Γ-limits and relaxations for rate-independent evolutionary problems

Abstract

This work uses the energetic formulation of rate-independent systems that is based on the stored-energy functionals ${\mathcal{E}}$ and the dissipation distance ${\mathcal{D}}$ . For sequences $({\mathcal{E}}_k)_{k\in {\mathbb{N}}}$ and $({\mathcal{D}}_k)_{k\in {\mathbb{N}}}$ we address the question under which conditions the limits q _∞ of solutions $q_k : [0, T]\to {\mathcal{Q}}$ satisfy a suitable limit problem with limit functionals ${\mathcal{E}}_\infty$ and ${\mathcal{D}}_\infty$ , which are the corresponding Γ-limits. We derive a sufficient condition, called conditional upper semi-continuity of the stable sets, which is essential to guarantee that q _∞ solves the limit problem. In particular, this condition holds if certain joint recovery sequences exist. Moreover, we show that time-incremental minimization problems can be used to approximate the solutions. A first example involves the numerical approximation of functionals using finite-element spaces. A second example shows that the stop and the play operator converge if the yield sets converge in the sense of Mosco. The third example deals with a problem developing microstructure in the limit k → ∞, which in the limit can be described by an effective macroscopic model.