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\(\Gamma\)-limits and relaxations for rate-independent evolutionary problems. (English) Zbl 1302.49013

Summary: 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 \(q_{\infty}\) 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 \(\Gamma\)-limits. We derive a sufficient condition, called conditional upper semicontinuity of the stable sets, which is essential to guarantee that \(q_{\infty}\) 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 \(k \rightarrow \infty\), which in the limit can be described by an effective macroscopic model.

MSC:

49J40 Variational inequalities
49S05 Variational principles of physics
35K90 Abstract parabolic equations
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