A mathematical framework for generalized standard materials in the rate-independent case

Lecture Notes in Applied and Computational Mechanics

Volumn 2006, Issue 28, 2006, Pages 399-428

  Mielke, Alexander ^a,b  

^a WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

^b HUMBOLDT UNIVERSITY   (Germany)

Author keywords

Energetic formulation; Gamma convergence; Magnetostriction; Piezoelectricity; Rate independence; Relaxation; Shape memory material

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; CRYSTALLOGRAPHY; PIEZOELECTRIC MATERIALS; SHAPE MEMORY EFFECT;

ELASTIC ENERGY DENSITY; ENERGETIC FORMULATIONS; FLOW LAW; GAMMA CONVERGENCE; GENERALIZED STANDARD MATERIALS; MATHEMATICAL FRAMEWORKS; RATE INDEPENDENCE; RATE INDEPENDENTS; RELAXATION; SHAPE-MEMORY MATERIALS;

PIEZOELECTRICITY;

EID: 33751257196     PISSN: 16137736     EISSN: None     Source Type: Book Series    
DOI: 10.1007/978-3-540-34961-7_12     Document Type: Article

References (80)

1. J. Alberty and C. Carstensen. Numerical analysis of time-depending primal elastoplasticity with hardening. SIAM J. Numer. Anal., 37:1271-1294 (electronic), 2000.
2. L.-E. Andersson and A. Klarbring. A review of the theory of static and quasistatic frictional contact problems in elasticity. R. Soc. Lond. Philos. Trans. Ser. A, 359:2519-2539, 2001.
3. S. Aubry, M. Fago, and M. Ortiz. A constrained sequential-lamination algorithm for the simulation of sub-grid microstructure in martensitic materials. Comput. Methods Appl. Mech. Engrg., 192:2823-2843, 2003.
4. F. Auricchio, A. Mielke, and U. Stefanelli. A rate-independent model for the isothermal quasi-static evolution of shape-memory materials. In preparation, 2006.
5. F. Auricchio and L. Petrini. Improvements and algorithmical considerations on a recent three-dimensional model describing stress-induced solid phase transformations. Int. J. Numer. Meth. Engng, 55:1255-1284, 2002.
6. J. Ball. Constitutive inequalities and existence theorems in nonlinear elastostatics. In Nonlinear Analysis and Mechanics: Heriot-Watt Symposium (Edinburgh, 1976), Vol. I, pages 187-241. Res. Notes in Math., No. 17. Pitman, London, 1977.
7. J. M. Ball and R. D. James. Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal., 100(1):13-52, 1987.
8. S. Bartels, C. Carstensen, K. Hackl, and U. Hoppe. Effective relaxation for microstructure simulations: Algorithms and applications. Comput. Methods Appl. Mech. Engrg., 193:5143-5175, 2004.
9. K. Bhattacharya. Microstructure of Martensite. Why it Forms and How it Gives Rise to the Shape-Memory Effect. Oxford University Press, New York, 2003.
10. M. Brokate, P. Krejčí, and H. Schnabel. On uniqueness in evolution quasivariational inequalities. J. Convex Analysis, 11:111-130, 2004.
11. M. Brokate and J. Sprekels. Hysteresis and Phase Transitions. Springer-Verlag, New York, 1996.
12. A. Chambolle. A density result in two-dimensional linearized elasticity and applications. Arch. Rat. Mech. Analysis, 167:211-233, 2003.
13. P. G. Ciarlet. Mathematical Elasticity. Vol. I: Three-Dimensional Elasticity. North-Holland Publishing Co., Amsterdam, 1988.
14. S. Conti and F. Theil. Single-slip elastoplastic microstructures. Arch. Rational Mech. Analysis, 178:125-148, 2005.
15. G. Dal Maso, G. Francfort, and R. Toader. Quasistatic crack growth in nonlinear elasticity. Arch. Rat. Mech. Anal., 176:165-225, 2005.
16. A. Dorfmann and R. W. Ogden. Some problems in nonlinear magnetoelasticity. Z. Angew. Math. Phys., 56:718-745, 2005.
17. M. Efendiev and A. Mielke. An energetical model for evolution in magnetostriction. In preparation, 2006.
18. M. Efendiev and A. Mielke. On the rate-independent limit of systems with dry friction and small viscosity. J. Convex Analysis, 13(1):151-167, 2006.
19. M. Efendiev, A. Mielke, R. Rossi, and G. Savaré. On a metric approach to a class of doubly nonlinear evolution equations and applications to rate-independent problems. In preparation, 2006.
20. G. Francfort and A. Garroni. A variational view of brittle damage evolution. Arch. Rational Mech. Anal., 2006. To appear.
21. G. Francfort and J.-J. Marigo. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids, 46:1319-1342, 1998.
22. G. Francfort and A. Mielke. Existence results for a class of rate-independent material models with nonconvex elastic energies. J. reine angew. Math., 2006. In print.
23. M. Frémond. Non-Smooth Thermomechanics. Springer-Verlag, Berlin, 2002.
24. M. Frémond and S. Miyazaki. Shape Memory Alloys. Springer-Verlag, Wien, 1996.
25. S. Govindjee, A. Mielke, and G. Hall. The free-energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis. J. Mech. Physics Solids, 50:1897-1922, 2002. Erratum and Correct Reprinting: 51(4) 2003, pp. 763 & IXXVI.
26. E. Gürses, A. Mainik, C. Miehe, and A. Mielke. Analytical and numerical methods for finite-strain elastoplasticity. In SFB404-Abschlussband. Springer-Verlag, 2006.
27. K. Hackl. Generalized standard media and variational principles in classical and finite strain elastoplasticity. J. Mech. Phys. Solids, 45(5):667-688, 1997.
28. K. Hackl and U. Hoppe. On the calculation of microstructures for inelastic materials using relaxed energies. In C. Miehe, editor, IUTAM Symposium on Computational Mechanics of Solids at Large Strains, pages 77-86. Kluwer, 2003.
29. B. Halphen and Q. S. Nguyen. Sur les matériaux standards généralisés. J. Mécanique, 14:39-63, 1975.
30. W. Han and B. D. Reddy. Plasticity (Mathematical Theory and Numerical Analysis), volume 9 of Interdisciplinary Applied Mathematics. Springer-Verlag, New York, 1999.
31. R. D. James. Configurational forces in magnetism with application to the dynamics of a small-scale ferromagnetic shape memory cantilever. Contin. Mech. Thermodyn., 14:55-86, 2002.
32. C. Johnson. Existence theorems for plasticity problems. J. Math. Pures Appl. (9), 55(4):431-444, 1976.
33. M. Kamlah. Ferroelectric and ferroelastis piezoceramics-modelling of electromechanical hysteresis phenomena. Continuum Mech. Thermodyn., 13(4):219-268, 2001.
34. D. Knees and A. Mielke. Energy release rate for cracks in finite-strain elasticity. Math. Methods Applied Sciences, 2006. Submitted (WIAS Preprint 1100).
35. M. Kočvara, A. Mielke, and T. Roubíček. A rate-independent approach to the delamination problem. Math. Mech. Solids, 2005. In print, online June 2005.
36. M. A. Krasnosel'skiǐ and A. V. Pokrovskiǐ. Systems With Hysteresis. Springer-Verlag, Berlin, 1989.
37. P. Krejčí. Evolution variational inequalities and multidimensional hysteresis operators. In Nonlinear differential equations (Chvalatice, 1998), volume 404, pages 47-110. Chapman & Hall/CRC, Boca Raton, FL, 1999.
38. M. Kružík, A. Mielke, and T. Roubíček. Modelling of microstructure and its evolution in shape-memory-alloy single-crystals, in particular in cualni. Meccanica, 40:389-418, 2005.
39. M. Kružík and M. Luskin. The computation of martensitic microstructure with piecewise laminates. J. Sci. Comp., 19:293-308, 2003.
40. M. Kružík and T. Roubíček. Mesoscopic model of microstructure evolution in shape memory alloys with applications to NiMaGa. Technical Report no. 2003, IMA Minneapolis, 2004.
41. H. Le Dret and A. Raoult. Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results. Arch. Ration. Mech. Anal., 154(2):101-134, 2000.
42. A. Mainik. A rate-independent model for phase transformations in shape-memory alloys. PhD Thesis, IADM, Universität Stuttgart, 2005.
43. A. Mainik and A. Mielke. Existence results for energetic models for rate-independent systems. Calc. Var. PDEs, 22:73-99, 2005.
44. J. A. C. Martins, A. Pinto da Costa, and F. M. F. Simões. Some notes on friction and instabilities. In Friction and instabilities (Udine, 2000), volume 457 of CISM Courses and Lectures, pages 65-136. Springer, 2002.
45. G. A. Maugin. The Thermomechanics of Plasticity and Fracture. Cambridge University Press, Cambridge, 1992.
46. A. Mielke. Estimates on the mixture function for multiphase problems in elasticity. In A.-M. Sändig, W. Schiehlen, and W. Wendland, editors, Multifield Problems, pages 96-103, Berlin, 2000. Springer-Verlag.
47. A. Mielke. Deriving new evolution equations for microstructures via relaxation of variational incremental problems. Comput. Methods Appl. Mech. Engrg., 193:5095-5127, 2004.
48. A. Mielke. Evolution in rate-independent systems (ch. 6). In C. Dafermos and E. Feireisl, editors, Handbook of Differential Equations, Evolutionary Equations, vol. 2, pages 461-559. Elsevier B.V., 2005.
49. A. Mielke. Temperature-induced phase transformations in shape-memory alloys. In preparation, 2006.
50. A. Mielke and M. Ortiz. A class of minimum principles for characterizing the trajectories of dissipative systems. In preparation, 2006.
51. 3 AS Math. Models Methods Appl. Sci., 2006. To appear (WIAS Preprint 1078).
52. A. Mielke, R. Rossi, and G. Savaré. On a metric approach to a class of rate-independent problems. In preparation, 2006.
53. A. Mielke and T. Roubíček. A rate-independent model for inelastic behavior of shape-memory alloys. Multiscale Model. Simul., 1:571-597, 2003.
54. A. Mielke and T. Roubíček. Numerical approaches to rate-independent processes and applications in inelasticity. In preparation, 2005.
55. 3 AS Math. Models Methods Appl. Sci., 16:177-209, 2006.
56. A. Mielke, T. Roubíček, and U. Stefanelli. Γ-limits and relaxations for rate-independent evolution equations. In preparation, 2006.
57. A. Mielke and F. Theil. A mathematical model for rate-independent phase transformations with hysteresis. In H.-D. Alber, R. Balean, and R. Farwig, editors, Proceedings of the Workshop on "Models of Continuum Mechanics in Analysis and Engineering", pages 117-129. Shaker-Verlag, 1999.
58. A. Mielke and F. Theil. On rate-independent hysteresis models. Nonl. Diff. Eqns. Appl. (NoDEA), 11:151-189, 2004. (Accepted July 2001).
59. A. Mielke, F. Theil, and V. I. Levitas. A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Rational Mech. Anal., 162:137-177, 2002.
60. A. Mielke and A. Timofte. An energetic material model for time-dependent ferroelectric behavior: Existence and uniqueness. Math. Meth. Appl. Sciences, 2005. In print (WIAS preprint no. 1014).
61. A. Mielke and A. M. Timofte. Modeling and analytical study for ferroelectric materials. Mech. Advanced Materials Structures, 2006. In print.
62. A. Mielke and A. M. Timofte. Two-scale homogenization for evolutionary variational inequalities via the energetic formulation. In preparation, 2006.
63. J.-J. Moreau. On unilateral constraints, friction and plasticity. In New Variational Techniques in Mathematical Physics (Centro Internaz. Mat. Estivo (C.I.M.E.), II Ciclo, Bressanone, 1973), pages 171-322. Edizioni Cremonese, Rome, 1974.
64. J.-J. Moreau. Application of convex analysis to the treatment of elastoplastic systems. In P. Germain and B. Nayroles, editors, Applications of Methods of Functional Analysis to Problems in Mechanics, pages 56-89. Springer-Verlag, 1976. Lecture Notes in Mathematics, 503.
65. S. Müller. Variational models for microstructure and phase transitions. In Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996), pages 85-210. Springer, Berlin, 1999.
66. M. Ortiz and E. Repetto. Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids, 47(2):397-462, 1999.
67. L. Prigozhin. On the bean critical-state model in superconductivity. European J. of Appl. Math., 7:237-248, 1996.
68. J. Rice. Inelastic constitutive relations for solids: An internal-variable theory and its applications to metal plasticity. J. Mech. Phys. Solids, 19:203-240, 1971.
69. H. Romanowski and J. Schröder. Coordinate invariant modelling of the ferro-electric hysteresis within a thermodynamically consistent framework. A mesoscopic approach. In Y. Wang and K. Hutter, editors, Trends in Applications of Mathematics to Mechanics, pages 419-428. Shaker Verlag, 2005.
70. T. Roubíček. Models of microstructure evolution in shape memory alloys. In P. Ponte Castaneda, J. Telega, and B. Gambin, editors, Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, pages 269-304. Kluwer, 2004. NATO Sci. Series II/170.
71. T. Roubíček and M. Kružík. Mircrostructure evolution model in micromagnetics. Zeits. angew. Math. Physik, 55:159-182, 2004.
72. T. Roubíček and M. Kružík. Mesoscopic model for ferromagnets with isotropic hardening. Zeits. angew. Math. Physik, 56:107-135, 2005.
73. P. Rybka and M. Luskin. Existence of energy minimizers for magnetostrictive materials. SIAM J. Math. Anal., 36:2004-2019, 2005.
74. F. Schmid. A local existence result for a finite-dimensional model in dry friction. In preparation, 2006.
75. F. Schmid and A. Mielke. Vortex pinning in super-conductivity as a rate-independent process. Europ. J. Appl. Math., 2005. In print.
76. A. Souza, E. Mamiya, and N. Zouain. Three-dimensional model for solids undergoing stress-induced phase transformations. Europ. J. Mech., A/ Solids, 17:789-806, 1998.
77. P.-M. Suquet. Sur les équations de la plasticité: Existence et régularité des solutions. J. Mécanique, 20(1):3-39, 1981.
78. F. Theil. Relaxation of rate-independent evolution problems. Proc. Roy. Soc. Edinburgh Sect. A, 132:463-481, 2002.
79. A. Visintin. Differential Models of Hysteresis. Springer-Verlag, Berlin, 1994.
80. H. Ziegler and C. Wehrli. The derivation of constitutive relations from the free energy and the dissipation function. In Advances in Applied Mechanics, Vol. 25, pages 183-237. Academic Press, Orlando, FL, 1987.