Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 3, p. 477-504

We consider a viscoelastic solid in Kelvin-Voigt rheology exhibiting also plasticity with hardening and coupled with heat-transfer through dissipative heat production by viscoplastic effects and through thermal expansion and corresponding adiabatic effects. Numerical discretization of the thermodynamically consistent model is proposed by implicit time discretization, suitable regularization, and finite elements in space. Fine a-priori estimates are derived, and convergence is proved by careful successive limit passage. Computational 3D simulations illustrate an implementation of the method as well as physical effects of residual stresses substantially depending on rate of heat treatment.

DOI : https://doi.org/10.1051/m2an/2010063
Classification:  35K85,  49S05,  65M60,  74C05,  80A17
Keywords: thermodynamics of plasticity, Kelvin-Voigt rheology, hardening, thermal expansion, adiabatic effects, finite element method, implicit time discretization, convergence
@article{M2AN_2011__45_3_477_0,
     author = {Bartels, S\"oren and Roub\'\i \v cek, Tom\'a\v s},
     title = {Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {3},
     year = {2011},
     pages = {477-504},
     doi = {10.1051/m2an/2010063},
     zbl = {1267.74037},
     mrnumber = {2804647},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2011__45_3_477_0}
}
Bartels, Sören; Roubíček, Tomáš. Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 3, pp. 477-504. doi : 10.1051/m2an/2010063. http://www.numdam.org/item/M2AN_2011__45_3_477_0/

[1] C. Agelet De Saracibar, M. Cervera and M. Chiumenti, On the formulation of coupled thermoplastic problems with phase-change. Int. J. Plasticity 15 (1999) 1-34. | Zbl 1054.74035

[2] J. Alberty, C. Carstensen and S.A. Funken, Remarks around 50 lines of Matlab: short finite element implementation. Numer. Algorithms 20 (1999) 117-137. | MR 1709562 | Zbl 0938.65129

[3] S. Bartels and T. Roubíček, Thermoviscoplasticity at small strains. ZAMM 88 (2008) 735-754. | MR 2488604 | Zbl 1153.74011

[4] L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data. J. Funct. Anal. 147 (1997) 237-258. | MR 1453181 | Zbl 0887.35082

[5] L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87 (1989) 149-169. | Zbl 0707.35060

[6] L. Boccardo and T. Gallouët, Summability of the solutions of nonlinear elliptic equations with right hand side measures. J. Convex Anal. 3 (1996) 361-365. | MR 1448062 | Zbl 0869.35108

[7] B.A. Boley and J.H. Weiner, Theory of thermal stresses. J. Wiley (1960), Dover edition (1997). | MR 112414 | Zbl 1234.74001

[8] S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Springer, second edition, New York (2002). | MR 1894376 | Zbl 1135.65042

[9] O. Bruhns and J. Mielniczuk, Zur Theorie der Verzweigungen nicht-isothermer elastoplastischer Deformationen. Ing.-Arch. 46 (1977) 65-74. | Zbl 0357.73034

[10] M. Canadija and J. Brnic, Associative coupled thermoplasticity at finite strain with temperature-dependent material parameters. Int. J. Plasticity 20 (2004) 1851-1874. | Zbl 1066.74515

[11] C. Carstensen and R. Klose, Elastoviscoplastic finite element analysis in 100 lines of Matlab. J. Numer. Math. 10 (2002) 157-192. | MR 1935965 | Zbl 1099.74544

[12] G. Dal Maso, G.A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity. Arch. Rational Mech. Anal. 176 (2005) 165-225. | MR 2186036 | Zbl 1064.74150

[13] G. Dal Maso, A. Desimone and M.G. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials. Arch. Ration. Mech. Anal. 180 (2006) 237-291. | MR 2210910 | Zbl 1093.74007

[14] C. Eck, J. Jarušek and M. Krbec, Unilateral Contact Problems. Chapman & Hall/CRC, Boca Raton (2005). | MR 2128865 | Zbl 1079.74003

[15] 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) 55-91. | MR 2244798 | Zbl 1101.74015

[16] P. Hakansson, M. Wallin and M. Ristinmaa, Comparison of isotropic hardening and kinematic hardening in thermoplasticity. Int. J. Plasticity 21 (2005) 1435-1460. | Zbl 1229.74027

[17] S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis. Kluwer, Dordrecht, Part I (1997), Part II (2000).

[18] D. Knees, On global spatial regularity and convergence rates for time dependent elasto-plasticity. Math. Models Methods Appl. Sci. (2010) DOI: 10.1142/S0218202510004805. | MR 2735915 | Zbl 1207.35083

[19] G.A. Maughin, The Thermomechanics of Plasticity and Fracture. Cambridge Univ. Press, Cambridge (1992). | MR 1173212 | Zbl 0753.73001

[20] C. Miehe, A theory of large-strain isotropic thermoplasticity based on metric transformation tensor. Archive Appl. Mech. 66 (1995) 45-64. | Zbl 0844.73027

[21] A. Mielke, Evolution of rate-independent systems, in Handbook of Differential Equations: Evolut. Diff. Eqs., C. Dafermos and E. Feireisl Eds., Elsevier, Amsterdam (2005) 461-559. | MR 2182832 | Zbl 1120.47062

[22] A. Mielke and T. Roubíček, Numerical approaches to rate-independent processes and applications in inelasticity. ESAIM: M2AN 43 (2009) 399-428. | Numdam | MR 2527399 | Zbl 1166.74010

[23] A. Mielke and 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. Farwing Eds., Shaker Ver., Aachen (1999) 117-129.

[24] A. Mielke and F. Theil, On rate-independent hysteresis models. Nonlin. Diff. Eq. Appl. 11 (2004) 151-189. | MR 2210284 | Zbl 1061.35182

[25] A. Mielke, T. Roubíček and U. Stefanelli, Γ-limits and relaxations for rate-independent evolutionary problems. Calc. Var. PDE 31 (2008) 387-416. | MR 2366131 | Zbl pre05236593

[26] T.D.W. Nicholson, Large deformation theory of coupled thermoplasticity including kinematic hardening. Acta Mech. 142 (2000) 207-222. | Zbl 1027.74010

[27] P. Rosakis, A.J. Rosakis, G. Ravichandran and J. Hodowany, A thermodynamic internal variable model for the partition of plastic work into heat and stored energy in metals. J. Mech. Phys. Solids 48 (2000) 581-607. | Zbl 1005.74004

[28] T. Roubíček, Nonlinear Partial Differential Equations with Applications. Birkhäuser, Basel (2005). | Zbl 1270.35005

[29] T. Roubíček, Thermo-visco-elasticity at small strains with L1-data. Quart. Appl. Math. 67 (2009) 47-71. | MR 2495071 | Zbl 1160.74011

[30] T. Roubíček, Rate independent processes in viscous solids at small strains. Math. Methods Appl. Sci. 32 (2009) 825-862. | MR 2507935 | Zbl 1194.35226

[31] T. Roubíček, Thermodynamics of rate independent processes in viscous solids at small strains. SIAM J. Math. Anal. 42 (2010) 256-297. | MR 2596554 | Zbl 1213.35279

[32] A. Srikanth and N. Zabaras, A computational model for the finite element analysis of thermoplasticity coupled with ductile damage at fonite strains. Int. J. Numer. Methods Eng. 45 (1999) 1569-1605. | Zbl 0943.74073

[33] Q. Yang, L. Stainier and M. Ortiz, A variational formulation of the coupled thermo-mechanical boundary-value problem for general dissipative solids. J. Mech. Phys. Solids 54 (2006) 401-424. | MR 2192499 | Zbl 1120.74367

[34] H. Ziegler, A modification of Prager's hardening rule. Quart. Appl. Math. 17 (1959) 55-65. | MR 104405 | Zbl 0086.18704