Mixed analytical-numerical relaxation in finite single-slip crystal plasticity

Continuum Mechanics and Thermodynamics

Volumn 20, Issue 5, 2008, Pages 275-301

  Carstensen, Carsten ^a   Conti, Sergio ^b   Orlando, Antonio ^c  

^a HUMBOLDT UNIVERSITY   (Germany)

^b UNIVERSITY OF DUISBURG ESSEN   (Germany)

^c SWANSEA UNIVERSITY   (United Kingdom)

Author keywords

Crystal plasticity; Quasiconvexity; Relaxation

Indexed keywords

CRYSTAL PLASTICITIES; CRYSTAL PLASTICITY; EFFICIENT; ELASTOPLASTIC; ENERGY DENSITIES; FINE STRUCTURES; LIMITING CASES; LOWER BOUNDS; MODELING; NONCONVEX OPTIMIZATION PROBLEMS; NONCONVEX VARIATIONAL PROBLEMS; NUMERICAL EXAMPLES; POLYCONVEXITY; QUASICONVEX; QUASICONVEXITY; RELAXATION; RELAXATION ALGORITHMS; SLIP SYSTEMS; SPECIAL STRUCTURES; STANDARD MODELS; TWO DIMENSIONS;

ABS RESINS; BOOLEAN FUNCTIONS; CONTROL THEORY; CRYSTALLOGRAPHY; ELASTICITY; OPTIMIZATION; PLASTICITY; PLASTICS; POWDERS;

SINGLE CRYSTALS;

EID: 54149087079     PISSN: 09351175     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00161-008-0082-0     Document Type: Article

References (43)

1. J. Adams S. Conti A. DeSimone G. Dolzmann 2008 Relaxation of some transversally isotropic energies and applications to smectic A elastomers Math. Mod. Meth. Appl. Sci. 18 1 20
2. Allaire, G.: Shape optimization by the homogenization method, vol. 146 of Applied Mathematical Sciences, Springer, New York (2002)
3. R.J. Asaro 1983 Micromechanics of crystals and polycrystals Adv. Appl. Mech. 23 1 115
4. S. Aubry M. Fago M. Ortiz 2003 A constrained sequential-lamination algorithm for the simulation of sub-grid microstructure in martensitic materials Comput. Meth. Appl. Mech. Eng. 192 2823 2843
5. J.M. Ball 1977 Convexity conditions and existence theorems in nonlinear elasticity Arch. Rat. Mech. Anal. 63 337 403
6. J.M. Ball R.D. James 1987 Fine phase mixtures as minimizer of energy Arch. Rat. Mech. Anal. 100 13 52
7. J.M. Ball R.D. James 1992 Proposed experimental tests for the theory of fine microstructures and the two-well problem Philos. Trans R. Soc. Lond. A 338 389 450
8. S. Bartels 2005 Reliable and efficient approximation of polyconvex envelopes SIAM J. Numer. Anal. 43 363 385
9. S. Bartels C Carstensen K. Hackl U. Hoppe 2004 Effective relaxation for microstructure simulations: algorithms and applications Comput. Meth. Appl. Mech. Eng. 193 5143 5175
10. Bartels S., Carstensen C., Conti S., Hackl K., Hoppe U., Orlando A.: Relaxation and the computation of effective energies and microstructures in solid mechanics. In: Mielke, A. (eds) Analisys, Modeling and Simulation of Multiphase Problems, pp. 197-224. Springer, Berlin (2006)
11. K. Bhattacharya G. Dolzmann 2001 Relaxation of some multiwell problems Proc. R. Soc. Edin. A 131 279 320
12. Carstensen, C.: Numerical analysis of microstructure. In: Blowey, J.F., Coleman, J.P., Craig, A.W. (eds.) Theory and Numerics of Differential Equations, Durham 2000. Springer, New York (2001)
13. C. Carstensen K. Hackl A. Mielke 2002 Non-convex potentials and microstructures in finite-strain plasticity Proc. R. Soc. Lond. A 458 299 317
14. S. Conti A. DeSimone G. Dolzmann 2002 Soft elastic response of stretched sheets of nematic elastomers: a numerical study J. Mech. Phys. Solids 50 1431 1451
15. S. Conti 2005 Microstructure and relaxation in single-crystal plasticity Oberwolfach Rep. 2 3000 3003
16. Conti, S.: Relaxation of single-slip single-crystal plasticity with linear hardening. In: Gumbsch, P. (ed.) Proceedings of Multiscale Material Modeling Conference. Freiburg, 18-22 September (2006) 30-35
17. S. Conti M. Ortiz 2005 Dislocation microstructures and the effective behavior of single crystals Arch. Ration. Mech. Anal. 176 103 147
18. S. Conti F. Theil 2005 Single-slip elastoplastic microstructures Arch. Ration. Mech. Anal. 178 125 148
19. S. Conti P. Hauret M. Ortiz 2007 Concurrent multiscale computing of deformation microstructure by relaxation and local enrichment with application to single-crystal plasticity Multiscale Model. Simul. 6 135 157
20. Dacorogna B.: Direct Methods in the Calculus of Variations, vol. 78 of Applied Mathematical Sciences. Springer, USA (1989)
21. A. DeSimone G. Dolzmann 2002 Macroscopic response of nematic elastomers via relaxation of a class of SO(3)-invariant energies Arch. Rat. Mech. Anal. 161 181 204
22. Dolzmann G.: Variational Methods for Crystalline Microstructure-Analysis and Computation, Lecture Notes in Mathematics 1803. Springer, Berlin (2003)
23. G. Dolzmann 1999 Numerical computation of rank-one convex envelopes SIAM J. Numer. Anal. 36 1621 1635
24. Hackl, K., Hoppe, U.: On the calculation of microstructures for inelastic materials using relaxed energies. In: Miehe Ch. (ed.) IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains. Kluwer, Dordrecht (2002)
25. R.V. Kohn 1991 The relaxation of a double-well energy Contin. Mech. Thermodyn. 3 193 236
26. Kohn, R.V., Strang, G.: Optimal design and relaxation of variational problems I, II, III, Comm. Pure Appl. Math. 39, 113-137, 139-182, 353-377 (1986)
27. M. Kruzik M. Luskin 2003 The computation of martensitic microstructures with piecewise laminates J. Sci. Comput. 19 293 308
28. C. Miehe M. Lambrecht 2003 Analysis of microstructure development in shearbands by energy relaxation of incremental stress potentials: large strain theory for standard dissipative solids Int. J. Numer. Meth. Eng. 58 1 41
29. C. Miehe M. Lambrecht E. Gürses 2004 Analysis of material instabilities in inelastic solids by incremental energy minimization and relaxation methods: evolving deformation microstructures in finite plasticity J. Mech. Phys. Solids 52 2725 2769
30. Mielke, A.: Relaxation via Young measures of material models for rate-independent inelasticity. In: Miehe Ch. (ed.) IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains. Kluwer, Dordrecht (2002)
31. Mielke, A.: Finite elastoplasticity, Lie groups and geodesics on SL(d). In: Newton, P., Weinstein, A., Holmes, P.J. (eds.) Geometry, Dynamics, and Mechanics. Lecture Notes in Mathematics 1278. Springer, New-York (2002)
32. Mielke, A.: Evolution in rate-independent systems (Chap. 6). In: Dafermos, C., Feireisl, E. (eds.) Handbook of Differential Equations: Evolutionary Equations. vol. 2. Elsevier, Amsterdam (2005)
33. C.B. Morrey 1952 Quasi-convexity and the lower semicontinuity of multiple integrals Pacific J. Math. 2 25 53
34. Müller, S.: Variational models for microstructure and phase transitions. In: Hildebrandt S., Struwe M. (eds.) Calculus of Variations and Geometric Evolution Problems. Lecture Notes in Mathematics 1713. Springer, Berlin (1999)
35. S. Müller V. Šverák 1999 Convex integration with constraints and applications to phase transitions and partial differential equations J. Eur. Math. Soc. 1 393 442
36. Nocedal J., Wright S.: Numerical Optimization. Springer, New York (2006)
37. M. Ortiz E.A. Repetto 1999 Nonconvex energy minimization and dislocation structures in ductile single crystals J. Mech. Phys. Solids 47 397 462
38. M. Ortiz E.A. Repetto L. Stainer 2000 A theory of subgrain dislocation structures J. Mech. Phys. Solids 48 2077 2114
39. Pardalos P.M., Romeijn H.E.: Handbook of Global Optimization, Vol. 2. Springer, Berlin (2002)
40. Pedregal P.: Parametrized Measures and Variational Principles. Birkhäuser, Basel (1997)
41. Roubíček T.: Relaxation in Optimization Theory and Variational Calculus. Walter de Gruyter, Berlin (1996)
42. K. Schittkowski 1985 NLQPL: A FORTRAN-subroutine solving constrained nonlinear programming problems Ann. Oper. Res. 5 485 500
43. V.P. Smyshlyaev J.R. Willis 1999 On the relaxation of a three-well energy Proc. R. Soc. A 455 779 814