Thermodynamics of rate-independent processes in viscous solids at small strains

SIAM Journal on Mathematical Analysis

Volumn 42, Issue 1, 2010, Pages 256-297

  Tomas, Roubíček ^a,b  

^a CHARLES UNIVERSITY   (Czech Republic)

^b INSTITUTE OF THERMOMECHANICS   (Czech Republic)

Author keywords

Damage; Doubly nonlinear variational inequalities; Energetic solution; Enthalpy transformation; Generalized standard materials; Heat equation; Magnetostriction; Plasticity; Shapememory alloys

Indexed keywords

DAMAGE; ENERGETIC SOLUTION; GENERALIZED STANDARD MATERIALS; HEAT EQUATION; SHAPE MEMORY ALLOY; VARIATIONAL INEQUALITIES;

CONTINUUM MECHANICS; DIFFERENTIAL EQUATIONS; ELASTICITY; ENTHALPY; MAGNETOSTRICTION; MAGNETOSTRICTIVE DEVICES; MATERIALS SCIENCE; MODELS; PHASE TRANSITIONS; PLASTICITY; STANDARDS; VARIATIONAL TECHNIQUES;

NONLINEAR EQUATIONS;

EID: 77955835989     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/080729992     Document Type: Article

References (76)

1. H.-D. ALBER, Materials with Memory, Lecture Notes in Math. 1682, Springer, Berlin, 1998.
2. F. AURICCHIO, A. MIELKE, AND U. STEFANELLI, A rate-independent model for the isothermal quasi-static evolution of shape-memory materials, Math. Models Methods Appl. Sei., 18 (2008), pp. 125-164.
3. F. AURICCHIO AND L. PETŘINI, Improvements and algorithmical considerations on a recent three-dimensional model describing stress-induced solid phase transformations, Internat. J. Numer. Methods in Engrg., 55 (2002), pp. 1255-1284.
4. F. AURICCHIO, A. REALI, AND U. STEFANELLI, A three-dimensional model describing stressinduced solid phase transformation with permanent inelasticity, Int. J. Plasticity, 23 (2007), pp. 207-226.
5. F. AURICCHIO, R. L. TAYLOR, AND J. LUBLINER, Shape-memory alloys: Macromodelling and numerical simulations of the superelastic behaviour, Comput. Methods Appl. Mech. Engrg., 146 (1997), pp. 281-312.
6. W. BALTENSPERGER AND J. S. HELMAN, Dry friction in micromagnetics, IEEE Trans. Mag., 27 (1991), pp. 4772-4774.
7. E.J. BARBERO, F. GRECO, AND P. LONETTI, Continuum damage-healing mechanics with application to self-healing composites, Int. J. Damage Mech., 14 (2005), pp. 51-81.
8. S. BARTELS AND T. ROUBÍČEK, Thermoviscoplasticity at small strains, Z. Angew. Math. Mech., 88 (2008), pp. 735-754.
9. S. BARTELS AND T. ROUBÍČEK, Thermoviscoelasticity with Rate-independent Plasticity in Isotropic Materials Undergoing Thermal Expansion, M2AN Math. Model. Numer. Anal., submitted.
10. A. BERGQVIST, Magnetic vector hysteresis model with dry friction-like pinning, Phys. B. Condensed Matter, 233 (1997), pp. 342-347.
11. G. BERTOTTI, Hysteresis in Magnetism, Academic Press, San Diego, 1998.
12. M. BERTSCH, P. PODIO GUIDUGLI, AND V. VALENTE, On the dynamics of deformable ferromagnets I. Global weak solutions for soft ferromagnets at rest, Ann. Mat. Pura Appl. (4), 179 (2001), pp. 331-360.
13. D. BLANCHARD AND O. GUIBÉ, Existence of a solution for a nonlinear system in thermoviscoelasticity, Adv. Differential Equations, 5 (2000), pp. 1221-1252.
14. L. BOCCARDO, A. DALL'AGLIO, T. GALLOUËT, AND L. URSINA, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147 (1997), pp. 237-258.
15. L. BOCCARDO AND T. GALLOUËT, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), pp. 149-169.
16. G. BOUCHITTÉ, A. MIELKE, AND T. ROUBÍČEK, A complete-damage problem, Z. Angew. Math. Phys., 60 (2009), pp. 205-236.
17. J.G. BOYD AND D.C LAGOUDAS, A thermodynamical constitutive model for shape memory materials. Part I. The monolithic shape memory alloys, Int. J. Plasticity, 12 (1996), pp. 805842.
18. C CARSTENSEN, Numerical analysis of the primal problem of elastoplasticity with hardening, Numer. Math., 82 (1999), pp. 577-597.
19. K. CHELMIŃSKI, Coercive approximation of viscoplasticity and plasticity, Asympt. Anal., 26 (2001), pp. 115-135.
20. G. DAL MASO, A. DESIMONE, AND M.G. MORA, Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Ration. Mech. Anal., 180 (2006), pp. 237-291.
21. G. DAL MASO, A. DESIMONE, M.G. MORA, AND M. MORINI, Globally stable quasistatic evolution in plasticity with softening, Netw. Heterog. Media, 3 (2008), pp. 567-614.
22. G. DAL MASO, G.A. FRANCFORT, AND R. TOADER, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176 (2005), pp. 165-225.
23. C ECK, Existence of solutions to a thermo-viscoelastic contact problem with Coluomb friction, Math. Models Methods Appl. Sci., 12 (2002), pp. 1491-1511.
24. C ECK, J. JARUŠEK, AND M. KRBEC, Unilateral Contact Problems, Chapman & Hall/CRC, Boca Raton, 2005.
25. M. EFENDIEV AND A. MIELKE, On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Anal., 13 (2006), pp. 151-167.
26. E. FEIREISL AND J. MÁLEK, On the Navier-Stokes equations with temperature-dependent transport coefficients, Differ. Equ. Nonlinear Mech., (2006), 14pp, Art. D 90616.
27. G. FRANCFORT AND A. MIELKE, An existence result for a rate-independent material model in the case of nonconvex energies, J. Reine Angew. Math., 595 (2006), pp. 55-91.
28. M. FREMOND, Non-Smooth Thermomechanics. Springer, Berlin, 2002.
29. M. FREMOND AND B. NEDJAR, Damage, gradient of damage and principle of virtual power, Internat. J. Solids Structures, 33 (1996) pp. 1083-1103.
30. S. GovINDJEE AND CH. MIEHE, A multi-variant martensitic phase transformation model: Formulation and numerical implementation, Comput. Methods Appl. Mech. Engrg., 191 (2001), pp. 215-238.
31. H. HAHN, Über Annäherung an Lebesgue'sehe Integrale durch Riemann'sche Summen, Sitz. Math. Phys. Kl. K. Akad. Wiss. Wien, 123 (1914), pp. 713-743.
32. B. HALPHEN AND Q.S. NGUYEN, Sur les matériaux standards généralisés, J. Mécanique, 14 (1975), pp. 39-63.
33. W. HAN AND B.D. REDDY, Convergence of approximations to the primal problem in plasticity under conditions of minimal regularity, Numer. Math., 80 (2000), pp. 283-315.
34. D. HELM AND P. HAUPT, Shape memory behaviour: Modelling within continuum thermomechanics, Internat. J. Solids Structures, 40 (2000), pp. 827-849.
35. K. HORMANN AND J. ZIMMER, On Landau theory and symmetric landscape for phase transitions, J. Mech. Phys. Solids, 55 (2007), pp. 1385-1409.
36. R.D. JAMES AND M. WUTTIG, Magnetostriction of martensite, Phil. Mag. A, 77 (1998), pp. 1273-1299.
37. Y. JUNG, P. PAPADOPOULOS, AND R.O. RITCHIE, Constitutive modeling and numerical simulation of multivariant phase transformation in superelastic shape-memory alloys, Internat. J. Numer. Methods Engrg., 60 (2004), pp. 429-160.
38. D. KNEES, A. MIELKE, AND C ZANINI, On the inviscid limit of a model for crack propagation, Math. Models Methods Appl Sei., 18 (2008), pp. 1529-1568.
39. M. KRUŽÍK, Periodic solutions to a hysteresis model in micromagnatics, J. Convex Anal., 13 (2006), pp. 81-99.
40. M. KRUŽÍK AND F. OTTO, A phenomenological model for hysteresis in poly cry stalinne shape memory alloys, Z. Angew. Math. Mech., 84 (2004), pp. 835-842.
41. E. KUHL AND G.A. HOLZAPFEL, A continuum model for remodeling in living structures, J. Mater. Sci., 42 (2007), pp. 8811-8823.
42. C. LEXCELLENT, S. MOYNE, A. ISHIDA, AND S. MIYAZAKI, Deformation behaviour associated with the stress-induced martensitic transformation in Ti-Ni thin films and their thermodynamical modelling, Thin Solid Films, 324 (1998), pp. 184-189.
43. J. LUBLINER AND F. AURICCHIO, Generalized plasticity and shape-memory alloys, Internat. J. Solids Structures, 33 (1996), pp. 991-1003.
44. A. MIELKE, Evolution of Rate-independent Systems, Handb. Differ. Equ., C Dafermos, E. Feireisl, eds., Elsevier, Amsterdam, 2005, pp. 461-559.
45. A. MIELKE, A mathematical framework for generalized standard materials in rate-independent case, in Multifield Problems in Fluid and Solid Mech., R. Helmig et al. eds., Springer, New York, 2006, pp. 491-529.
46. A. MIELKE, Complete-damage evolution based on energies and stresses, Discrete Contin. Dyn. Syst., to appear
47. A. MIELKE AND M. ORTIZ, A class of minimum pinciples for characterizing trajectories and the relaxation of dissipative systems, ESAIM Control Optim. Cale Var., 14 (2008), pp. 494516.
48. A. MIELKE, A. PETROV, AND J.A.C MARTINS, Convergence of solutions of kinetic variational inequalities in the rate-independent quasistatic limit, J. Math. Anal. Appl., 348 (2008), pp. 1012-1020.
49. A. MIELKE AND R. RossI, Existence and uniqueness results for general rate-independent hysteresis problems, Math. Models Meth. Appl. Sci, 17 (2007), pp. 81-123.
50. A. MIELKE AND T. ROUBÍČEK, Rate-independent damage processes in nonlinear elasticity, Math. Models Methods Appl. Sci., 16 (2006), pp. 177-209.
51. A. MIELKE AND T. ROUBÍČEK, Numerical approaches to rate-independent processes and applications in inelasticity, Math. Model. Numer. Anal., 43 (2009), pp. 399-428.
52. A. MIELKE, T. ROUBÍČEK, AND U. STEFANELLI, T-limits and relaxations for rate-independent evolutionary problems, CaIc Var. Partial Differential Equations, 31 (2008), pp. 387-416.
53. A. MIELKE, T. ROUBÍČEK, AND J. ZEMAN, Complete damage in elastic and viscoelastic media and its energetics, Comput. Methods Appl. Mech. Engrg., in print, doi:10.1016/ j.cma.2009.09.020.
54. A. MIELKE AND F. THEIL, A mathematical model for rate-independent phase transformations with hysteresis, in Models of Continuum Mechanics in Analysis and Engineering, H.-D. Alber, R. Balean, and R. Farwig, eds., Shaker Ver., Aachen, 1999, pp. 117-129.
55. A. MIELKE AND F. THEIL, On rate-independent hysteresis models, Nonlinear Differ. Equ. Appl., 11 (2004), pp. 151-189.
56. A. MIELKE, F. THEIL, AND V.I. LEVITAS, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162 (2002), pp. 137-177.
57. A. MIELKE AND A. TIMOFTE, Two-scale homogenization for evolutionary variational inequalities via the energetic formulation, SIAM J. Math. Anal., 39 (2007), pp. 642-668.
58. J.M. MCNANEY, V. IMBENI, Y. JUNG, P. PAPADOPOULOS, AND R.O. RITCHIE, An experimental study of the suprelastic effect in a shape-memory Nitinol alloy under biaxial loading, Mech. Mater., 35 (2003), pp. 969-986.
59. V. NOVÁK, P.ŠITTNER, G.N. DAYANANDA, F.M. BRAZ-FERNANDES, AND K.K. MAHESH, Electric resistance variation of NiTi shape memory alloy wires in thermomechanical tests: Experiments and simulation, Mater. Sei. Engr. A, 481-482 (2008), pp. 127-133.
60. E. PATOOR, A. EBERHARDT, AND M. BERVEILLER, Micromechanical modelling of shape memory alloys behaviour, Arch. Mech. (Arch. Mech. Stos.), 40 (1988), pp. 775-794.
61. P. PODIO GUIDUGLI, T. ROUBÍČEK, AND G. TOMASSETTI, A thermodynamically-consistent theory of the ferro/paramagnetic transition, Arch. Ration. Mech. Anal., to appear.
62. T. ROUBÍČEK, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2005.
63. 1-data, Quart. Appl. Math., 67 (2009), pp. 47-71.
64. T. ROUBÍČEK, Rate independent processes in viscous solids at small strains, Math. Methods Appl. Sci., 32 (2009), pp. 825-862.
65. T. ROUBÍČEK AND G.TOMASSETTI, Thermodynamics of shape-memory alloys under electric current, Z. Angew. Math. Phys., in print, DOI: 10.1007/s00033-009-0007-l.
66. A. SADJADPOUR AND K. BHATTACHARYA, A micromechanics inspired constitutive model for shape-memory alloys, Smart Mater. Structures, 16 (2007), pp. 1751-1765.
67. N. SlREDEY, E. PATOOR, M. BERVEILLER, AND A. EBERHARD, Constitutitve equations for polycrystaline thermoelastic shape memory alloys. Part I: Intragranular interactions and behavior of the grain, Internat. J. Solids Structures, 36 (1999), pp. 4289-1315.
68. M. SOFONEA, W. HAN, AND M. SHILLOR, Analysis and Approximation of Contact Problems with Adhesion or Damage, Chapman & Hall/CRC, Bocea Raton, 2006.
69. V. STOILOV AND K. BHATTACHARYA, A theoretical framework of one-dimensional harp phase fronts in shape memory alloys, Acta Mater., 50 (2002), pp. 4939-1952.
70. P. THAMBURAJA AND L. ANAND, Poly cry stalinne shape-memory materials: Effect of crystalographic texture, J. Mech. Phys. Solids, 49 (2001), pp. 709-737.
71. M. THOMAS AND A. MIELKE, Damage of nonlinearly elastic materials at small strains: Existence and regularity results, Z. Angew. Math. Mech., to appear.
72. R. TICKLE, Ferromagnetic Shape-Memory Materials, Ph.D. thesis, University of Minnesota, Minneapolis, 2000.
73. A. VISINTIN, Modified Landau-Lifshitz equation for ferromagnetism, Phys. B, 233 (1997), pp. 365-369.
74. A. VISINTIN, Homogenization of the nonlinear Kelvin-Voigt model of viscoelasticity and of the Prager model of plasticity, Contin. Mech. Thermodyn., 18 (2006), pp. 223-252.
75. A. VIVET AND C LEXCELLENT, Micromechanical modelling for tension-compression pseudoelastic behaviour of AuCd single crystals, Eur. Phys. J. A.P., 4 (1998), pp. 125-132.
76. S.R. WHITE, N.R. SOTTOS, P.H. GEUBELLE, J.S. MOORE, M.R. KESSLER, S.R. SRIRAM, E.N. BROWN, AND S. VISWANATHAN, Autonomic healing of polymer composites, Nature, 409 (2001), pp. 794-797.