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.
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Research partially supported by LC06052 (MŠMT), MSM21620839 (MŠMT), A1077402 (GAČR), by the Deutsche Forschungsgemeinschaft under MATHEON C18 and under SFB404 C7, by the European Union under HPRN-CT-2002-00284 Smart Systems, and by the Alexander von Humboldt-Stiftung. Both, TR and US gratefully acknowledge the kind hospitality of the WIAS, where this research was initiated.
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Mielke, A., Roubíček, T. & Stefanelli, U. Γ-limits and relaxations for rate-independent evolutionary problems. Calc. Var. 31, 387–416 (2008). https://doi.org/10.1007/s00526-007-0119-4
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DOI: https://doi.org/10.1007/s00526-007-0119-4