Γ-limits and relaxations for rate-independent evolutionary problems

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.