Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion*
Institut für Numerische Simulation,
Rheinische Friedrich-Wilhelms-Universität Bonn, Wegelerstraße 6,
53115 Bonn, Germany. email@example.com
2 Mathematical Institute, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic.
3 Institute of Thermomechanics of the ASCR, Dolejškova 5, 182 00 Praha 8, Czech Republic.
Revised: 17 June 2010
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.
Mathematics Subject Classification: 35K85 / 49S05 / 65M60 / 74C05 / 80A17
Key words: Thermodynamics of plasticity / Kelvin-Voigt rheology / hardening / thermal expansion / adiabatic effects / finite element method / implicit time discretization / convergence
The authors warmly thank Professor Alexander Mielke and Dr. JiříPlešek for many fruitful discussions, and an anonymous referee for many helpful comments on an earlier version of the manuscript. S.B. acknowledges support by “Nečas center for mathematical modeling” LC 06052 (MŠMT ČR). T.R. acknowledges the hospitality of SFB 611 “Singular phenomena and scaling in mathematical models” of the University of Bonn, as well as partial support from the grants A 100750802 (GA AV ČR), 201/09/0917, 106/09/1573, and 201/10/0357 (GA ČR), and MSM 21620839 (MŠMT ČR), and from the research plan AV0Z20760514 (ČR).
© EDP Sciences, SMAI, 2010