×

Global existence result for thermoviscoelastic problems with hysteresis. (English) Zbl 1238.35157

Summary: We consider viscoelastic solids undergoing thermal expansion and exhibiting hysteresis effects due to plasticity or phase transformations. Within the framework of generalized standard solids, the problem is described in a three-dimensional setting by the momentum equilibrium equation, the flow rule describing the dependence of the stress on the strain history, and the heat transfer equation. Under appropriate regularity assumptions on the data, a local existence result for this thermodynamically consistent system is established, by combining existence results for ordinary differential equations in Banach spaces with a fixed-point argument. Then global estimates are obtained by using both the classical energy estimate and more specific techniques for the heat equation introduced by L. Boccardo and T. Gallouët [J. Funct. Anal. 87, No. 1, 149–169 (1989; Zbl 0707.35060); Nonlinear Anal., Theory Methods Appl. 19, No. 6, 573–579 (1992; Zbl 0795.35031)]. Finally a global existence result is derived.

MSC:

35Q74 PDEs in connection with mechanics of deformable solids
35B65 Smoothness and regularity of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
74H20 Existence of solutions of dynamical problems in solid mechanics
74N30 Problems involving hysteresis in solids
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Halphen, B.; Nguyen, Q. S., Sur les matériaux standards généralisés, J. Mécanique, 14, 39-63 (1975) · Zbl 0308.73017
[2] Alber, H.-D.; Chełmiński, K., Quasistatic problems in viscoplasticity theory. I. Models with linear hardening, (Operator Theoretical Methods and Applications to Mathematical Physics. Operator Theoretical Methods and Applications to Mathematical Physics, Oper. Theory Adv. Appl., vol. 147 (2004), Birkhäuser: Birkhäuser Basel), 105-129 · Zbl 1280.74017
[3] Mielke, A.; Theil, F., On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11, 151-189 (2004), Accepted July 2001 · Zbl 1061.35182
[4] Mielke, A., Evolution in rate-independent systems, (Dafermos, C.; Feireisl, E., Handbook of Differential Equations, Evolutionary Equations, vol. 2 (2005), Elsevier B.V: Elsevier B.V Amsterdam), 461-559, (Chapter 6) · Zbl 1120.47062
[5] Francfort, G.; Mielke, A., Existence results for a class of rate-independent material models with nonconvex elastic energies, J. Reine Angew. Math., 595, 55-91 (2006) · Zbl 1101.74015
[6] Mielke, A.; Roubíček, T., Rate-independent damage processes in nonlinear elasticity, Math. Models Methods Appl. Sci., 16, 177-209 (2006) · Zbl 1094.35068
[7] Mielke, A.; Rossi, R., Existence and uniqueness results for a class of rate-independent hysteresis problems, Math. Models Methods Appl. Sci., 17, 81-123 (2007) · Zbl 1121.34052
[8] Mielke, A., A model for temperature-induced phase transformations in finite-strain elasticity, IMA J. Appl. Math., 72, 644-658 (2007) · Zbl 1128.74035
[9] Mielke, A.; Petrov, A., Thermally driven phase transformation in shape-memory alloys, Gakkōtosho Adv. Math. Sci. Appl., 17, 667-685 (2007) · Zbl 1138.49014
[10] Mielke, A.; Roubíček, T.; Stefanelli, U., \( \Gamma \)-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations, 31, 387-416 (2008) · Zbl 1302.49013
[11] Agelet de Saracíbar, C.; Cervera, M.; Chiumenti, M., On the formulation of coupled thermoplastic problems with phase-change, Int. J. Plast., 15, 1-34 (1999) · Zbl 1054.74035
[12] Srikanth, A.; Zabaras, N., A computational model for the finite element analysis of thermoplasticity coupled with ductile damage at finite strains, Internat. J. Numer. Methods Engrg., 45, 11, 1569-1605 (1999) · Zbl 0943.74073
[13] Rosakis, P.; Rosakis, A.; Ravichandran, G.; Hodowany, J., A thermodynamic internal variable model for the partition of plastic work into heat and stored energy in metals, J. Mech. Phys. Solids, 48, 3, 581-607 (2000) · Zbl 1005.74004
[14] Canadija, M.; Brnić, J., Associative coupled thermoplasticity at finite strain with temperature-dependent material parameters, Int. J. Plast., 20, 1851-1874 (2004) · Zbl 1066.74515
[15] Håkansson, P.; Wallin, M.; Ristinmaa, M., Comparison of isotropic hardening and kinematic hardening in thermoplasticity., Int. J. Plast., 21, 7, 1435-1460 (2005) · Zbl 1229.74027
[16] Bartels, S.; Roubíček, T., Thermoviscoplasticity at small strains, ZAMM Z. Angew. Math. Mech., 88, 9, 735-754 (2008) · Zbl 1153.74011
[17] Roubíček, T., Thermodynamics of rate-independent processes in viscous solids at small strains, SIAM J. Math. Anal., 42, 1, 256-297 (2010) · Zbl 1213.35279
[18] Bartels, S.; Roubíček, T., Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion, Math. Modelling Numer. Anal., 45, 29-55 (2011) · Zbl 1267.74037
[19] L. Paoli, A. Petrov, Global existence result for phase transformations with heat transfer in shape-memory alloys, WIAS Preprint No. 1609, 2011.; L. Paoli, A. Petrov, Global existence result for phase transformations with heat transfer in shape-memory alloys, WIAS Preprint No. 1609, 2011.
[20] Brezis, H., Opérateurs Maximaux Monotones et Semi-Groupes de Contractions Dans les Espaces de Hilbert (1973), North-Holland Publishing Co: North-Holland Publishing Co Amsterdam, North-Holland Mathematics Studies, No. 5. Notas de Matemática (50) · Zbl 0252.47055
[21] Dal Maso, G.; DeSimone, A.; Mora, M. G., Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Ration. Mech. Anal., 180, 2, 237-291 (2006) · Zbl 1093.74007
[22] Souza, A.; Mamiya, E.; Zouain, N., Three-dimensional model for solids undergoing stress-induced phase transformations, Eur. J. Mech., A/Solids, 17, 789-806 (1998) · Zbl 0921.73024
[23] Auricchio, F.; Petrini, L., Improvements and algorithmical considerations on a recent three-dimensional model describing stress-induced solid phase transformations, Internat. J. Numer. Methods Engrg., 55, 1255-1284 (2002) · Zbl 1062.74580
[24] Auricchio, F.; Petrini, L., A three-dimensional model describing stress-temperature induced solid phase transformations: thermomechanical coupling and hybrid composite applications, Int. J. Numer. Methods Eng., 61, 5, 716-737 (2004) · Zbl 1075.74598
[25] Mielke, A.; Theil, F., A mathematical model for rate-independent phase transformations with hysteresis, (Alber, H.-D.; Balean, R.; Farwig, R., Proceedings of the Workshop on Models of Continuum Mechanics in Analysis and Engineering (1999), Shaker-Verlag: Shaker-Verlag Aachen), 117-129
[26] Mielke, A., Estimates on the mixture function for multiphase problems in elasticity, (Multifield Problems (2000), Springer: Springer Berlin), 96-103 · Zbl 1058.74067
[27] Hall, G.; Govindjee, S., Application of the relaxed free energy of mixing to problems in shape memory alloy simulation, J. Intell. Mater. Syst. Struct., 13, 773-782 (2002)
[28] Govindjee, S.; Mielke, A.; Hall, G. J., The free-energy of mixing for \(n\)-variant martensitic phase transformations using quasi-convex analysis, J. Mech. Phys. Solids, 50, 1897-1922 (2002), Erratum and Correct Reprinting: 51(4) (2003) 763 & I-XXVI · Zbl 1116.74399
[29] Mielke, A.; Theil, F.; Levitas, V. I., A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162, 2, 137-177 (2002) · Zbl 1012.74054
[30] Govindjee, S.; Hackl, K.; Heinen, R., An upper bound to the free energy of mixing by twin-compatible lamination for \(n\)-variant martensitic phase transformations, Contin. Mech. Thermodyn., 18, 7-8, 443-453 (2007) · Zbl 1160.74393
[31] Roubíček, T., Thermo-visco-elasticity at small strains with \(L^1\)-data, Quart. Appl. Math., 67, 1, 47-71 (2009) · Zbl 1160.74011
[32] Kondrat’ev, V. A.; Oleinik, O. A., Boundary-value problems for the system of elasticity theory in unbounded domains. Korn’s inequalities, Russian Math. Surveys, 43, 5, 65-119 (1988) · Zbl 0669.73005
[33] Duvaut, G.; Lions, J.-L., Inequalities in Mechanics and Physics (1976), Springer-Verlag: Springer-Verlag Berlin, Translated from the French by C. W. John, Grundlehren der Mathematischen Wissenschaften, 219 · Zbl 0331.35002
[34] Valent, T., (Boundary Value Problems of Finite Elasticity. Boundary Value Problems of Finite Elasticity, Springer Tracts in Natural Philosophy, vol. 31 (1988), Springer-Verlag: Springer-Verlag New York), Local theorems on existence, uniqueness, and analytic dependence on data · Zbl 0648.73019
[35] Cartan, H., Cours de Calcul Différentiel (1990), Hermann: Hermann Paris
[36] J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1. (French) Travaux et Recherches Mathématiques, No. 17 Dunod, Paris 1968 xx+372 pp.; J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1. (French) Travaux et Recherches Mathématiques, No. 17 Dunod, Paris 1968 xx+372 pp.
[37] Simon, J., Compact sets in the space \(L^p(0, T; B)\), Ann. Mat. Pura Appli., 146, 65-96 (1987) · Zbl 0629.46031
[38] Boccardo, L.; Gallouët, T., Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87, 1, 149-169 (1989) · Zbl 0707.35060
[39] Boccardo, L.; Gallouët, T., Strongly nonlinear elliptic equations having natural growth terms and \(L^1\) data, Nonlinear Anal., 19, 6, 573-579 (1992) · Zbl 0795.35031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.