lays in the same plane with a waves propagation be developed that is capable of describing large soil deformation direction vector. Rayleigh waves in a typical elastic body have no at high excitation frequencies. At these frequencies, the dynamic dispersion; in the asymmetrical Cosserat medium they possess a soil behaviour is mainly governed by the effects of dry friction dispersion caused by microrotations; Rayleigh waves dispersion between soil particles. This implies that at this frequency band, in geological mediums are resulted from the rocks elastic prop- the soil deformation is due to the stick-slip motion of the parti- erties variation while earth depth. Moreover, in the latter case cles, which can hardly be captured by a continuum model. there is also the loss of oscillation energy (dissipation) that has Therefore, in this contribution, a two-dimensional lattice an effect on changes of fluctuation amplitude. model is proposed, which describes the discrete nature of the stick-slip phenomenon. Next to the dry-friction elements, the lat- As a rule, we have an opportunity to observe Rayleigh waves tice is composed of discrete masses, springs and dashpots. The in solids and geological mediums only indirectly, construing the masses and springs are introduced to allow for wave propagation data of vibrations measuring, represented as seismograms. How- in the lattice, whereas the dashpots account for soil viscosity. ever, not only Rayleigh waves are offered on these seismograms, The focus of this contribution is placed on the dynamic re- and that is the reason why do we need the qualitative procedures sponse of the lattice to a pulse load applied to a surface particle of a filtration. One of possible methods of Rayleigh waves fil- of the lattice. The response is studied as a function of the du- tration is based on the polarizing analysis, and its purpose is to ration, frequency content and magnitude of the load. The main determine the parameters, which characterize signal ellipticity. attention is paid to the energy dissipation accompanying propagation of the pulse. It is shown that the experimentally observed frequency-independent soil damping can be reproduced by the The majority of the works developed the polarizing analysis proposed lattice. methods are based on the Fourier-analysis. But Rayleigh waves J. Hoving, Delft University of Technology, Faculty of Civil Engi- are rather complicated object for studying; their dispersion and neering, Structural Mechanics Section, P.O. Box 5048, NL 2600 ellipticity in real medium are described not only by one vari- GA Delft, Netherlands able, but also by function of frequency. Because of this fact the spectrum-time representation approach seems more substantial for us. It is remarkable, in particular, in opportunity of sig- ELECTROCONVECTION OF POORLY CONDUCTING nals distinguishing not only by their spectrum components, but LIQUID IN THE MODULATED ELECTRIC FIELD AT also by regularities of spectrums changes in time. The spectrum- PRESENCE OF CHARGE INJECTION time approach, known today as the wavelet-analysis, has received powerful development and by right has become one of the major VLADIMIR A. IL’IN methods of the signals numerical analysis. [email protected]

In the given research the number of methods based on the Electroconvection of poorly conducting liquid in the modu- wavelet-analysis is offered. Presented methods allow defining lated electric field of the horizontal capacitor is investigated at all ellipticity parameters of surface waves in the form of time- unipolar charge injection. Injection essential influences on dy- frequency spectrums. The presented methods effectively work namics of a liquid in the case when voltage on electrodes exeed both with two- and three-component signal and allow exactly some threshold value. Free charge arising in a liquid interacts identifying the Rayleigh wave in seismograms. with an external electric field therefore the liquid can come in movement even in weightlessness. It is assumed that a injection of charge is directly proportional This work is supported by a grant from the Deutsche to intensity of an electric field in the capacitor. The electrohydro- Forschungsgemeinschaft (DFG) within the framework of the pri- dynamic (EHD) approximation is used. Diffusion of a charge is ority program SPP 1114 “Mathematical methods for time series not taken into account. The main aim of this work is to investigate analysis and digital image processing”. stability of liquid and supercritical regimes of electroconvection. Distributions of charge density, potential, intensity of an elec- Mikhail Kulesh, University of Potsdam, Am Neues Palais, 10, tric field are calculated in an one-dimensional case, which is char- Potsdam, 14469, Germany acterized equilibrium of the liquid. The linear stability of equi-

47 librium is studied. Maps of stability are constructed. Processes of carry of charges through volume of a dielectric liquid are investigated. In nonlinear case the problem is solved with the help of finite difference method. Influence of amplitude and modulation frequency of electric field on dynamics of supercritical regimes is investigated. Dependence of current density in the capacitor on a degree of supercriticallity is obtained for various amplitudes and Figure 2: The model of the rod after the elements of frac- modulation frequencies of external electric field. The time evo- ture have been operated lution of electroconvective regimes are investigated with help of Fourier analysis. The research described in this publication was made possible of material particles that are not connected with each other, but in part by Award No. PE-009-0 of Civilian Research Develop- interact with the elements of the first component, Fig.2. ment Foundation (CRDF) and by Program of preparation of con- In the paper we consider the temperature impact at the end of ducting scientific schools NSh - 1981.2003.1. the rod. The temperature field is assumed to be independent on V.A. Il’in, Perm State University, Bukirev str., 15, Perm, 614990, stress-strain field in the rod. Approximate continual equations of Russia the model could be written as following —until the elements of fracture have been operated

INFLUENCE OF THE KINETICS OF PROCESSES ON 2 ∂ u − ·· =( + ) ∂θ THE BEHAVIOR OF THIN STRUCTURES IN A E 2 ρu E1α1 E2α2 (1) CONTINUOUS MEDIUM ∂x ∂x — after the elements of fracture have been operated DMITRY A. INDEITSEV 2 ∂ u1 ·· · · ∂θ [email protected] E1 − ρ1u1 − R1 (u1 − u2) − R2 u1 − u2 = E1α1 ∂x2 ∂x The formation kinetics and subsequent evolution of thin-film ·· · · ∂θ −ρ2u2 + R1 (u1 − u2)+R2 u1 − u2 = E2 α2 structures are investigated. Models for films growth via surface ∂x diffusion are analyzed. Influence of dynamic effects to diffu- (2) sional processes in thin-film structures is considered in the frame- Where E = E1 + E2,ρ = ρ1 + work of continuum mechanics. Nonlinear effects involved in the ρ2,E1,E2,ρ1,ρ2,R1,R2,l,α1,α2,λ — are the parame- growth of continuous thin structures are examined. ters of the model. D.A. Indeitsev, Institute for Problems in Mechanical Engineer- The theoretical problem was solved numerically. Physical pa- ingRAS, Bol’shoy pr. V.O., 61, St.Petersburg, 199178 Russia rameters of the structure could be find by the comparison of the results of simulation and data received in the experiments. The parameters obtained by this way permit us to extrapolate theo- THE TEMPERATURE IMPACT AT THE END OF THE retical model to the another cases of temperature loadings. The ROD OF COMPLICATED STRUCTURE qualitative coincidence of the numerical results with the experimental data permit to make the conclusion that this model is suit- DMITRY A. INDEITSEV NIKITA F. MOROZOV able for the investigation of the processes in solids of complicated VIKTOR N. NAUMOV ELENA I. PROKURATOVA structure under temperature impact. IPME RAS, St. Petersburg, Russia

We used the one-dimension rod of the complicated structure with the elements of fracture as the object of the modelling. NON-CLASSICAL TERMS IN THE EQUATIONS OF BALANCE FOR A GROWING BODY

HANS IRSCHIK [email protected]

Recently, an increasing interest has been devoted in the literature to the correct form of the equations of balance for growing bodies, e.g. in continuum mechanics based formulations for the growth of biological materials. In such formulations, non- Figure 1: The model of the rod before the elements of frac- classical growth terms do arise. In the present paper, we derive ture have been operated relations that must necessarily hold between the growth terms in the fundamental balance equations (mass, momentum, spin and In the case when elements of fracture do not operate the be- energy) and the growth terms in derived balance equations, such havior of the rod is similar to the behavior of the one-dimension as balance of moment of momentum and the equations of bal- elastic one, Fig.1. When elements of fracture have been operate ance of kinetic and internal energy, respectively . We then sug- we consider the second component as one-dimension continuum gest a corresponding continuum model of growth in extension of

48 the formulations of the Russian scientist Meshchersky for mass- either quicker than the cavity or considerably slower. This phe- points with a variable mass. Using this model, we are able to nomenon is of resonant nature and is observed only in narrow classify some recent biomechanical formulations and to resolve intervals of relative frequency of vibrations. The intensive vibra- an appearent discrepancy. tions of the solid accompany it. The vibrations dont influence H. Irschik, Johannes Kepler University of Linz, Altenbergerstr.69, significantly the behaviour of the system out of these intervals. Linz, A 4040, Austria The amplitude variation doesnt change the frequency of resonant excitation. With increase (decrease) of the amplitude the intensity of the motion of the solid also increases (decreases) respectively. THE PROBLEMS OF THE MATHEMATICAL The work is supported by Russian Foundation for Basic Re- MODELLING OF SOME TUBULAR BONES search (grant RFBR 03-01-00552) and Perm State Pedagogical EUGENE N. IVANOV University (grant PSPU 01-05). N.V. Kozlov, Perm State Pedagogical University, 24, Sibirskaya [email protected] av., 614600, Perm, Russia In the field of biomechanics, conducting direct experiments is very difficult and in some cases impossible. The use of mathematical modeling in the contemporary study of bone strength helps to carry out precise preoperative planning and treatment. LIFT FORCE ACTING THE SOLIDS IN LIQUID UNDER It is a well-known fact that bones have very complex geometry VIBRATIONS and anisotropic inner structure. Taking into account the com- ALEVTINA A. IVANOVA VICTOR G. KOZLOV plex structure of bones, finite elements method (FEM) enables to carry out various researches in biomechanics. The purpose of the [email protected] research was the construction of finite elements models of various tubular bones, such as tibia, radial and metacarpal bones. The report presents the new experimental results concern- Construction of models was based on scans of cross-section cuts ing the lift force acting the solid inclusions in the cavity filled of skeletal bones, and mechanical properties of skeletal bones. with incompressible liquid and subject to vibrations. Different The bones are considered as non-homogeneous anisotropic lin- kinds of vibrations are considered: translational and combined ear elastic body. The finite elements program Ansys is applied translational-rotational. It is shown, that near the walls of the to solved the problem. Results of the mathematical modeling of cavity, at the distance of viscous interaction, the body oscilla- a persons bones enable to define optimal geometrical parameters tions result in excitation of repulsion force. The direction of this of clamps, which are used at osteosynthesis and to choose an op- lift force is opposite to the known one of inviscid nature force of timal treatment. attraction to the wall of the cavity (in the area of high frequencies E.N. Ivanov, Rostov State University, Zorge, 5, Rostov-on-Don, at the distance from the wall comparable to the body size). The 344090, Russia vibrational force of viscous nature is found to be very significant (it can provide the heavy body lifting in gravity field) but man- INFLUENCE OF VIBRATION ON BEHAVIOUR OF ifest itself only at relatively small distance comparable with the LIGHT SOLID IN ROTATING CAVITY FILLED WITH thickness of Stokes boundary layer. LIQUID The investigation of vibrational interaction, when viscosity of liquid plays a determining role, is carried out in wide interval of ALEVTINA A. IVANOVA NIKOLAI V. KOZLOV dimensionless frequencies. In the area of high frequencies the [email protected] dependence on the distance of body interaction with the cavity wall has a minimum: near to the border the force of repulsion The behaviour of a solid cylinder in intensively rotating cav- dominates but quickly decreases with distance, outside the area ity, filled with liquid and subject to vibration in the direction nor- of viscous interaction the force of attraction appears and at first mal to the rotation axis, is studied experimentally (the horizontal grows with distance, then quickly decreases. In low-frequency cylindrical cavity is rotating around its axis). The water is used as area when the size of inclusion is less or comparable to the Stokes liquid; the solid is a hermetical void cylinder made of plexiglass. layer thickness, the only force of repulsion is found, which de- We are studying the motion of the solid versus the rotation fre- creases exponentially with the distance. quency, frequency and amplitude of vibration. The observation is In case of translational vibrations the lift force does not qual- performed in stroboscopic illumination. itatively change with relative density both light and heavy in- The solid takes a steady position in the center of the cavity clusion are removed from the cavity walls. The account of vis- under the influence of centrifugal force; it happens in a threshold cous vibrational interaction is important also in case of nontrans- way as the rotation frequency is being increased. In the absence lational vibrations at the distance from the border, comparable of vibrations the solid is lagging in the frame of the cavity, i.e. in with Stokes layer thickness, it plays the determining role. the laboratory coordinate system the solid is rotating slower than Acknowledgements the cavity. This retrograde motion is caused by the gravitational force; its intensity decreases with the rotation frequency increase. The work is supported by Russian Foundation for Basic Re- As the experiment shows, the vibrations may provoke an in- search (grant RFBR-Ural 04-01-96055) tensive movement of the solid in the cavity frame at certain fre- A.A. Ivanova, Perm State Pedagogical University, 24, Sibirskaya quencies ratio. Depending on the frequency, the solid can rotate av., Perm, 614000, Russia

49 The work was supported ﬁnancially by the RFBR (04-01- NANOMECHANICS: CONTINUUM AND DISCRETE 00398) and “Advanced Scientiﬁc School of Russia” (Sc.Sch- APPROACHES 2000.2003.1). A.V. Karapetyan, Moscow State Lomonosov University, Leninskie ELENA A. IVANOVA ANTON M. KRIVTSOV Gory, Moscow, 119992, Russia NIKITA F. MOROZOV BORIS N. SEMENOV ALLA D. FIRSOVA [email protected] KINETIC NATURE OF THE INCUBATION TIME

Advances in high technologies using nanometer-size struc- A.V. KASHTANOV Y.V. P ETROV tures requires calculation of mechanical properties for the objects [email protected], of the nanosize scale level. Majority of the theoretical mechan- [email protected] ical models for nanoobjects is based on the macroscopic equations of theory of elasticity. The values of the elastic moduli are The incubation time approach provides an effective way to ex- usually taken from macroscopic experiments. However, a lot of plore various problems of dynamic fracture. It supposes the dis- researchers have noted inconsistency between the values of the crete structure of both spatial and temporal scales and allows us elastic moduli obtained from micro- and macroexperiments. The to consider the discrete nature of dynamic fracture processes. The presented paper is devoted to theoretical investigation of the in- corresponding “size” of the temporal scale is called by incubation fluence of the scale effects on Young modulus, Poisson’s ratio and (structural) time and it is carried by a physical sense of charac- the bending stiffness of a nanocrystal, which is extended in one teristic time of the relaxation processes during material micro- direction and has a limited number of atomic layers in another fracturing. direction. It is shown that reduction in the number of the layers Using the same principles the equivalent continual approach to leads to decreasing the Poisson’s ratio and the bending stiffness the dynamic fracture description is proposed. The current state of and increasing the Young modulus of the crystal. The analysis of fracturing sample is defined by the specific differential equation dynamics of rod, plate, cylindrical and helical shells, presented in of continual mechanics with natural initial condition and fracture the paper, may be a theoretical foundation for experimental test- criterion: ing of the applicability of the continuum theory to the calculation dθ = 1 ( ( ) − ( − )) ( )= F (t) of mechanical characteristics of nano-objects and the experimen- dt τ q t q t τ ,qt Fc (1) tal determination of the bending stiffness of nano-shells. Idea of θ(t0)=0 the experiments are based on comparison of eigenfrequencies of θ(t∗)=1 the objects having different sizes. where t is the time, θ(t) ∈ [−∞, 1] is the state function defining N.F. Morozov, St. Petersburg State University, IPME RAS, St. the current level of material damaging (fracturing), τ is some pos- Petersburg, Russia itive constant (the incubation time), t0 is the moment of applying the load to initially “nonfractured” material (that is q(t0)=0), t∗ is the time to rupture, F (t) is the local intensity of force INVARIANT SETS OF DYNAMICAL SYSTEMS WITH field, Fc is the critical value of F (t) which is defined experimen- FIRST INTEGRALS: THEORY AND APPLICATIONS tally. In particular, F (t) coincides with the field of local stresses ( )= ( ) ( ) ALEXANDER V. K ARAPETYAN F t σ t for defectless materials and F t is accepted to be equal to the local stress intensity factor F (t)=K1(t) for [email protected] solids with microcrack. In that way the fracture is considered as some evolutional (temporal) process of material transition from Dynamical systems with first integrals are considered. Any set the state θ =0into the state θ =1during the time t∗ in the in the phase space of the system is an invariant (a stable invariant) accordance with equation (1). set if it gives a stationary (an extremal) value to one integral for Proposed approach generalizes the criterion of critical stress fixed levels of other integrals. as well as Irwin criterion of critical stress intensity factor for the Dynamical systems admitting first integrals and a function that cases of non-momentary events of fracture. That is for the cases does not grow along motions are considered too. Any set in the when a time period (which is called as incubation or relaxation phase space of the system is an invariant (a stable invariant) set time) is needed to produce rupture after the moment of reaching if it gives a stationary (minimal) value to an ungrowing function a critical value by the local stress field. for fixed levels of integrals. It is shown that the approach proposed is the differential form Sufficient conditions of instability of invariant sets are given of incubation time criterion of fracture. As such as the incubation for both cases. time approach it operates with two material parameters only. And Note that zero-dimensional invariant sets correspond to steady these parameters define both static and dynamic material strength motions of the system. In general case motions of the system and they can be measured easily in experiments on static material belonging to invariant sets depend on time and represent periodic, strength. But in contrast to an integral form the suggested ap- elliptic etc. functions of time. proach describes the state function determining the level of ma- As examples there are considered a number classical problem terial damaging during the fracture process. of rigid body dynamics: a heavy rigid body with a fixed point, a Some applications of the incubation time approach are consid- heavy rigid body containing viscouse liquid with a fixed point, a ered. The behavior of dynamic fracture toughness is investigated rigid body in liquid, a heavy rigid body on a plane with friction for the crack problems and estimation of temporal dependences etc. of strength is conducted for different loading conditions.

50 Arseny V. Kashtanov, St.-Petersburg State University, Universitet- DEFORMATIONS CONCENTRATORS AND FRACTURE sky pr., 28, St.-Petersburg, 198504, Russia OF PLASTIC BODIES

A.I. KHROMOV A.A. BUKHANKO S.L. STEPANOV SOME GAS FLOWS WHICH ARE SOLUTIONS OF GENERALIZED CAUCHY PROBLEM The approach to research of areas of abrupt change of the form ALEXANDER L. KAZAKOV as strains concentrators is offered. Definition of strains fields in [email protected] the neighbourhood of the concentrator is carried out on the basis of the ideal rigid plastic-bodies theory and at flat deformation Some time-dependent gas flows with shock waves are con- is reduced to integration of the ordinary differential equations. sidered. This flows mathematically describing by generalized A number of solutions for concentrators as V-notched bar are re- Cauchy problem posed on two initial surfaces. In generalized ceived. Dependence of strains on change of the form and position Cauchy problem the different initial conditions are posed on dif- of plastic area is investigated during plastic flow. Nonuniqueness ferent initial surfaces. In papers of V.M. Teshukov (see, for ex- of plastic flow (is shown within the framework of completeness ample, [1]) some of this problems are investigated. of the solution). Deformation and power criteria of a choice of We consider gas flows with shock waves which diverge from preferable plastic flow are offered. The problem about fracture of an axis or center of symmetry. The compression wave due to V-notched bar pulled in tension is considered. smooth motion of an impenetrable piston in a gas at rest is fo- On the basis of the received solutions the approach to research cused on the axis or the center of symmetry. The shock wave of fracture processes for more complex bodies models is formu- converged from the axis or the center of symmetry has the fi- lated: elasto-plastic, strengthened plastic bodies, etc. Thus it is nite inconstant velocity. The solution of this problem represent offered to consider the material in the neighbourhood of vertex an extension of Sedov’s self-similar solutions of the case of two crack as rigid-plastic material. It allows to carry out the analyt- independent variables [2]. Gas flows constructed can used by ical description of fields of deformations the neighbourhood of simulation of the powerful gas compression [3]. New theorems vertex crack and to apply new deformation criteria of fracture. of existence and uniqueness are proved. Anastasia A. Bukhanko, Institute of Machining and Metallurgy of FEB RAS, 1, Metallurgov Str., Komsomolsk-on-Amur, 681005, Notion “generalized Cauchy problem” for the non-linear par- Russia tial differential equations is used at the basis of the results of Ch.Riquier [4], N.M.Gunter, S.L.Sobolev [5], N.A.Lednev [6], S.P.Bautin [7] and other mathematicians. COMPUTATIONAL SIMULATION OF IRREVERSIBLE Investigation supported by RFFI, project 04-01-00205. DYNAMIC DEFORMING AND FRACTURE OF DAMAGEABLE SOLIDS AND STRUCTURES REFERENCES A.B. KISELEV O.V. NEKHAEVA [1] Teshukov V.M. Regular reflection of a shock wave from a rigid wall (in Russian) // Prikladnaya matematika i [email protected] mekhanika, 1982, No 2, 225-234. Thermomechanical processes, which proceed in deformable [2] Sedov L.I. Methods of similarity and dimension in mechan- solids under intensive dynamic loading, consist of mechanical, ics (in Russia). M.: Nauka, 1987.- 432 c. thermal and structural ones, which correlate themselves. The structural processes involve the formation, motion and interac- [3] Zababakhin E.I., Zababkhin I.E. Phenomena of Unlimited tion of defects in metallic crystals, phase transitions, the break- Cumulation. M.: Nauka, 1988.- 173 c. ing of bonds between molecules in polymers, the accumulation of microstructural damages (pores, cracks), etc. Irreversible defor- [4] Riquier C. Les syste`mes d’e´quations aux de´riveés par- mations, zones of adiabatic shear and microfractures are caused tielles. Paris, Gauthier-Villars, 1910. by these processes. Dynamic fracture is a complicated multistage [5] Sobolev S.L. Analytical solution for systems of equations process including an appearance, evolution and confluence of mi- with partial derivatives with two independent variables (in crodefects and a formation of embryonic microcracks, pores their Russian) // Matematicheskiy sbornik, 1931, No 1-2, 107- grow up to the break-up of a bodies with division into separate 147. parts. The present paper include new results in the next scopes: [6] Lednev N.A. New method of solving differential equations 1) development the thermodynamically correct mathematical with partial derivatives (in Russian) // Matematicheskiy models of damageable thermoelastoviscoplastic medium (mi- sbornik, 1948, No 2, 205-266. crofracture); 2) development the methods for determination of “nonstandart” [7] Bautin S.P. Cauchy problem with initial data on different constants of medium models, connected with microfracture of surfaces for quasi-linear analytical system (in Russian) // material; Differencialniye uravneniya, 1996, No 6, 804-813. 3) numerical simulation of destruction (fragmentation) of constructions (macrofracture); A.L. Kazakov, Ural State University of Railway Transport, Kol- 4) numerical investigation of some problems for damageable mogorov st., 66, Ekaterinburg, 620034, Russia solids and structures (dynamical deforming and fracture of thick-

51 walled cylindrical and spherical shells under explosion; dynami- Natalya Sevodina, Institute of Continuous Media Mechanics, Ak. cal deforming and fracture of thick-walled two-layer shell, filled Korolev str., Perm, 614013, Russia with liquid, under impact and high velocity penetration). Some previously obtained results in consideration domains are published in the papers [1 - 4]. DEFINITION AND OPTIMIZATION OF THE Russian Foundation for Basic Research (grant No. 03-01- DISSIPATIVE CHARACTERISTICS OF 00127) and The Program for support of Leading Scientific ELECTROELASTIC SHELLS Schools of Russian Federation (grant Sc.Sch.-19.2003.1) are ac- knowledged for financial support. E.P. KLIGMAN V.P. M ATVEYENKO N.A. YURLOVA REFERENCES [email protected] [email protected] [email protected] [1] A.B. Kiselev, M.V. Yumashev and O.V. Volodko O.V., “De- forming and fracture of metals. The model of damageable thermoelastoviscoplastic medium”, J. of Materials Process- In recent years, the concept of itellifibers - discrete piesoele- ing Technology, Vol. 80-81, pp. 585-590, (1998) . ments and piesofibers manufactured on the basis of piesoelectric materials has been introduced and applied to the development [2] A.B. Kiselev, “Mathematical modeling of dynamical defor- of structural components. At present, piesoelectric elements are mation and combined microfracture of a thermoelastovis- finding wide use in practice, for example, in dampening struc- coplastic medium”, Moscow Univ. Mech. Bull., Vol. 53, ture vibrations under loading. In this case, they are usually built No. 6, pp. 32-40, (1998). in at special points of the system subjected to deformation and [3] A.B. Kiselev and A.A. Lukyanov, “Mathematical modeling connected together or to the point of zero potential with pas- of dynamic processes of irreversible deforming, micro- and sive RLC-circuits. The principle of vibration damping is based macrofracture of solids and structures”, Int . J. of Forming on the transformation of the mechanical energy into electric one, Processes, Vol. 5, No. 2-3-4, pp. 351-362, (2002). which is then dissipated in the external RCL-circuits in the form [4] A.B. Kiselev, A.A. Lukyanov and Thiercelin M., “Numeri- of heat and electromagnetic radiation. In this process the dissipa- cal simulation of dynamic propagation of curvilinear cracks tion power depends on the parameters of the external circuit. of hydraulic fracturing”, Moscow Univ. Mech. Bull., Vol. The dynamic properties of electroelastic systems can be de- 59, No. 1, pp. 36-41, (2004). fined based on the families of the ampitude-frequency characteristics obtained from different electromechanical excitation tests Alexey B. Kiselev, Moscow M.V. Lomonosov State University, or by solving homogeneous spectral problem. Leninskie Gory, 1, Moscow, 119992, Russia In the framework of the finite element approach the problem of constructing the amplitude-frequency characteristics of the elec- NATURAL VIBRATIONS OF ELASTIC SHELL WITH troelastic system is reduced to numerical solution of the matrix FLUID equation for different values of frequency ω : E.P. KLIGMAN T.I. KLIGMAN N. SEVODINA 2 [K ]) − ω [M ]+[G(ω)] {X } = {F } (1)

Here [K ], [M ] , [G(ω)] are the stiffness and mass matrixes and The filling of shell structures with fluid significantly changes the matrix of “stiffness”, in which the elements depend on the pa- their dynamic properties. This phenomenon is more pronounced rameters and the form of the external circuit; X is the vector of in the case when the spectrum of natural vibrations of the elas- vibration amplitude; F is the vector of loads. The optimization tic shell intersects that of the fluid in the gravity field. Of special problem reduces to a search of external circuit parameters capa- concern is the case of the free surface fluid. The dynamic and dis- ble of reducing to a minimum the vibration amplitude in a given sipative properties of the system are defined during the solution of frequency band. the homogeneous spectral problem. The ?shell-fluid? mathemat- The problem of optimization of structure dynamic properties ical model is constructed in terms of the linear equations of the has been considered for a conical shell, which is partially cov- anisotropic shell theory and the flow equation of fluid. On the as- ered with a piesoelectric layer polarized in the direction the of sumption of small-amplitude vibrations, the linear equations are the layer thickness and shunted with the resistors R and the in- used and the shell surface moistened with fluid is taken to be un- ductance coils L. To suppress asymmetric vibration modes the changed. It is also assumed that the fluid flow is vortex free and, piesoelectric layer was subdivided into 16 sections, each being therefore, it can be described by the Laplace equation. In the flow plated and fitted with electrodes. The number of sections is spec- equation, the shell effect is represented by the boundary surface ified by the number of circumferential half-waves generated at displacement, and the fluid effect, in the shell equation, by the lowest frequency of natural modes. normal pressure. Combining both equations and representing the shell displacement and fluid pressure by the harmonic time func- The numerical simulation made for this particular structure has tion, we arrive at the spectral problem of continuum mechanics. shown that application of the R-circuits reduces the resonance As an example, the vibrations of a hinged orthotropic parabolic amplitude by half and the connection of additional inductance shell with different degrees of filling with fluid have been consid- coils (RL-circuits) results in a fourfold decrease. ered. The problem has been solved within the variational state- The work was supported by the RFBR under grant 03-01- ment in terms of a specially developed model of the finite element 00557. method. This work was supported by the Russian Foundation for Nataliya A. Yurlova, Institute of Continuous Media Mechanics Basic Research under project 04-01-96028 RFBR-URAL. Ural Branch of RAS, 1, Korolev str., Perm, 614013, Russia

52 σx + σy s = A MICRO-POLAR THEORY FOR PIEZOELECTRIC 2 MATERIALS The corner µ is connected to a corner of inclination φ between a direction of main strain σ1 and an axis o˜ ratio tan 2µ = YAROSLAV KOLPAKOV PAVEL ZHILIN λ tan 2φ. [email protected] If the material isotropic (λ =1) corners µ and φ will be equal. We shall believe, that the anisotropic loose environment passes Theory of the piezoelectric materials had been developed in a limiting condition if in a point of environment there is such ∗ many years ago. There exist a several theories of the piezoelec- platform along which normal a component tensor of strains σx ∗ tricity. All of them lead to the very complicated equations. The and the tangent of a component tensor strains τxy satisfies to a exact solutions of these equations may be found only for very par- condition of dry friction Kulon ticular cases. By this reason it is not easy to compare theoretical ∗ ∗ and experimental results. At the present time it seems to be possi- τxy =(H − σx)tanρ. (3) ble to say that there is no qualitative discrepancies between theory We shall note, that in a static of loose environments value H and experiments. From the pure theoretical point of view in the refers to as the resulted factor of coupling, and ρ — a corner of theory of the piezoelectricity there are some serious problems. It internal friction. They are constants of a material and are deter- was supposed that the stress state of the piezoelectric material can mined experimentally. Substituting in the equation (3) compo- be described by means of the symmetrical stress tensor. However nents of strains (2) we shall receive, that T =(H − s)sinρ, some piezoelectric materials are the dipole crystals. In such a and corner µ∗ we shall find from equation sin 2µ∗ cos ρ + case the rotation degrees of freedom must be taken into account. λ cos 2µ∗ sin ρ =1. Thus the criterion of fluidity similar (1) will It means that the theory of the piezoelectric materials must be become constructed on the base of the micro-polar continuum. The the- 2 ory of such a kind is presented in the report. The basic equations 2 (σx − σy) σx + σy 2 2 τxy + =(H − ) sin ρ, (4) are derived from the fundamental laws of Eulerian mechanics and 4λ2 2 contain two unsymmetrical stress tensors. The theory presented and the equations similar (2) will be written down so: in the report differs from conventional theory very significantly. However under some assumptions this theory may be reduced to σx = s + λ(H − s)sinρ cos 2µ, the classical one. The theory was tested on some simple problems and results were compared with classical ones. σy = s − λ(H − s)sinρ cos 2µ, (2 ) Ya. Kolpakov, Department of Theoretical Mechanics, St. Peters- τxy =(H − s)sinρ sin 2µ. burg State Polytechnical University, RUS-195251, St. Petersburg, Politechnicheskaja, 29, Russia. Substituting last expressions in the known equations of bal- ( − )= tan z tan ρ P.A. Zhilin, Department of Dynamics of Mechanical Systems, ance, entering replacement H s H ρe , we shall Institute for Problems in Mechanical Engineering of Russian receive: Academy of Sciences, RUS-199178, V.O. Bolshoy 61, St. Peters- ∂z ∂z burg, Russia. (1 − λ sin ρ cos 2µ) − sin ρ sin 2µ+ ∂x ∂y ∂µ ∂µ +2λ cos ρ sin 2µ − 2cosρ cos 2µ =0, THE THEORY OF LIMITING BALANCE OF THE ∂x ∂y ANISOTROPIC LOOSE ENVIRONMENT ∂z ∂z ALEKSANDRA A. KOMAROVA IGOR V. S HIRKO sin ρ sin 2µ − (1 + λ sin ρ cos 2µ)+ ∂x ∂y komarova [email protected], ∂µ ∂µ [email protected] +2cosρ cos 2µ +2λ cos ρ sin 2µ =0 (5) ∂x ∂y In the theory of plasticity of metals at the flat deformed condi- Believing, that the criterion (4) represents plastic potential, the tion components tensora of strains satisfy to criterion of fluidity equation for speeds we shall receive as Mises-Hill’s [1]: ∂u ∂u ∂v 2 λ sin 2µ =(cos2µ + λ sin ρ)( + ), 2 (σx − σy) 2 τxy + = T (1) ∂x ∂y ∂x 4λ2 ∂v ∂u ∂v thus the material is supposed ideally rigid-plastic and ortotropic, λ sin 2µ =(− cos 2µ + λ sin ρ)( + ), (6) ∂y ∂y ∂x axes x, y coincide with the main axes of anisotropy, T — value of a limit of fluidity at pure shift in this system of coordinates, λ These equations belong to hyperbolic type and have two fam- — parameter of anisotropy. ilies of characteristics which can be written down so: The condition of plasticity (1) is satisfied identically if σx, σy dy du = − =tan(ψ ± α), and τxy present in the following kind dx dv 2cos ( 22 + 2 sin2 2 ) dy σx = s + λT cos 2µ dz ρ cos µ λ µ dx = , ( sin + cos 2 )+ dy sin 2 ( − (1 − 2)sin cos 2 ) σy = s − λT cos 2µ (2) dµ λ ρ µ dx µ λ λ ρ µ = sin 2 tan 2 = tan 2 cos 2 = − λ sin ρ cos 2ψ τxy T µ Where ψ λ µ, α cos 2µ .

53 The system of the equations (5), (6) contains 4 required func- brittle fracture properties. The investigation is done within the tions z, µ, u, v and allows to determine completely an in- terms of the Synthetic theory of strength: plastic deformation is tense condition and a field of speeds u and v of rigidly plastic realized due to the breaking of the solid into pieces by the sliding anisotropic loose environment. surfaces, the formation of regular block structures, the mutual slip Using approaches of work [2], with the help of the received and turn to a certain angle. The condition of deformation limit results it is possible to develop the theory of elastic - plastic de- allows to keep the continuality of the original material. Brittle formation of the anisotropic loose environment. fracture mechanics formulates fracture criteria in the form of in- Acknowledgements variant relations of critical values of strain macro-characteristics. Work is executed at support of the grant of the Ministry of At present, it is basically founded on physical methods of solids higher education of the Russian Federation under the program and modern branches of mechanics, which take into account ma- “Basic researches in the field of natural and the exact sciences” terial microstructure at strain. Various known experimental data, E-02-4,035 directed at examining the physical side of material strain at brittle REFERENCES fracture, show that loading, on condition of given displacements, [1] Hill R. The Mathematical theory of plasticity M. 1956 enables us to record a so-called descending leg of the diagram ”maximum tangential stress and principal shear” (or beyond-limit [2] Shirko I.V., Kondratjev D.S., Stetsenko P.V. The defor- behaviour of material or postpeak strength). As a result, they mation theory in mechanics of the granulated environ- (above-mentioned experiments) reveal a non-symmetric scheme ments// Mechanics of componental materials and designs of deformation and strain localization in a number of materials Vol.8,4,2002 (rock, loose ground). In originally homogeneous isotropic mate- A.A. Komarova, Moscow Institute of Physics and Technology, In- rial there appear size and shear resistance anisotropy. Such con- stitute pereulok, 9, Dolgoprudniy, 141700, Russia tinuum properties can be analytically explained (illustrated) by introducing a new set of stressed state invariants in mathematical simulator, which differs from the classical one. The analysis PHASE TRANSITIONS IN DAMAGED MATERIAL of the real position of units of the inner layers of the sliding line intersection gives the picture of the existence of the deformation VLADIMIR I. KONDAUROV field jumping lines in the plastic zone and their extension for the ANASTASSIYA N. LYUBICHEVA problem on determination of displacement values near the open- [email protected] ing in an infinite plate. N. Kontchakova, Voronezh State University, University Sq. 1, A first kind phase transition in homogeneous isotropic mate- 394006, Voronezh, Russia rial, which occupying spherical area is considered. It is supposed, that inclusion of a new phase can have the spherical form or form of a spherical layer. The size of this inclusion is beforehand un- ON NUMERICAL IMPLEMENTATION OF NON-LOCAL known and will be determined from the decision of the problem. FRACTURE CRITERION TO INHOMOGENEOUS Own deformation of phase transformation and its latent energy MEDIA are taken into account. A stress state is defined by value of defor- VADIM F. KOSHELEV ALEXANDER M. LINKOV mation and damage. Processes of healing of defects are neglected in the model. [email protected] [email protected] Conditions, at which process of phase transformation is quasistatic and proceeds at constant temperature, are investigated. It The paper aims to develop technique for studying fracture was shown, that solution of the problem with power boundary nucleation and propagation in the vicinity of (multi-) wedge conditions is unstable and realizes as dynamic process. For the points in inhomogeneous media. Singularities being generated problem with kinematic boundary conditions quasistatic phase by wedges with arbitrary angles, the classical crack theory is not transformation is possible. All two phase configurations are applicable. Thus, non-local criteria of Neuber-Novozhilov’s type energy-wise equivalent when there is no preference point. Practi- become preferable. Their advantages are that they (i) may be used cal appendices of this problem are connected to forecasting defor arbitrary sources of singularities and (ii) provide classical re- struction of elements of designs in which after mechanical or sults in many particular cases considered in the crack theory. thermal influence the inclusions of a new phase which distin- For a chosen criterion we need to evaluate average stresses guished in density of weight and elastic properties have appeared. along small intervals at various directions in a close vicinity of a A.N. Lyubicheva, Institute for Problems in Mechanics, pr. Ver- singular point. This requires accurate calculation of stresses: oth- nadskogo, 101-1, Moscow, 119526, Russia erwise, the results are unstable and unreliable. In this paper we present a computational method for accurate solving the problem for singular points, which are common apexes of a system of THE LINES OF DEFORMATION FIELD JUMPING IN wedges (grains, inclusions, blocks of material). THE PLASTIC REGION IN THE GALIN’S PROBLEM The method includes: (i) new efficient numerical procedure for evaluation of the ex- NATALIA KONTCHAKOVA ponents characterizing asymptotics of stresses with physically [email protected] admissible singularities; (ii) incorporation of the found asymptotics into recurrence The problem of displacement field determination is considered quadrature rules and robust procedures for finding generalized in the case of elastic - plastic model of solid. The material has SIFs and for accurate evaluation of stresses within wedges and at

54 their contacts; Low is proposed (iii) efﬁcient searching procedures, which ﬁnd the most per- 0 0 spective direction of fracture nucleation (both within wedges and Φ1 = (σ − C ) · (σ − C )dΩ=0 along their contacts) by using the calculated stresses in the chosen Ω fracture criterion. 0 Here σ — stress tensor, — strain tensor. It is following from Numerical examples illustrate accuracy of the method devel- the equation that the component of tensors σ and C0 are equal oped. in the region Ω except maybe some points of null set. Similarly, V.F.Koshelev, IPME RAS, 61, Bolshoi pr., St.-Petersburg, 199178, other two functionals are introduced Russia 0 0 Φ2 = ( − ) · ( − )dΩ=0 Ω AN ANALYSIS OF THE COMPUTER MODEL FOR THE 0 0 UNILATERAL MULTIBODY SYSTEMS DYNAMICS Φ3 = (σ − σ ) · ( − )dΩ=0 Ω IVA N I. KOSSENKO where σ0 = C0 and σ = C. The integral formulation of the [email protected] Hook‘s low gives the possibility to reduce the linear elasticity problem to variational.

Computer modeling and simulation of dynamics for multibody ∗ ∗ Φi(σ ,u )=minΦi(σ, u)=0,i=1, 2, 3 systems consisting of rigid bodies with unilateral constraints σ,u (MBSUC) is not an easy problem. When developing the models, one encounters several obstacles difficult to overcome. In This work was supported by RFBR (02-01-00157, 02-01- the current talk an approach to modeling the MBSUC dynamics 00252, 03-01- 96586), Leading Scientific Schools Grant NSh- based on Modelica language is described. 1627.2003.1 and Grant NWO (047.014.007). Vasily V. Saurin, pr. Vernadskogo 101-1, Moscow, 119526, Russia First of all so called acausal modeling approach is applied here because MBSUC is represented in this case via the hybrid automata having a large number or even very large number of states. It turns out one can avoid the model structural complexity growth NUMERICAL ANALYSIS OF LINEAR ELASTICITY on this way, at least on the Modelica level of the description. PROBLEMS BASED ON INTEGRO-DIFFERINTIAL The problem of the regularization for transitions between the APPROACH states of the unilateral constraint is under resolution. The transitions being simulated are the following ones: between relative G. KOSTIN V. S AURIN flight and contact, and for the case of contact between sliding and [email protected], [email protected] rolling. The model of dry friction is taken into account. Impacts distributed throughout the MBSUC are also implemented. The work is devoted to numerical analysis of the variational The comprehensive verification of the regularized model in linear elasticity problems formulated in the presentation on the compare with the model of the exact hybrid automata is carried Conference by the authors (”Integro-Differential Approach in the out. An example of the heavy ellipsoid on the rough plane is used Linear Theory of Elasticity”). The numerical algorithm is devel- for this purpose. The ellipsoid supposed to have a possibility of oped and 2-D linear elasticity problems for which the algorithm jumps, slipping, and rolling. Some known qualitative examples has successfully been applied are considered. The research is re- of the rigid body dynamics on the rough plane are also under con- stricted by the following conditions: an elastic body is occupied sideration. a convex polygonal region, loaded statical forces, and clamped This work was prepared with partial support of Russian Foun- on a fixed part of the boundary, and volume forces are equal to dation for Basic Research, grants 02-01-00196, SS-2000.2003.1. zero. The algorithm consists of specified steps. In the fist step I.I. Kossenko, Moscow State University of Service, Moscow, Rus- the independent variables (stresses σ and displacements u) are sia approximated by finite polynomial series.

nσ k kl l k−l INTEGRO-DIFFERENTIAL APPROACH IN THE σij = σij x y ,i,j=1, 2 LINEAR THEORY OF ELASTICITY k=0 l=0

nu k G. KOSTIN V. S AURIN kl l k−l ui = ui x y ,i=1, 2 [email protected], [email protected] k=0 l=0 After that, equilibrium equations and boundary conditions are In the paper we consider some possible modiﬁcations of gov- satisﬁed. As the result the unconstrained minimization problem erning equations of the linear theory of elasticity. To this purpose for quadratic functional arises. the algebraic formulation of the Hooks Low is changed on the in- T w Kw +2bw → min tegral one. The resulted integro-differential boundary value prob- w∈RN lem can be reduced to a variational problem. In order to take into account uncertainty of the elastic moduli‘s tensor on the bound- Here w — vector of design parameters (stresses and displace- aries of bodies the following integral formulation of the Hooks ments); K ∈ RN×N — symmetric positive matrix; ); b ∈ RN

55 — vector of boundary condition. To avoid undesirable computa- information about specific base sequences along the double he- tional errors all calculations have been performed analytically us- lix. The estimation of real values of rigidity characteristics sugar- ing the ”Maple” software. The obtained results are discussed and phosphate chains has allowed to improve the model of DNA and compared with the classical variational approach and strength of to show, that in a double chain of DNA there can be four types materials theory. The bilateral integral error estimation criterion topological solitons. Interaction between the solitons is investi- is developed and applied to evaluation of the algorithm conver- gated as well as their interaction with the chain inhomogeneities gence. and stability of the solitons with respect to thermal oscillations. It This work was supported by RFBR (02-01-00157, 02-01- is shown, that thermal oscillations promote the soliton time of life 00252, 03-01- 96586), Leading Scientific Schools Grant NSh- in inhomogeneous sequence of the bases. Along side with topo- 1627.2003.1 and Grant NWO (047.014.007). logical solitons, localized nonlinear normal modes (breathers) are Georgii V. Kostin, pr. Vernadskogo 101-1, Moscow, 119526, Rus- also investigated. Using transition to complex variables and mul- sia tiple scale expansion techniques, we have found analytical representation of breathers. Then analytical results were compared with the data of computer simulation and their good accordance THE GRAVITATION CORRECTION FOR THE was shown. We studied also the interaction of two breathers that ACOUSTICS EQUATIONS turns out to be elastic. Some applications of the obtained results to physical processes in DNA macromolecule are discussed. This DANIIL P. KOUZOV work was supported by the Russian Foundation of Basic Research [email protected] (Grants No. 04-02-17306 and 04-03-32119). Natalya A. Kovaleva, N.N. Semenov Institute of Chemical The gravitation correction for the acoustics equations is intro- Physics, Russian Academy of Sciences, ul. Kosygina 4, Moscow, duced. The comparison of wave processes submitting traditional 119991, Russia and improved equations is carried out for elementary model of ideal gas isothermal oscillations in a gravity field. NONLINEAR SINUSOIDAL AND VARICOSE D.P. Kouzov, Institute of Mechanical Engineering of Russian INSTABILITY IN THE BOUNDARY LAYER (REVIEW) Academy of Sciences, Vasilievsky Ostrov, Bolshoy Prospect, 61, St.Petersburg, 199178, Russia V.V. K OZLOV YU.A. LITVINENKO L.L. LOEFDAHL V.G. CHERNORAI G.R. GREK

THE GRAVITATION CORRECTION FOR THE [email protected] ACOUSTICS EQUATIONS Results of studying sinusoidal and varicose instabilities of DANIIL P. KOUZOV streaky structures at the nonlinear stage of the laminar-turbulent process in shear flows are presented. The flow behavior in the [email protected] course of spatial evolution of streaky structures with a secondary high-frequency disturbance generated on them is discussed. Var- The gravitation correction for the acoustics equations is intro- ious scenarios of origination and development of coherent vortex duced. The comparison of wave processes submitting traditional structures examined in physical and numerical experiments are and improved equations is carried out for elementary model of considered. Specific features of the development of sinusoidal ideal gas isothermal oscillations in a gravity field. and varicose cases of destruction of the steady streamwise streaky D.P. Kouzov, Institute of Mechanical Engineering of Russian structure are demonstrated, such as transverse and streamwise Academy of Sciences, Vasilievsky Ostrov, Bolshoy Prospect, 61, modulation of the structure by the secondary-disturbance fre- St.Petersburg, 199178, Russia quency, appearance of new streaky structures in the downstream direction, and emergence and evolution of unsteady lambda-type structures localized in space in both cases. NONLINEAR TWIST DYNAMICS OF DNA Victor V. Kozlov, Institute of Theoretical and Applied Mechanics, MACROMOLECULE Siberian Branch of Russian Academy of Sciences, Institutskaya 4/1, Novosibirsk, 630090, Russia N.A. KOVALEVA L.I. MANEVITCH A.V. SAVIN [email protected], [email protected], EXPERIMENTAL STUDY OF STABILITY OF [email protected] VIBRATIONAL LIQUID FLOW IN A ROTATING CYLINDER Functioning of a DNA macromolecule (in particular reading VICTOR G. KOZLOV DENIS A. POLEZHAEV of the information) is substantially defined by opening of its double helix. In the present work the elementary model allowing to [email protected] describe this process is considered. One type of internal movements of DNA, namely rotational motions of the bases about The results of experimental investigation of vibrational motion sugar-phosphate chains is taken into account, thus each base is of centrifuged liquid in rapidly rotating cylinder are presented. considered as a twist pendulum and a chain as a line of connected As it is shown in [1], normal to the axis of rotation vibrations nonlinear pendulums. Our model for DNA takes into account the result in the resonant excitation of azimuth wave on the liquid

56 surface. The wave generates intensive averaged azimuth liquid A.N. Sal’nikova, Perm state pedagogical university, 24, motion in the cavity in the direction of its propagation. In de- Sibirskaya av., 614600, Perm, Russia pendence on parameters of vibration the outdrive or retrograde motion is excited. It is found that in the domain of resonant excitation of outdrive motion the liquid flow can be unstable. The SIMULATION OF EARTH–MOON SYSTEM visualization of flow with aluminum powder detects in overcriti- DEVELOPMENT cal domain the regular spatial vortexes periodical along the axis of cylinder and located near the solid boundary. The shape of the ANTON M. KRIVTSOV ERIK M. GALIMOV vortexes (rings or helical ones) depends on the problem parame- MIKHAIL S. LEGKOSTUPOV ALEXEY V. Z ABRODIN ters. The loss of stability is of threshold type. The vortexes size [email protected] is of the same order of magnitude as the thickness of liquid layer. The intensification of vibrational motion does not influence upon The EarthÐMoon system is exceptional among other planets the shape of vortexes and their structure, which do not transform and satellites in Solar system due to the great size of the satellite till the critical value of liquid velocity, when vortexes collapse (Moon) comparing to the size of the planet (Earth). To explain and regular structure disappears. The systematic experimental this phenomenon and other important properties of the EarthÐ study of spatial period of vortexes and thresholds of liquid mo- Moon system different hypothesis for the EarthÐMoon system tion stability in dependence on vibrational parameters in a wide formation were suggested. The most widely accepted hypothe- range of dimensionless frequency of rotation is performed. sis for the Moon formation is impact extrusion of the protolunar Acknowledgements material from the Earth due to its collision with a planetary body The work is supported by Russian Foundation for Basic Re- of the Martian size. However this hypothesis failed to explain search (grant RFBR 03-01-00552). the recent geochemical observations. In the presented lecture we REFERENCES argue that under certain conditions the Moon could have been [1] Ivanova A.A., Kozlov V.G., Polezhaev D.A. Vibrational dy- formed in the process of the rotational collapse of gas-dust cloud. namics of centrifuged liquid layer // Izv. RAN. Mekh. Zhid. To model dynamics of the protoplanet cloud the particles dy- i Gaza. 2005 N2. P. 133-142) namics method is used. The initial gas-dust cloud is simulated as a system of solid particles, which are partly evaporated due to adiabatic compression of the accumulation. We assume the fol- D.A. Polezhaev, Perm State Pedagogical University, 24, lowing addendums of the particle interaction: long-range gravita- Sibirskaya Str., Perm, 614000, Russia tional attraction, short-range viscoelastic repulsion at collisions, and gasdynamic repulsion due to gas evaporation from the surface of the particles. It can be shown that gasdynamic repulsion BEHAVIOR OF TWO-LIQUID SYSTEM IN A ROTATING is an effective counterbalance to gravitational attraction. Also the CAVITY SUBJECT TO LONGITUDINAL VIBRATIONS initial distribution of the particles in the original cloud, the velocity dispersion and the kinetic-potential energy ratio are analyzed VICTOR G. KOZLOV ANASTASIYA N. SAL’NIKOVA as factors determining the protoplanet cloud evolution. [email protected] Based on the described model, the computer simulation of the EarthÐMoon system formation is presented, which demonstrates The behavior of system of two immiscible liquids of different a possibility of separation of the Moon-size satellite in the pro- density in the horizontal rotating cylinder under the translation cess of the rotational collapse of the gas-dust accumulation. The axial vibrations is investigated experimentally. Researches are peculiarities of the planet-satellite formation process are studied carried out in a wide range of vibrational and rotational parame- under wide range of system parameters; in particular dependence ters. The shape of liquids interface, laws of transitions of a dense on the moment of momentum of the protoplanet cloud is investi- liquid in a centrifugal state at increase of rotation frequency is gated. studied. Attention is devoted to the wave processes proceeding A.M. Krivtsov, St.Petersburg State Polytechnical University, on the interface and average effects, caused by vibrations: vi- IPME RAS, St.Petersburg, Russia brational flows, deformation of the cylindrical shape of liquids interface. At increase of rotation frequency centrifugation occurs: un- APPLICATION OF BIOMECHANICALLY PROVED der action of centrifugal inertial force more dense liquid is uni- OSTEOSYNTHESIS IN SURGERY OF ANTERIOR formly distributed along walls, the interface gets the cylindrical SEGMENT OF FOOT shape. Vibrational influence results in excitation of inertial waves D. KUBASOV V. S IKILINDA A. POLODJISHNIKOVA of different type (depending on vibrational parameters and speed of rotation), helical and axially symmetric, propagating along an [email protected] axis of rotation. The structure of average flows near to the solid borders excited by standing inertial waves is investigated. Existing methods of treatment of transverse sliced into lay- It is shown that at low frequency of cavity rotation intensive ers foot, demand long unloading and immobilization of the limb. vibrations result in deformation of the cylindrical shape of the The operative treatment included mobilization of a capsule of 1 interface, appearance of a periodic along the axis relief on the metatarsal-phalanx joint and the basement of the basic phalanx interface is observed. of 1 toe; removal of bone exostosis and its medial bag; abduction The work is supported by Russian Foundation for Basic Re- of distal part of 1 metatarsal bone, leading to restore an axis of 1 search (grant RFBR 03-01-00552). metatarsal bone and 1 toe; adduction of distal part of 1 toe with

57 plastics. In case of significant slice into layers of anterior segment the gas filtration in a porous medium by P.Ya. Kochina and G.I. of foot, techniques were applied that approached 1 metatarsal Barinblath. However exact solutions of boundary problems were bone to bones of foot with the purpose of reduction of slicing found only for special cases. into layers of anterior segment of foot. Ways of achieving the cor- This talk is devoted to presentation of approximate solutions rect position stabilization of 1 metatarsal bone are rather various. of several boundary problems for the nonlinear heat conductivity. Thus spokes, plates, lavsan tapes, auto-or allo- sinews, endopros- The basic idea of our approach is to use the finite velocity of thesis of 1 metatarsal-phalanx joint are used traditionally. The nonlinear conductivity. The second idea is to take the symmetry purpose of the research. To improve technics of an osteosynthesis of nonlinear heat equation into account. Approximate solutions and to improve results of treatment and to refuse from plaster im- of several important problems are given. Convergence of these mobilization in patients with transverse sliced into layers foot and solutions is proved. pplication of these solutions in practice is valgus deviation of 1 toe. Material and methods. During 2003 discussed. and 2004 we began to use the ambassador chevron osteotomy fix- N.A. Kudryashov, Moscow Engineering Physics Institute (State ing of a fragment of 1 metatarsal bone with miniscrews of 1,5 and University), Kashirskoe shosse, 31, Moscow, 115409, Russia 2,7 mm diameter or clamps with automatically adjusting compression. For an osteosynthesis of 1 metatarsal bone were used clamps with automatically adjusting compression of loopy type, NON-LINEAR ANALYSIS OF DYNAMIC STABILITY OF flat, wavy and runic. The way of metal construction installation AN OFFSHORE PIPE CONVEYING FLUID consist in the following. A pattern with two sharp awllike ledges was placed on a bone and using nitinol drill 1,75 mm in diam- GUIDO L. KUIPER ANDREI V. M ETRIKINE eter bored apertures under 5 corner through the pattern aperture [email protected] strictly fixed on the distance exceeding 1/4-1/5 distance between legs of a wavy clamp. For the necessary form giving for the metal This study deals with dynamics of a submerged cantilever pipe constructions installation special clip-deformator was used. Be- conveying fluid. In the near future, the cantilever pipe will be fore the installation the construction was cooled up to 6-7. Then used in the offshore industry to facilitate a new concept of lique- it was placed in clip-deformator and, having compared legs of a fying natural gas (LNG concept). In accordance with this concept construction with apertures in a bone, carried out metal osteosyn- the gas is liquefied offshore on a ship. To facilitate the gas and thesis. The received results. After operation we did not apply condensate treatment on the ship, a great volume of cooling wa- plaster immobilization in patients. But the there were some pecu- ter is required. At some locations, to be cold enough, this water liarities in rehabilitation period. The patient moved with a super- has to be pumped up from at least 100 meters below the sea level. fluous back inflection of foot with full loading on a calcaneal seg- The pumping takes place through a set of steel, free-hanging ris- ment of foot and moderate loading on a forward segment of foot. ers, whose top is attached to the ship, whereas the tip is free (can- We had been executed 29 traditional operations with a resection tilever pipe). This type of risers has not been used in the offshore of the basis of 1 toe, plastics of joint, fixing by spokes and with industry in the past and, accordingly, there is almost no literature use of plaster immobilization. 14 operations using new technol- on dynamics of such risers. ogy, applying miniscrews and metal constructions with memory There exist a number of problems associated with dynamics effect of the form have been executed. No displacement of bone of risers, such as vortex- and wave-induced vibrations, axial dy- fragments or migration of metal constructions or necrosis of post- namics, dynamics of deep-water risers, etc. This study focuses on operative wound were noted. Movements in 1 metatarsal-phalanx a less enlighten problem of flow-through-the-riser-induced insta- joint were restored fully. There were no movements difficulty or bility. Studying this phenomenon in application to a vertically pain syndrome after new method comparing with postoperative suspended, fully submerged pipe conveying water, researchers outcomes after traditional methods of treatment. Conclusions: have found a contradiction between theoretical predictions and Osteosynthesis using constructions with automatically adjusting experiments. Theory predicts instability at small internal fluid compression allows to carry out stable fixing during all period velocities, while experiments did not show such instability. An of consolidation of fragments, that allows to refuse from plaster attempt to explain this contradiction have been made by Padous- immobilization in surgery of anterior segment of foot. sis [1], suggesting that the theoretical prediction is wrong because During application of new metal constructions is necessary to of an improper description of the negative pressurisation of water follow technology exactly, apply patterns and special toolkit. at the inlet of the pipe. In a recent paper [2], it is shown theoreti- Dimitri O. Kubasov, Rostov State Medical University, cally that the negative pressurisation influences the stability only Nachichevanski, 29, Rostov-on-Don, 344022, Russia slightly and cannot explain the contradiction. What could explain it is the hydrodynamic drag caused by surrounding water, which is shown to be an essential stabilizing factor. NONLINEAR HEAT EQUATIONS: FIFTY YEARS LATER In the present contribution, stability of the cantilever pipe aspirating fluid is studied using a non-linear description of the hy- NIKOLAI A. KUDRYASHOV drodynamic damping. As shown in small-scale experiments the [email protected] drag coefficient of the Morison equation depends strongly on frequency and amplitude of the vibrating pipe. This dependence is Problems of nonlinear heat conductivity were first considered taken into account in the analysis. The study is accomplished more than fifty years ago by Ya.B. Zeldovich and A.S. Kom- semi-analytically using the Galerkin procedure. In this way the paneyets. They found exact solutions of problems in the case critical speed of the flow through the pipe can be computed and of the problems for nonlinear heat conductivity were studied at compared with experiments. By taken into account the non-linear that time as well. These problems are also studied at solution of description of the hydrodynamic drag it has been shown that the

58 flow velocity, used in the experiments in the past was not high a skateboard, known as the Snakeboard. The Snakeboard allows enough to induce the instability. the rider to propel himself forward without having to make con- REFERENCES tact with the ground. The Snakeboard consists of two wheel- [1] Paidoussis, M.P. (1999) Aspirating pipes do not flutter at based platforms upon which the rider is to place each of his feet. infinitesimally small flow. Journal of Fluids and Structures These platforms are connected by a rigid crossbar with hinges (13), p. 419-425. to allow rotation about the vertical axis. To propel the Snake- [2] Kuiper, G.L. and Metrikine, A.V. (2005) Dynamic stability board, the rider first turns both of his feet in. By moving his torso of a submerged, free-hanging riser conveying fluid. Journal through an angle, the Snakeboard moves through an arc defined of Sound and Vibration (280), p. 1051-1065. by the wheels angles. The rider then turns both feet so that they point out, and moves his torso in opposite direction. By continu- G. Kuiper, Delft University of Technology, Faculty of Civil Engi- ing this process the Snakeboard may be propelled in the forward neering, Stevinweg 1, 2628 CN Delft, Netherlands direction without the rider having to touch the ground. Equations of motion of the Snakeboard model are derive in RHEOLOGICAL INSTABILITY AND FRACTURE OF Appells form and their analytical and numerical investigation is ELASTOPLASTIC SPECIMENS UNDER TENSION performed. The main types of Snakeboard motion (forward motion, rotate motion, transverse motion) are found. All the ob- VLADIMIR N. KUKUDZHANOV tained theoretical results are confirmed by the numerical experiments. [email protected] This work was supported by RFBR (04-01-00398) and ”Ad- vanced Scientific School of Russia” (AS-2000.2003.1). The methods and models for numerical modeling of rheolog- A.S. Kuleshov, Moscow State Lomonosov University, Leninskie ical instability under tension are developed. The influence of Gory, Moscow, 119992, Russia strain localization in shear bands and damage on the process of necking and failure of specimens are investigated. The results for tree different types of stress-strain states were considered and compared: 1) tension of an axisymmetric cylindrical specimen; ON A STRETCHING-TORSION OF A NATURALLY 2) tension of a thick plate in plane strain state and 3) a thin plate TWISTED ROD tension in plane stress state conditions. The models of material NATALYA V. K URBATOVA taking into account the generation of microdefects as well as their growth were considered (GTN-models). It was shown that for ad- [email protected] equate descriptions of the process of stability and localization in shear bands it is necessary to take into account damage of ma- By means methods of spectral theory of operators Saint- terial. The process of instability under tension is possible only Venant solution for pseudocylinder were represented as linear for the material with softening diagram or when for elastoplastic combination of twelve elementary solutions, for determine them material Drucker postulate is satisfied. were obtained two-dimensional problems on a cross-section and In the cylindrical or in square cross-section elastoplastic speci- formulated variational principle for a stretching-torsion and bend- mens the shear bands do not form and the fracture occur in planes ing pseudocylinder [1]. orthogonal to the direction of the tension. In ideal-plastic material Due to this became possible to apply FEM for solving prob- in post critical state the periodic structure of necking is formed. lem of a stretching-torsion naturally twisted rod with rectangular When the porosity of damage material is taking into account then cross-section. The geometry has defined the rectangular form of the fracture surfaces take cup-cone shape that corresponds to ex- a finite element with four nodal points and bilinear basic function perimental results. For the thick plate under plane strain for GTN [2]. material the very intensive shear bands were inclined about 45 de- Features of a problem are those, that the system of the alge- grees to the tension direction. For the thin plate the angle of shear braic equations received as a result of discretization of a problem bands was 35-40 degrees. It should be mentioned that for all con- is non-negative. sidered elastoplastic models the postcritical states were unstable As a result of research the behaviour factors of rigidity of a and mesh-sensitive. But they can be regularized if the elastoplas- rod from extent of twisting is analysed and the tensely-deformed tic matrix would be replaced by elastoviscoplastic matrix. state of a cross-section is demonstrated for a set of parameters. This work was carried out under financial support of the Rus- Acknowledgements The author is deeply grateful to Yu.A. sian Foundation of Basic Research (Grant 03-01-00701) and Fun- Ustinov for the inspiration. damental Research Program of RAS (OEMMPU-14). V.N. Kukudzhanov, Institute for Problems in Mechanics of Rus- REFERENCES sian Academy of Sciences, Vernadskogo pr-t, 101, b.1, Moscow, [1] Ustinov Yu. A. Saint-Venant problems for a pseudocylinder. 119526, Russia Moscow 2003. p. 128. [2] Ustinov Yu. A. Kurbatova N.V. Saint-Venant problems for ON THE SNAKEBOARD DYNAMICS a rods with physical and geometrical spiral anysotropy. // Russian Izvestiya, North-Caucas. Region. Ser. Natural Sci- ALEXANDER S. KULESHOV ences. Math. Model. Spec. Issue. 2001. P. 154-157. [email protected]

We consider a simple mathematical model of a derivative of N.V. Kurbatova, Rostov State University, Rostov-on-Don, Russia

59 is shown that using of complex values leads to two-waved pro- NONLINEAR INTERACTION OF NORMAL ELASTIC cesses in these cases corresponding to physical character of the WAVES IN SOLIDS problem. Selective character of action of magnetic field to the string’s vibrations exhibits in forms of obtained solutions (some KRISTINA KURENNAYA of them are performed as the only function and the others - as [email protected] series that depends from conditions for the sizes and position of active part of the string). In the paper it is considered and inves- In resent years there is observed the intensive development of tigated the case of stimulated vibrations using the same method the ultrashort elastic waves diapason, where the nonlinear interac- too. tions are quite effective. The problem of nonlinear effects anal- N.F.Kurilskaya, Altay State Technical University, 46 Lenin Street, ysis while propagation of harmonical elastic waves in lowsym- Barnaul, 656000, Russia metrical anisotropic crystalline bodies belongs to the number of actual and not investigated in many fundamental and applied aspects problems of deformed solids dynamics. So just in this ON MAGNETOSTRICTIVE VIBRATIONS OF sphere it is possible to create the new class of parametrical de- CONDUCTIVE STRING IN MAGNETIC FIELD vices, which effetely use the acoustic effects of the second harmonics, and by that to make possible the calculation of construc- NATALYA F. KURILSKAYA tive parameters of acoustic-electronics devices with quite more [email protected] precision. Also the nonlinear effects are used while the development of the method of power hypersound generation with the It is considered equations of magnetostrictive vibrations of help of quantum generators. conductive string in homogeneous non-stationary magnetic field. In presented work the problem of nonlinear distortion while Some solutions of the problem are obtained in closed form for interaction of normal SH- and P-SV-waves propagation along natural vibrations of the string with classical and non-classical elasticequivalent direction in cubic system crystalline lamina has boundary conditions as in form single function as in form of se- been considered. The investigation is based on nonlinear rep- ries under complementary restrictions for sizes of active part of resentations of elastic potential and elastic deformations tensor the string and the character of non-stationarity of magnetic field components. The displacement vector components are repre- by modified method of Fourie. The case of stimulated mag- sented as a sum of linear waves and nonlinear distortions, pro- netostrictive vibrations of conductive string is considered too. portional to the acoustic Mach number first degree. An analytical It is investigated the opportunity of damping of magnetostric- representation for second harmonics of interacting normal waves tive effect by the special choice of function of external excita- has been found, and the contribution of combinative type section. However there is no complete damping of parametric action ond harmonics to the summarized nonlinear distortion has been of magnetic field because magnetostrictive vibrations are three- investigated. It was found out that combinative type harmonics dimensional and damping of magnetostrictive effect exhibits only contribution to the summarized nonlinear distortion is leading. in one plane. So and in this case action of magnetic field to the Also the more general problem of nonlinear SH-waves excitation string’s vibrations remains as forced as parametrical. near the elastic layer edge has been solved, that allows to esti- N.F.Kurilskaya, Altay State Technical University, 46 Lenin Street, mate real magnitude of nonlinear effects. The method allows to Barnaul, 656000, Russia analyze nonlinear waves propagation along arbitrary directions in anisotropic elastic waveguides. K. Kurennaya, Potsdam University, Am Neues Palais, 10, Pots- UNSTEADY DYNAMICS OF EDGE WAVES ABOVE A dam, 14469, Germany SHELF SLOWLY VARIED ALONG SHORE

A.A. KURKIN O.E. POLOUKHINA I.F. NIKOLKINA VIBRATIONS OF CONDUCTIVE STRING WITH [email protected], [email protected] CLASSICAL AND NON-CLASSICAL BOUNDARY CONDITIONS IN MAGNETIC FIELD The sea and ocean waters are in the constant dynamics. The waves that influence greatly all the physical processes in the NATALYA F. KURILSKAYA world ocean are considered to be the most popular waves in the [email protected] dynamics. Therefore the studying of wave dynamics is important for the building the theoretical base. More other, the researchers Integro-differential equation in partial derivatives with classi- of this kind are important for the reality. The list of the practical cal and non-classical boundary conditions describing of vibra- problems that have to be studied in the connection of the theory tions of conductive string in homogeneous stationary magnetic of the wave dynamics includes navigation, the protection against field acting on given part of string is investigated by modified tsunami, the non-contact methods of ocean studying, the usage of method of Fourie. Integral addend of equation of vibrations char- wave energy and others [1]. acterizes action of magnetic field and essentially changes proper- The present paper deals with the problem of unsteady dynam- ties of system. It is suggested to look for the solution of initial ics of edge waves above the shelf slowly varied along the shore. equation in complex form with complementary addend depend- The edge waves influence the zone which is economically im- ing only from time. Such approach allows us to obtain some ex- portant, as a result the intensive research into this sphere have act solutions of the problem for the string with different types been done recently. The problem of unsteady dynamics of edge of boundary conditions as for classical as non-classical ones. It waves is connected with the fish industry, oil industry, projecting

60 buildings along the shore. The problem of unsteady dynamics of REFERENCES edge waves in the cylindrical pool is solved numerically in the [1] Galimov E.M., Krivtsov A.M. Non-impact concept of the present paper. Exponential shelf and shelfstep testing give ap- origin of the Earth-Moon system. Geochemical constraints propriative results. The analysis of the amplitude characteristics and dynamical simulation. 2005. Paper in press. is based on the adiabatic theory of linear edge waves above the V.A. Kuzkin, St. Petersburg State Polytechnical University, St. slope bottom slowly varied along the shore [2]. The asymptotic Petersburg, Russia theory approves the idea that the wave amplitude could be found from the energy flow conservation law. Three bottom profiles are considered: the constant slope beach, exponential shelf and ON THE FEM APPROACH FOR A BENDING OF A shelfstep. The amplitude modification along the shore is calcu- NATURALLY TWISTED ROD lated for the cases when the parameters of the shelf zone change slowly towards the coast. The characteristic shelf width influence NATALYA M. KUZNETSOVA NATALYA V. K URBATOVA the amplitude modification of the edge waves, and its increasing leads to amplitude decreasing above the exponential shelf. The kuznet [email protected], [email protected] even decreasing of three parameters of the shelfstep along the coast leads to double increasing of the zero mode amplitude of The three-dimensional problem of the theory of elasticity for a the edge wave. The characteristic shelf width end width does not naturally twisted rod, with the stress-free surface side was consid- influence the zero mode amplitude, the increasing of slope and ered. This investigation is based on monograph results [1], where depth in the infinity of the concave exponential shelf leads to the this problem was reduced to a set of two-dimensional boundary order decreasing of the amplitude of the edge wave. value problems and the variational principle. This paved the way for construction the FEM solution of the bending of naturally We can conclude that slowly changing bathymetry intensify twisted rod with rectangular cross-section. the edge waves and with the help of dispersive focusing it leads Rectangle with four nodal points and six unknowns for each to the appearance the anomalous edge wave along the shore [3]. node is considered as a finite element. Unknown functions on REFERENCES element is found as generalized displacements. The solution is formed as linear combinations of unknown displacements with [1] Kononkova G.E., Pokazeev K.W. The sea waves dynamics. bilinear base functions, which are defined on a cross-section. M.: Moscow State University, 1985. With using FEM technology the system of algebraic equations [2] Kurkin A., Pelinovsky E. Shallow-water edge waves above for local finite element and for whole grid is obtained. an inclined bottom slowly varied in along-shore direction As opposed to a problem of stretching-twisting, the problem of // European Journal of Mechanics B/Fluids. 2003. V. 22. P. bending is reduced to quadratic functionals, with complex items. 305 316. It strongly complicates a problem. In result the algorithms of FEM discretization of problem and [3] Kurkin A., Pelinovsky E. Focusing of edge waves above the construction of solution on a cross-section have been ob- sloping beach // European Journal of Mechanics B/Fluid. tained. 2002. V. 21. P. 561 577. Acknowledgements The authors are deeply grateful to Yu.A. Ustinov for the inspi- Irina F. Nikolkina, Nizhny Novgorod State Technical University, ration. Minin St. 24, Nizhny Novgorod, 603600, Russia REFERENCES [1] Ustinov Yu.A. Saint-Venant problems for a pseudocylinder. Moscow 2003. p. 128. ANALITICAL MODELING OF PROTOPLANET CLOUD FRAGMENTATION N.M. Kuznetsova, N.V. Kurbatova, Rostov-on-Don, Russia VITALIY A. KUZKIN

[email protected] FINDING OPTIMAL PARAMETERS OF A BEAM ON ELASTIC BASE UNDER VARIOUS LOADING The paper is devoted to qualitative analysis of results obtained by computer modeling of the development of the Earth-Moon A.V. KVANIN G.V. KOSTIN V.V. S AURIN system. This modeling was performed in paper [1] by method [email protected], [email protected] of particles dynamics. Main hypotheses is based on the assumption of origin of the The mechanical system consisted of a beam embedded into Earth-Moon system in the process of rotational collapse of a elastic medium and loaded by static forces is considered. The gas-dust cloud, followed by its fragmentation in two protoplan- elastic properties of the system are known. To find the stress- ets. The moment directly before fragmentation is being consid- deformed state of the system the decomposition method is ap- ered. In this moment cloud is represented by two touching bodies, plied. This approach leads to solving a plane elasticity prob- revolving about their common center of mass. The bodies are as- lem (estimation of the modulus of subgrade reaction) and one- sumed to have the form of globes or disks (two main variants are dimensional bending problem of the beam on elastic (Winkler’s) considered) staying in equilibrium of the gravitational and iner- base. The sensitivity analysis of cross-section beam shape, geo- tial forces. Then dependence of the moment of momentum and metrical and rigidity parameters of the system on its integral char- critical rotational velocity on the mass ratio of the protoplanets acteristics (maximal beam flexure, elastic energy, etc.) is carried is studied. These dependences show that the most probable ratio has the same order as the current Earth-Moon system. 61 out. The results of numerical modelling for various loading such For examples are high-speed pressed, make a exact hole in sur- as distributed load, a point force, periodically distributed load are face of constructional materials, welding of explosion and etc.. presented. The modeling of this technological process have a mathemati- This work was supported by RFBR (02-01-00157, 02-01- cal difficulties. There are not only in process of calculation, but 00252, 03-01-96586), Leading Scientific Schools Grant NSh- much more difficulties are in process of description quantitative 1627.2003.1 and Grant NWO (047.014.007). peculiarities of intensity deformational process. Mathematical Alexei V. Kvanin, pr. Vernadskogo, Moscow, 119526, Russia modeling statement of boundary problems are impossible without knowledge about peculiarities of spreading boundary shock indignation by deformable medium. Quantity and kind of ori- THE PROPERTIES OF THE PLANE-STRAIN gin surfaces of strong breaks are unknown, but it is necessary QUASI-STATIC EQUATIONS FOR A RIGID-PLASTIC to know for practical technologic process by impulse and shock MATERIAL WITH ANISOTROPIC HARDENING influence to deformable materials for control of quality technological works. TATIANA B. LAV ROVA In really paper consider peculiarities of construction mathe- [email protected] matical modeling for solution non-automodel boundary problems of nonlinear dynamics in cylindrical coordinates. Method of solu- The process of quasi-static deformation of rigid-plastic mate- tion considered and illustrated for solution of technological prob- rial with anisotropic hardening is considered. Additionally it is lems about shock deformation thinck-tube from uncompressible adopted the kinematical restriction of plane strain. As a yield- high-elastic materials ing condition it is taken a translational-isotropic generalization N.F. Lebedeva, Far Eastern State Technical University, Pushkin- of the Tresca condition, where the principal stresses are replaced skaya 10, Vladivostock, 690000, Russia by those of the Kadashevich-Novozhilov active stress tensor. A certain characteristic size of the yielding surface is specified by a material function that depends on the hardening parameter [1]. Its ANOMALOUS CONTINENTAL SHELF WAVES variation in the active-loading process is determined completely by the choice of the yielding condition [2]. The plane-strain prob- A.K. LEKANOV O.E. POLOUKHINA A.A. KURKIN lem for such a material is statically undeterminate. The whole [email protected], [email protected] system of equations consists of two equilibrium equations and the strain-compatibility one. The system is shown to be hyperbolic, equations of characteristics having been obtained [3]. The group In the present work we deal with 2D topographically trapped analysis of the system is performed. To lower the order of the oceanic waves known as continental shelf waves; their frequen- system, the method of power-series expansion in loading param- cies are small compared to Coriolis parameter; their characteristic eter is used. The features of suggested procedure of the analysis feature is also multi-modal structure. Here we study the forma- of above-mentioned system are illustrated by the example of the tion of large amplitude shelf waves. The linear effect of disper- problem for extension of a strip weakened with symmetric trian- sive focusing for these waves is demonstrated for single-mode gle cuts. and multi-modal waves. The optimal conditions for focusing are discussed as well as characteristics of wave field far from the fo- REFERENCES cal area. The influence of initial conditions on the process of [1] Bykovtsev G.I., Lavrova T.B. A model of solid with anomalous shelf wave formation is investigated. anisotropic hardening and different responses for extension and compression (in Russian)// Izv. AN SSSR. Mech. Anton K. Lekanov, Nizhny Novgorod State Technical University, Solids. No.2. 1989. P. 146-151. Minin St. 24, Nizhny Novgorod, 603600, Russia [2] Bykovtsev G.I., Lavrova T.B. The properties of singular yield surfaces in strain space / In: Applied problems of con- ON INFLUENCE OF FORM OF HOMOGENEOUS tinuum mechanics. Vladivostok. 1991. P. 3-20. POROUS FUEL ELEMENT ON COOLING PROCESS [3] Lavrova T.B. The properties of equations of plane-strain problem for rigid-plastic material with anisotropic harden- V.A. LEVIN N.A. LUTSENKO ing / In: VIII Russian congress of theoretical and applied [email protected], [email protected] mechanics. Volume of abstracts. Perm. 2001. P. 389. T.B. Lavrova, Samara State University, Acad. Pavlov street, The gas flow through solid porous homogeneous motionless Samara, 443011, Russia medium with heat evolution is considered. Such movement may arise from cooling of fuel elements. Similar model was suggested in [1, 2] for description of the cooling process of the destroyed ANALITICAL MODELING SHOCK WAVES IN CURVED block of Chernobyl atomic power station. COORDINATES The porous fuel element is supposed to be homogeneous and motionless, it is bounded from sides by non-heat-conducting NATALIA F. LEBEDEVA-KARLSSON walls, and it is opened on top and underneath. Cool gas under [email protected] pressure is blown from beneath, it moves from bottom to top through porous medium and becomes warm in result of heat ex- Modern technological processing of materials include meth- change, and then it flows out to open space with given pressure. ods combined by high-speed of deformation of procurements. The cooling model is founded on the model of two interactive

62 continuums [3]. The heat-evolution process in solid phase is sup- interaction with microcracks, which grows due to large plastic posed to result from process of chemical reaction, and at the same deformation (in GTN porous material model). time volume and mass of solid phase change insignificantly and The prediction of fracture based on model which does not tak- we may neglect the same changing. Heat-evolution decreases ex- ing into account material damage leads to unrealistic fracture sur- ponentially, intensity of interphase heat exchange is proportional faces. The model with microdamage gives better results for frac- to temperature difference of phases in the considered point of ture surfaces, but it should be mentioned that calculation algo- medium. Gas viscosity depends on temperature by Sutherland rithms are very rough and far from perfections, and should be formula. improved. The numerical method, which is combination of explicit and This work was carried out under financial support of the Rus- implicit finite difference schemes, is proposed for solving un- sian Foundation of Basic Research (Grant 03-01-00701) and Fun- steady plane problems of gas movement through homogeneous damental Research Program of RAS (OEMMPU-14). porous fuel element. The plane unsteady problem about begin- A.L. Levitin, Institute for Problems in Mechanics of Russian ning of forced filtration in the moment of heat-evolution starting Academy of Sciences, Vernadskogo pr-t, 101, b.1, Moscow, in solid phase is considered for elements of three different forms: 119526, Russia smoothed convergent form, step convergent form and for element with stagnation zones. New steady-state conditions of cooling or unlimited heating of fuel element are shown to be established in COMPARISON OF DISCRETE AND CONTINUUM that case. The most heating zones of those elements are deter- MODELING FOR 2D NANOCRYSTAL STRIPE mined. The process of cooling of the fuel element is shown to VIBRATIONS be affected strongly by the form of the element. Narrowing of the element in the upper part makes worse the total cooling of the OLGA S. LOBODA ANTON M. KRIVTSOV element. Zones with slow movement of gas (stagnation zones) loboda [email protected] result in intensive local heating. It may disturb stability even if heat-evolution in solid phase is a little. In this work we investigate bending vibrations of a 2D REFERENCES nanocrystal stripe having a) cubic, b) hexagonal lattice. The vi- [1] Maslov V.P., Myasnikov V.P., Danilov V.G. Mathemati- bration spectrum and dispersion relations are obtained. On the cal simulation of the destroyed block of Chernobyl atomic other hand the spectrum is determined by the continuum theory power station. Nauka, Moskva, 1987. (in Russian) of rods where longitudinal and bending vibrations are consid- [2] Maslov V.P. Overheating effects in filtration mediums. Dok- ered. Then eigenfrequencies are expressed in the terms of elastic lady AS. 1992. Vol.326. 2. pp.246-250. (in Russian) moduli. Comparing eigenfrequencies being found using discrete [3] Nigmatulin R.I. Foundations of the mechanics of the het- and continuum approaches the conditions for elastic moduli de- erogeneous medium. Nauka, Moskva, 1978. (in Russian) termination are obtained. The obtained elastic characteristics for nanostripe are compared with the same characteristics obtained Nickolay A. Lutsenko, Institute of Automation and Control Pro- earlier from the static approach. cesses, FEB RAS, 5, Radio St., Vladivostok, 690041, Russia O.S. Loboda, IPME RAS, Bolshoy pr. V.O., 61, St. Petersburg, 199178, Russia THE INFLUENCE OF NOTCHES AND CUTS ON THE SPECIMENS STABILITY UNDER TENSION OSCILLATIONS IN A HARDSPHERE MODEL UNDER GRAVITY ALEXANDER L. LEVITIN alex [email protected] NADEJDA LOGUINOVA

The damage problem under tension is significant for many technological processes of metal forming (stretch, dragging, Evolutionary processes in granular system confined in a closed rolling). Clear understanding of origins and conditions for lo- box under gravity are established to be oscillatory in nature [1]. calization in shear bands is important. For a threeÐdimensional model of granular system we have nu- In this work loss of stability and postcritical deformation of merically investigated its transition from a given initial state in elastoplastic material and damage of specimens under tension range of hardÐsphere approximation. are investigated. Three models of elastoplastic material are con- In the present study we inquire into the system parameters in- sidered: 1) the classical ideal-plastic model; 2) the Gurson- fluence on the oscillations characteristics. We have established Tvergaard-Needleman model (GTN-model) of effective porous how the oscillations period, amplitude and damping factor de- material with ideal-plastic matrix; 3) the GTN porous model with pend on the system parameters. To simplify the problem under macrodamage. In the 2)- and 3)-models under the deformation consideration we have converted our system through dimension- process a nucleation and a growth of pores are taking into ac- less units. Some analytical dependencies result directly from the count. In the last model after the local damage criterion is reached change of variables used to reduce the system. the damaged material (FEM element) removed from the calculation and the free surface formed instead it. REFERENCES The notches and cuts influence on stress-strain state on the [1] N.B. Loguinova, G.J.F. van Heijst, P.P.J.M. Schram, stability and postcritical state is investigated. The thin cuts and S.A. Trigger, and Yu.P. Vlasov, PHYSICA A 323 (2003) notches were using for modeling material imperfections and their 155.

63 Nadejda B. Loguinova, Moscow State University of Transport The buffer represents a screw cylindrical spring made of steel Communications, Obrazstova Street 15, Moscow, 127994, Rus- with circular cross section, which basis is embedded in the plate sia and does not lose its stability during deformation. Deformation of the spring is directly proportional to the force applied. As a method of the decision the ray method and method of APPLICATION OF THE STRESS AND STRAIN splicing asymptotic expansion received for small times in a con- CONTROL BY EIGENSTRAIN TO BIOMECHANICAL tact area and outside of it are used. In the present work, the pro- SYSTEMS cedure proposed in [1] for the analysis of transverse impact of a VALERIY LOKHOV YURIY NYASHIN solid body upon an elastic buffer positioned on an elastic isotropic plate, is generalized to the case of shock interaction of a solid [email protected], [email protected] body with a elastic buffer positioned on an elastic orthotropic plate. The general theory of linearized continuum mechanics with During the interaction of the sphere with the buffer and, hence, eigenstrain is considered in relation to biomechanical systems. the plate, quasilongitudinal and quasitransverse waves, which are The general term “eigenstrain” was introduced by H. Reiss- the surfaces of strong discontinuity, begin to propagate.In a plate ner in 1931 where it was supposed that total strain can be rep- of the surface of strong discontinuity represent cylindrical sur- resented as a sum of elastic strain related to stress by Hookes faces - strip, whose generators are parallel to the normal to the law and inelastic strain (eigenstrain). As eigenstrain it is possible median surfaces and guides locating in the median surface are to consider thermal strain, plastic strain, creep strain, strain due ircumferences extending with the normal velocities. Behind the to phase transitions (in particular hyperelastic strain in materials wave fronts, the solution is construction in terms of ray series rep- with shape memory), growth and remodeling strains in living tis- resenting power series, whose coefficients are the different order sues, etc. It turns out that there are important properties of solu- discontinuities in the time-derivatives of the required functions tion of continuum mechanics problems that are independent from and the variable is the time passed from the moment of arrival of origin of eigenstrain. a wave to the given points of the plate. To determine the ray series In 1987 T. Mura introduced the concept impotent eigenstrain coefficients for the desired function, it is necessary to differentiate as an eigenstrain not producing stress. In 2001 H. Irschik and the governing equations with respect to time, to take their differ- F. Ziegler introduced the concept of nilpotent eigenstrain as an ence on the different sides of the wave surface, and to apply the eigenstrain not producing total strain at every point of a system. condition of compatibility. As a result of the procedure described, The fundamental theorem is proved that any eigenstrain ex- we are led to the system of recurrent differential equations, which isting in a body can be uniquely decomposed into impotent and solution gives us the discontinuities in time-derivatives of the de- nilpotent constituents. The proof of theorem is based on concepts sired values within arbitrary constants. The arbitrary constants of functional analysis, in particular, on theory of Hilbert spaces. are determined at splicing on border of contact area of the solu- It is shown that these constituents are orthogonal in this space. tion for required function inside a contact disk and outside of it. The application of theorem on decomposition and its corol- The found jumps allow us to write down the required functions laries allows constructing algorithms of independent control of as the ray series with the coefficients expressed algebraically. stress and deformations, displacements as well (problem of shape control of bodies). The obtained algorithms in many cases sim- The simple and compact analytical expressions for contact plify considerably the solution of control problems, as they do not force are defined. The carried out numerical researches allow to demand repeated solution of appropriate boundary value prob- make the conclusion about influence of parameters of a construc- lem. tion, including elastic orthotropic properties, on dynamic char- In this work developed methods are applied to biomechanical acteristics of interaction. Five-term truncated ray series for the systems where growth strain of living tissues and eigenstrain due desired functions have allowed one to calculate with the given to phase transitions in the appliance made of shape memory al- accuracy the stresses in the contact area of the plate and dynamic loys take place. sag of the plate. Valeriy A. Lokhov, Perm State Technical Unversity, Komsomolskii prospect, Perm, 614000, Russia REFERENCES

DYNAMIC STABILITY OF AN ORTHOTROPIC PLATE [1] Rossikhin Yu.A., Shitikova M.V., and Loktev A.A., The SUBJECTED IMPACT EXCITATION analysis of thin-walled building structures subjected to impact excitation, Proc. 4th International Ph.D. Symposium in ALEXEY A. LOKTEV YURIY A. ROSSIKHIN Civil Engineering. Sept. 19th - 21th. 2002, Munich, Ger- MARINA V. S HITIKOVA many, P.Schiebl, N. Gebbeken, M. Keuser, and K. Zilch [email protected] (eds.) Vol.1. - P. 487-492.

Mathematical modelling of the impact of a solid body upon a elastic buffer positioned on an elastic orthotropic plate, whose dy- A.A. Loktev, Yu.A. Rossikhin, M.V. Shitikova, Voronezh State Uni- namic behaviour is described by the Uﬂyand-Mindlin equations versity of Architecture and Civil Engineering, Department of The- taking the rotary inertia and shear deformations into account, is oretical Mechanics, st. 20-letija Oktjabrja 84, Voronezh 394006, investigated. Russia

64 conception of this method is based upon the known qualities of DEFORMATION OF ELASTIC MATERIAL WITH glass materials and the methods of improvement their static and STRESS-STATE-DEPENDENT PROPERTIES AT THE dynamic strength. Glass material is the part of sandwich compos- CRACK TIP UNDER LONGITUDINAL SHEAR ite in which it is a layer between stretched metal panelings. De- CONDITIONS pending on the purpose of usage different quality types of glass with different metal paneling stretched up to the necessary sizes EVGENY V. L OMAKIN HANS-JURGEN CHRIST are used. Therefore the reductions of glass mass takes place in the [email protected], process of its formation what prevents the formation of surface [email protected] micro fissures and leads to the compression of the glass mass. Due to more intensive size contraction of metal paneling during Many materials contain microdamages, such as microcracks, the cooling process of the glass mass reduction takes place. The inclusions, pores, reinforcing elements and others, which behave reduction measure is regulated by the difference in the tempera- in different ways depending on the type of external forces. There- ture expansion coefficient of the composite layers by the special fore, the effective deformation properties of the materials depend admixtures in the liquid glass mass phase. The liability of com- not only on the bulk concentration of heterogeneities but on the posite layers junction is assisted by a certain temperature level type of the stressed state in solids, too. The influence of the dam- and conditions of metal paneling surface in the period of the glass ages on the stress, strain and displacement fields at the crack tip formation of the inner layer. The main problem of the model is to can be simulated by the corresponding dependencies for effective define displacements and stresses originating in making process deformation properties of a material. Since the stress state near of composite shell depend upon temperature and time conditions. the end of a crack is inhomogeneous one, the deformation proper- The hemispherical environment consists of external metal cov- ties are the functions of co-ordinates and this effect must be taken erings inside which glass melt is placed. Prepared beforehand into account to analyze the crack problem. hemispherical metal facings are established in the form, heated The processes of shear and volumetric deformation are interre- up to temperature providing reliable connections with glass melt, lated in the materials under consideration. Therefore it is impos- space between them is filled with glass hot melt and then there sible to separate the fields corresponding to shear and bulk strain is a cooling of all the composition. Losses of heat at cooling of a material. The constitutive equations are proposed to simulate a composite much more surpass that amount of heat, which is the dependence of the effective elastic properties of the materials formed at its deformation. Therefore the general task of defor- on the type of the stress state and to describe the relationship be- mation composite environments is an incoherent problem and the tween the shear strains and volumetric deformation of the mate- temperature task and then deformation are solved separately. rials. In these equations the ratio of hydrostatic component of the Determination of displacements and stresses in making pro- stresses to the stress intensity is used as the stress-state-parameter. cess of composite shell we give on base of main hypotheses the- It is shown that for the solution of the problems under lon- ory of shells, hypotheses Kirhgofa in glass and hypotheses Tim- gitudinal shear conditions, the traditional approach based on the oshenko in metallic jackets. hypothesis of antiplane strain can not be used because the longi- 1.Patent Russian Federation N 2067060. The processes of tudinal shear causes the displacements in other directions. The manufacture the solid body shell of underwater apparatus. Pikul stress function can not be used, too. Therefore the corresponding V.V.// Inventor Bulletin.1996.N 27 representation for the displacement field is proposed. In spite of O. Lubimova, Far Eastern State Technical University, Vladivos- non-linearity of the constitutive equations, the displacements can tok, Russia be represented in corresponding simple forms which give the possibility to study the singular parts of the stresses and the strains. The method for the numerical study of stress, strain and dis- ANALYSIS OF ROTATING DISKS UNDER CYCLIC placement fields is proposed. The calculations for various mate- THERMAL AND MECHANICAL LOADS rials functions are performed and compared with the values corresponding to the linear elastic solids. Under the condition of H. MAHBADI M.R. ESLAMI M. KAMRAN out-of-plane shear the opening of the crack faces is observed. h [email protected], [email protected] This work was supported by INTAS (grant N 03-51-6046). Evgeny V. Lomakin, MGU, Faculty of Mechanics and Mathemat- Cyclic loading of structures result into either structural shake- ics, Vorobjovy Gory, Moscow, 119992, Russia down or ratcheting. The prediction of ratcheting and shakedown behavior may depend on the hardening theory which used to plastic analysis of the structure. Two general hardening theories are DEFINITION OF RESIDUAL PRESSURE DURING available for plastic analysis of structures namely, isotropic and MAKING THREE-LAYERS OF SHELL ON BASE GLASS kinematic hardening theories. The isotropic hardening model AND METAL always predict shakedown behavior, if creep is not considered (Naderan et.al [1], Mahbadi and Eslami [2,3]). In this article OLGA N. LUBIMOVA cyclic loading analysis of the rotating disk under mechanical and [email protected] thermal loads is investigated. Various kinematic hardening models such as Prager [4], Armstong-Frederick [5] and Chaboche We dive consideration to mathematical model of making hemi- [6,7], are used to evaluate the structure behavior in the plastic spherical joint shell on base of glass and metal. The hemispher- region. The disk material is assumed to obey nonlinear strain ical joint shell consists of new sandwich-type material in which hardening curve. An effective numerical method is proposed to glass material is confined between stretched metallic jackets. The calculate cyclic loading results of the rotating disk structure. The

65 results of the proposed method are compared with those of the The goal of this contribution is the investigation of the gap ANSYS computer program. It is concluded that the Prager kine- soliton analogue and its stability in the modulated chain frag- matic hardening model always predicts shakedown to reversed ment consisting of four anharmonic oscillators with alternation plasticity for cyclic loading analysis of the rotating disk. The of normal frequency parameters. Nonlinear monochromatic os- Armstong-Frederick kinematic hardening model predicts ratchet- cillations of this system have been studied analytically and nu- ing for load controlled cyclic loading and shakedown to reversed merically, and dependences of frequencies of the oscillations on plasticity for deformation controlled cyclic loading of the struc- the integral of the number of states are calculated. These depen- tures. The same results are obtained when the Chaboche kine- dences determine uniquely the quasi-classical energy spectra of matic hardening model is used to evaluate the cyclic stress-strain the system since the relationship ωi(N)=∂Ei/∂N between curve. The analysis data are validated against the experimental the frequency, the energy and the integral of the number of states data wherever these data were available. holds. REFERENCES The results of the investigation are the following. The prob- [1] Naderan-Tahan, K., Eslami, M.R. and Mahbadi, H., ”A lem of monochromatic oscillations of four oscillators has been Cyclic Loading Analysis of Structures of Strain Hardening reduced to four independent ones, namely the problems of sta- Material”, ISME2000 Conf., Sharif Univ. of Tech., Tehran, tionary states of two different coupled nonlinear oscillators. A 2000, pp. 67-74. bifurcation diagram of the corresponding solutions and a composite structure of the frequency spectra for arbitrary value of normal [2] Mahbadi, H., and Eslami, M.R., ”Load and Deforma- frequency parameters ratio γ have been obtained. The principal tion Controlled Cyclic Loading of Beams, Based on the nonlinear oscillations and the bifurcations on their spectral de- Isotropic Hardening Model”, Accepted for publication. pendences have been found analytically. The process of appear- Trans. ISME. ance of analogue of the gap soliton has been investigated in de- [3] Mahbadi, H., and Eslami, M.R., ”Load and Deformation tail. The critical value γc , above which the gap soliton analogue Controlled Cyclic Loading of Thick Vessles, Based on arises, has been found. Transformation of the gap soliton solution the Isotropic Hardening Model”, Accepted for publication. into the out-gap-soliton one has been demonstrated evidently for Trans. ISME. the case of a large difference between oscillator frequencies. [4] Prager, W., A New Method of Analyzing Stresses and The stability problem of the solitonic solutions has been for- Strains Work-Hardening Plastic Solids, Journal of Applied mulated and solved. The analysis has included a study of the Mechanics, December 1956, pp. 493-496. solution stability with respect to small deviations and numerical [5] Armstrong, P.J. and Frederick, C.O., “A Mathematical Rep- simulations of the gap soliton dynamics under the perturbations. resentation of the Multiaxial Bauschinger Effect”, CEGB As a result of the consideration eigenfrequencies of the corre- Report No. RD/B/N 731, 1966. sponding linearized equations have been found and frequency regions of a stability of the gap soliton analogue have been deter- [6] Chaboche, J.L., Dang-Van, K. and Cordier, G. Modelization mined. It is shown that the gap soliton loses its stability after of the Strain Memory Effect on the Cyclic Hardening of 316 transformation into the out-of-gap soliton solution. These results Stainless Steel Proceedings of the International Conference have been confirmed by direct numerical simulations of the gap on SMiRT, 1979, Div. L, Berlin, Germany. soliton evolution. [7] Chaboche, J.L. Time Independent Constitutive Theories for E.Yu. Malyuta, Scientific and Technological Center of Electro- Cyclic Plasticity, International Journal of Plasticity, Vol. 2, physics of NAS of Ukraine, 4a Chaikovsky Str., Kharkov 61002, 1986, pp. 149-188. Ukraine Hossein Mahbadi, Tehran, Iran M.M. Bogdan, B.Verkin Institute for Low Temperature Physics and Engineering of NAS of Ukraine, Lenin Ave. 47, Kharkov, 61103, Ukraine STABILITY OF GAP SOLITON MODES IN MODULATED NANOCLUSTERS CROSS-SECTIONAL STABILITY OF ALUMINIUM E.YU.MALYUTA M.M. BOGDAN EXTRUSIONS [email protected] J. MENNINK F. SOETENS N.A. KUTANOVA In the last decade in studying nonlinear dynamics of discrete [email protected], modulated systems a great attention is paid to discussion of ex- [email protected], istence of the so-called gap solitons. It is known that a spec- [email protected] trum of a modulated system has a frequency gap in which linear waves cannot exist. However due to nonlinearity the solitonic Aluminium extrusions applied in daily practice are often thin- waves can be excited in this frequency region. They are spatially walled with complex cross-sectional shapes. These shapes are localized up to a lowest edge of the gap of a linear wave spec- based on a variety of demands that are in general non-structural. trum. Then they are transformed into the out-of-gap solitons, As a result, several types of instability may occur, including over- which have partly non-vanishing asymptotics. Diatomic anhar- all and cross-sectional instability modes as well as mode inter- monic chains, coupled modulated nonlinear optical waveguides, actions. Research on overall buckling is usually based on sim- magnetic molecular nanoclusters (magnetic molecules) are typi- ple and symmetrical cross-sections, whereas cross-sectional in- cal physical examples of nonlinear modulated systems in which a stability is simplified to buckling of individual plates. It is there- presence of the gap soliton excitations is theoretically predicted. fore highly unlikely that these design rules provide an accurate

66 description of the actual buckling behaviour of arbitrary cross- ration, since they may propagate relatively large distances inside sections. As predicted failure modes not necessarily agree with most solid materials without causing any permanent change in actual ones, the outcome of the results may be overly conserva- the material as a result of their passage. These waves are usu- tive but could be unsafe as well. ally used in order to measure the thickness of the solid stratum, In order to investigate the actual cross-sectional stability be- to locate flaws, to measure the size and shape of different inclu- haviour of aluminium extrusions, a large experimental program is sions. When an elastic wave propagating inside a solid body en- executed at Eindhoven University of Technology. This program counters an interface separating two different strata, then reflec- consists of aluminium extrusions under uniform axial compres- tion and refraction accompanied by mode conversion takes place. sion. Test specimens with rectangular hollows (SHS), U-sections The presence of different dislocations, in general, complicates the (US), as well as very complex cross-sectional shapes (CS) have distribution of wave characteristics considerably, and the prob- been tested. A detailed investigation is made into the influence lem has not been solved analytically. That is why the numerical of the test set-up, initial imperfections, as well as the material modelling of the stress-strain state near dislocations under low characteristic. This results in is a large set of test data on the amplitude waves should be an indispensable stage in solving of actual buckling behaviour of aluminium extrusions, including lo- mechanics problems for cracked solids. cal, distortional, flexural and flexural-torsional buckling, as well The present work aimed at solving of a three-dimensional frac- as mode interaction. ture dynamics problem for a stratified medium with a plane in- To support the findings of the experiments, a numerical pro- terfacial crack under time-harmonic loading. The reflection and gram using the finite element (FE) method is executed. Most ex- the refraction of the wave on the interface between strata stra- periments are simulated using the actual geometry, material, and tum were studied. In order to solve the problem numerically the imperfections. Comparison of the experimental and numerical method of boundary integral equations was used. The distribu- results shows that an accurate prediction is achieved. Further- tion of the displacement discontinuity vector on opposite faces more, the FE-analyses allow a detailed investigation of specific of the crack was investigated for different materials and different aspects like the bifurcation load, the influence of imperfections wave numbers. and materials, the test set-up, and mode interaction. V.O. Menshykov, S.P.Timoshenko Institute of Mechanics, Nesterov The FE-results enable the development of a new and general str. 3, Kiev, 03057, Ukraine prediction model for the local buckling behaviour of aluminium extrusions. Based on the actual local buckling behaviour of cross- sections, it is derived for uniformly compressed aluminium extru- MODEL TESTING OF OFFSHORE STRUCTURES sions with arbitrary cross-sections consisting of flat plates. As such, it allows an accurate and conservative prediction of the NICAT MESTANZADE GOKHAN YAZICI strength and stiffness of a large range of commercial extrusions. [email protected], [email protected] The promising results of this model may result in design rules that enable more economical designs and are able to include dis- Offshore gravity platforms and tension-leg platforms are tortional buckling and mode interaction. This combination of strategically critical structures and they are exposed to a variety experimental, numerical (FE), and analytical work results in a of different loading conditions during their service life. The dy- thorough investigation on the actual local buckling behaviour of namic response of an offhore gravity platform and a tension leg aluminium extrusions with arbitrary and complex cross-sections. platform were investigated analytically and the theoretical results Jeroen Mennink, TNO Building and Construction Research, Van were confirmed by testing scaled models of the structures. Mourikbroekmanweg 6, Delft, 2600 AA, The Netherlands Offshore gravity platforms are one of the most exceptional structures in marine and coastal engineering. These platforms are built from pre-stressed reinforced concrete and are primarily de- INTERFACIAL PLANE CRACK UNDER signed to be used in the northern regions. The natural frequency TIME-HARMONIC LOADING formulation of the gravity platform with the added water-mass was obtained by Rayleighs discrete method. In order to deter- VASYL MENSHYKOV OLEKSANDR MENSHYKOV mine the magnitude and the distribution of the added water mass [email protected], [email protected] along the height of the structure and to check the results of the analytical research, a scaled model of the platform was built and Nowadays, the wide use of high-strength materials make it tested. The scale of the model was 1:50. The model was 4 meters possible to reduce considerably the weight of designed construc- high and weighed 40 kN. The model was tested on a rigid floor tions. It is a common knowledge that all modern structural mate- and on displaceable cushions in the Khazar (Caspian) Sea. rials contain different defects (cracks, shells, etc.). Such defects Tension Leg Platforms (TLP) unlike semi-submersibles are appear due to the process of material manufacture, creation of used for drilling and production of oil and gas in the fields. There- elements of the construction and its further exploitation. If the fore, they are used in small deposit fields where the use of fixed stress-strain relationship was determined regardless of the possi- platforms is not economical and they offer the possibility of trans- bility for cracks to occur and grow, then it might result in sudden portation to different field locations. The geometric form of the collapse of structures under considerably low stresses. Therefore, structure is a conical shell surface. The dynamic behavior is de- it is extremely important to study the load capacity of structural pendence on the buoyancy of main structure and tension in the materials with existing and incipient cracks under dynamic load- legs. To determine the dynamic behavior of the structure, a phys- ing. ical model of the structure was tested at Istanbul Technical Uni- Moreover, the low amplitude waves are almost ideally suited versity marine engineering laboratory. The scale of model was for application to non-destructive testing and geophysical explo- 1:100. Four sensors (Displacement Transducer CMP16) recorded

67 the displacements along X and Y directions. The tensile forces 31 (5-6), 423-464. on the legs were connected to special tension bolts and registered [2] Metrikine, A.V., Popp, K. (1999) Vibration of a periodically by dynamometer. supported beam on an elastic half-space, European Journal G. Yazici, Istanbul Kultur University, A23A D:27 7-8.kisim of Mechanics A/Solids 18(4), 679-701 (1999). Atakoy, Istanbul, 34750, Turkey [3] Vostroukhov A.V., Metrikine A.V., (2003) Periodically supported beam on a visco-elastic layer as a model for dynamic A DISCONTINUOUS-STRIP METHOD FOR analysis of a high-speed railway track, International Journal CALCULATION OF THE DYNAMIC RESPONSE OF of Solids and Structures 40(21), 5723-5752. PERIODICALLY-INHOMOGENEOUS 3D STRUCTURES TO MOVING LOADS A. Metrikine, Delft University of Technology, Faculty of Civil En- gineering, Stevinweg 1, 2628 CN Delft, Netherlands ANDREI V. M ETRIKINE [email protected] STRESS TENSION OF PIECE-WISE STRIP WITH COLLINEAR SYSTEM OF CRACKS OR THIN To model the dynamic response of a ballasted railway track to INSERTIONS DURING ANTIPLANE AND PLANE a moving train, one has to take into account that the rails and the DEFORMATIONS ground interact through the sleepers, which are discretely and periodically positioned beneath the rails. This implies, in particular, M.S. MKRTCHYAN S.M. MHKITARYAN that the ground surface has to be considered free everywhere but S.V. VERLINSKY underneath the rectangular areas beneath the sleepers, where the contact conditions are to be imposed. The rails can be modelled as beams on periodically positioned discrete supports. Thus, the model to be composed has to be periodically-inhomogeneous and This investigation examines problems of stress state of piece- three dimensional (3D). The dynamic response of periodically- wise strip during antiplane and plane deformations, whereas the inhomogeneous 3D models of the ballasted track has been anal- strip consists of two heterogeneous materials of different height, ysed by only a few researchers because of its relative complex- when collinear system has arbitrary number of cracks present on ity. The common approach to the problem is described by Popp the line of connections or when there are absolute rigid thin in- et al. [1]. This is based on the transient Boundary Element sertions. The edges of the cracks and borders are loaded by point Method, according to which the dynamic reaction of the 3D and distributed forces. ground to a motion of each sleeper is calculated in the time do- Transformation of Fourer enables determining singular inte- main. Then, accounting for interaction between the sleepers both gral and other systems of equations of the set problems from trough the ground and through the rails, an explicit time integra- which location densities characteristic values of displacements tion is applied. The major drawback of this approach is that only on cracks faces, races contact stresses on insertions, expansible a relatively small number of sleepers can be considered to avoid stresses beyond the system of cracks or insertions, stress intensity lengthy calculations. Consequently, the approach becomes un- factors, and concentrations of these stresses, opening of cracks reliable at high train speeds, when long waves are generated in are developed. In particular cases, we received closed solutions the rails and a large number of the sleepers are involved in the and investigated effects of relations between the cracks or inser- dynamic process. In this contribution, an alternative approach is tions. applied, which was proposed and developed by the author [2, 3]. Sergey Verlinsky, State Engineering University of Armenia, 105 This approach is based on application of the periodicity condi- Teryan str., 375009, Yerevan, Republic of Armenia tion, which governs the steady-state response of infinitely long, periodically inhomogeneous systems to uniformly moving loads. Employing this condition, the dynamic contribution of infinitely VELOCITIES OF WAVE PROPAGATION IN FLUID many sleepers can be accounted for. The original aspect of this MIXTURES contribution lies in application of a so-called discontinuous strip- method, which allows solving the problem with properly formu- LEV A. MOLOTKOV lated contact conditions between the sleepers and the ground. Ac- [email protected] cording to this method, the contact area under each sleeper is decomposed into a number of rectangular elements, within which In order to investigate wave propagation in fluid mixtures, it is the stresses are considered as uniform. Then, the periodicity con- useful to approximate such mixtures by block fluid media. These dition is applied to each sequence of the respective elements of media consist of the same cells, which contain four blocks in two- infinitely many sleepers. This is followed by solution of a matrix dimensional case and eight blocks in three-dimensional case. The equation, the rank of which is equal to the number of elements blocks can be filled by different fluids. For such block fluid me- within one contact area. The proposed method requires transfor- dia, the effective models are established by the method of matrix mation of the problem into the frequency domain and, therefore, averaging. This method uses the passage to the limit under con- its applicability is limited to linear statements of the problem and dition, that all blocks are narrowed to points and this passage cor- to the steady-state regimes. responds to fluid mixtures. In general case the obtained effective REFERENCES models are anisotropic fluids described by wave equations. [1] Popp, K., Kaiser, I., Kruse, H. (1999) Vehicle-track dynam- For the wave velocities the effective densities along coordi- ics in the mid-frequency range, Vehicle System Dynamics nate axes, explicit expressions are derived. These expressions

68 differ from corresponding expressions in contemporary publica- the stabilization problem for steady motions of non-holonomic tions. Our expressions of velocities in mixtures contain the ef- systems with time-independent constraints. fective densities but expressions in other papers contain the mean The specific structure of examined class of systems make it densities. possible to formulate the controllability and observability criteria We consider two partial cases, that two blocks filled by differ- for nonholonomic systems of the special class. ent fluids alternate and that all blocks except one block are filled The presented theoretical results were successfully applied for by one fluid but one block contains another fluid. The compari- investigation of stability and stabilization of the model of three- son of these cases shows that the relative positions of blocks in wheels vehicles which can be written by non-holonomic system the cell influences essentially on the effective densities and ve- of the mentioned special class. locities. For velocities of waves estimating inequalities are es- The research was partial by supported by Russian Foundation tablished. The velocity in a fluid mixture can not be greater than for Basic Research (03-01-00194) and the “Universities of Rus- the greatest velocity in mixed fluids but can be less than the least sia” Programme. velocity in mixed fluids. Victor M. Morozov, Institute of Mechanics, Michurinskiy pr., 1, The work was supported by RFFI (grant 03-05-65015). Moscow, 119899, Russia Lev A. Molotkov, PDMI, Fontanka, 27, St. Petersburg, 191011, Russia MODEL OF A STRUCTURAL ELEMENT IN ELASTOMERIC COMPOSITES FILLED WITH STABILITY AND STABILIZATION OF NANOPARTICLES NONHOLONOMIC SYSTEMS OF SPECIAL CLASS V.V. M OSHEV S.E. EVLAMPIEVA V.M. MOROZOV V.I. KALENOVA M.A. SALMINA [email protected] [email protected], [email protected], [email protected] The model is based on two major premises. The first takes into account the geometrical peculiarities. It assumes that the size of The problems of stability and stabilization of steady-state mo- filler particles is so small and their number is so great that the tions of the specific class of nonholonomic systems are examined. mean spaces between the neighboring particles become apprecia- Suppose that it is possible to chose the general coordinates for the bly shorter than the contour length of rubber macromolecules. considered systems so that nonholonimic constrains can be rep- Under such condition one macromolecule can be adsorbed by resented as two groups several adjacent particles forming links between them. The second premise is of physical nature. It considers the behavior of l rubber chain as a non-Gaussian one with the upturns due to finite ˙ = 1 ( )˙ ( = +1 ); qχ bχr q qr χ l ,...,m chain extensibility. r=1 Hence the geometry of a unit cell of the model in the initial l state is represented as the length of a spring, simulating a rubber ˙ = 2 ( )˙ ( = +1 ) qρ bρr q qr ρ m ,...,n molecule lying along two substrates, simulating the size of the r=1 adjacent filler particles and the bridge between them. and next conditions are hold It is assumed that the spring is retained on the surface of sub- 1 1. Coefficients bχr are functions of the coordinates strates by the adhesive forces throughout its length with the ends 2 ql+1,...,qm only, the coefficients bρr can depend on the coordi- fixed to the outer positions of the substrates. Applying some ten- nates q1,...,ql,ql+1,...,qm. sile effort to substrates results in stretching the elastic bridge with 2. Potential forces acting onto system are the derivatives of some additional portion of the spring slid off the substrates into the force function and the U(q) also depends on the coordinates a gap between them. Further increase in the tensile force induces ql+1,...,qm only, i.e. U = U(qχ). enlarged slipping off the spring from the substrates into the bridge 3. The coefficients ars in the expression for the reduced ki- between them until the sliding area reaches the outer ends of the n netic energy Θ and the expression µ=l+1 Θµpνµrs depend on model. Then both dragged and proper elastic resistances of the the coordinates ql+1,...,qm only. spring come into play till the most loaded gap part of the spring The specific property of the systems of considered class is that breaks down. all coordinates qr are cyclic ones in the absence of the control Transition from the simple model to the three-dimensional is forces (Qr =0,Qχ =0); the coordinates qχ(χ = l+1,...,m) realized in the form of an isometric, hexagonal prism with a are positional ones. The next statement can be formulated nanoparticle in its center. Now it allows us to calculate the con- Theorem 1. If nonholonomic system with mentioned condi- centration dependences of composites in terms of the simple one- tions 1-3 posses a manifold of S-SM, then S-SM is stable (un- dimensional model. stable) if all of roots of the characteristic equation corresponding An illustrative problem has been solved to verify the model. to the linearized system, except but zero roots, have the nega- Rubber was selected with freely jointed segments of 2 nm long. tive real parts (at least one root has the positive real part). In the It was filled with carbon black of maximum filling ϕmax =0.33 stable case, any perturbed motion sufficiently close to the un per- and particle diameter of 40 nm. The three-dimensional model turbed motion will tend to one of the possible steady motions in has allowed us to calculate the concentration dependences for the the aforementioned manifold as t →∞. initial modulus, the dependence of the initial modulus on the par- The linearized equations of motion of nonholonomic system ticle size and the influence of the matrix bonding with the filler and the equations of measurements are fundamental for solving on the shape of tensile curves. The obtained results agree fairly

69 well with experimental data. MODELLING OF ICE FLOW OF GLACIERS The proposed model of elastomeric composite, obtained in the SUBJECTED TO CLIMATIC CHANGES framework of the elasticity theory, can be taken as a basis for consideration of time dependent properties, which allow us to predict OLEG V. NAGORNOV YURI V. KONOVALOV creep and relaxation behavior. This work is supported by RFBR and Administration of the [email protected] Perm area under grant “Ural - 2004” N 04-01-96038. Two-dimensional flowline model is developed to study Svetlana E. Evlampieva, Institute of Continuous Media Mechan- changes of velocity and stress field of glacier subjected to sea- ics RAS, Koroleva Street, 1, Perm, 614013, Russia sonal climatic changes. The model, which takes into account the transverse change of glacier width, is tuned by data on measured shape of glacier, the surface velocity at the glacier front, and the THERMOELASTIC PHOTOACOUSTIC MICROSCOPY mass balance measurements. Mathematical model to determine OF VICKERS INDENTED METALS velocity in glacier includes the mass balance equation for incompressible ice, Navie-Stokes equation, the Glen law and the heat K.L. MURATIKOV A.L. GLAZOV conduction equation. System of diagnostic non-linear equations [email protected], for steady-state flow was solved by iteration procedure. The ve- [email protected] locities are calculated at each time layer with temperature determined by the heat conduction equation. The mass balance equa- The problem of residual stress detection is an important prob- tion allows for determination of new position of the glacier sur- lem of modern mechanics and material science. Various methods face. Solution of the non-linear integral-differential equations de- such as the hole drilling method and its modifications, X - ray scribing velocity components is determined by the iterations too. and neutron diffraction, ultrasonic, optical, Raman spectroscopy, The value of the fitting parameter in the constitutive equation is stress pattern analysis by the measurement of thermal emission found out by condition of steady-state flow and by comparison of are widely used for residual stress detection at present. Recently, measured and calculated velocities at the glacier front. Flows of we have reported a number of theoretical and experimental in- several Central Asia glaciers were studied. It was found out that vestigations in the field of the thermoelastic photoacoustic effect the flow velocity near the front of glacier is decreased due to two application to the problem of residual stress detection [1-5]. reasons. The first is negative mass balance that results in thin- In these publications the model of the thermoelastic photoa- ning of the glacier tongue. The second is connected with deeper coustic effect in solids with residual stresses was proposed. It penetration of cool temperatures from the surface to the bed be- was successfully used for the explanation of the photoacoustic cause of thinning of glacier in its low part. As a result the inter- signal behavior near radial crack tips in Vickers indented ceram- nal heating becomes smaller in this part of glacier that moderates ics. However, the analytical solution of this problem was ob- the flow. Seasonal variations of temperature moderate/accelerate tained only for one dimensional case. In this work three dimen- the horizontal flow of ice near the surface, and significant addi- sional solutions are presented for the thermoelastic photoacous- tional longitudinal stresses arise in places where the surface slope tic signal in inhomogeneous solids which were obtained in qua- has great gradients. The arising stresses can exceed the compres- sistatic approximation by using perturbation theory. The experi- sive/tensile strength and produce crevasses. The derived changes mental results obtained for different metals including nanonickel, of the glacier shape show the degradation and decrease of glacia- nanocopper, shape memory alloys are also presented. A good tion at several mountain glaciers. correspondence between experimental and theoretical results is O.V. Nagornov, Moscow Engineering Physics Institute, demonstrated. Kashirskoe Shosse 31, Moscow, 115409, Russia This research was supported by the Russian Foundation for Basic Research under grant number 04-02-17622. STATICS AND DYNAMICS OF RODS WITH SKIN REFERENCES FRICTION [1] K.L.Muratikov, A.L.Glazov, R.N.Rose, and D.E.Dumar, LEV V. N IKITIN J.Appl.Phys., 88, 2000. [2] K.L.Muratikov, A.L.Glazov, R.N.Rose, and D.E.Dumar, In extention of the corresponding studies of the professor High Temp.- High Press., 33, 286, 2001. Ya.G.Panovko to the memory of whoom the present confernce is devoted static and dynamic problems of elastic rods interact- [3] K.L.Muratikov, A.L.Glazov, R.N.Rose, and D.E.Dumar, ing with evironments according to the Coulomb dry friction law Rev.Sci.Instrum., 74, 3531, 2003. are considered. Qualitative analysis of the strictly nonlinear in partial derivatives equations of the problem is performed. Ex- [4] K.L.Muratikov, A.L.Glazov, Technical Physics, 48, 1028, act solutions of problems of damping of pulses, periodical forced 2003. oscilations, pile driving, action of shock waves on buried pipe [5] K.L.Muratikov, A.L.Glazov, in Proceedings of the XXXII lines, dynamics of the pressed to each other rods and other ap- Summer School “Advanced Problems in Mechanics”, plied problems are considered. A new problem of static bending St.Petersburg, 2004, 313. of an elastic beam pressed to a rigid rough foundation is posed and considered. Kyrill L. Muratikov, Ioffe Physical-Technical Institute, Poly- L.V. Nikitin, Institute of physics of the Earth, RAS, B. Gruzinskaya tekhnicheskaya 26, St.Petersburg, 194021, Russia 10, 123995, Moscow, Russia

70 ACOUSTOELASTICITY PHENOMENON USING FOR MICROMECHANICS OF MOLECULAR DIFFUSION IN-PLANE STRESS EVALUATION OF CIVIL THROUGH MULTISCALE “BRICK-AND-MORTAR” ENGINEERING STRUCTURES COMPOSITE MATERIALS, WITH APPLICATION TO THE PERMEABILITY OF HUMAN SKIN NADEZHDA YE.NIKITINA JOHANNES M. NITSCHE TSUO-FENG WANG [email protected] GERALD B. KASTING [email protected], [email protected], [email protected] Stresses arising in an engineering material due to temperature changes, body weight; redistribution of loadings leads to This talk addresses the effective transport properties of inter- reduction of durability of separate elements, and, moreover, all esting multiscale structures motivated by the study of perme- construction as a whole. Because of complexity of products ability of human skin. A two-dimensional two-phase compos- and wide range of loadings, to which their components are ex- ite material is considered comprising water-like cells (”bricks”) posed during manufacturing and exploitation, the exact account surrounded by lipid layers (”mortar” or ”cement”). Each of of acting stresses is a very difficult problem for theoretical study- these phases has a substructure, the cells comprising a mixture ing. Therefore problem of an experimental estimation of the real of water and protein, and the lipid being organized into struc- stressed state of machines and engineering structures is rather ur- tured sheets. How molecules pass through this material involves gent in various industries. an intriguing interplay between solubility and diffusion in both phases, strongly affected by the fact that the lipid phase is highly anisotropic. Two different topologies of the lipid phase are considered in order to understand the range of transport behavior pos- It is possible to solve this problem without destruction of a sible in this type of structure. Numerical results are presented material by the way of exiting of waves of small amplitude in showing in detail how the effective permeability depends on the comparison with stress value and evaluation (with a relative error physicochemical properties of the cell and lipid phases. These at least 0,01their propagation in comparison with the initial ma- results are corroborated by an asymptotic analysis that yields ex- terial. Relative cheapness and safety of an acoustical method in plicit formulas for the permeability in four limiting cases, and comparison with x-ray, more wide opportunities at the choice of sheds light on the mechanisms of transport. Application of the materials and practical problems in comparison with usual ten- results to predict permeability of the outermost layer of human sometry or magnetic methods make the acoustoelastic effect at- skin (the ”stratum corneum”) is demonstrated with reference to tractive enough to achievement of these purposes. the transport of water and three important drugs. J. M. Nitsche, University at Buffalo, State University of New York, Furnas Hall, Buffalo, NY (State of New York), 14260-4200, USA The results of researches connected to determination of stressed state of large-sized multiple-sectioned engineering structures of special assignment are generalized. The problems of the MAXILLODENTAL SYSTEM AS A SPECIALIZED acoustoelastic phenomenon using for investigation of the stressed BIOMECHANICAL SYSTEM state of wider class of industrial objects are covered within the framework of own ”conversional program”. The main principles YURIY NYASHIN EUGENIYA SIMANOVSKAYA of the acoustoelastic effect application for biaxial stresses evalua- VICTOR TVERIER tion with the help of bulk waves propagated normally to the plane [email protected] of stress acting are considered. Biomechanical analysis of functions of human masticatory apparatus in norm and in various pathological processes is carried An interesting practical example of the acoustoelasticity phe- out. Human maxillodental system is considered as a specialized nomenon using for in-plane stress evaluation is described. The multiunit biomechanical system where the masticatory muscula- closed pipe cutted from the linear part of large-diameter magis- ture, the temporomandibular joints, the dentofacial articulation tral pipeline was exposed to inner pressure of water. The precise play a leading part. The role of the masticatory pressure as a measurements of time-of flight of shear and longitudinal waves basic mechanical factor causing development and functioning of propagated across the plane of stress acting were made before the system is analyzed. Research of transformation of the mas- and during the loading. Then, the values of stresses acting along ticatory pressure allows carrying out analysis of a structure and and across to the pipe axes were evaluated. Theoretical solution activity of teeth (incisors, canines, premolars, molars) and also of a question is known as a ”Lame problem” and was founded in the periodontal ligament. Theoretically and experimentally it was 19th century. The observed data for ”ring” and ”axes” stresses are shown that in the periodontium the tracts for migration of peri- quite closed to Lame predictions. In closed pipe - closed results. odontal fluid are available which is capable to move under pressure. It has been found experimentally that the periodontium acts as an absorber of a mechanical load and the evaluation of a per- N.Ye. Nikitina, Blagonravov Mechanical Engineering Research meability coefficient for the periodontium is given. The mathe- Institute of Russian Academy of Sciences, Nizhny Novgorod matical model for deformation of elastic medium with voids sat- branch, Belinskogo 85, 603024 Nizhny Novgorod, Russia urated with free fluid was developed. The problem of redistri-

71 bution of fluid in the periodontium was solved at a translational L. Mitutsova, K. Delchev, V. Vitkov, Institute of Mechanics BAS, movement of teeth. Acad.G. Bonchev str., bl.4, 1113 Sofia, Bulgaria It is shown that the dentofacial articulations can be considered as the specialized multiunits being active organs of the masticatory apparatus. Analysis of transformation of the masticatory ANISOTROPIC ELASTO VISCO PLASTICITY OF pressure allows establishing the association of a dentoalveolar ALUMINUM ALLOY unit pathology with the temporomandibular joints. The clinical ALEXANDER I. OLEINIKOV data convincingly testify that at congenital pathological processes in range of the upper jaw (the palate and lip cleft) a second true [email protected] both temporomandibular joints are involved in process. The authors elaborated complex therapy of children with a congenital Strain features of samples plates and sheets from a high- cleft of the labium and the palate immediately right after deliv- strength aluminium alloy in delivery condition at various ery. It prevents serious complications and provides beneficial and temperature-speed modes are analyzed. Behaviour of a material well-timed requirements for growth, development of the facial in various directions in a plate (sheet) plane, on a normal to it and skeleton and formation both of the dentoalveolar unit and tem- on a direction under a corner to a normal is considered. poromandibular joints at children. Efficiency of various approaches to description of elastic, plas- Y.I. Nyashin, Perm State Technical University, Komsomolskii tic and steady creep anisotropy of the alloy and their dependence prospect, Perm, 614000, Russia on a loading kind is investigated. Development of the approach offered in [1, 2] is supposed. Applied given the governing relationship in CAD/CAE-modeling of process forming are reported. Active form-building phase and elastic restoration are ana- A BIOMECHANICAL STIMULATOR FOR lyzed. The role of the big turn corners is marked at calculation SCIENTIFIC-EXPERIMENTAL STUDY OF THE form-building processes. REGENERATION OF SPINAL CORD LOCOMOTION CAPABILITIES AFTER TRAUMATIC BREAK REFERENCES [1] Oleinikov A.I. Fundamental General Relationships For A D. OKHOTSIMSKY O. NIKITIN Y. G ERASIMENKO Model Of Isotropically Elastic Heteromodular Medium.// J. N. SERBENJUK L. MITUTSOVA K. DELCHEV Appl. Maths. Mech. 1993, Vol. 57, 5, pp. 913-920. V. V ITKOV A. PLATONOV V.YAROSHEVSKY [2] Oleinikov A.I. The Description of Deformation of Hetero- [email protected] geneously Resisting Materials.// Doklady Physics. 1998, Vol. 43, 8, pp. 517-518. As medical practice shows, the nerve-muscle regulation suf- fers during a spinal cord disease of different etiology or under A.I. Oleinikov, 681013, r.220-3, Lenin str., 27, Komsomolsk-na- a spinal cord trauma. This yields a serious functional upset of Amure, Russia the human support-locomotion apparatus. One of the methods of its recuperation is to act upon the spinal cord neurons by using natural communication channels, i.e. through the limb receptor THE OPTIMAL BIOMECHANICS OF THE FOOT system. Note that the conditions of treatment should be similar PROSTHESIS WITH THE VARIABLE THICKNESS to those of the natural locomotion activity. LEAVES An experimental biomechanical stimulator is developed for laboratory use. The present report it presents the methods of M.A. OSIPENKO YU.I. NYASHIN S.I. BRYNSKIKH study and device technical design. The new device has been de- [email protected] signed in the Institute of Mechanics, Sofia, and has been manufactured and successfully tested in the experimental department The elastic element of the foot prosthesis represents the stack of the Keldysh Institute of Applied Mathematics, Moscow. In of the cantilever beams (leaves) of equal width but different the paper it illustrates the integration of the device within a tech- lengths and variable thicknesses. The leaves fit each other closely nical complex designed in the Laboratory of Movements Phys- without loading. There is no friction between the leaves. The iology at the Institute of Physiology ”I. S. Pavlov”, St. Peters- given concentrated force is applied perpendicular to the lower- burg. Initially, the method and equipment are employed to study most leaf. Given the deflection and the length of the lower- the neuro-muscle regulation of a spinal animal (a cat). The effi- most leaf, the elastic element with the minimum principal stress ciency of a simultaneous application of epidural (electrical) and value is considered optimal (sometimes the lengths of some other biomechanical stimulation is verified. A long-term scientific- leaves are given). Optimum is attained by means of the selection experimental study of the regeneration of the spinal cord post- of the lengths and variable thicknesses (profiles) of the leaves. traumatic locomotion capabilities is performed. Proving the ef- The following principle is often used for the search of the opti- ficiency of the approach, it will be applied for the design of a mal elastic element parameters: if the equal-stress element can be rehabilitation complex for spinal patients. constructed under given conditions then it is optimal. It is shown D. Okhotsimsky, N. Serbenjuk, A. Platonov, V.Yaroshevsky, that the single-leaf element under strong bending has the ellipti- Keldysh Institute of Applied Mathematics RAS, 4 Miusskaya Sq., cal profile. This result is the generalization of the well-known fact 125047 Moscow, Russia that the single-leaf element under weak bending has the parabolic O. Nikitin, Y. Gerasimenko, Pavlov Institute of Physiology RAS, profile. The following results for the two-leaf element are ob- Makarova str., 6, 199034, Sankt Peterburg, Russia tained under the assumptions that the bending is weak and the

72 stress sign is constant on the leaves surfaces: the optimal leaves in a constant homogeneous magnetic field is described by system of equal length have the parabolic profiles, but the equal-stress of two quasi-linear equations of a parabolic kind for an average elastic element with the leaves of different lengths does not exist. number of ferroparticles in aggregates and a parameter of an ef- The leaves profiles for the optimal but not equal-stress element fective magnetic field. At the certain values of fluid parameters with the leaves of different lengths are found numerically. If the and magnetic field strength the curve of local connection for this assumption that the stress sign is constant on the leaves surfaces is system has -like form, the average number of particles in aggre- cancelled then the equal-stress element with the leaves of differ- gates is ”fast” variable, and parameter of an effective magnetic ent lengths can be constructed for the lengths ratio greater than field - ”slow” variable. In this case there exists the solutions such 0.74. The lowermost leaf of such element has the inner hinge. as hot and cold static autosolitons, having the steady form and The further investigation is required to establish whether this el- zero velocity both, and travelling autosolitons moving without ement is optimal and whether the equal-stress element exists for attenuation with constant velocity. Static autosolitons are real- the lengths ratio less than 0.74. ized as sub-layer of more or less aggregated continuum in a layer Svetlana I. Brynskikh, Department of Theoretical Mechanics, of a liquid between two parallel nonmagnetic plates. Travelling Perm State Technical University, 29a, Komsomolsky Prospect, autosolitons represent the wave structures such as one or sev- 614600, Perm, Russia eral parallel strips of more aggregated continuum moving along plates. The received solutions describe the processes of ferroparticles aggregation, which are observed experimentally in films of BIFURCATIONS OF CYLINDRICAL PRECESSIONS OF a magnetic fluid. AN UNBALANCED ROTOR Nickolai F. Patsegon, Kharkov National University, 4 Svoboda sq., Kharkov, 61077, Ukraine INNA A. PASYNKOVA [email protected] LARGE EDDY SIMULATION AND PDF FOR A statically unbalanced rigid rotor is supposed to be supported TURBULENT COMBUSTION FLOWS with non-linear elastic bearings. The rotor is considered as a me- SREEBASH C. PAUL MANOSH C. PAUL chanical system with four degrees of freedom. It is assumed that THEODOSIUS KORAKIANITIS WILLIAM P. J ONES the rotor angular velocity is constant. The rotor is under external and internal friction forces. It is known that among the steady- [email protected] state rotor motions the direct synchronous cylindrical precession can take place. The turbulent flow and species concentration with temperature The one-parameter bifurcation problem of the cylindrical pre- during the combustion process are investigated within a cylindri- cession with the rotor angular velocity as a parameter is investi- cal combustor. Propane (C3H8) and air are injected into the com- gated. One can observe different types of the stability loss. For bustion chamber, where the inlet air temperature is kept at 500C. some range of variation of the parameter, jump phenomenon and To solve the equations governing the turbulent flow, the Large bi-stability occur but the steady-state motion remains to be the Eddy Simulation (LES) technique has been employed where as direct synchronous cylindrical precession. In this case the bifur- the Probability Density Function (PDF) approach has been im- cation is hard. For some other values of the parameter there is plemented for the combustion modelling. LES seeks the solution the domain of ”instability in 3D” due to the conical swinging of by separating the flow into two parts: large scale eddy which car- the shaft. The stability loss is accompanied by separation of two ries the majority of the energy and resolved directly and small stable regimes of relative equilibrium and the bifurcation is soft. scale eddy which has been modelled. This separation has been The precession motion becomes the hyperboloidal one as the ro- made by applying a filtering approach to the mass conservation, tor axis traces a one-sheet hyperboloid surface. the momentum conservation and the mixture fraction equations. At last, when internal friction forces are significant, the self- The obtaining result is compared with the experimental result. centering regime can become unstable and the stable self-excited S. C. Paul, M. C. Paul, T. Korakianitis, Department of Mechanical vibrations can appear. The approximate equations for these vi- Engineering, University of Glasgow, Glasgow G12 8QQ, UK brations are given and compared with integration results. W. P. Jones, Department of Mechanical Engineering, Imperial I.A. Pasynkova, Saint-Petersburg State University, Universitetsky College London, London SW7 2AZ, UK pr., 28, Saint-Petersburg, Petrodvorets, 198504, Russia

MULTILEVEL MODEL OF STRENGTH AND AUTOWAVES IN MAGNETIC FLUID FRACTURE TOUGHNESS OF NANOCOMPOSITES

N.F. PATSEGON L.N. POPOVA MIKHAIL N. PERELMUTER [email protected] [email protected]

The processes of static and travelling autosolitons generation To model the shear strength of nanocomposites we supposed in a layer of a magnetic ﬂuid, which is a colloidal suspension of that the zone of the adhesive interaction between nanotubes and magnetic particles in the liquid carrier are considered. The model a polymeric matrix is a thin interface layer which has resistance of continuum is used, which microstructure can transform due to only in the relation to action of shear stresses and has the given formation and destruction of the ferroparticles aggregates in the curve of deformation. The stress state of nanotubes and a poly- magnetic ﬁeld. Evolution of a magnetic state of such continuum meric matrix is determined in the assumption, that the nanotube

73 is a cylindrical fibril with the straight axis, embedded in a infi- the purpose of establishing a model for the dynamics of such nat- nite polymeric matrix, and the displacement along the axis of the ural phenomena as the complex interaction between heart-beat, nanotube under the action of the external loading along this di- respiration and the oscillatory process with a frequency close rection are much more than others components of the nanotube to 0.1 Hz which has been discovered in various medical stud- and matrix displacements. The analytical solutions for the axial ies. Phase synchronization phenomena for two processes in the displacement and normal stress in the nanotubes and the shear presence of a third one are considered in both a continuous time stress in the interface layer for a case of the bilinear deformation system and in a system of stochastic circle maps. The theoreti- curve of an intermediate layer with elastic and hardening or soft- cal results are compared with the real data collected from various ening branches are obtained. From the experimental observations groups of patients at rest and under anaesthesia. Based on the has been found that the crack bridging mechanisms is very im- outcome, a new model of cardiovascular system dynamics is pro- portant during nanocracks formations and fracture of nanocom- posed. posites. In most cases observed in the references the sizes of A. Pershakova, Lancaster University, Lancaster, LA1 4YB, United the nanocrack bridged zones were comparable with the whole Kingdom crack size. The two-parametric fracture criterion for a problem of quasi-static growth of a crack with the bridged zone of a large scale is proposed and considered. Its« supposed that the bonds INITIAL DAMAGING IN KACHANOV–RABOTNOV’S constraining the crack opening exist between the crack surfaces. KINETICS THEORY The energetic characteristics of the large scale bridged crack are introduced. These characteristics are the strain energy release Y.V. P ETROV A.V. KASHTANOV G.I. LIHACHEV rate (ERR) associated with the crack tip stress intensity factors [email protected] (SIF) and the rate of the energy dissipation by the bonds (RED) associated with the energy necessary to create a unit of new sur- The influence initial damaging of a material on dependence of face of the bridged zone. In this model the ERR and the RED parameter damaging from time is investigated. According to re- depend on the bridged zone size and the bonds characteristics. sults received in [1], it is possible to believe, that at the appendix The condition of the crack tip limit equilibrium is the equality of loading damaging in a material instantly (within the framework of the ERR and the RED values (the first condition of fracture). of the static theory) the size of the enclosed constant loading , and The second condition of fracture is the condition of the bond limit - strength of a material grows down to size, where - strength of stretching at the trailing edge of the bridge zone. Based on these a material. The experimental data for various constructional ma- two fracture conditions the regimes of the bridged zone and the terials are considered and is shown, that the account initial dam- crack tip equilibrium and quasi-static growth are considered: 1) aging allows correctly describe experiments on long durability the crack tip propagates and the size of the bridged zone increases when the dependence between time before destruction and load- without rupture of the bonds; 2) the size of the bridged zone de- ing in logarithmic coordinates is not linear. The constant equa- creases due to rupture of the bonds without the crack tip propa- tions of Kachanov - Rabotnov for such materials are determined. gation; 3) the crack tip propagates with simultaneous the bonds REFERENCES rupture at the trailing edge of the bridged zone. The last case corresponds to quasi- static crack growth if both fracture condi- [1] A.V. Kashtanov, J.V.Petrov - Fractal generalization of tions are fulfilled. The equilibrium size of the bridged zone and power balance in the linear mechanics of destruction, SPb the critical external stress are determined from these two condi- 2001. tions. In the general case of the bonds deformation law (which Grigory Lihachev, St.Petersburg State University, Russia came from the analysis of the shear stress between nanotubes and matrix) the problem of the bond stress and the energetic characteristics determination is transformed to the numerical solution of THERMAL FRACTURE OF A BIMATERIAL the singular integral- differential equations. Parametric analysis CONTAINING INTERNAL DEFECTS of the proposed model under the different properties of materials, nanofibers and bridged zone size was performed. Work was VERA E. PETROVA KLAUS HERRMANN partly supported by RFBR, grant number is 05-01-00191 and by [email protected], grant for leading scientific school of the Russian Federation, the [email protected] grant number is 1849-2003.1. M.N. Perelmuter, pr. Vernadskogo 101-1, Moscow, 119526, Rus- Bimaterial compounds are often used in different engineering sia structures, which are often subjected to thermal loading. As a consequence interface thermal fracture is commonly observed in these structures. Numerous investigations reveal that the interface STOCHASTIC COUPLED CIRCLE MAPS WITH strength influences on the strength of a bimaterial as a whole. In APPLICATIONS TO MODELLING OF THE HUMAN a number of previous authors works the thermoelastic fracture CARDIOVASCULAR SYSTEM. problems have been considered for a bimaterial with an interface crack and internal defects. The bimaterial was subjected to ther- ALEXANDRA PERSHAKOVA DMITRI LUCHINSKII mal loading caused by a heat source or by a uniform heat flux, ANETA STEFANOVSKA PETER V.E. MCCLINTOCK applied at infinity, normal to the interface. The papers presented [email protected] a method for the solution of the problems. This method is based on the theory of complex potentials, analytical continuation the- A system of two coupled stochastic circle maps is derived for orem and superposition technique and leads to singular integral

74 equations for the problems in hand. Due to the superposition with a transport of approximation domain, and the time integra- principle the main temperature and thermoelastic problems de- tion algorithm of the Newmark type as applied to such dynamic compose into a series of simple subproblems each of them has shell equations. The quadrilateral shell finite elements with 4-, simple geometry. The subproblems are solved separately, to do 9,- and 16-nodes and six degrees of freedom in each node have this the known complex potentials for the crack/singularity prob- been developed, [2]. Using these elements many numerical test lem can be used or new potentials can be constructed. examples have been performed and some results of this analysis The primary interest of the present study is thermal fracture of for complex, multifold, elastic shell structures undergoing unre- interface in a bimaterial containing an interface crack and internal stricted deformation are presented and analyzed. defects. A bimaterial is subjected to a heat flux or a heat source. Summarizing, the lecture presents the global view on the non- The previously obtained method is applied. Effects of heat con- linear theory and numerical analysis of the piecewise regular ductance of bimaterial interface as well as heat conductance of an as well as irregular, multi-fold shell structures undergoing un- interface crack surface on the stress-strain state on the interface restricted deformation and motion. and on the stress intensity factors are investigated in the presence of internal defects. Influence of the thermoelastic parameters of the bimaterial and location and orientation of different systems of REFERENCES internal cracks on the stress intensity factors at the interface crack [1] Libai A., Simmonds J.G.: The Nonlinear Theory of Elastic tip are also studied. Shells, 2nd ed. Cambridge Univ. Press, Cambridge 1998. V.E. Petrova, Voronezh State University of Architecture and Civil Engineering, ul. 20-let Oktjabrja 84, Voronezh, 394006, Russia [2] Chro«scielewski« J., Makowski J., Pietraszkiewicz W.: Stat- ics and Dynamics of Multi-fold Shells: Non-linear Theory and Finite Element Method (in Polish). Wyd. IPPT PAN, DYNAMICALLY AND KINEMATICALLY EXACT Warszawa 2004. THEORY OF IRREGULAR SHELL STRUCTURES AND ITS NUMERICAL IMPLEMENTATION [3] Eremeyev V.A., Pietraszkiewicz W.: The non-linear theory of elastic shells with phase transitions. J. Elasticity 74 WOJCIECH PIETRASZKIEWICZ (2004), 67-86. [email protected] [4] Makowski. J., Pietraszkiewicz W.: Thermomechanics of Shells with Singular Curves. Wyd. IMP PAN, Gdansk« 2002. Basic relations of the general, six-fields, non-linear theory of shells are reviewed and their numerical implementation in nonlinear problems of statics, stability, and dynamics of multi-fold Acknowledgements. shell structures is presented. This research was supported by the Polish State Committee for Two-dimensional (2D) balance laws of the linear and angu- Scientific Research under grant KBN No 5 T07A 008 25. lar momentum for shells are formulated using an exact through- W. Pietraszkiewicz, Polish Academy of Sciences, Institute of the-thickness integration of appropriate 3D balance laws of con- Fluid-Flow Machinery, ul. Gen. J. Fiszera 14, Gdask, 80-952, tinuum mechanics [1,2]. The corresponding shell kinematics is Poland then established on the base surface as an energetically exact dual structure from the 2D integral identity. As a result, the work-averaged translation vector and rotation tensor, describing the gross deformation of the shell cross-sections, are the only in- ON THE PROBLEM OF DRIVING A SOFT SHELL dependent field variables of the resulting initial-boundary-value ALONG A RIGID BODY ENVELOPED BY THE SHELL problem (IBVP) of the Cosserat surface type. All 2D shell relations are formulated on multi-connected, EKATERINA POLYAKOVA VICTOR CHAIKIN multi-fold base surface with stationary or moving singular curves. OLEG ZUEV The stationary singular curves can model folds, branches, or self- [email protected], v [email protected], intersections of the base surface, stepwise thickness or material [email protected] changes, reinforcement by bars, curvilinear junctions, etc. The moving singular curves may model coherent or incoherent phase boundaries [3], propagating wave fronts, cracks, etc. [4]. To com- The problems concerning driving of a soft shell along a sur- plete the IBVP we develop new dynamic and kinematic continu- face of a rigid body enveloped by the shell arise at designing of ity conditions at the singular curves and analyze several forms of clothing, engineering of technical, household and medical goods. kinetic constitutive equations for the linear and angular momen- In this paper it is supposed that the shell envelops some part of tum densities. Different ways of modelling the shell junctions are indefinitely long axially symmetric body. The shell is assumed to discussed as well. be pulled along its surface in the direction of its symmetry axis In the numerical part we present several aspects of numerical by some forces applied to the edges of the shell supposed to be analysis of the IBVP for complex shell structures performed on orthotropic and linearly elastic. The conditions of losing contact the configurational space including the rotation group SO(3), [2]. between the shell and the body and the conditions of stability loss In particular, we discuss problems of local and global parame- of the shell form are investigated. terization of SO(3), aggregation of rotations an their increments, Ekater Polyakova, Saint-Petersburg State University of Technol- definitions of virtual and real velocities and accelerations in mate- ogy and Design, 18, Bolshaya Morskaya str., Saint-Petersburg, rial and spatial representation, indirect C0 interpolation on SO(3) 191186, Russia

75 platform and a dynamometer. The foot is being put on the plat- FORMATION OF ROGUE WAVES AND HOLES IN form and fixed with the belts. The platform has a rotator mech- OCEAN anism; the axis of rotation coincides with the axis of the ankle joint enabling foot position changes. Excitation of mechanical A.V. PORUBOV I.V. LAVRENOV oscillation is brought about by the electro-dynamical vibrator by [email protected] means of a stock with a tip and is perpendicular to the anatom- ical axis of the shank. Vibration amplitude and frequency are The problem of two-dimensional non-linear modeling of ab- being changed by means of a sound generator and power am- normal waves in ocean is studied. The main idea is to consider plifier. Vibration signal measurement is fulfilled by means of a these waves as two-dimensional localized waves. First, the rogue vibroreceiver-accelerometer which is fixed on the shank surface waves are studied. Despite numerous works on the subject, many either under the Achilles tendon or under the tibia. questions remain unclear, in particular, a mechanism of the rogue The equipment records amplitude-frequency characteristics of waves formation. It is suggested to consider a formation of them the human tendon-muscular complex or the tibia automatically as a result of a resonant crested waves interaction in framework of for different kinds of vibration excitation and for various posi- the Kadomtsev-Petviashvili equation [1]. Recently another kind tions of the foot. A resonance curve depends on physical and of abnormal waves was detected - the so-called deep holes. Two mechanical parameters of the investigated object conditioned by models are proposed to explain their formation. First is based on the real tissue state around the injury. an influence of a wind, the second one accounts for an influence of an ocean stratification. Results and Discussion REFERENCES [1] A.V. Porubov, H.Tsuji, I.V. Lavrenov, and M.Oikawa, For- The examinations using the developed equipment were made mation of the rogue wave due to nonlinear two-dimensional in the War Veteran hospital in Ivanovo. Resonant amplitudes and waves interaction, Wave Motion (2005), in press. frequencies were used as informational parameters. The statistical analysis of the data showed the following parameters of the Alexey V. Porubov, A.F. Ioffe Physical Technical Institute of the first resonance for healthy pushing and non-pushing extremities RAS, Polytekhnicheskaya st., 26, Saint-Petersburg 194021, Rus- respectively: soft tissues (Achilles tendon connected with triceps) sia resonant frequency 43.1 ± 2.24 and 45.1 ± 1.12 Hz; maximal amplitude 14.05 ± 0.77 and 12.63 ± 0.16 m/s2 correspondingly. These parameters reflect stiffness of tendon-muscular complex FORCED VIBRATION TECHNOLOGY FOR and its interaction with surrounding tissues. An acute tear of the NON-DESTRUCTIVE TESTING IN ORTHOPAEDY Achilles tendon leads to the visible changes of the measured resonant frequencies and amplitudes, and the resonant curves shape V.V. P OTEKHIN A.V. ZINKOVSKY L.B. MASLOV as well that gives a possibility to make a conclusion about soft tis- V.I. SHAPIN S.E. LVOV E.V. BLESKIN sues physiological state around injury. Those of the tibia were de- slava [email protected], termined also. The resonant frequency equals to 125.5±2.61 Hz; 2 [email protected] maximal amplitude equals to 0.87 ± 0.03 m/s for both extremities. Permanent control of these parameters during patient re- Introduction covery allows inspecting strength and stability of the callus. Let us conclude that the method could be proposed for non-invasive testing and control in orthopaedy. A technology of active vibration diagnostics has been widely utilized for non-destructive testing of machines and mechanisms REFERENCES for many years [1]. Since the 80th of the last century biomedi- [1] Korablev S.S., Shapin V.I. and Philatov J.E. (1989): Vi- cal engineers have made attempts to convert this technology to brational diagnostics in Instrument-Constructions. Hemi- biomechanical area. For instance, vibration methods for the tibia sphera, USA. fracture testing [2] and diagnostics of the hip implant stem loos- [2] Christensen A.B., Ammitzboll F. and al (1986): Assess- ing [3] were proposed. ment of tibial stiffness by vibration testing in situ. Journal A patent method of the resonance vibration diagnostics has of Biomechanics 19(1): 53Ð60. been developed in the Laboratory of Biomechanics (ISPEU, Rus- [3] Georgiou A.P. and Cunningham J.L. (2001): Accurate di- sia) for analysis of the mechanical defects of soft and hard tissues agnosis of hip prosthesis loosening using a vibrational tech- of the human shank [4]. This approach permits a researcher to nique. Clinical Biomechanics 16(4): 315Ð323. define a physiological state by means of analysis of investigated system response on local vibration effort. [4] Maleshev I.V., Nozdrin M.A. and al (1997): Research complex for vibrational diagnostics of the Achilles tendon. Patent of Russian Federation No 2077266. Materials and Methods [5] Zinkovsky A.V., Maslov L.B. (2003): Resonant properties of the shank biological tissues. Research Bulletin of Saint- Experimental equipment for vibratory examination of the hu- Petersburg State Polytechnical University 4: 83Ð88. man shank locomotion elements such as the Achilles tendon and a tibia was developed. The device is composed of a platform with V.V. Potekhin, A.V. Zinkovsky, Department of Biomechanics and a firmly fixed vibrator, a splint for extremity fixation in the mid- Valeology. St.Petersburg State Polytechnical University. Poly- physiological position, which is located on a firmly fixed table, a teckhnicheskaya 29, 195251, St.Petersburg, Russia

76 L.B. Maslov, V.I. Shapin, Department of Theoretical and Applied vessel which includes rigid perforated baffle plates was consid- Mechanics. Ivanovo State University of Power Engineering. Rab- ered. The presence of rigid perforated bafflers leads to a sig- fakovskay 34, 153003, Ivanovo, Russia nificant geometrical complication. In this case determination of S.E. Lvov, E.V. Bleskin, Department of Traumatology and Or- frequencies and modes of liquid oscillations is of great mathe- topaedy. Ivanovo State Medical Academy. Engels 8, 153462, matical and calculation difficulties. But for thin-walled bafflers Ivanovo, Russia with large number of small holes exact formulation of a problem can be replaced by a more simple problem with averaged conjugation conditions for velocity potential and its normal derivatives ON THE STABILITY OF CYLINDRICAL PRECESSION on bafflers. OF DYNAMICALLY SYMMETRIC SATELLITE ON THE Full mathematical statement of limiting problem was given, ELLIPTIC ORBIT solution of which is asymptotic limit of initial problem solutions if the hole number increases simultaneously with decreasing of ALEXANDER N. PROKOPENYA hole sizes and distances between them. Variational and operator [email protected] statement of limiting problem was given. It was shown that for the limiting problem fundamental qualitative properties of fre- In the present paper we study the stability of stationary mo- quency spectrum and modes of free normal oscillations are con- tion of dynamically symmetrical satellite in the newtonian grav- served. itational field. It is considered the case of cylindrical precession Oscillations of ideal fluid in rectangular vessel with vertical [1] when a center of mass of the satellite moves in elliptic or- and horizontal bafflers and the same problem in cylindrical ves- bit while the satellite rotates uniformly about the axis of its dy- sel of circular cross-section with horizontal radial and circular namical symmetry being perpendicular to the orbit’s plane. A perforated bafflers was considered as examples illustrating com- particular case of the satellite’s motion without rotation was in- mon conclusions. vestigated in detail in [2]. Here we consider the resonance case Procedure of numerical analysis for frequencies and modes of β = T0/Trot =3/2, where T0 and Trot are periods of the orbital normal oscillations in arbitrary shape vessels with arbitrary hole motion and rotation, respectively. number was offered. Proposed method is, in essence, a realiza- We have found the domains of instability of cylindrical pre- tion of Ritz method based on a variational formulation of the cession of the satellite in the Liapunov sense and domains of its problem under consideration. Calculations of main frequencies linear stability in the parameter space. Using the infinite deter- and modes of free normal oscillations for the number of different minant method [3], we have calculated the equations for the sta- vessels were carried out. bility boundaries in the form of power series in the eccentricity Y.I. Rudnev, Kharkov National University, Svobody sq. 4, of satellite’s orbit. For the domains of linear stability we have in- Kharkov, 61077, Ukraine vestigated the influence of nonlinear terms in the equations of the disturbed motion. With the help of the computer algebra system Mathematica we constructed analytically a canonical transforma- CONVECTIVE FLOW IN A SPHERICAL BACKLASH IN tion normalizing the Hamiltonian of the system. Then the prob- CONDITIONS OF MICROGRAVITY lem of cylindrical precession stability in the Liapunov sense is solved in the framework of KAM-theory on the basis of Arnold- M.A. RYAKINA E.L. TARUNIN Moser and Markeev theorems [4]. [email protected], [email protected] REFERENCES [1] V.V.Beletsky. Motion of a satellite about its center of mass At space stations the gauge of natural convection in the cylin- in gravitational field. Moscow, MSU, 1975. drical form - DACON [1] is already tested. There is a hope, that [2] A.P.Markeev, T.N.Chekhovskaya. On the stability of cylin- parameters of the gauge can be improved due to change of geom- drical precession of a satellite on the elliptic orbit. Prikl. etry of the gauge with a spherical backlash as in this case natural Math. Mech. (1976), V.40, 1040-1047. convection arises at any orientation of a gravity. For finding-out [3] R.Grimshaw. Nonlinear ordinary differential equations. of such opportunities numerical experiments are carried out. In CRC Press, 2000. the work numerical methods used equations in the approach of Bussinesk [2] in the spherical backlash at small values of number [4] A.P.Markeev. Libration points in Celestial Mechanics and Grashoph. The internal sphere is heated more up. Flow is sup- Cosmic Dynamics. Moscow, Nauka, 1978. posed symmetrical concerning the vertical axis directed against gravity. For the decision of the problem a method of grids for A.N. Prokopenya, Brest State Technical University, Moskowskaya the equations in variables of function of a flow and a whirlwind st. 267, Brest, 224017 Belarus of speed is used. Small intensity of convection allows choosing the scheme with the central differences with the second order approximations without effect of accounting viscosity. Thermal SMALL OSCILLATIONS OF IDEAL FLUID IN VESSELS convection is investigated in the assumption, that acceleration of WITH PERFORATED BAFFLERS free falling is periodic function of time. Parameters of the problem are - number Prandtle, number Grashoph, the attitude of ra- YURI I. RUDNEV DMITRY I. BORISOV dius of internal sphere to radius of external sphere, frequency [email protected] and amplitude of a pulsation g(t). The purpose of these calculations is definition of dependence of the response natural con- Small oscillations problem of ideal fluid in a partially filled vection (its integrated and local characteristics) at various typical

77 frequencies of modulation of gravity at space stations. The nu- portant biological molecules as DNA and peptides, the confor- merical decision of the problem was carried out till reception of mational dynamics of the latter deserves special attention. The the established mode of the flow. Calculations have shown, that important problem is setting the temperature of the system, or for the period of fluctuations equal of 0.01 dimensionless times, to be more general and precise consideration of the interaction the establishing mode of fluctuations comes after 10 fluctuations. between the system and the medium, characterized by some ther- In the case of modulation of number Grashoph under formula modynamic parameters. In real molecular simulations large sys- G = 500(1 + cos ωt) and an internal radius 0.5 a maximum tems are considered, it leads to impossibility of setting the in- of stream function of the flow in the establishing mode changed teraction directly. For this purpose the simplified models of so from 7.5 up to 13.5. Dependences different values testimonials called ”mechanical thermostats” are often used. The correspon- from parameters of periodic function g(t) are found. By results dence of distribution functions of the systems characteristics to of these researches the border of frequencies at which excess the Gibbs canonical ensemble could be the criteria of the thermostat gauge natural convection “does not work” is found out. efficiency. In fact, the problem of modeling Gibbs canonical dis- REFERENCES tribution is one of the fundamental for modern statistical physics. [1] On The Measurement Of Low-frequency Microaccleration Various thermostat models were introduced in the method of On Board Of An Orbital Station By The Convection Sensor Molecular Dynamics (see [1], and references therein), their effi- ”DACON” // Adv. Space Res., 2003, Vol. 32, No.2, Pp. 199- ciency depends on the simulated system. The most widely used 204. models nowadays are the Berendsen and the Nose-Hoover« ones. [2] Gershuni G.Z., Zuchovitsky E.M. Convective stability of in- The dynamics of some systems, confined to the first one was compressible fluid // Moscow, Nauka, 1972, 392p. studied in [2]. As the development of the research, described in [3], the dynamics of some special systems was analyzed for the Maria A. Ryakina, Perm State University, Bukireva, 15, Perm, Nose-Hoover« thermostat, using its most widely applied in numer- 614990, Russia ical experiments model, which is realized by introducing nonlinear dissipative terms (nonlinear alternating friction) into the right hand sides of the equations of motion. AN IDEA AND THEORY OF SHEAR DEVICE FOR INVESTIGATING THE LOCALIZATION PHENOMENA It is known that for simple systems (ideal gas) the Nose-« Hoover thermostat doesnt lead to correct, from the point of view EVGENIY I. RYZHAK of thermodynamics, distribution functions, but it is generally accepted that the situation becomes better with the increase of com- e i [email protected] plexity and dimension of the simulated system. Suggested is an idea of the lamination shear device for inves- Employing the method of averaging and direct computer sim- tigating the localization phenomena in bulk or soft plastic media. ulations it has been shown that for an ensemble of harmonic os- The device consists of a pile of laminae with a cavity of rather cillators (which is the generalization of a harmonic lattice) for arbitrary shape cut in it. The cavity is filled with a medium exam- some parameters of the thermostat oscillatory motion around the ined which forms the specimen of corresponding shape. The pre- stationary solution with the frequency, dependent on the values of scribed uniform shearing of the pile induces the basic shear pro- thermostat parameters, is observed. Or for their larger values one cess in the specimen, which is also uniform unless its stability is can observe the process of chaotisation. (See [3] for details). For violated. By means of author’s modifications of the well-known this case also the distribution of energy against degrees of free- Van Hove’s theorem, the stability is proved to be preserved up dom has been obtained numerically, and for some situations the to violatoin of strong ellipticity for the material deformed, which distribution is incorrect. violation stipulates the onset of localized instability. Thus, the Besides the resonance effects for the harmonic systems have localization pattern arises on the background of a homogeneous been studied. In case of parametric excitation of the system, the state of the specimen and is influenced mostly by its shape. Such parametric resonance, resulting in the increase of energy ampli- a combination of theoretically substantiated properties enables in tude, can be easily observed. Its conditions were found with the principle to investigate the dependence of the localization pattern help of Rayleighs method. And the subject of the very recent on the shape of domain of deformation. The question stated is research is systems with constraints. still open, being of great interest for mechanics of the localization phenomena. E.I. Ryzhak, Institute of physics of the Earth, RAS, B. Gruzinskaya REFERENCES 10, Moscow, 123995, Russia [1] M.D’Alessandro, A.Tenenbaum, and A.Amadei, J.Physical chemistry B106, 5050 (2002). NONLINEAR DYNAMICS OF THE SYSTEMS IN THE NOSE-HOOVER ENVIRONMENT [2] V.L.Golo, K.V.Shaitan, Biofizika, 47, 611 (2002)

VLADIMIR N. SALNIKOV [3] V.L.Golo, Vl.N.Salnikov, K.V.Shaitan, Phys. Rev. E70, [email protected] 046130 (2004)

At present time Molecular Dynamics is one of the most wide spread methods of modeling and computer simulations of molec- V.N. Salnikov, Moscow State University, Vorobjevy gory, 1, ular systems. Among the objects of modeling there are such im- Moscow, 119992, Russia

78 [3] Karapetyan A.V., Sharakin S.A., On the stationary motions ON THE STABILITY OF LINEAR POTENTIAL of two mutually gravitating bodies and their stability, Her- GYROSCOPIC SYSTEMS ald of Moscow University, Ser.1, Mathematics, Mechanics, pp. 42-48, (1992) TATIANA V. S ALNIKOVA [email protected] T.V. Salnikova, Moscow State Lomonosov University, Leninskie Gory, Moscow, 119992, Russia Let us consider the small oscillations of the dynamical system in the region of an equilibrium position, described by the differ- SOME PROBLEMS OF STATICS, THERMOELASTICITY ential equation with n degrees of freedom: AND DYNAMICS OF MICROPOLAR BARS n Mz¨ + Gz˙ + Kz =0,z∈ R (1) SAMVEL H. SARGSYAN ANAHIT J. FARMANYAN MANYAK N. MUTAFYAN SONA A. VARDANYAN The matrix of kinetic energy M = M T is positive definite, K = KT is the matrix of potential energy, GT = −G - is the matrix [email protected], [email protected] of gyroscopic forces, rank G =2k. Theorem. The sufficient conditions of the stability of the triv- The progress in micro and nanotechnology in the construction ial solution of equation (1) is that matrices A and B are positive of new materials and in the study of stress-deformable state in definite, where the bodies from these materials contributed to the actuality of re- −1 A = K − GM G/4, searches in asymmetric (momental, micropolar) theory of elasticity, or in other words in Cosserats theory. In this respect, natu- −1 2 B =(GM G + γ M)/4 − K, rally, the problem of the construction of the theory of micropolar γ2 = −λ2, λ is the nonzero root of the characteristic equation elastic thin bars, plates and shells gains actuality. In the work [1] the asymptotic method is developed for the transition from det(G − λM)=0 the three-dimensional problem of asymmetric theory of elasticity in the three-dimensional thin sphere of plate (in the cylinder of Notice. The theorem is true for the case, when G reads: the small thickness) to the two-dimensional equations. There is also constructed the interior iterational process and the micropo- T 01 lar boundary layers, and the problem of their matching with the G = γS IkS, Ik = diag(I,...,I,0),I= −10 interior problem are also studied. On the basis of the interior iterational process the general applied two-dimensional theory of S is the non-degenerate (n × n) matrix. When n =3any non- micropolar elastic thin plates is constructed. In current report the trivial skew-symmetric matrix G has this form for k =1. asymptotic approach stated in work [1] is developed for the study Let us consider the problem of the motion of two mutually of the boundary value problem of the two-dimensional theory of gravitating bodies; both of them are modeled by two equal mass- asymmetric elasticity in thin rectangular sphere. Two known vari- points conjoined by weightless inextensible rods. The flat motion ants of asymmetric theory of elasticity (continuum and pseudo- is defined by the coordinates r,ϕ, ϕ1, ϕ2, where r is the distance continuum of Cosserat) are considered. The main purpose of the between the centers of mass of the bodies with masses m1 and research is the asymptotic transition from two-dimensional prob- m2 and lengths of rods a1 and a2 respectively, ϕ is the angle lem of the asymmetric theory of elasticity to one-dimensional between the straight line, connecting the centers of mass and the equations. In present work the applied two-dimensional the- fixed straight line in the motion plane, ϕ1 and ϕ2 - the angles ory of micropolar bars-strips is considered (both on the basis of between the rods and the straight line, connecting the centers of Cosserats continuum and on the basis of Cosserats pseudocon- mass. The equilibria of the reduced three-dimensional system, tinuum) for problems of statics, thermoelasticity and dynamics. described by the Routh equations correspond to the stationary Here the theory of micropolar boundary layer is constructed. The motions r = const, ϕ˙ = const,ϕ1 = const,ϕ2 = const existence of two types of boundary layers force (plane and an- of the initial system, so we can apply the theorem for studying the tiplane) and momental (plane and antiplane) is shown in case of problem of gyroscopic stabilization for some specific systems. asymmetric theory of elasticity with the independent fields of dis- 2 The stationary motions ϕ1 = ϕ2 = π/2,r = r(k ), where k placements and rotations as the basis. The general properties of is the cyclic integral constant corresponding to the cyclic coordi- boundary layer solutions are studied, the formula of the gener- nate ϕ are always unstable in Lyapunov’s sense. For the set of alized orthogonality type is proved, the general solutions of the points of the bifurcation diagram for which the degree of insta- boundary value problems are constructed in form of series. The bility equals two we can verify the gyroscopic stabilization. In problem of the matching of asymptotic expansions of the interior particular for the system of twin asteroids Ida-Dactyl the stability problem and boundary layers is studied, in the result of which is proved. the interior problem (consequently the applied, one-dimensional theory) is separated as an independent boundary value problem. REFERENCES Separate boundary value problems are also formulated for the [1] Kozlov V.V., Gyroscopic stabilization and parametric res- boundary value problems. The problem of matching of the men- onance, J.Appl. Maths Mechs, Vol. 65, No.5, pp.715-721, tioned asymptotic expansions is studied also on the basis of the (2001) variational principle of the two-dimensional theory of asymmet- [2] Huseyin K., Hagedorn P., Teschner W. On the stability of ric elasticity with the independent fields of displacements and ro- linear conservative gyroscopic systems, ZAMP, V.34, 6, pp. tations, in the result of which the unknown coefficients are de- 807-815, (1983) termined in the series of the general solutions of the boundary

79 value problems. Analogically, the applied one-dimensional the- HARMONIC VIBRATION OF RIGID PUNCH ON THE ory of micropolar bars-strips is constructed (in the case of prob- FREE SURFACE OF A POROUS ELASTIC HALF-PLANE lems of statics, thermoelasticity and dynamics) on the basis of the two-dimensional asymmetric theory of elasticity with constrained ANTONIO SCALIA MEZHLUM A. SUMBATYAN rotations as the basis. Here, the boundary value problems, the properties of the functions of boundary layer type and their gen- [email protected], [email protected] eralized orthogonality are studied in detail. The problem of the matching of the interior problem (of applied one-dimensional the- Investigation of dynamic properties of various elastic solids ory) with the boundary value problems is studied and the separate under contact conditions is a very important problem in the prac- boundary conditions for the mentioned problems are taken out. tice of ultrasonic inspection of materials, vibrations of engineer- Here, there is also realized a huge numerical research. In the case ing structures on elastic foundations, in soil mechanics, in seis- of the problem of statics and thermoelasticity all calculated values mology and many other fields. Usually, such materials can be cor- (force and momental stresses, displacements and rotations) are rectly described by classical dynamic equations of linear isotropic defined both in the interior sphere of thin rectangular and along elastic solids. However such materials like soils, composite ma- the boundary of the contact zone near the edges of the rectan- terials, granular materials, etc., show a specific characteristic regular. In the case of the dynamic problem the frequencies and sponse to applied dynamic load. the forms of natural vibration of bars-strips are defined and the There are a number of theories, to describe the dynamic prop- forced vibrations are studied. Here, we are also comparing the erties of porous materials, the most classical one is a Biot’ con- results of the theory with the theory with constrained rotations solidation theory of fluid-saturated porous solids. Typically, these and with the classical theory. There are also carried out conclu- theories reduce to ordinary elasticity when the pore fluid is ab- sions concerning the micropolarity of the materials of bars-strips sent. This is why Cowin and Nunziato proposed a new theory and concerning the applied theory. which was also developed in detail for the linear case. Some REFERENCES static contact problems for such materials were studied in our re- [1] Sargsyan S. H. On Some Interior and Boundary Effects in cent works. Puri and Cowin have studied various types of plane Thin Plates Based on the Asymmetric Theory of Elasticity// waves which can propagate in these linear elastic materials with Lecture Notes in Applied and Computational Mechanics. voids. Recently, Chandrasekharaiah studied propagation of the Vol. 16. Theories of Plates and Shells. Critical Review and Rayleigh surface waves in such porous materials. New Applications. Springer. 2004. P. 201-210. In the present work we develop a semi-analytical approach to a dynamic contact problem about a rigid punch oscillating on the S.H. Sargsyan, Sayat-Nova 2, apt.11, Gyumri, 377501 Armenia free surface of the porous elastic half-plane without friction. By using Fourier transform, the problem is reduced to an explicit THE DYNAMICAL REACTION OF A 3-LAYERED form in an integral equation with a convolution kernel holding VISCOELASTIC SHELL over the contact area. We then apply a collocation technique to solve this equation and study the distribution of the contact pres- YUSUP N. SAYFUTDIONOV SERGEY A. LYCHEV sure and dynamic compliance of the foundation, comparing these [email protected], [email protected] physical characteristics with classical linear elastic case. M.A. Sumbatyan, Rostov State University, Zorge Street, 5, The analytical solution of dynamic problem of a 3-layered Rostov-on-Don, 344090, Russia spherical viscoelastic shell, with elastic support on the boundary is found. It is supposed that the outer layers are thin and un- ONE-MODE SH-WAVE PROPAGATION THROUGH A dergo the membrane tension. The inner one undergoes shear and PERIODIC ARRAY OF INTERFACE CRACKS its cross-sections possess rotation inertion. The present analys constrained by mutual layer sliding prohibition. The material of E. SCARPETTA M.A. SUMBATYAN outer layers are suppose to be elastic and orthotropic, whereas that of inner one being isotropic viscoelastic. The constitutive [email protected], [email protected] law is specified by integral operator with the exponential kernel. The support is elastic with respect to normal and tangential dis- In the context of wave propagation through damaged or com- placements and cross-sections shell rotations. The dynamics is posite (elastic) solids, we develop an analytical approach to study described by a system of integrodifferential equations, which can the penetration of a (scalar) SH-plane wave into a periodic array be transformed in to the initial problem with operator coefficients. of collinear cracks. We assume that such an array separates two The solution of abovementioned problem is obtained by means half regions of the solid which have different material properties, of integral transformations with conjugated matrix kernels, which described by wave speeds c1 and c2. can be constructed as a linear combinations of egen-functions of Starting from suitable Fourier expansions of the wave field in conjugate polynomial operator pencils, derived from problem un- these two regions, we impose natural boundary and continuity der consideration. The representation of solution over the sys- conditions so as to reduce the problem to an integral equation (of tem of egen-functions and adjoined functions, which complete- Fredholm type) based on the opening between adjacent cracks. ness and minimality are proved. The preposed representations Thus, by means of a mild approximation holding uniformly in form the theoretical background for the computer simulation of the given (one-mode) regime of propagation, we get an integral dynamical reactions of oil and gas tanks. equation free of frequency which is analytically solved following Yu.N. Sayfutdionov, S.A. Lychev, Samara State University, previous papers. This directly leads to full-explicit representa- Samara, Russia tions for the wave field and the relevant scattering parameters.

80 Our approach is based on a so-called ?one-mode? approx- sion force for fixation - 5-15 H, stiffness coefficient - 200-300 imation, when the principal non-homogeneous mode of the H/mm. The use of these constructions allows to save prolonged structure is taken into consideration exactly, but all other non- compression between fragments without threat of eruption of a homogeneous decaying modes are treated asymptotically. Cer- patella spongy bone and loss of stability. tainly, the only homogeneous mode is also considered without Vladimir Sementzov, Rostov State Medical University, any approximation. Such a method permits an explicit represen- Nachichevanski, 29, Rostov-on-Don, 344022, Russia tation of the reflection and transmission coefficients in terms of frequency parameter. Some figures are finally set up to compare our results with the numerical ones obtained by some direct nu- BUOYANCY LOSS BY A COVERED OBJECT DUE TO merical simulation. BENDING OF ITS FLEXIBLE SHELL SUBJECTED TO Edoardo Scarpetta, University of Salerno, Via Ponte don Melillo, DISTRIBUTED PRESSURE Fisciano (Salerno), 84084, Italy ALEXANDER D. SERGEYEV [email protected] TO THE BIOMECHANICS OF PATELLA FRACTURES FIXATION Our aim is to obtain what happens if a strongly deformable body penetrates into inertia medium in a global external field V.A. SEMENTZOV V.A. SERGAL’ more and more deeply. So we consider buoyancy conditions for A.A. POLOISHNIKOVA R.S. ROMANOV a flexible object submerged into medium with a gradient of pressure along the depth when pressure inside the body is assumed to be constant. For the body bounded by a flexible elastic shell that According to some authors, from 30 up to 70 percent of all phenomenon may be well-definedly analyzed only for big non- fractures of lower extremities fall on knee-joint injuries, that quite linear deformations. Therefore to get the qualitative analytical often become the cause of a lingering incapacity for work of the resultes we investigate the 2-D case, when the buoyant force de- patients. 3.5-8.3 percent of them are patella fractures. Patella pends on the square of the object only and the role of the flexible fractures are most often met among young and middle-aged peo- shell is played by a flexible contour. As the square is associated ple. The frequency of poor results to cure these types of fractures with a form of its flexible contour, therefore, the object square is remains high and is from 16.4 up to 50 percent. may be deformed by external acting on the contour. To vary the Variety of osteosynthesis techniques in operation testifies to depth of diving we use a mass influx into two tanks inside the the pendency of the patella fixation problem and it requires rigid object in the model of a simulated system. Due to bending flexi- fixation with permanent fragment compression. bility of the contour the initial square of the floating body changes Nowadays several patella fracture classifications exist, the under interior pressure and two concentrated forces, cased by the most famous are Speck, Regazzoni (1994) and Rogge, and Goose tanks. We show the influence of magnitudes of the shell elas- (1985) classifications. At the heart of patella fracture classifi- tic parameters and geometric size of the submerged object both cations is the displacement principle (without displacement, dis- on the situation when interior pressure inside the square has no placement to 2 mm and displacement to more than 3 mm) and ability to keep floating of the object. In that case it is observed fracture (longitudinal, transverse and multifragmental fractures). irreversible engulfing of the elastic body by medium. Gunter and his co-authors (2000) classify fractures according to A.D. Sergeyev, Research Institute for Problems in Mechanical En- the displacement degree and the fragments’ size. So, three de- gineering, Russian Academy of Sciences, Bolshoy 61, V.O., St. grees of a fragment displacement, full-grown as a result of the in- Petersburg, Russia jury of a patella supporting apparatus. According to the fragment size one distinguishes macro- (more than 10 mm) and micro- fragmentaries (less than 10 mm). STABILITY OF A RECTANGULAR PLATE UNDER Since 1982 to cure patella fractures fixators with form’s mem- BIAXIAL TENSION ory are used. The peculiarity of these constructions is the pos- DENIS N. SHEIDAKOV LEONID M. ZUBOV sibility of producing even prolonged compression, independent on the fracture line and with the strength not causing bones erup- [email protected], [email protected] tion. It’s very important because the patella is a spongy bone and this provides the possibility for early functional treatment, In majority of researches on stability of elastic bodies equi- that decreases the periods of a temporary disability of the injured. librium buckling under compressive loads is considered. But in- These constructions are most often used to cure transverse large stability of deformed bodies under tension stresses is also possi- fragmentary fractures. We considered transverse and large frag- ble. In this communication a problem of stability for a rectan- mentary fractures (the fragment value is 10 mm and more), the gular plate under homogeneous deformation of biaxial tension in presence of the diastasis for more than 2 mm between the frag- a plane of a plate is considered by means of three-dimensional ments and the damage of the congruence of a patella articular equations of nonlinear elasticity. Both faces of a plate are free surface with the ”step” for more than 2 mm to be the reason for and there are special boundary conditions on lateral bounds of a the osteosynthesis. plate. These special conditions allow separation of variables in We used undulating fixators, produced at the engineering- linearized equilibrium equations. Parameters of deformation are medical center “MATI-Medtech” with the following character- stretch ratios of a plate in the unperturbed state. Neutral equilib- istics: the form change temperature is no more than 10 degrees rium equations for a case of isotropic incompressible material of a C, the reconstruction temperature - 35+/-2 degrees C, compres- general kind are derived. Separation of variables reduces problem

81 to a homogeneous linear boundary-value problem for a system of mechanical systems by increasing or decreasing effective stiff- ordinary differential equations. This problem is solved numeri- nesses, and influences on the stability of the system. The simplest cally. For two models of material critical curves and instability examples of such phenomena can be seen in the stabilisation and region are plotted in the plane of deformation parameters. Buck- destabilisation of the transversal oscillations of Bernoulli-Euler ling modes are defined and their qualitative features are analyzed. beams, and can be caused by small longitudinal vibrational ex- It is founded that instability region is included in nonconvexity citations [1]. Another interesting effect in this category is the region of the potential energy of a plate, which formulated as a Indian magic rope trick, in which a wire with a naturally unstable function in terms of deformation parameters. Numerical results vertical orientation can be stabilized due to vertical vibration of shows that instability region in a case of tension stresses weakly the bottom support point [2, 3]. depends on geometry of a plate. Based on obtained facts sim- The paper considers the influence of longitudinal vibrational ple approximate criterion of elastic plate stability under tension excitations on the stability of transversal oscillation within rect- stresses is formulated. angular plates. The problem is solved by means of the method This work was supported by Russian Foundation of Basic Re- of direct separation of motions (MDSM) [4]. This approach was searches (grant No 05-01-00638). chosen from amongst various possible solution procedures [4, 5] D.N. Sheidakov, Rostov State University, Zorge str., 5, Rostov-on- because in this case, MDSM allows the derivation of explicit an- Don, 344090, Russia alytical formulae for the “vibrational correction” to the critical load. The results that are obtained allow one to control the stability of the system by choosing appropriate excitation parameters. STABILITY OF OSCILLATORY FLOW IN LIQUID Effects concerned with the influence of excitations on the sta- COLUMN bility of mechanical systems are of potentially significant practical interest. Thus, vibrational stabilisation and destabilisation V. S HEVTSOVA A. MYALDUN D. MELNIKOV could either be useful, or alternatively even dangerous, when [email protected], [email protected],e present in machines and structures, so a unifying understanding [email protected] of such phenomena is very important. The vibrational control of the dynamic properties and stability of mechanical systems is The appearance and development of thermoconvective oscilla- an important, and ongoing problem. The problem investigated in tory flows are investigated experimentally with numerical support the paper can also be considered as a simple modelling problem in a cylindrical column with a free lateral surface (so-called liq- which may be helpful for describing more complex systems. uid bridge) filled with 5 and 10 cSt silicone oils. In this system a fluid volume is held by surface tension between two differen- Acknowledgements tially heated horizontal flat co-axial disks. The convective flow is initiated by thermocapillary forces even for very small-applied The authors are very grateful to the Royal Society and the Russian temperature differences and evolves under the combined action Foundation for Basic Research for partial financial support. of two effects; buoyancy and thermocapillarity. REFERENCES The experiments with high Prandtl number fluids were carried [1] Blekhman I.I., Gavrilov S.N., Shishkina E.V. On application out under terrestrial conditions for a wide range of volumes of of the method of direct separation of motions to the inves- liquid bridges. The convective flow becomes oscillatory when tigation of distributed systems // Izvestiya VUZov. Severo- the temperature difference increases and achieves certain critical Kavkazskii region. Estestvennye nauki (Notices of Univer- ∆ =∆ value, T Tcr. sities. South of Russia. Natural sciences). Special volume Depending upon the volume of liquid between the rigid walls ”Nonlinear Problems of Continuum Mechanics” - 2003. P. the column may have different shapes in normal gravity condi- 113Ð123. tions; the relative volume of column (with respect to the cylindrical case V = πR2h) can vary from 0.7 until 1.3 depending upon [2] Blekhman I.I., Dresig H., Shishkina E. On the Theory of the aspect ratio and static Bond number. Indian Magic Rope // Selected Topics in Vibrational Me- It is found that periodic oscillatory convective flows are much chanics. Chapter 8. World Scientific, 2004. P. 139Ð149. more stable with respect to external disturbances in the case of [3] Champneys A.R., Fraser W.B. The “Indian rope trick” for 1 1 under filled liquid zone, V< , than in zone with V> . a parametrically excited flexible rod: linearized analysis // The evolution of the various harmonics amplitudes, phase Proc. Roy. Soc. Lond. 2000. A456. P. 553Ð570. planes, return maps with increasing of ∆T have been thoroughly analyzed. [4] Blekhman I.I. Vibrational Mechanics. Singapure, New Jer- Valentina Shevtsova, Dept. of Chemical Physics, University of sey, London, Hong Kong: “World Scientific”, 2000. 509 p. Brussels, 50, av. F.D.Roosevelt, Brussels, B-1050, Belgium [5] Cartmell M.P., Ziegler S.W., Khanin R., Forehand D.I.M. Multiple scales analyses of the dynamics of weakly nonlinear mechanical systems // Appl. Mech. Rev. 2003. V. 56, N VIBRATIONAL STABILISATION OF ELASTIC PLATES 5. P. 455-492. EKATERINA V. S HISHKINA MATTHEW P. C ARTMELL E.V. Shishkina, Institute for Problems in Mechanical Engineering shishkina [email protected], of Russian Academy of Sciences, Bolshoy pr.V.O.,61, St. Peters- [email protected] burg, 199178, Russia M.P. Cartmell, Department of Mechanical Engineering, James Vibrational excitations can change the dynamic properties of Watt Building, University of Glasgow, Glasgow G12 8QQ

82 Marina I. Shmurak, Perm State Technical University, 29a, Kom- BIOMECHANICAL MODEL OF THE MAMMA somolsky Prosspect, 614990, Perm, Russia M.I. SHMURAK V.M. TVERIER E.Y. SIMANOVSKAYA Y.I. NYASHIN PARTICLE TRANSPORT METHOD FOR [email protected], CONVECTION-REACTION PROBLEMS [email protected] ANTON SMOLIANSKI OLGA SHIPILOVA Now majority of children is artificial fed. However modern HEIKKI HAARIO researches have proved, that artificial feeding considerably dif- [email protected] fers from breast-feeding. Character of feeding in many respects determines development of the dentofacial system, its muscu- loskeletal apparatus, and of some vital functions such as sucking, The paper is focused on an explicit technique, the Particle chewing, swallowing, breath, hence development of the child as Transport Method (PTM), for solving convection-reaction prob- a whole. There is necessity of modeling of breast-feeding and lems. This work is a logical extension of the approach, presented bottle-feeding. in the framework of the Conference “Advanced Problems in Me- chanics” at 2004. PTM is a characteristic-based unconditionally The model of functioning of the women mamma during feed- stable technique that, despite its Lagrangian nature, allows for the ing which completely describes process of breast-feeding is con- accurate approximation of the solution on a fixed Eulerian mesh. structed. The glandular tissue of the mamma represents lobules-alveolar Following meshless concept, a system of independent moving system. Alveoli are united in lobules, and then in larger forma- particles is used to transport exact solution. The transfer from the tions - lobes having elastic excretory ducts. Large channels be- particle system to the fixed mesh is done with a special, mono- fore nipple have small expansions, so-called chylocysts or milky tone projection technique. The adaptivity procedure for the par- sinuses and come to an end thin exit channels. ticle system makes PTM a monotone high-resolution numerical scheme which can successfully work on structured and unstruc- Milk is formed in secretory cells of alveoli, its development tured meshes. occurs constantly and in small amount. Then milk enters the system of channels chylocysts so-called ”depositories” of milk The performance of the presented algorithm is tested on sev- where it is collected prior to the beginning of feeding. eral convection-reaction problems involving the 1st-order reac- To the moment of feeding the system of channels of the tion. mamma is filled with milk. For getting of milk it is necessary O. Shipilova, Lappeenranta University of Technology, Skinnari- that the child overcomes only sphincter resistance in a nipple. lankatu 34, Lappeenranta, FIN-53850, Finland With the beginning of feeding the sinuses and large channels are released first of all and the active current of milk from alveoli begin. COSSERAT CONTINUUM AS A MODEL OF The model of the mamma is constructed founding on physiol- MESOSCALE ogy and series of the known results of experimental researches. IGOR SMOLIN The following assumptions are made in model construction: 1) all large channels and chylocysts are incorporated into one [email protected] tank with elastic walls; 2) the system of excretory channels in the nipple is represented In the physical mesomechanics of materials the hierarchical as a rigid tube. Hydraulic resistance of the rigid tube is great; approach to the description of deformation of solids is developed. 3) elasticity and resistance for each group of channels are Three scale levels are at the least considered: macro, meso and change in time; micro. The approaches of the generalized continua are ideally 4) transient processes of an establishment of flow of milk are suitable for the mesoscopic level, where the material still can be not examined; considered as a continuum, but at the same time the discontinuity 5) the system of channels always is filled by milk. of its structure, internal heterogeneity manifests itself already. Function of change of tank pressure, function the volume rate Here it is proposed to use a model of elastoplastic Cosserat of flow of milk in the beginning of excretory channels in a nipple, continuum for the mesoscale level. The physical ground for in- function of pressure in the child oral cavity are known results of troduction of the additional rotational degree of freedom there- experimental researches. Function of tank flexibility is suggested with is the presence of an internal motion as a result of the defect based on the physiological data. ensemble evolution at the microscale. This results in acting of The model allows determining time dependence for the vol- couple stress at the mesoscale and non-symmetry of force stress. ume rate of milk flow from alveoli and hydraulic resistance of Only symmetrical force stress exists at the macroscale owing to the nipple. averaging over all the heterogeneities. We construct mathematical model of normal functioning of the Since the evolution of the defect ensemble takes place mainly mamma. Such model allows to study and to explain the process at the plastic flow stage, the parameters of the constitutive equa- of breast-feeding of the baby. tions for couple stress are suggested to be not constants but func- Obtained results for example hydraulic resistance of the nip- tions of plastic strain accumulated. Two kinds of elastoplasticity ple can be used for modeling of bottle-feeding and further, for are considered. The first one uses a coupled criterion that in- improvement of parameters of the nipple and bottle (form, size cludes both force stress and couple stress. In the second case two etc.). independent criterions for each kind of stresses are used.

83 The influence of couple stresses on strain localization at the power quality. For this reason, the design and optimization of the mesoscale and their contribution to the averaged macroscopic control system required for variable speed operation is a complex stress-strain curve are discussed. and demanding task and no single, globally acceptable controller I. Smolin, Institute of Strength Physics and Materials Science SB structure and parameter tuning methodology is available. RAS, Pr. Academicheskii, 2/1, Tomsk, 634021, Russia In the present paper, a novel controller is applied to a stall regulated variable speed wind turbine. The emphasis is placed on the reduction of the mechanical stresses, expressed by the variability A NEW MODEL OF ORTHOTROPIC of the torque in the drive train, without compromising signifi- INCOMPRESSIBLE NONLINEAR ELASTIC MATERIAL. cantly the energy yield. The simulation of the wind turbine operation is performed using a relatively simple mechanical equivalent ARCADY SOKOLOV for the drive train and the static aerodynamic power relations for arcady [email protected] the wind turbine rotor. The dynamics of the electrical generator and power converters have been ignored, since their time scale is A new model of orthotropic incompressible elastic material, much shorter compared to the mechanical time constants or the more complex then well-known Mooney-Rivlin model, is pro- frequencies where the wind speed variability occurs. Hence, the posed. In this model the strain energy function is the quadratic electrical generator is modeled as a “torque actuator”, the torque form of components of Cauchy strain tensor. It is obtained as a reference being determined by the wind turbine control system. result of Taylors expansion with respect to incompressibility re- Benchmark for the evaluation of the controller performance is the lation. This material is characterized by ten material parameters. operation of the wind turbine controlled via the standard optimal In the case of small deformation the relation between the new torque-speed characteristic of the rotor. model and Hookes law in the linear theory of elasticity of or- Argiris Soldatos, National Technical University of Athens, De- thotropic material is established. And relations between material partment of Electrical and Computer Engineering, Division parameters of this new model and classic parameters of Hookes of Signals, Systems and Robotics, Zographou, 15773, Athens, law are obtained. Greece This model is used for solving several problems on finite uni- K.G. Arvanitis, Agricultural University of Athens, Department of form deformations of orthotropic body. Inflation and torsion of Agricultural Engineering, Laboratory of Agricultural Machinery circular cylinder is also considered. and Automation, Iera Odos 75, Botanikos, 11855, Athens, Greece A. Sokolov, Rostov State University, 5 Zorge str., Rostov-on-Don, S. Papathanassiou, A. Kladas, National Technical University 344090, Russia of Athens, Department of Electrical & Computer Engineering, Electric Power Division, Laboratory of Electrical Machines and Power Electronics, 9, Iroon Polytechneiou Street, 15780 Athens, CONTROL FOR VIBRATION ATTENUATION IN Greece VARIABLE SPEED WIND TURBINES

A.G. SOLDATOS K.G. ARVANITIS CONSTRUCTION OF ANALYTICAL-NUMERIC S. PAPATHANASSIOU A. KLADAS SOLUTIONS OF BORDER PROBLEMS OF PLANE [email protected] STATIONARY FLOWS OF INCOMPRESSIBLE LINEAR VISCOUS FLUID USING FINITE ANALYTICAL The use of deterministic control in conjunction with customary ELEMENTS METHOD variable speed control strategies in order to assure small torsional MIKHAIL B. SOLOVIEV axis loading in variable speed wind turbines, is proposed in this paper. This is done in order to protect the turbine axis from me- [email protected] chanical strains thus guaranteeing proper and safe operation al- beit at some expense in the power produced to be delivered. The A new method of obtaining of approximate solutions of non- proposed method requires control force application only at the linear Navier-Stokes equations for stationary plane flows of in- generator level and is well within the capabilities of the genera- compressible linear viscous fluid is being considered in this work. tor. The efficacy of the scheme is illustrated by simulations on a Proposed method is based on introduction of complex structure in prototype wind turbine of the stall type. the spaces of coordinates and values of sought functions. Sought Variable speed wind turbines employ power electronics con- complex-valued functions and their complex conjugations is rep- verters to decouple the speed of rotation of the electrical gener- resented in form of special decompositions by powers of com- ator from the practically constant frequency of the grid, where plex conjugate variable with coefficients in form of holomor- the wind generator is connected. For this purpose, a variety of phic functions. The character of offered method is that func- configurations have been investigated and applied in practice for tions complex conjugate to sought ones are represented in the the electrical part of the machine. In all cases, the main objec- same form as sought functions, but with different holomorphic tives of variable speed operation have been the increase of energy functions. The substitution of that decompositions in complex- yield, along with the reduction of the mechanical stresses and the ified Navier-Stokes equations and condition of incompressibil- improvement of the wind turbine output power quality. These ity results in appearing of a system of interlocking differential control targets are in principle conflicting, since maximization equations concerning holomorphic functions representing sought of the energy output requires fast tracking of wind speed varia- complex-valued fields and their complex conjugations. This tions, which in turn results in strong variations of the generator equations set may be written down in form of recurrence rela- and drive train torques and therefore deterioration of the produced tions. This gives possibility to define two independent holomor-

84 phic functions. The definition of these functions defines the solu- contact take place. There are some works that have investigated tion of considered equations in form of iterative process. this problem from different positions with some limitations [5, For obtaining of approximate solution of border problem in 6]. simpliest simply connected domain a method of minimization We try to solve this problem (to find optimal profile) in sug- of mean-square distance from given boundary conditions, which gestion of parallel shift of profile when wear occur. This hypoth- gives possibility to find two initial holomorphic functions cor- esis is in accordance with experiment in some period of exploita- responding to a solution satisfying given boundary conditions is tion, and allow reduce nonlinear evaluation problem to differen- being proposed. This method was tested on border problems with tial equation for one variable. In this case we reduce optimization kinematic, force and mixed boundary conditions. Results of cor- process to solving of differential equation: responding numeric experiments are being shown. For obtaining of approximate solutions of border problems in 2 domains with more complex geometry and domains that are not d d − α f (ξ)+β =1/2 A 1+ f (ξ) × simply connected a method of finite analytical elements is being dξ dξ proposed. The main idea of this method consists of that initial a(ξ) ωR(ξ) γ −1 −1 functions are being sought as analytical functions, that is, as sets × p (ξ,z) |(λ (ξ,z)) | dzπ (R (ξ)) − b(ξ) of analytical elements. To test proposed method numeric experi- ωR(ξ) ments were carried out. Their results are shown in this work. M.B. Soloviev, Dorodnicyn Calculation Centre, Vavilova street The research was carried out under the financial support of the 40, Moscou, 117967, Russia Russian Foundation for Basic Research (grant No 04-01-00766). REFERENCES [1] IHHA’99 STS Conference. Simulation of Wheel/Rail Con- EVOLUTION OF THREE-DIMENSIONAL ELASTIC tact and Wear in Curved Track. Gogyacheva I.G., Dobychin BODIES PROFILES IN ROLLING CONTACT M.N., Soldatenkov I.A.

SERGEJ SOSHENKOV [2] Kalker J.J. Three-Dimensional Elastic Bodies in Rolling contact. Kluwer Academic Publishers, 1990. [email protected] [3] Kogan A.. Ya. Track Dynamics and its Interaction with Rilling Stock (in Russian). M., Transport, 1997. Problem of Three-Dimension interaction of rolling bodies is well known [1, 2, 4]. Nevertheless a lot of aspects of this problem [4] Specht W. New Particulars of Wear of Heavy Railway Car- remain unresolved. The most interesting contact problem is when riage Wheels. Glasers Annalen, V.9,1987, pp.271-280. rail and wheel interacts. In this work we try to solve this problem [5] Wear 253 (2002) 100-106. Zakharov S., Zharov I. Simula- in two aspects: point and conformal contact of profiles. tion of mutual wheel/rail wear. Point contact problem consists of some sub goals. Contact [6] Zakharov S.M., Komarovsky I.A., Zharov I.A.. Wheel/Rail problem - definition of contact parameters such as pressure, con- Wear Simulation. Wear, V. 215, 1998, pp.18-24. tact zones. As usually we neglect the influence of tangential stresses on normal pressure distribution. In fact, this assump- Maria A. Ryakina, Perm State University, Bukireva, 15, Perm, tion is acceptable on rail-wheel contact [2]. Then go problems 614990, Russia of tangential stresses and slippage zones. Key problems are wear law and dynamic of rolling-stock. The last problem is beyond the scope of actual report. But dynamic problem is necessary for evo- ON THE EFFECT OF THE INTERFACE CONDITIONS lution problem. Obtained by all existing theories of elasticity and ON THE DYNAMIC RESPONSE OF A BEAM ON A viscous elasticity creep curves are not agreed with wheel-rail ex- HALF-SPACE TO A MOVING LOAD perimental data. The inclination of theoretical creep curve in lin- MICHAEL J.M.M. STEENBERGEN ear region is more than two or even four times greater. So in practice especially when large torsion torque is applied on wheelset [email protected] there are no any benefits to use complicated time-taking methods. So the slippage in contact zone is greater than it is assumed. In When studying the dynamic response of modern railway slab accordance with [5] wear law may be presented as function of track systems to a high-speed train, the dynamic slab-soil interac- squared creep and local pressure. So for practice wheel-rail con- tion has to be accounted for. The higher the train speed and/or the tact evolution problem more actual is a method of instantaneous higher the frequency of the train vibrations, the more important axis with given transverse slippages because wear occur in high it is to model the soil as a tree-dimensional elastic (visco-elastic, creep zones. For numerical computations we use fast-computing visco-elasto-plastic, etc.) continuum. The simplest example of dynamic program, based on integral transformations [3]. We find such continuum is an elastic half-space, which disregards both graphs of wheel profiles evolution for different initial wheel pro- the railway track superstructure and variation of the soil proper- files. ties, but accounts for wave propagation in the soil. The slab of During wear profiles may become more conformal, i.e. contact the track can be sufficiently well modelled as a beam of a finite zones considerable part of profile may occur. In this situation width. common methods are unacceptable. This contact is non-Herzian In this contribution, the dynamic response of the beam on an and half-space approximation is impossible. Conformal contact elastic half-space is investigated to a uniformly moving load. The problem is important for numerical dynamic computations and focus of the study is placed on the effect of the description of for choosing optimal profile in curved tracks where conformal the interface conditions between the beam and the half-space on

85 this response. Three conventional approaches to this description ence, SB, RAS, 2/1, Academicheskii pr., Tomsk, 634021, Russia are addressed. According to the first approach, the stresses are assumed to be uniformly distributed across the beam, whereas the continuity of the beam and half-space displacements is required only along the centre line of the beam. In the framework of the STABILITY PROBLEMS RELATED TO THE SPACE second approach, the stresses are still considered to be uniformly ELEVATOR distributed across the beam, whereas the continuity is required between the beam displacement and the average displacement of A. STEINDL H. TROGER the half-space under the beam. The third approach is the classical approach of the contact mechanics. The slab vertical displacements as predicted by the three afore- One of the main problems of modern space exploration and mentioned approaches are compared in this contribution. A con- space technology is the high cost of sending a payload from the clusion is drawn as to the train velocities at which the simplified surface of the Earth into space by rockets. Hence for a long time descriptions of the interface conditions (the first two approaches) there have been other ideas around for a cheaper way of transport- are applicable. As an intermediate result, expressions are derived ing payloads into orbit. In 1960 the Russian scientist Yu. Artsu- for the dynamic stiffness of the half-space as a function of the tanov ([1]) proposed the space elevator. Its idea is to use a string frequency and wavenumber of the waves in the beam. forming a connection from the surface of the Earth to a satel- M. Steenbergen, Delft University of Technology, Faculty of Civil lite in geostationary orbit (approximately 35800 km altitude), as Engineering, Section of Railway and Road Engineering, P.O. Box track of an elevator. It could provide easy access to the geosyn- 5048, NL 2600 GA Delft, Netherlands chronous orbit and beyond ([4]), because in order to compensate for the weight of the downhanging string, the string must be extended beyond the geostationary radius. NUMERICAL INVESTIGATION OF FRACTURE AND This idea makes use of the fact that a massive string moving on STRAIN LOCALIZATION IN GEOMATERIALS a circular orbit around the Earth, under the action of gravitational and centrifugal forces, has a radial relative equilibrium position. YURII P. S TEFANOV It can be shown that in this equilibrium position the string is under [email protected] tension ([2]). In order to obtain a minimum weight design the shape of the string must be tapered such that it is thickest at the Description of the behavior of geomaterials still remains to be point of highest tension which is at the geosynchronous radius an urgent problem in understanding the processes occurring in and thinnest where the tension is lowest (at its ends). them and in predicting their behavior in various conditions. The Until 1991 these ideas were purely academic, since no material macroscopic behavior of these materials can be brittle or near- was available to realize such a project. However, at that time so- plastic, depending on loading conditions. These features are an called “carbon nanotubes” were discovered, which are cylindrical influence of pressure on inelastic pseudoplastic deformation and macromolecules composed of carbon atoms. Forming nanoropes pronounced changes in the specimen volume in the course of from nanotubes a theoretical strength of 100 times higher than deformation. Fracture of rocks is, first of all, associated with steel can be expected theoretically but with only one-sixth of the the presence of pores and cracks of different scales. Under any weight of steel. Experiments show that about half of the theo- type of loading the stress-strain state is inhomogeneous. Near retical strength can be achieved ([5]). Moreover, besides their pores and cracks tensile stresses arise, which leads to tensile extreme strength carbon-nanotubes also allow large strains up to crack growth. In this work the processes of localized deforma- 16-24 %. The ratio between tensile strength and density is crucial tion band formation and crack generation in brittle-plastic ma- for the taper ratio of the string, that is, the ratio between the cross terials are simulated numerically in various loading conditions. section at the geosynchronous orbit to the cross sectional area of The patterns of cracking and strain localized band for a sand- the cable at Earth. For example the taper ratio required for steel 33 8 stone specimen in compression and are obtained. To describe would be 1.7 × 10 , for Kevlar 2.6 × 10 and for carbon nan- inelastic behavior of pressure-sensitive materials, the mathemat- otubes 1.5. ical tools and concepts of plasticity theory are employed. In the A careful, practically relevant investigation of stability of the computations the Nikolaevskii model with a non-associated flow system’s relative equilibrium would have to take into account rule is used. In the course of ?plastic? deformation damages, are various perturbations such as the gravitational attraction of the accumulated. Macrocrack opening and material fracture are de- Moon, atmospheric drag and moving payloads up and down the scribed explicitly. For this purpose the node-splitting technique is cable. However, even the question of stability of a long unper- used and the free-surface conditions are given on newly formed turbed cable in the spherical symmetric Newtonian gravitational boundaries. This provides self-acting account of crack genera- field is still an open problem, because in [2] and [3] it is shown tion for the entire calculation area in the course of deformation. that a dumbell satellite, which is a system of two point masses The use of the considered approaches in describing the behavior connected by a massless rigid rod, possesses a stable radial rel- of a rock specimen as an ideal homogeneous medium allows a ative equilibrium position only if the distance between the two good agreement of fracture patterns as well as of various types masses is not too large. Here by large we understand a distance of stress-strain dependences with the experimental findings. In of the order of the radius of the orbit. order to describe in detail the behavior peculiarities of heteroge- Hence we will investigate whether and under which conditions neous porous and cracked materials, it is necessary to give due a continuous massive tapered string, necessary for the realization consideration for their structural features and crack growth. of the skyhook, has a stable radial relative equilibrium. Further Yu.P. Stefanov, Institute of Strength Physics and Materials Sci- we will study what is the influence of fixing the string’s end at

86 the surface of the Earth on the stability of the radial equilibrium length of the cracked body) are found. The asymptotic fields ob- position. tained allow to find and to study the geometry of the totally dam- REFERENCES aged zone where the damage parameter reaches its critical value [1] Artsutanov, Yu., Into Space on an Electric Locomotive, and all the stress tensor components are equal to zero. Komsomolskaya Pravda, 31 July 1960. Together with the eigenfunction method a finite difference method in a creepÐdamage coupled Mode III crack problem is [2] Beletsky, V. V., Levin, E. M., Dynamics of Space Tether applied. Systems, Advances of the Astronautical Sciences, 83, 1993. All the equations of the considered boundary value problem [3] Krupa, M., Steindl A., Troger, H., Stability of Relative are then formulated as the difference equations resulting in the Equilibria. Part II: Dumbell Satellites, Meccanica 35:353Ð linear algebraic system of equations. The left difference scheme 371, 2001. has been used for the approximation. The shape and the charac- [4] Pearson, J., The orbital tower: a spacecraft launcher using teristic length of the totally damaged zone are found as a part of the Earth’s rotational energy, Acta Astronautica, 2, 1975, the solution. It is shown that the geometry of the totally dam- 785Ð799. aged zone obtained by the finite difference method is in a good agreement with the one given by the asymptotic analysis. [5] Ruoff, R. S., Quian, D., Liu, W. K. ‘Mechanical properties REFERENCES of carbon nanotubes: Theoretical predictions and experi- [1] Zhao J., Zhang X. The Asymptotic Study of Fatigue Crack mental measurements’, C.R.Physique 4 (2003) 993Ð1008. Growth Based on Damage Mechanics. Engn Fracture Me- Hans Troger, Technische Universitat¬ Wien chanics. 1995. V. 50. P. 131-141. [2] Zhao J., Zhang X. On the Process Zone of a Quasi-Static Growing Tensile Crack With Power-Law Elastic-Plastic ON CREEP–DAMAGE COUPLED BOUNDARY VALUE Damage. Int J. of Fracture. 2001. V. 108. P. 383-395. PROBLEMS IN FRACTURE MECHANICS [3] Murakami S., Hirano T., Liu Y. Asymptotic Fields of Stress LARISA V. S TEPANOVA and Damage of a Mode I Creep Crack in Steady-State Growth. Int J. Solids Structures. 2000. V. 37. P. 6203-6220. [email protected] [4] Murakami S., Liu Y., Mizuno M. Computational Methods Many works have been done to analyze the near crack tip fields for Creep Fracture Analysis by Damage Mechanics. Com- coupled with elastic, elastic-plastic fatigue damage [1,2], creepÐ put Methods Appl Mech Engrg. 2000. V. 183. P. 15-33. damage [3,4] and elastic-plastic-viscoplastic damage [5]. Some [5] Lu M., Lee S.B., Kim J.Y. and Mai H.C. An asymptotic of the essential aspects of the considered set of problems and the analysis of a tensile crack in creeping solids coupled with results obtained can be highlighted. damage. Part I. Small damage region around the crack tip. 1. The damage gives significant influence on the stress and Int. J. Solids Structures. 1997. V. 34. P. 3163-3178. strain (strain rate) fields near the crack tip. [6] Riedel H. Fracture at High Temperature. Springer, Berlin. 2. The mathematical structure of governing equations is af- 1987. fected by the modelling of damage. 3. While the Hutchinson-Rice-Rosengren (HRR)-field of non- L.V. Stepanova, Samara State University, Akad. Pavlov, 1, linear fracture mechanics always shows the stress singularity at Samara, 443011, Russia the crack tip for any finite value of the stress exponent, the pre- ceding material damage in front of the crack tip decreases the singularity, and may give non-singular stress field. UNSTEADY BEHAVIOR OF AN ELASTIC BEAM 4. The totally damage and (or) active damage zone (process FLOATING ON THE SURFACE OF DEEP WATER zone) need be modelled in the crack tip region. IZOLDA V. S TUROVA In the present work the asymptotic stress, strain rate and continuity fields in the vicinity of the Mode I and Mode III crack in [email protected] damaged materials are obtained using the generalized self-similar variable initially proposed by Riedel [6]. The form of the simi- Results of studies of the hydroelastic behavior of floating bod- larity solution has been introduced by Riedel. However, a gen- ies (ice floes, breakwaters, and floating platforms) have broad ap- eral solution for the problem has not been constructed yet. The plications. Mat-type structures are usually considered as floating present contribution is an attempt to obtain the stress, strain rate airport. Typical dimensions necessary as a floating airport could and damage distribution near the crack tip as the dimensionless be 5 km long, 1 km wide, and only a few meters deep, so that the functions of the similarity variable under the assumption of the flexural rigidity is relatively small. Therefore elastic responses existence of a totally damaged zone near the crack tip. The shape are more important than the rigid-body motions. and the characteristic length of the totally damaged zone are not In this paper, a simplified model is used, in which an elongated known and should be found as a part of the solution. The forth Ð rectangular plate is replaced by an elastic finite beam. The fluid term asymptotic expansion of the continuity parameter and three is assumed to be incompressible, inviscid and its motion irrota- Ð term asymptotic expansion of the stresses for a Mode III crack tional. The depth of the fluid is infinite. The edges of the beam and the three Ð term asymptotic expansion of the continuity pa- are free. A combined motion of the elastic beam and the fluid is rameter and the two Ð term asymptotic expansion of the stresses considered within the linear theory. for a Mode I crack at large distances from the crack tip (at large The transient elastic deformations of a beam is computed by distances as compared with the crack length, the characteristic means of the time-domain differential equation method, with

87 elastic deflections expressed by a superposition of modal func- REFERENCES tions with time-dependent unknown amplitudes. The memory ef- [1] K. Visscher, M. J. Schnitzer, and S. M. Block, Nature, 400, fects in hydrodynamic forces are taken into account. This method 184 (1999). can be applied to any transient problem. The time histories of the [2] J. Christensen-Dalsgaard, Rev. Mod. Phys., 74, 1073 impacted force and the position and velocity of an airplane during (2002). landing are modelled. The behavior of the beam under different loads is investigated and it is shown that the bounded dimension [3] P. Congdon, ”Bayesian statistical modelling”, Wiley series of the elastic beam has substantial effect on its unsteady behavior. in probability and statistics, (Wiley, Chichester, 2001). This research work was supported financially by the grant for Y. Stylianos, D.G. Luchinsky, Newstead Mission Critical Tech- “Leading Scientific Schools” (Program No. SS-902.2003.1). nologies Inc., 9100 Wilshire Blvd. Suite 540 East Beverly Hills, I.V. Sturova, Lavrentyev Institute of Hydrodynamics, CA 90212-3437 pr.Lavrentyeva 15, 630090, Novosibirsk, Russia V.N. Smelyanskiy, NASA Ames Research Center, Mail Stop 269-2, Moffett Field, CA 94035, USA RECONSTRUCTION OF NONLINEAR DYNAMICS FROM NOISY INCOMPLETE MEASUREMENTS CONVECTIVE STABILITY IN HORIZONTAL LAYER YORGOS STYLIANOS DMITRY G. LUCHINSKY OF MICROPOLAR FLUID WITH FREE SURFACE VADIM N. SMELYANSKIY DENIS A. SUKHOV [email protected], [email protected], [email protected] [email protected] Models of of fluids taking into account microrotations and In this talk I will briefly describe ongoing activities at Mission couple stresses are known as micropolar fluids. These models Critical Technologies Inc. (USA) related to NASA missions. In was proposed in pioneer works by E. Aero [1] and K. Eringen [2]. particular, I will focus on a novel technique of inferring parame- Within framework of micropolar fluid model every particle have ters of nonlinear stochastic dynamical systems. a six degrees of freedom of rigid body and we can take into ac- Nonlinear stochastic dynamical models are widely used in count the rotational interaction of a particles. In micropolar fluid studying complex (natural and man-made) phenomena; examples are also exist couple stresses. In [3] the general constitutive equa- range from molecular motors [1] to coupled matterÐradiation sys- tions of viscoelastic micropolar fluid is proposed. For proposed tems in astrophysics [2]. The central problem of reconstructing model of viscoelastic micropolar fluid a property of so-called ori- (i.e. inferring) nonlinear stochastic dynamical models from noisy entational elasticity similar to liquid crystals may be described. measurements has accordingly received much attention in the sta- In this paper the infinitesimal instabilities of plane horizon- tistical physics community [3]. tal layer with free boundary of viscoelastic micropolar fluid un- We consider the following typical situation in applied stochas- der uniform heating are investigated. The recent investigations of tic nonlinear dynamics. We observe a L-dimensional time series convective instabilities of plane layers of viscoelastic micropolar ( ) of signals y t measured at each h seconds starting at some time fluid are presented in [4]Ð[6]. Now the influence of free bound- t0. Signal includes measurement errors. The actual state of the ary is investigated. For this problem the a-priory unknown free dynamical system is described by N-dimensional (N>>L) boundary should be determined alongside with fields of linear ( ) state vector x t , which follows stochastic dynamics. In a great and angular velocities. The linearized initial boundary problem ( ) number of important problems the vector field, f x , is also un- is deduced and its solutions are obtained. The neutral curves known. The question is: How and with what accuracy can we are presented. The viscoelasticproperties influence on the critical ( ) ( ) determine x t and f x ? We propose a novel solution of this values of Prandtl numbers and Grashof numbers are investigated. problem based on the application of the path-integral approach to Also the case Newtonian fluid with free boundary which prop- the full Bayesian inference. The key features of the suggested al- erties are determined by the surface energy dependent on curva- gorithm are the following: (i) a likelihood function written in the ture of a surface is considered. form of a path-integral over stochastic system trajectories, properly accounting for the Jacobian prefactor term that optimally REFERENCES compensates for dynamical noise; (ii) a parametrization of the [1] Aero E.L., Bulygin A.N., Kuvshinskiy E.V. Asymmetric hy- unknown vector field that reduces the problem of nonlinear dy- dromechanics (in Russian) // PMM. 1965. Vol. 29. No 2. namical inference to an essentially linear one. These features al- Pp. 297Ð308. lowed us to write an efficient Bayesian inference algorithm for [2] Eringen A.C. Theory of micropolar fluids // J. Math. Mech. nonlinear noise-driven dynamical models. 1966. Vol. 16. 1. P. 1Ð18. We analyze a number of examples and compare our results to [3] Eremeyev V.A., Zubov L.M. Theory of elastic and viscoelas- the results of earlier research to show that our method is faster, tic micropolar fluids // PMM. 1999. Vol. 63. No 5. Pp. 801Ð more accurate and is stable on a broad range of dynamical sys- 815. tems. Solution of the problem will open a new venue for extract- ing dynamical and statistical information from experimental ob- [4] Eremeyev V.A., Sukhov D.A. Convective instability of plane servations in physics, chemistry, biology, physiology, engineer- layer of viscoelastic micropolar fluid with free boundaries ing, econometrics. In particular, it can be applied to the monitor- (in Russian) //Izvestia Vuzov. Sev.-Kavk. Region. (Notices ing of the state of the space shuttle engine and the monitoring of of Universities of South Russia) Natural sci. 2003. No 4. cardiovascular responses of astronauts. Pp. 24Ð27.

88 [5] Eremeyev V.A., Sukhov D.A. Convective instabilities in under cyclic loading. It has been found experimentally that the plane layer of viscoelastic micropolar fluid (in Russian) reinforcement effect of elastomeric materials is basically speci- //Izvestia Vuzov. Sev.-Kavk. Region. (Notices of Universi- fied by the properties of the resulting elastomeric network. The ties of South Russia) Natural sci. 2004. Special issue. Math- proposed mathematical model is used to study the elastomeric ematics and Mechanics of Continuum Mechanics. Pp. 101Ð network with chains of different length. On loading the lengths 109. of active chain parts vary differently. The change in the length of [6] Eremeyev V.A., Sukhov D.A. On the convective instabilities the shorter chain parts starts well before the change of the longer of viscoelastic micropolar fluid// Proc. APM’2004. Saint- parts and is much more pronounced. Petersburg, 2004. P. 128Ð134. The elastomeric structure mostly depends on the fraction volume of the filler. In the presence of the filler, the active and pas- D.A. Sukhov, Rostov State University, Rostov-on-Don, Russia sive elastomeric chain parts of different length are formed in the material. On the filler particle surface there occurs a viscoelastic layer. As the material extends, the chain parts slip inside the layer. RESONANCE SIREN BASED ON FORWARD AND Calculations show that under deformation the length of short ac- BACKWARD RUNNING WAVES AND ITS MICROWAVE tive chain parts increases essentially in the spaces between neigh- ANALOGUE boring particles, whereas that of long chain parts remains almost unchanged. This results in a more uniform distribution of load- EUGENE D. SVIAGENINOV ing along the chains, the softening effect and the manifestation of [email protected] viscoelastic properties. Calculations have been performed under the assumption that the material volume is constant on micro and The main idea is to combine a radial siren as the most power- macrolevel. ful exciter of acoustical vibrations and resonance phenomena as Numerical results obtained for cyclic loading indicate that the the way to accumulate the vibration energy. The best composi- proposed approach can adequately describe the experiments with tion for this is surrounding a siren with a cylindrical cavity. But filled elastomers. Furthermore, the model can take account of the traditional siren generates axially symmetrical vibrations having recovery of the original properties of the material after long rest too high frequencies so the thickness of the relevant resonator or thermostatting. The material recovers its properties due to the must be too great. For decreasing excitation frequency of a siren ability of the network to regain its original state, which leads to a new type of rotor/stator perforation is proposed. As a result a reduction of the residual deformations prior to next loading. one-pole non- axially symmetrical wave can be easily produced. This work was supported by RFBR and Perm Regional Ad- This wave is running one instead of standing wave produced by ministration (grant 04-01-96058) and performed in the frame of the traditional siren. Moreover the direction of wave revolution the Program “Structural mechanics of materials and construction can be easily changed at the same angular frequency of the ro- elements. Interaction of nano-micro-meso- and macro scales at tor. Three main results are achieved with this approach. First, deformation and fracture” of the Power Engineering, Machine new siren by itself can be used as the angular view indicator in Building, Mechanics and Control Processes Department of RAS sonics. Secondly, it opens the way of usage resonance in siren de- and under the Integration Project of fundamental researches ful- vices. Thirdly, it allows to obtain resonant running waves in the filled in collaboration with scientists of UB RAS, SB RAS and both opposite directions with the same revolving shaft. The last is FE RAS. most important for mixing and pounding gas or fluid media. The Lyudmila A. Komar, Institute of Continuous Media Mechanics invention is intended for oil-gas-processing, chemical, medical, RAS, Akademic Koroleva, 1, Perm, 614013, Russia pharmaceutical, food and other branches of industry. Using physical analogy between mechanical and electrodynamical oscilla- ASYMMETRICAL OSCILLATIONS PECULIARITIES tions the similar microwave analogue is composed. It includes the REVIEW set of uniformly disposed at angular direction multivibrators surrounded with suitable cylindrical endovibrator. This device can EUGENE L. TARUNIN perform the same functions as magnetron (using in microwave oven, for instance) or other microwave generators but much eas- [email protected] ier, more economical and reliable as compared with them. E.D. Sviageninov, IPME RAS, Bolshoj, 61, St.Petersburg, Here we summarize results of analytical and numerical solu- 199178, Russia tions of movement a body caused by asymmetrical force [1-6]. Some of them have unfamiliar properties and that is why such results must be known engineers and scientists. Part of problems MODELING OF THE BEHAVIOR OF FILLED has relation to the devices with help of which some people wanted ELASTOMERIC MATERIALS WITH REGARD FOR to remove classical mechanics [3]. It is strange that there are his THE INFLUENCE OF A FILLER VOLUME FRACTION followers till today. Some problems have relation to the old classical problem formulated in 1886 by the Russian scientist N.E. A.L. SVISTKOV L.A. KOMAR Zhukovsky about disperse of grains that are placed on a moving [email protected], [email protected] plane surface [2]. Let us consider that movement of a plane surface in one direction may be described by the function: A structural model of the elastomeric network consisting of freely jointed chains is developed and presented to describe the R cos ω1t, 0 ≤ t ≤ T1 = π/ω1 x(t)= (1) behavior of the elastomeric material filled with small particles −R cos ω2 (t − T1) ,T1 ≤ t ≤ T

89 The asymmetry of the function (1) indebted to not equal times STABILITY AND DESTRUCTION OF THE MATERIAL of movements the surface forward and back. So we have a new = T −T1 WITH MICROSTRUCTURE parameter - ξ T1 . A body is connected with the surface by any friction force. In the case of a dry friction movement of the PAVEL V. T KACHEV body in the non inertial system can be known from solutions of the nonlinear equation: pavel [email protected] d2x dx A lot of facts give evidence about crucial role of the internal + b · sign = dt2 dt structure in the processes which attend deformation. That is why an influence of the material structure to its mechanical behavior a1 cos 0.5(1 + ξ)t, 0 ≤ t ≤ 2π/(1 + ξ)=T1 = requires a serious attention. Most of classic continuum methods − cos 0 5(1 + −1)( − ) ≤ ≤ 2 (2) a2 . ξ t T1 ,T1 t π have serious problems describing strong deformation and fracture, because the material is not really continuous in this case. The right part of (2) is the second derivative of (1). Equation is −1 The microstructure can be considered by the three ways. The written in a non dimensional form (R is a unit for a distance, ω first way is to describe microstructure and receive stability cri- is a unit for a time). Solutions of the equation (2) were discussed terion of crystal cell. The second way is to simulate defor- in [3-4]. A case of non linear viscosity force was considered in mation and fracture processes using molecular or particle dy- [5]. A case when the second part of (2) was placed by an elastic namic method. The third way is to consider the microstructure in force was investigated in [6]. Solutions of (2) were found by the classic continuum mechanic equations, in other words make analytical and numerical methods. transfer from the microstructure to the continuum mechanic. It was proved that when ξ =1(symmetrical oscillations) in a First the material is represented as an infinite crystal. Transfer settled periodical regime of movement a mean value of velocity from microstructure to continuum mechanics is made using long- of the body equals zero. For ξ =1.0 − 2.51 the body has two wave approximation. This allows obtaining nonlinear continuum intervals of the dry friction b with a positive mean velocity V in equations of the infinite crystal under finite uniform deformation. a settled periodical regime and a one interval with a negative V . These equations are found without limitation to the space dimen- Change ξ for ξ−1 changes the sign of V . There are different sion. For the stability verification a small deformation is super- types of movement in every interval. It was found values of main posed to a finite deformation of the crystal lattice being described distinctive points of the function - V (b, ξ). by the obtained nonlinear macroscopic equations. Criterions of It was found that in the case of a non linear viscosity force the material stability and their relevance to the crystal structure

[5] a mean velocity V in a settled periodical regime is always are obtained. negative when ξ>1 and its absolute value is very big for a small Second the spall fracture was simulated. The particle dynamic friction and a big deviation ξ of unit. method is convenient to analyze kinetic processes in the fracture A resonance curve for asymmetrical force [6] has several un- zone. This analysis is very difficult to perform using continuum usual features. Main of them is a reduction of an amplitude in the methods. In the presentation a 2D problem of a plate impact in- =1 place where the maximum must be for ξ . teraction is solved using the particle dynamic method. An ideal REFERENCES single-crystal particle packing with Lennard-Jones pair potential [1] Blekhman I.I. Vibration mechanics. - Moscow: Nayka, is used. Dependence of the spall velocity on the system geom- 1994, 398 p. etry is investigated. Relation between the spall velocity and the dissociation velocity is found. Influence of the crystal lattice ori- [2] Tarunin E.L. Versions of N.E. Zhukovsky’s problem about entation on the fracture scenario is investigated. disperse // Problems of mechanics and control (Non linear Thirdly comparison of values of critical distances has been dynamic systems).-Perm: Perm University, is.34, 2002, p. made. 101-111 This work was supported by Russian Foundation for Basic Re- [3] Tarunin E.L. Influence of a friction on moving of inerthoid search, grant No 02-01-00514. The author is grateful to A. M. // Problems of mechanics and control (Non linear dynamic Krivtsov for the statement of the problem and D. A. Indeitsev, P. systems).-Perm: Perm University, 1996, p.195-223 A. Zhilin for the discussions. P.V. Tkachev, Institute for Problems in Mechanical Engineering [4] Tarunin E.L. Peculiarities of asymmetric oscillations with a Russian Academy of Sciences, Molecular and Particle Dynamics dry frictions // Problems of mechanics and control (Non lin- Simulation Group, St. Petersburg, Russia ear dynamic systems).-Perm: Perm University, iss.33, 2003, St. Petersburg State Polytechnical University, Dep. of Theoretical p.167-183 Mechanics, St. Petersburg, Russia [5] Tarunin E.L. Peculiarities of asymmetric oscillations with a nonlinear viscous friction // Bulletin of Perm Polytechnic University Applied mathematics and mechanics, 2003, p.9- ELECTROMAGNETIC METHOD OF 15 VIBRODIAGNOSTICS OF COMPOSITE BEAM INTERNAL STATE [6] Tarunin E.L. Resonance in a case of asymmetrical force // Problems of mechanics and control (Non linear dynamic ALEXANDER K. TOMILIN systems).-Perm: Perm University, iss.36, 2004, p.180-190 [email protected] E.L. Tarunin, Perm State University, Bukireva, 15, Perm, 614990, Russia The diagnostics of vibrating constructions internal state is an

90 important issue. It is common knowledge that at the initial state length and amplitude, the weight of the upper body, the founda- of material destruction, some micro-cracks appear, which are in- tion parameters, the friction value and so on. visible. One of the possible nondestructive methods for control Supported by RFBR, grant 04.01.00327. of material internal state may become an electromagnetic vibro- P.E. Tovstik, St.Petersburg State University, St.Petersburg, Russia diagnostics, which theoretical bases are described in the present paper. I have considered a composite beam, consisting of several ON THE INFLUENCE OF FRICTION ON THE CELT longitudinal layers with different characteristics. One of the lay- RATTLEBACK MOTION ers is electrically conducting, its ends are closed-loop in electric circuit, which contains frequency analyzer (FA). A homogeneous TATIANA P. TOVSTIK stationary magnetic field operates in the active section; its induc- tion is directed transversely to plane of beam vibration. On basis [email protected] of beam vibration differential equations analysis, the formulae have been deduced, which determine partial damped frequencies The Celt rattleback motion is one of the classical examples of dependency on stated coefficient of internal resistance. Forma- the nonlinear motions of rigid bodies [1]. It is assumed that the tion of micro-cracks in any beam layer reflects inevitably on the body moves on the horizontal plane without jumping and sliding. value of general internal resistance coefficient. In turn, it leads In the previous works [2,3] the body motion on the rough hori- to the change of partial damped frequencies values. Thus, it is zontal plane was examined. The system is conservative and the reflected on the frequency analyzer readings. body spins in one direction and then reverses during the infinite time. In this work the friction influence is investigated. 1. Babakov I.M. Theory of vibrations.-M.: Nauka, 1968. 560 The body has the three-axes ellipsoid of inertia, one of the axes p. in the equilibrium state is vertical, and the directions of two other 2. Biderman V.L. Theory of mechanical vibrations. axes do not coincide with the directions of the principal curva- M.:Vysshaya shkola. 1980. 408 p. tures of the body surface at the tangent point. The various kinds 3. Tomilin A.K. Vibration of electromechanical systems with of resistance are introduced in the examined system of equations, distributed constants. Ust-Kamenogorsk, 2004.-272 p. namely the roll friction, the rotating friction, and the viscous fric- 4. Vasilenko N.V. Theory of vibrations. Kiev: Naukova tion on the air. dumka. 1992. 430 p. The motion near the equilibrium state is investigated ana- A.K. Tomilin, East Kazakhstan State Technical University named lytically and numerically. This nonlinear system contains two after D. Serikbaeb, 12 Burova Street, apt.45, Ust-Kamenogorsk, quickly changing vibrations p1(t)=p10(t)sin(ν1t+α1(t)) and Kazakhstan, 492000 p2(t)=p20(t)sin(ν2t + α2(t)) and one slow motion Ω3(t). In the case without friction the analytical investigation was ful- ON THE MARINER FIXED OFFSHORE PLATFORM filled by Markeev [1]. After averaging the quick vibrations with =2 =2 DYNAMICS UNDER RANDOM WAVE FORCES periods T1 π/ν1 and T2 π/ν2 he find the periodic trajectory of amplitudes in the three dimensional phase space PETR E. TOVSTIK TATIANA M. TOVSTIK {p10,p20, Ω3}. This trajectory lies on the surface of an ellipsoid. VJACHESLAV A. SHEKHOVTSOV We find the large period T of this trajectory (T {T1,T2}). In the presence of friction we investigate the evolution of the trajec- [email protected] tory {p10,p20, Ω3} by averaging again the nonlinear system with the period T . The dependence of this evolution of the kinds of The simplest mathematical model of the mariner fixed offshore friction is investigated and it is compered with numerical results. deepwater platform is presented. The model consists of the upper REFERENCES rigid body, of the supporting mainframe, and of the foundation. [1] Markeev A.P. Dynamics of the body contacting with the Firstly the real platform is replaced by the equivalent beam and rigid surface.Ð Moscow, Nauka, 1992., 336 p. (in Russian) then by the system with one degree of freedom. The non-linear forced vibrations under action of the wave ex- [2] Tovstik T.P. Dynamics of the Kelt rattleback // XXX Sum- citation are investigated. It is supposed that the waves are planar mer School. Advanced Problems in Mechanics. APM 2002, and they have a small amplitude. The harmonic waves and the St.Petersburg. stationary random waves are studied. The generalized wave force [3] Tovstik T.P. Analytical research of the Celt stone dynamics consists of two parts depending on the relative water acceleration // XXXI Summer School. Advanced Problems in Mechan- and velocity respectively. The main non-linearity of the system ics. APM 2003, St.Petersburg. consists on the velocity part of the wave excitation and on the T.P. Tovstik, Institute of Problems of Mechanical Engineering visco elastic ground. The geometric non-linearity is inessential Russian Academy of Sciences, Bolshoi Prospect 61, RUS-199178, because the vibration amplitudes are comparatively small. St.Petersburg, Russia The main aim of discussion is to compare the numerical solution and the approximate analytical solution. For the analytical solution the averaging method is used. For the numerical solution TRANSFORMATION MATRIX OF VIBRATION WAVES the canonical expansion of stationary processes is used to simu- ON JUNCTION OF ROUND CYLINDRICAL RODS late the random wave excitation for waves with the given spectral NATALYA A. TSIKOVA density. Under these conditions by numerical experiments the upper [email protected] body amplitudes are found in dependence on the waves and on the construction parameters. Among them there are the waves In view of calculation of low-frequency vibration waves in

91 ship’s constructions and engineering structures, the dynamic induced by a non-zero rotation of the cavity. The analysis is moti- equations of thin plates and rods are used instead equations of vated by attempts to use ring-lasers to measure terrestrial gravito- space elasticity theory. magnetism or the Lense-Thirring effect produced by the rotation Dynamic equations of thin plates and rods contain formulae of the Earth. for finding of wave field. When all main types of vibration are Robin Tucker, Lancaster University, Lancaster, LA1 4YB, United accounted, these equations may be rather complicated. Besides, Kingdom various wave modes in rods generate different energy flows at equal amplitudes. Consequently, probability of error is very high. Therefore some questions appear about the choice of rational INFLUENCE OF THE MOLECULAR wave transmission calculation method. CHARACTERISTICS OF MATTER ON DETONATION As a calculation method it is proposed to use the matrix of PHENOMENA vibration waves transformation on junction of round cylindrical rods. In that paper the matrix finding algorithm, general proper- ANDREY V. U TKIN IGOR F. GOLOVNEV ties of that matrix and control results organization are described. VASILY M. FOMIN N.A. Tsikova, Saint-Petersburg Marine Technical University, St. Petersburg, Russia

Investigation into the influence of the thermal effect of chem- NONLINEAR INTERNAL WAVES GENERATED BY ical reaction, Q, on the physical characteristics of detonation TOPOGRAPHY IN A STRATIFIED FLUID wave, and also the establishment of general regularities in variation of Q in a broad range of values are of considerable interest. HIDEKAZU TSUJI MASAYUKI OIKAWA This matter is hard to examine experimentally because differ- [email protected] ent Q refer to different explosives (with different densities, different chemical composition, etc.). For this reason, the depen- Internal waves which propagate in stratified fluid have been dence of detonation-wave characteristics on the thermal effect of studied for many researchers mainly to explain various phenom- chemical reaction cannot be measured directly in an actual ex- ena observed in the atmosphere and ocean. One of the important periment. At the same time, the molecular-dynamics method al- topic is about the generation of the waves due to localized distur- lows easy modification of reaction exothermicity by varying the bance (ex. topography). In this study, for simplicity, we consider shape of the potential surface. Since such a modification leaves long waves in the two-layer fluid which includes a layer which all other properties of the system unchanged, there arises a pos- has the vertical scale larger than the horizontal scale. sibility to establish a unique relation between the thermal effect Linear analysis fails when the phase speed of the wave almost of the chemical reaction and the structure of the detonation wave equals to the flow speed. In this case weakly nonlinear analysis that consists of the shock front and the reacting zone, whose rear shows that the system is described by a governing equation — boundary is defined by the Chapman-Jouguet point. forced Benjamin-Ono(fBO) equation. It takes the forcing effect As the physical system, we consider a three-dimensional hy- due to topography in the BO equation. pothetical molecular crystal in which exothermic reactions are Recently Choi and Camassa(1996) proposed the reduced equa- possible. The reaction was initiated by an impulse of an external tions for the propagation of the waves without any assumption force applied to the left side of the crystal and acting on it during a about the amplitude. In this study the equations which describe certain time interval. In the numerical experiment, the crystal was both the generation and propagation of the waves are derived. placed in an adiabatic shell that restricted the dispersion of deto- By solving the equations numerically, we investigate the na- nation products in the plane normal to the propagation direction ture of the generation and propagation of the full nonlinear waves of the detonation wave. A detailed description of the intermolec- in this system. The results are compared not only to the fBO ular potential and mathematical model was reported elsewhere equation but also to the hydrostatic equations which describe the [1]. full nonlinear waves in a single-layer system. In the present study, we examine how the thermal effect of H. Tsuji, Research Institute for Applied Mechanics, Kyushu Univ., chemical reaction affects processes in detonating molecular crys- 6-1 Kasuga-Koen, Kasuga, Fukuoka JAPAN tal (traveling velocity and structure of the detonation wave, delay time of the chemical reaction, and energy macrocharacteristics of the crystal in the disturbed region). A comparison of molecular- TWISTED RING LASER ACCELEROMETERS dynamic data with predictions of the continual theory of detonation is given, including check of fulfillment of the Chapman- R. W. TUCKER D. BURTON Jouguet condition. [email protected] REFERENCES A new approximation scheme, designed to solve the covariant [1] V.M. Fomin, I.F. Golovnev, and A.V. Utkin, Relation Maxwell equations inside a rotating hollow slender conducting between the atomistic picture and continuum mechanics cavity (modelling a ring-laser), is constructed. It is shown that description of detonating solid-state explosives // Shock for well-defined conditions there exist TE and TM modes with Waves. - 2003. V.13. N.2. - P. 155-165 respect to the longitudinal axis of the cavity. A twisted mode spectrum is found to depend on the integrated Frenet torsion of A.V. Utkin, Institute of Theoretical and Applied Mechanics SB the cavity and this in turn may affect the Sagnac beat frequency RAS, 4/1 Institutskaya str., Novosibirsk, 630090, Russia

92 the chute inclination and the flow rate, and correlated these fea- NUMERICAL SIMULATIONS OF HEAT PROPAGATION tures with the efficiency of the mixing process. FROM DEEPEN HEAT SOURCES UNDER NON-PLANE A. Valance, Groupe Matiere Condensee et Materiaux, CNRS, SURFACE WITH USING PARALLEL COMPUTERS Universite de Rennes 1, Bat 11A, Campus Beaulieu, Rennes, 35042, France NATALIA A. VAGANOVA MIKHAIL YU.FILIMONOV [email protected] ON TIME DOMAIN IDENTIFICATION METHOD FOR A problem of constructing thermal fields from a deepen heat LINEAR OSCILLATION MECHANICAL SYSTEMS source (for example from a pipeline) in heterogeneous media with C. VALLEE« S. CHARLES S.YA.STEPANOV accounting solar radiation on earth surface is considered. In con- A.A. BUROV trast to [1] surface is not suggested to be plane. A channel is included into the mathematical model. In addition the angle of [email protected] solar radiation is taken into account. It leads to nonlinear boundary conditions for boundary-initial problem for linear termocon- This contribution relates to the general linear mechanical oscil- ductivity equation in complex three-dimensional domain. Paral- lation models. Initially it was inspired by the problem of dynam- lel multiprocessor using allowed to obtain detailed thermal fields ical behavior of a high-speed rotor in the annular seals. There on small-scales grids for described phenomena. is no contact between the stator and the shaft, so leakage is in- The work is supported by Russian Foundation for Basic Re- evitable and its occurrence is strongly influenced by the position search (04-07-90120) and a program of integrative projects of of the rotor in the stator. The hydrodynamic forces generated by UrB RAS, SO RAS and DVO RAS. a pressure distribution are the nonlinear functions of positions, REFERENCES velocities and accelerations of the shafts. The dynamic response [1] N.A.Vaganova. Simulation of thermal fields of a deepen of the seal can be linearised in the vicinity of a steady rotation isolated pipeline with accounting radiation. Proc. of APM- of the rotor. The coefficients matrices M of added masses, C of 2003 damping and K of stiffness are to be obtained. One of the typical identification methods is based on the least-square method in the N.A. Vaganova, Institute of Mathematics and Mechanics UrB frequency domain and on the Fourier transformation. Another RAS, 16, S.Kovalevskaya str., Ekaterinburg, 620219, Russia approach, the so called time domain method, becomes more and more useful nowadays. This is because the new computer types of test apparatus appear, they are able to measure and record the SHEAR INSTABILITIES IN GRANULAR MIXTURES system coordinates and external forces as functions of the con- UNDER GRAVITY tinuous time. Due to this instruments the evaluation of the co- ALEXANDRE VALANCE CAROLINE PASCOT efficients M, C and K by means of the least-square in the time RENAUD DELANNAY domain becomes more simple, more easy and more precise in contrast to the frequency domain method. [email protected] The linear identification system of the algebraic equations for specifying the unknown elements of matrices M, C and K is Many geophysical processes, such as rock avalanches, debris evaluated in the special tensor form. In some special cases it is flows, erosion, transport and deposition of sediments, involve possible to evaluate the determinant of matrix G of this system granular matter. In natural granular flows, grain size dispersion is in the analytical way. It gives the rule for the appropriate choice important and mixing or demixing phenomena can occur during of the parameters of the identification method. The experimental the flow and the deposition process. and numerical tests of this method were carried out and demon- The physical mechanisms leading to segregation or mixing are strated the very good accuracy even in the presence of the high still poorly understood. We investigate them through a simple noise. The method is adapted to any scientific experiments char- experiment of granular chute flows down inclines. We let two acterized by the various types of linear evolutionary equations. populations of grains of different sizes flow side by side down Sergey Yakovlevitch Stepanov, Computing center of RAS, in a rough inclined chute. The grains of each stream are colored. Moscow, Russia Our set-up allows to control the inclination angle of the chute and the flow rate of each stream of grains. Our purpose is to characterize the mixing process occurring SIMULATION OF THE PROTOPLANETS SYSTEM at the interface between the two streams of grains. A simple GROWTH way to do this is to analyze the deposit on the chute once the SERGEY VASILYEV ANTON M. KRIVTSOV flow has stopped. Indeed, the region where mixing has operated is characterized by a much thinner deposit. We show that the [email protected] mixing region (which is located close to the interface) increases linearly in size with the downstream distance. Moreover, our ex- The numerical modelling of the protoplanets system develop- periment results suggest that the destabilization of the interface ment in frames of one of the cosmogonical hypothesis is pre- can enhance the efficiency of the mixing process. We observe sented. The presented model is based on the idea of initial exis- that the interface destabilizes when increasing the rate of shear tence of two rotating bodies (protoplanets) surrounded by a dust between the two streams. This instability is the analog of the cloud. The dust cloud feeds the system of growing protoplanets. Kelvin-Helmholtz instability in fluids. We have analyzed the am- In frames of our simulation, it was obtained that the relative plitude and wavelength of the interfacial waves as a function of growth of the protoplanets may be developed in different ways

93 according to initial body’s mass ratio. It is shown that there is of locations of displacements on crack faces is determined. Ex- such initial ratio that provides a stationary growth, i.e. protoplan- plicit formulae are deduced for normal and shearing stresses out ets don’t grow relatively. of the system of cracks on the line of their disposition. These for- Application of multi-processor computation systems allowed mulae immediately determine these stresses on the basis of the obtaining more smooth functional dependence of growth and to solution of the defining SIE. realize the analysis of the convergence by different parameters of The defining SIE of the problem in question is efficiently the problem. solved through well-known numerical-analytical methods of the S. Vasilyev, St. Petersburg State Polytechnical University, Politec- SIE solution that are based on the quadrature formulae of Gauss nicheskaya st., 29, St. Petersburg, 195251, Russia with Chebyshev and other knots. The patterns of determining basic mechanic and thermal values are studied on quite a large range of typical parameters. MATHEMATICAL MODEL OF DYNAMICS TWO-MASS Sergey Verlinsky, State Engineering University of Armenia, 105 VIBRATION MACHINE Teryan str., 375009, Yerevan, Republic of Armenia

GALINA V. VASILYEVA SERGEY A. RUMYANTSEV THE LAGRANGIAN MECHANICS OF ELASTIC [email protected], SHELLS AND ITS NUMERICAL APPLICATION [email protected] YURY M. VETYUKOV VLADIMIR V. E LISEEV In the message the mathematical model of the two-mass vibrational transport machine with self-synchronized unbalanced mass [email protected] drive, allowing to describe not only the steady-state movements, but also transient dynamic processes, which are accompany the Thin shells find usage in broad range of technical applications, start of machine from the rest and impact loads is represented. which leads to the need of reliable methods of computation of The system of the differential equations of movement of the their deformed state. Questions of stability and supercritical be- two-mass vibrational transport machine with independently ro- havior of shells require solution of geometrically nonlinear prob- tating vibration exciters allowing to describe non-stationary dy- lems and are of high theoretical and practical interest [1]. namics enough a wide class of machines with various schemes of Most of commercial software allowing for finite element anal- a suspension and interaction of moving parts is considered. ysis of mentioned problems utilizes numerical schemes of shell Results of numerical modelling of transient and steady-state analysis, which are based upon the general 3-dimensional the- dynamics of the two-mass machine are represented. ory of elasticity. Initially an approximation of displacements and deformations is considered, which includes changes of the val- The information is submitted in graphic form which allows to ues with respect to the thickness (3rd coordinate). The governing analyze transient dynamic processes. equations of the shell element are developed by means of more G.V. Vasilyeva, Ural State University of Railway Transport, Kol- or less strict mathematical and logical constructs, allowing one to mogorov st., 66, Ekaterinburg, 620034, Russia get rid of the 3rd coordinate using additional kinematical characteristics. This way leads to applicable models; however their trustworthiness can be doubted due to many approximate equal- STRESS-DEFORMATION TENSION OF PIECE-WISE ities and assumptions accepted during the development of the HOMOGENEOUS ELASTIC HALF-PLANE WITH model (see e.g. [2]). Although there exist more correct models, SYSTEM OF COLLINEAR CRACKS OR RIGID which are based upon the increased order of interpolation with INSERTION ALONG WITH TEMPERATURE IMPACT respect to the thickness of the shell (p-version of finite element method, see [3]), the equivalence of asymptotic behavior of the SERGEY V. V ERLINSKY finite element model and in the original 3-dimensional formula- [email protected], [email protected] tion with the thickness tending to zero is still an open question. Probably the problem of qualitative difference between the shell Within the framework of the plane theory of elasticity, the construction and the volumetric construction must be solved not problem of stress-deformation state of piece-wise elastic half- on the level of numerical schemes, but on the level of reliable plane when the collinear system of cracks or absolutely rigid original equations of mechanics of shells. insertions are on the line of connection heterogeneous materials In the present work a modern formulation of the Kirchhoff at permanent temperature field is investigated. Also considering shell model is considered. Such a shell consists of material nor- the problem when elastic piece-wise half-plane consists of up- mals: each point of a shell possesses three translational and two per strip and lower half-plane with different elastic and thermal rotational degrees of freedom [4]. The tensors serving as mea- constants. In advance, corresponding problems of the heat con- sures of bending deformation and plane deformation, as well as duction theory are solved. As a result, thermal functions are de- the corresponding force factors, are defined by means of the vir- termined at permanent thermal regime, characterizing the values tual work principle. The Kirchhoff hypothesis of orthogonality of temperature in the points of strip and half-plane. It is required of the deformed normal to the deformed surface is used in the to determine the cracks opening, normal and shearing stresses out variation procedure. The resulting equations have quite a simple of the system of cracks and stress intensity factors. form both for the case of linear deformation and for geometrically With the help of integral transformation of Fourier, the deter- nonlinear case. mining system of singular integral equation (SIE) of the set prob- Expressing the strain energy of the shell via the introduced lem is deduced, from which the complex combination of densities deformation tensors, we come up with the problem of minimiza-

94 tion, which is convenient for numerical analysis. A kind of Ritz Mikhail M. Vetyukov, St. Petersburg State Mining Institute, Vasi- method is considered, when the displacements are approximated lyevskiy ostrov, 21 liniya, 2, St. Petersburg, 199026, Russia by a set of non-finite functions, which are not equivalent to zero anywhere on the shell surface. A finite element software com- PARALLEL IMPLEMENTATION OF BURNS–HUT plex is developed, which is capable of analyzing the deforma- ALGORITHM FOR SIMULATION OF PLANET SYSTEM tion of shells of arbitrary form with various boundary conditions FORMATION and loadings conditions. Problems have been solved concerning loss of stability and supercritical behavior in axisymmetrical and ILYA B. VOLKOVETS ALEXANDER A. LE-ZAKHAROV other constructions. The comparison of the results with analytical solution of stability problems and the data, obtained with other software allows one to estimate the advantages of the developed numerical scheme for modeling the deformation of elastic shells. One of the methods for modeling of space objects formation is to compute dynamics of interacting particles. Since the grav- REFERENCES itational interaction is long-range we need to calculate force be- [1] Yakushev V. L. Nonlinear deformations and stability of 2 tween all particles in the considered system. So, we have O(n ) elastic shells. M., Nauka, 2004 (in Russian). computational complexity for every integration step, where n is [2] ABAQUS Theory Manual, http://www.abaqus.com. the total number of particles. However, there are some algorithms [3] Duester A. High order finite elements for three- of approximate calculation that allow avoiding direct force com- dimensional, thin-walled nonlinear continua. Aachen: putation and consequently cutting down the calculation complex- Shaker Verlag, 2002. ity. The presented poster is devoted to one of such algorithms — the BurnsÐHut algorithm, which is based on hierarchical cells [4] Eliseev V. V. Mechanics of elastic bodies. St.-Petersburg, space decomposing. For forces calculation the particles are com- St.-Petersburg State Polytechnical University publishing, bined in groups for each cell. Then the forces between cells and 2003 (in Russian). particles can be calculated instead of all forces between all particles in the system. To use this algorithm for parallel computation Yu.M. Vetyukov, St.Petersburg State Polytechnical University, ul. some modifications are required. Each processor calculates par- Polytechnicheskaya, 29, St. Petersburg, 195251, Russia ticles motion in some specified part of the computational aria. To share this region between all processors we need to take one of THE SIMULATION OF PERIODIC PROCESSES IN the levels in hierarchical space fragmentation and distribute cells VIBRATORY JAW CRUSHER ALLOWING FOR of this level between the processors. Type of the distribution de- INFLUENCE OF CRUSHED MATERIAL pends on the shape of the modeling configuration because it has a strong influence on the processors load balance. To cut down M.M. VETYUKOV R.F. NAGAEV computational complexity, processors exchange not the particles M.M. UTIMISHEV coordinates but some parameters that describe the cells. Appli- cation of the computational algorithms to the problem of planet [email protected] formation is considered. In particular, problem of EarthÐMoon formation as a result of rotational collapse of a dust cloud is be- A crush machine which consists of two same symmetrical mo- ing modeled. bile jaws oscillating simultaneously one against another in a sur- A.A. Le-Zakharov, St.Petersburg State Polytechnical University, face belongs to stereomechanical type. In such dynamical models Russia it is always supposed that jaws impact one each other directly, but the presence of material is ignored. According to it, the motion of material in working camera was not researched, and its influence SUBHARMONIC RESONANCE ARISING DUE TO THE on the movement of crushing jaws was left out of account. In CHANGE OF EFFECTIVE DISSIPATIVE FORCES WITH considered problem the joint motion of jaws and crushed mate- ACTION OF POLYHARMONIC EXCITATION rial is investigated. The vibrodisplacement of material in working IOSIF VULFSON camera of crush machine is described basing upon the differential relation, which connects time moments of falling of the material [email protected] and destruction of it in the given level. For the functions, which describe the running wave of finishing the destruction of material, In essence, all resonant phenomena can be considered as the beginning of its fall and the wave of beginning of the destruction vibrations supported by forces, that compensate the negative or clamp wave, the conditions are given, which permit the im- work of resistance. Such view of physical nature of a resonance possibility of undesirable accumulation of material in working enables to receive the effective analytical dependences, allowing camera. The formula, which defines the total time of passage of to estimate a peak of researched modes and conditions of their material throw machine, is obtained when forms of jaws surface existence. The special class of the problems connected to real- and law of their periodical vibrations are known. Dynamical anal- ization of the given approach is devoted to research of the action ysis shows that among present different stability periodic modes of the additional excitation on the change of effective dissipation of motion the most essential one is that involves intervals of sep- forces [1, 2, 3]. In the present paper it is established that the ad- arate motion of jaws and their standstill in one oscillation period. ditional high-frequency or low-frequency excitation exerts essen- In order to simulate the motion, the optimal profile of jaws is cal- tial effect on the level of dissipation, determining the conditions culated, which ensures the minimum value of the total time of of occurrence of subharmonic resonance’s. As applied to analyt- material stay in the working camera. ical investigation of this problem the efficiency of transformation

95 the initial differential equation to the modified type is established. to be taken into account. At big values of this coefficient of fric- As this takes place the additional excitation is replaced with en- tion the upper part of the tube is blocked and the force from the ergetically equivalent correction of dissipative forces. lower to the upper end is not transmitted. It was established that the effective dissipative force decreased This work was supported by Russian Foundation for Basic Re- with taking into account the action of additional excitation. Fore search under grant 04-01-96048, regional programm “Ural”. some characteristics of hysteresis loops the analytical solution are I.N. Wasserman, Institute of Continuous Media Mechanics RAS, received, allowing to predict the excitation of subharmonic reso- Acad.Korolev Str 1, Perm, 614013, Russia nance’s. The results of computer simulation are in good agreement with the results of analytical investigation. OPTIMAL DESIGN OF NOTCHED COMPONENTS REFERENCES AGAINST FRACTURE AND FATIGUE FAILURE [1] I.I. Vulfson. Determining the reduced parameters of dissipation with biharmonical vibrations (in Russian). Vibrotech- B. WILCZYNSKI« Z.MROZ« nica, 1968. N4(9). [email protected], [email protected] [2] I.I. Vulfson. Influence of the high frequency actions on the conditions of parametrical excitation of mechanisms. (in Mechanical failures have caused many injuries and much fi- Russian). Mashinovedenie, 1977. N5. nancial cost. There are many mechanical failure modes of metals. Excessive deformation and yielding are probably the most com- [3] I.I. Vulfson, M.N. Vulfson. Refined equivalent linearisation monly studied failure modes. The mechanical components gener- of the positional dissipative forces with polyharmonical vi- ally work under time variable oscillation and cyclic stresses. The bration. Journal of Machinery Manufacture and Reliability. fracture and fatigue phenomena are created by theses stresses and 2004, N4. are observed in geometrical irregularities, usually called notches. I. Vulfson, St.-Petersburg State University of Technology and De- To predict the fatigue life of a notched component, both crack sign, Bolshaya Morskaya St., 18. St.-Petersburg, 191186, Russia initiation stage (life to the formation of a crack of length of 1 mm) and crack propagation stage (life from the existing crack to fracture) are needed. The elastic stress concentration factor and INFLUENCE OF COULOMB FRICTION ON elastic-plastic notch root strain are often required for crack ini- OSCILLATIONS OF SUCKER ROD AND TUBING tiation analysis. On other hand, in the prediction of notch crack STRINGS IN OIL WELLS growth, the stress intensity factor plays very important role. The crack growth per cycle (Paris’s law) is proportional to the range IGOR N. WASSERMAN of the stress intensity factor. There are different ways to decrease the stress intensity factor. Varying the shape of a boundary of [email protected] the machine component, introducing stiffeners into the component, introducing the protective patches covering the crack area The considered system is a mechanical model of oil well and stop holes concept are widely discussed in literature. Gen- sucker rod pumping plant with the pump connected with the erally the value of the fatigue life (a number cycles to failure of lower end of the tubing string. It represents a heavy elastic multi- the notched component) depends on several factors like geome- diameter rod (sucker rod spring), located in an elastic pipe (tubing try of the component, material properties, history of the response string), which, in turn, is located in spatially curved channel (cas- of the structure to external loading, residual stress and boundary ing). The upper end of a rod performs periodic motion according conditions. to the specified law,and the lower end is subjected to the force, Optimization of notched components against fracture and fa- which depends on the direction of its movement relative to the tigue failure is discussed in this paper. The paper deals with two lower end of the pipe. The upper end of the pipe is fixed, and classes of problems. The first one is the stress intensity factor the lower end is subjected to the force opposite to that, acting on minimization that increases the fracture strength, and the second the lower end of the sucker rod string. As a result of interaction one is the fatigue life time maximization that increases the fatigue of the rod with walls of the tube the force of Coulomb friction strength. It can be assumed that cracks are to be present at some directed against their relative movement arises. Similar force of critical points around the boundary or inside the body. Hence, friction occurs between the tubing string and casing. The friction in general, the stress intensity factor minimization task can be coefficients for these two type of Coulomb friction forces may be treated as a minimization of maximum effective stress intensity different. factor or a mode I crack (if it is dominating) subject to some be- Using the developed model the analysis of influence of the co- havioral constraints (on displacements, stress concentration fac- efficientss of friction between a sucker rod and tubing stings (f1) tor, etc.). In this paper the attention is paid to the stress intensity and between tubing string and casing (f2) on dynamic behaviour factor reduction: (1) by changing the shape of the component of the sucker rod string in the near-real conditions is carried out. contour, (2) by introducing into the body stiffeners to stop the The first of these two coefficients (f1) exerts a significant influ- crack (stiffening of machine part), (3) by introducing the defense ence on the vibrations of both the sucker rod and tubing strings: hole system (weakening of structural component), and (4) the ap- at their upper ends the amplitudes of longitudinal forces consid- plication of stop holes concept at the crack tip to induce crack erably increase. arresting. The second class of problems is: for given bound- Friction between the tubing string and the casing (f2) has a ary conditions, external loading and material properties find such weak influence on the sucker rod string behavior. However, when shape of a notched part for which the minimal number of cycles we study dynamic behaviour of the tubing string, this factor needs to failure reaches its maximum value. The first class of problems

96 to discuss is a well known min-max approach, and the second is of boundary value problems of fracture mechanics for perforated max-min one. Using the bound formulation both problems are plates. converted to the simple min or max problems, respectively. The work is supported by Russian Fund of Basic Research To solve so stated optimization problems we use the Sequen- (grants RFBR-Urals 04-01-97503 and 04-01-96067). tial Linear Programming method with ’move limits’ enhanced by the CAGD method (Bezier curves, superellipses) for shape deﬁni- Valeriy E. Wildemann, Perm State Technical University, 29a, tion of components to be optimized and the numerical methods of Komsomolsky Ave., Perm, 614600, Russia analysis, the FEM and the BEM. A ’quarter-point’ FEM or BEM elements are used to compute the stress intensity factors and the BEM method coupled with the Mrz fatigue plasticity model or Neuber type rules (uniaxial and multiaxial) to compute the actual NUMERICAL SIMULATION OF strain-stress ﬁeld at the notch tip. SELF-SYNCHRONIZATION PHENOMENON OF B. Wilczynski,« Mechanical Department, Technical University of MECHANICAL VIBRO-EXCITERS Koszalin, Raclawicka 15/17, 75-620 Koszalin, Poland Z. Mroz,« Institute of Fundamental Technological Research, Polish NIKOLAY P. YAROSHEVICH Academy of Science, Swietokrzyska 21, 00-049 Warsaw, Poland

ELEMENTS OF STRENGTH ANALYSIS OF A problem of self-synchronization of unbalanced vibro- DEFORMABLE SYSTEMS ON THE BASIS OF FAILURE exciters located on a rigid platform with one degree of freedom KINETICS DESCRIPTION is considered.

V.E. WILDEMANN M.V. POLYAKOV Numerical integration of nonlinear differential equations was A.M. TSIPLYAKOV carried out by means of standard package of computer programs Maple V. [email protected] Test simulation of the problem on self-synchronization of two identical unbalanced vibro-exciters with positive and almost uni- Questions of complex advanced strength analysis including form partial angular velocities was carried out. load-bearing structure safety estimation based on the analysis of factors that influence the character of processes of initiation and Plot of change of integrated coordinate of platform displace- evolution of defects are considered. ment, plot showing phase difference between exciters and plots Advanced computation of structure elements with regard for of synchronous and partial angular velocities of exciters during failure mechanisms and kinetics enables to describe stages of the first twenty seconds after the start of electric motors are pre- equilibrium crack growth and to predict load-bearing capacity sented. The presented plots show the dynamics of synchroniza- and survivability of objects proceeding from loss of stability con- tion of vibro-exciters. ditions for the processes of defect development. Object analysis Numerical simulation of the problem on self-synchronization based on these ideas presupposes the working out of damage ac- of four almost identical vibro-exciters was carried out. cumulation and structural failure models, development of numerical methods for non-linear boundary value problems, study of A problem of double synchronization of vibro-exciters i.e. a basic regularities of deformation weakening of damaged materi- case when one of them rotates with double frequency was con- als and structures. sidered. It is shown that in case of rigorously multiple par- Questions of the numerical modeling of failure processes and tial frequencies of vibro-exciters their synchronization is always the efficiency of iteration procedure organization algorithms in achieved. It is proved by constancy of the value of the phase shift particular have been considered. The preference has been given between the vibro-exciters which in this case is close to and cor- to the algorithm taking into account non-simultaneity of element responds to theoretical value. failure during continuous loading. At the same time synchronization does not occur even at very Results of calculations of simplest structure elements perfo- small spread of partial frequencies (more than ) Using the device rated plates under soft (force) and rigid (kinematic) loading are in which one end of the spring is hinged to the exciter rotating presented. Dependence of limit damaged states on sizes of calcu- at principal velocity and the other end is fixed to the object to lation region with kinematic boundary conditions has been con- which vibration is imparted makes it possible to achieve stabil- firmed. This scale effect is explained by difference in proper- ity of multiple-synchronous rotation of vibro-exciters at consid- ties of loading systems with regard to invariable region of defect erably greater spread of frequencies. forming. Some other devices intensifying the tendency of vibro-exciters Improved estimation of strength reserve, analysis of surviv- to self-synchronization were investigated. The carried out nu- ability, load-bearing capacity reserves and deformation resources merical simulation verifies the existing theoretical propositions, under various types of loading, reliability estimation with regard shows the accordance between the obtained before analytical and for stage of main crack forming and evolution (for plates with numerical results and enables the determination of the parame- quasi-periodical displacement of holes with random technologi- ters of devices which are recommended for intensification of the cal deviations), examination of possibility to control failure pro- effect of multiple self-synchronization. cesses and to increase load-bearing capacity have been realized and illustrated within the framework of advanced strength anal- N.P. Yaroshevich, Lutsk State Technical University, 75 Lviska Str., ysis methodology adjustment on the basis of numerical solution Lutsk, 43018, Ukraine

97 Mechanics of Composite Materials and Structures Dept., Perm REGULARITIES OF MULTI-PARTICLE State Technical University, Komsomolskiy Ave., 29-a, 614990, INTERACTIONS IN RANDOM STRUCTURES AND Perm, Russia DESCRIPTION OF MICRO- AND MACROFAILURE OF UNIDIRECTIONAL FIBRE-REINFORCED COMPOSITES ORIENTATIONAL AND MAGNETIC STRUCTURE OF A ALEXEY V. Z AITSEV ANDREY V. L UKIN FERRONEMATIC WITH BISTABLE ANCHORING ON ANATOLIY A. TASHKINOV NICKOLAY V. T REFILOV THE BOUNDARIES OF A LAYER [email protected], [email protected] A.N. ZAKHLEVNYKH O.R. SEMYONOVA cemenova [email protected] Structural stochasticity of unidirectional composites is caused by randomness of the shape, mutual disposition and orientation as well as dispersion of fiber diameters. Principles of al- The transitions in a bistable cell of a ferronematic, i.e. di- gorithm construction for the computer generation of random- lute suspension of needle-like magnetic particles in nematic liq- structured unidirectional fibre-reinforced composites are formu- uid crystal, are studied. It is supposed, that on top boundary of lated, maximal void fractions of inclusions are defined and ef- a layer the director is directed along a normal to a surface (rigid fects of the self-organizing and self-regularization are detected homeotropic anchoring), while on lower boundary of a layer there and explained. The special researching of the structures with ran- is a degeneration of an axis of easy orientation, and so both domly distributed fiber diameters is allowed to determine that the homeotropic, and planar coupling of the director with the bound- dispersion of these diameters is the factor predetermining occur- ary can be stable. Planar and homeotropic states are divided by rence of the localization and lack of the periodic component in a potential barrier of finite height, which can be overcome due random fields. Computational experiments on transversal shear to soft anchoring by influence of an external field on the director and tension, homogeneous tension in the reinforcement direction and magnetic particles. and anti-plane (i.e. in orthogonally related planes which are par- Within the framework of continuum theory the influence of ex- allel to the generatrix of fibres) shear are shown, that the effective ternal magnetic field directed along a normal to a layer on switch- elastic modules of composites do not depend on the symmetry ing of orientation between homeotro-pic and homeotropic - pla- and asymmetry aspects of the distribution laws. But asymmetry nar states in a cell with bistable surface anchoring is studied. By aspect is significantly influenced on the fractional structure of the minimization of the free energy functional of a ferronematic, in- materials as well as on the character of multi-particle interaction cluding both volume, and surface contributions, the set of equa- at the scale distances from half up to second averaged fiber diam- tions describing the orientational and magnetic states of a fer- eters. These scales are predetermined character of the strain and ronematic with weak bistable surface anchoring was obtained. It stress heterogeneity in undamaged composites and significantly is shown, that in such a cell in the absence of the applied mag- influenced on the damage evolution scenario at the initial stage of netic field one of two possible ferronematic states can be stable. quasi-static loading. Using correlation analysis of generated uni- One of them corresponds to homogeneous homeotropic ordering, directional fibre-reinforced random-structure composites the new at which the director in a layer is orthogonal to both boundaries. criteria for definition of characteristically size of the representa- The other one corresponds to hybrid homeotropic - planar align- tive volume domain taking account of multi-particle interaction ment, at which there is a gradient of orientation in a volume of at the system of reinforcement aggregates is proposed. a cell from planar (or close to it) ordering on lower boundary of The non-linear two-level structurally-phenomenological a layer to homeotropic ordering on the top boundary. The ba- model of damaged unidirectional fibre-reinforced random- sic state of a ferronematic in the absence of a magnetic field is structure composites is represented. The model is allowed to studied and threshold values of anchoring energy are found, such describe the inclination and coarsening of defects in matrix as that for lower values only homeotropic ferronematic state is sta- multistage process and to define the moment of macrofailure as ble. The cases of absolutely rigid and soft coupling of molecules a result of the loss of stability of damage evolution. The iteration of a liquid crystal with magnetic particles of a ferronematic were procedure for the automatic selection of the quasi-static loading considered. The case of weak magnetic field was investigated step for a composite with elastic-brittle structural elements is analytically. In case of finite field strength the set of equations represented under FEM numerical solution of boundary-value was solved numerically. For weak fields a good conformity of problems. The procedure suggests that the minimum possible numerical and analytical results was obtained. number of matrix domains change their deformation properties The opportunity of first order transition from a hybrid resulted from the partial loss or recovery of the bearing strength homeotropic - planar phase in a homeotropic phase and return if the stress-strain state type changes on the structural level. For transition was investigated at various values of material parame- various schemes of combined triaxial proportional macrodefor- ters of a suspension. The critical values of material parameters mation of composites the main regularities of damaged matrix of a ferronematic, the anchoring energy, the thickness of a layer, domain evolution are defined. Qualitative coincidence of the and magnetic field strength are determined, at which there is a macrofailure character in computational experiments with the transition between homeotropic and homeotropic - planar states results of mechanical tests on anti-pain shear of the glass-epoxi of a ferronematic. plastics is shown. This work was partially supported by grant PE-009-0 from The research was made possible by Award No PEÐ009Ð0 of CRDF. the U.S. Civilian Research and Development Foundation for the Oksana R. Semyonova, Perm State University, Bukirev str. 15, Independent States of the Former Soviet Union (CRDF). Perm, 614990, Russia

98 and Al). Planetary mill is a set of cameras, rotating round their SOME PROBLEMS OF THE NONLINEAR ROD axes. These cameras are placed into a cylinder, which is also ro- THEORY tating. The axes of cylinder and cameras are parallel to gravity direction. Cameras are filled with milling balls interacting with PAVEL A. ZHILIN each other, powder particles and walls of camera. Due to of dif- [email protected] ficult cameras motion the acceleration of balls can reach values of hundreds g. This process is proceeding appreciably far from Nonlinear theory of the rods is the oldest and, maybe, the most equilibrium and dissipation is supposed to take place on the fol- important theory in continuum mechanics of solids. However lowing levels: on macroscopic level because of difficult vortex there are some theoretical problems which have no solution up balls motion, on mesoscopic level because of fracture and adhe- to now. The report is devoted to discussion of the dynamic theory sion of particles and on microscopic level because of diffusion of the thin spatially bent and naturally twisted rods. The sug- and evolution of dislocation structure inside powder particles. On gested theory includes all known variants of the theory of rods, the first stage of modeling of this process the numerical model but possesses wider branch of applicability. A new method of describing balls motion was built. All types of balls interactions construction of the elasticity tensors is offered and their structure are approximately reduced to impacts. The collisions of balls are is established. To this end a new theory of the tensor symmetry in considered with taking into account friction and loss of kinetic space with two independent orientations is essentially used. For energy because of inelastic interaction of balls. This approach let plane elastic curves all modules of elasticity are determined. The to describe balls motion as a sequence of pare balls collisions. significant attention in the report is given to the analysis of some Energy loss was described with using of recovery coefficient e (e classical problems, including those from them, solution of which = 0 corresponds to absolutely inelastic impact and e = 1 corre- leads to paradoxical results. In particular, it is in detail consid- sponds to elastic impact). Dependence of this coefficient on plas- ered well-known elastica by Euler and it is shown, that alongside tic properties (parameters of yield surface) of powder particles with known equilibrium configurations there are also dynamic was suggested. Estimations of energy dissipating on macroscopic equilibrium configurations. In this case the form of an elastic level and energy assimilating by particle were obtained. Aver- curve does not vary, but the bent rod makes rotations around of aged parameters of individual particle loading the loading paths a vertical axis. Energy of deformation in this case does not vary. and effort values were also obtained. These characteristics are Let us note that these movements are not movements of a rod using for solving the problem of dislocation structure formation as the rigid whole for the clamped end face of the rod remains and evolution of plastic properties of individual particle. For this motionless. From this it follows, that the bent equilibrium con- purpose plane dislocation selforganization model based on cellu- figuration in the Euler elastica is, in contrast to the conventional lar automata technique was used. Model describes an evolution point of view, unstable. On the other hand, this conclusion is not of edge dislocations of three equisloped slip systems; dislocation confirmed by experimental data. Therefore there is a paradoxi- lines are orthogonal to a simulation plane. This model demon- cal situation which demands the decision. The similar situation strates formation of dislocation cell structure, which parameters known under the name of the Nikolai paradox arises at torsion of depend on (nonproportional cyclic) loading path. a rod by the boundary twisting moment. In this case experiment Ivan Y. Zoubko, Komsomolsky ave. 29a, Perm, 614600, Russia shows that twisting moment produces stabilizing effect that is in the sharp contradiction with the theoretical data. In the report it is shown what to avoid the specified paradoxes it is possible at a special choice of the constitutive equation for the moment. It appears that the moment should depend in the special form on angular velocity. Last dependence is not connected with the presence (or absence) internal friction in the rod. P.A. Zhilin, Laboratory of Dynamics of Mechanical Systems, Institute for Problems in Mechanical Engineering RAS, St.Ð Petersburg, RUS-199178, V.O. Bolshoy 61, St. Petersburg, Rus- sia.

MILLING BALLS DYNAMICS IN THE MECHANICAL ALLOYING PROCESS AND SIMULATION OF DISLOCATION STRUCTURE FORMATION OF POWDER PARTICLES

I.YU.ZOUBKO P.V. T RUSOV [email protected]

Process of mechanical alloying is the way to produce a solid solution of different crystalline materials or their compounds by high energetic mechanical actions on particles of their powder in a planetary mill. The product of the process is a composite powder of homogeneous solution of all initial components. There is no other way to alloy some materials (for instance Ti