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Numerical Implementation of an Eulerian Description of Finite Numerical implementation of an Eulerian description of finite elastoplasticity Dissertation zur Erlangung des Grades Doktor-Ingenieurin (Dr.-Ing.) der Fakult¨atf¨urBau- und Umweltingenieurwissenschaften der Ruhr-Universit¨atBochum von M. Sc. Marina Trajkovi´c- Milenkovi´c aus Leskovac, Serbien Bochum 2016 Dissertation eingereicht am: 10.08.2016 Tag der m¨undlichen Pr¨ufung: 21.10.2016 Erster Referent: Prof. em. Dr.-Ing. Otto T. Bruhns Zweiter Referent: Vertr.-Prof. Dr.-Ing. Ralf J¨anicke To my little M. Summary The numerical implementation and validation of a self-consistent Eulerian consti- tutive theory of finite elastoplasticity, based on the logarithmic rate and additive decomposition of the stretching tensor, via commercial finite element software has been the objective of this thesis. The elastic material behaviour has been modelled by a hypo-elastic relation while for the plastic behaviour an associated flow rule has been adopted. In accordance with the objectivity requirement, in the constitutive relations the objective time derivatives have been used; the logarithmic rate, the Jaumann and Green-Naghdi rates as corotational rates and the Truesdell rate, the Ol- droyd rate and the Cotter-Rivlin rate as non-corotational rates. It has been shown that for the large elastic deformation analysis the reliability of the Jaumann and Green-Naghdi rates decreases with the increased influence of finite rotations while the non-corotational rates have to be excluded from the constitutive relations. It has been proved that the implementation of the Log- rate successfully solves the so far existing integrability problem and the residual stress occurrence at the end of closed elastic strain cycles. The constitutive relation for finite elastoplasticity based on the INTERATOM model has been developed and implemented. The obtained results have been compared with the experimental records. The proposed constitutive model has been proved as reliable for monotonic deformations while for cyclic deformations some improvements remain to be implemented. The directions for improvements have been given as well. Zusammenfassung Das Ziel dieser Arbeit sind die numerische Implementierung und die Validierung einer selbstkonsistenten Eulerschen Beschreibung der finiten Elastoplastizit¨at, die auf der logarithmischen Rate und additiven Zerlegung des Dehnungstensors basiert. Implementierung und Validierung erfolgen in eine kommerzielle Finite Elemente Software. Das elastische Materialverhalten wird durch eine hypo-elastisch Beziehung mod- elliert, w¨ahrendf¨ur das plastische Verhalten eine assoziative Fließregel angenom- men worden ist. Entsprechend der Forderung nach Objektivit¨atder konstitu- tiven Gleichungen sind objektive Zeitableitungen verwendet worden; und zwar die logarithmische Rate, die Jaumann Rate und die Green-Naghdi Rate als mitrotierende und die Truesdell Rate, die Oldroyd Rate und die Cotter-Rivlin Rate als nicht-mitrotierende Raten Zeitableitungen. Es wird gezeigt, dass f¨urgroße elastische Verformungen die Genauigkeit der Zeitableitungen nach Jaumann und Green-Naghdi mit zunehmendem Einfluss gr¨oßererRotationen abnimmt, w¨ahrenddie nicht-mitrotierenden Zeitableitun- gen als ungeeignet aus konstitutiven Beziehungen auszuschließen sind. Mit Hilfe der logarithmischen Rate konnte das bisher ungel¨osteIntegrabilit¨atsproblem gel¨ostund das Auftreten von Eigenspannungen am Ende geschlossener elastis- cher Beanspruchungszyklen vermieden werden. Das verwendete Materialgesetz f¨urdie finite Elastoplastizit¨atbaut auf dem IN- TERATOM Modell auf und wurde hier an ein vereinfachtes Materialverhalten angepasst und implementiert. Die erhaltenen Ergebnisse wurden mit experi- mentellen Ergebnissen verglichen. Das vorgeschlagene Materialgesetz kann f¨ur monotone Verformungen als zuverl¨assigangesehen werden. F¨urzyklische Verfor- mungen dagegen werden einige Verbesserungen als notwendig erachtet. M¨ogliche Verbesserungsvorschl¨agewerden gegeben. Acknowledgments I would like to express my sincere gratitude to Prof. em. Dr.-Ing. Otto T. Bruhns for giving me the privilege to work in this area and for his guidance throughout my work on this thesis. I really appreciate his warm hospitality dur- ing my research stays at the Institute of Mechanics of the Ruhr University in Bochum. I want to thank to Vertr.-Prof. Dr.-Ing. Ralf J¨anicke for his acceptance to be the co-reviewer of my thesis. I also want to thank to Prof. Dr.-Ing. R¨udigerH¨offerfor beeing the faculty reviewer of my thesis. My sincere gratitude to Prof. Dr Dragoslav Sumarac,ˇ my co-advisor, for his support and fruitful cooperation all these years. I also want to thank to Prof. Dr.-Ing. G¨unther Schmid, Prof. Dr.-Ing. R¨udiger H¨offerand Prof. Dr Stanko Brˇci´cfor their advice and motivation. Being the guest researcher at the Institute of Mechanics of the Ruhr University in Bochum was the extremely valuable experience for me. For co-financing my stays in Bochum I deeply appreciate the DAAD grants obtained through the SEEFORM programme. I want to thank to Prof. Dr. Heng Xiao and Dr.-Ing. Albert Meyers for very useful discussions while I was doing the first steps toward this goal. I want to thank to all members of the (former) Chair of Technical Me- chanics for friendly and pleasantly working atmosphere. My special thanks to our small Serbian community in Bochum for their help. I really appreciate the support of my friends and colleagues from the Faculty of Civil Engineering and Architecture in Niˇs.Many thanks for my colleagues from the Faculty of Mechanical Engineering in Niˇsfor fruitful discussions. Finally, I want to thank to my parents, my sister and her family, and mostly to my son and husband for their constant support, love and encouragement. Bochum, 2016 Marina Trajkovi´cMilenkovi´c Contents List of Figures iv List of Tables vii Conventions and Notations viii 1 Introduction1 1.1 Motivation..............................2 1.2 Aim of the thesis...........................4 1.3 Outline................................4 2 Deformation and motion5 2.1 Introduction..............................5 2.2 Kinematics of deformable body...................5 2.2.1 Configurations and body motion..............5 2.2.2 Lagrangian and Eulerian description............7 2.2.3 Material and spatial time derivative............9 2.3 Analysis of deformation....................... 10 2.3.1 Deformation gradient.................... 10 2.3.2 Polar decomposition..................... 11 2.4 Analysis of strain........................... 13 2.4.1 Generalized strain measures................. 13 2.5 Analysis of motion.......................... 15 2.5.1 Velocity gradient....................... 15 2.5.2 Rate of deformation..................... 17 2.5.3 Vorticity tensor........................ 17 2.6 Objectivity of a tensor field..................... 19 2.6.1 Lagrangian and Eulerian objectivity............ 19 2.6.2 Objective time derivatives.................. 20 2.6.3 Corotational and convective frame............. 21 2.6.4 Non-corotational rates.................... 23 2.6.5 Corotational rates...................... 25 2.7 Decomposition of finite deformation................ 29 3 Conservation equations and stress measures 32 3.1 Introduction.............................. 32 3.2 Conservation laws.......................... 32 i ii 3.2.1 Conservation of mass..................... 32 3.2.2 Conservation of linear momentum and Cauchy's first law of motion........................... 33 3.2.3 Conservation of angular momentum and Cauchy's second law of motion......................... 35 3.3 Stress measures and stress rates................... 36 3.3.1 Alternative stress measures................. 36 3.3.2 Lagrangian field equations.................. 38 3.3.3 Work rate and conservation of mechanical energy..... 39 3.3.4 Stress rates.......................... 41 3.4 Conjugate stress analysis....................... 44 3.5 Weak form of balance of momentum................ 48 3.5.1 Principle of virtual work................... 48 3.5.2 Rate of the weak form of balance of momentum...... 50 4 Constitutive relations 52 4.1 Introduction.............................. 52 4.2 Continuum thermodynamics..................... 53 4.2.1 First law of thermodynamics................ 54 4.2.2 Second law of thermodynamics and principle of maximum dissipation.......................... 57 4.3 Decomposition of the stress power................. 60 4.4 Hypo-elasticity............................ 61 4.4.1 Hypo-elastic relation for De - From small to finite strains 61 4.4.2 Hyper-elastic potential.................... 63 4.5 Plasticity............................... 65 4.5.1 Yield surface and loading condition............. 67 4.5.2 Priniple of maximum plastic dissipation and its consequences 69 4.5.3 Eulerian description of finite plastic deformations..... 71 4.5.3.1 Classical models from IA model......... 75 5 Numerical results 77 5.1 Introduction.............................. 77 5.2 Hypo-elasticity............................ 77 5.2.1 Finite simple shear...................... 78 5.2.2 Plate with a hole in finite tension.............. 83 5.2.3 Closed elastic strain path - Hypo-elastic cyclic deformation 85 5.2.3.1 Small rotations (ξ = 0:1)............. 86 5.2.3.2 Moderate rotations (ξ = 1)............ 89 5.2.3.3 Large rotations (ξ > 1).............. 93 5.3 Elastoplasticity............................ 101 5.3.1 Small cyclic elastoplastic deformations........... 101 5.3.2 Finite elastoplastic deformations.............. 107 5.3.2.1 Elastic - perfectly plastic............. 108 iii 5.3.2.2 Elastic -
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