Gilles Francfort, Alexander Mielke
Existence results for a class of rate-independent material models with nonconvex elastic energies
1. Introduction
In mechanics, rate-independent evolutionary problems have always played an important role, e.g., in Coulomb friction or in perfect plasticity. The intrinsic nonsmoothness made these models difficult to handle mathematically. Only the development of variational inequalities, see e.g. [G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, Springer-Verlag, Berlin 1976. Translated from the French by C. W. John, Grundl. Math. Wiss. 219 (1976).] paved a way for their treatment. In particular, the theory of linear elastoplasticity led to major advances, see [C. Johnson, Existence theorems for plasticity problems, J. Math. Pures Appl. (9) 55(4) (1976), 431–444.], [J.-J. Moreau, Application of convex analysis to the treatment of elastoplastic systems, in: P. Germain and B. Nayroles, eds., Applications of methods of functional analysis to problems in mechanics, Springer Lect. Notes Math. 503 (1976), 56–89.], [P.-M. Suquet, Sur les équations de la plasticité: existence et régularité des solutions, J. Mécanique 20(1) (1981), 3–39.]. However, despite a consistent mathematical formulation of general material models within the framework of standard generalized materials (cf. [P. Germain, Q. Nguyen, and P. Suquet, Continuum thermodynamics, J. Appl. Mech. (Trans. ASME) 50 (1983), 1010–1020.]), the theory remained restricted to the case of linear evolutionary variational inequalities which are usually written in the form
where X is a Hilbert space with scalar product
, A is a bounded, symmetric and positive definite operator and
is the exterior forcing term. The dissipation functional
is assumed to be homogeneous of degree 1, lower semi-continuous and convex. Rate independence means that if y is a solution for the loading ℓ, then for each strictly monotone time reparametrization α the function y ○ α solves (1.1) for the loading ℓ ○ α. To see this, note that the left-hand side in (1.1) is homogeneous of degree 1 in (? v).
Journal fur die reine und angewandte Mathematik (Crelles Journal), Walter de Gruyter
Print ISSN: 0075-4102
Volume: 2006, 06/2006
Pages: 55 - 91
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