A mathematical framework for standard generalized materials in the rate-independent case
- Mielke, Alexander
2010 Mathematics Subject Classification
- 35K85 49S05 74C15 74N15
- Rate independence, energetic formulation, Gamma-convergence, relaxation, shape-memory material, magnetostriction, piezoelectricity
Standard generalized materials are described by an elastic energy density and a dissipation potential. The latter gives rise to the evolution equation (flow law) for the internal variables. The energetic formulation provides a very weak, derivative-free form of this flow law. It is based on a global stability condition and an energy balance. Using time-incremental minimization problems, which allow for the usage of the rich theory in the direct method of the calculus of variations, it is possible to establish general, abstract existence results as well as convergence for numerical approximations. Applications to shape-memory materials and to magnetostrictive or piezoelectric materials are surveyed.
- Multifield Problems in Solid and Fluid Mechanics, R. Helmig, A. Mielke, B. Wohlmuth, eds., vol. 28 of Lecture Notes in Applied and Computational Mechanics, Springer, Heidelberg, 2006, pp. 351--379