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Title: [Gamma]-limits and relaxations for rate-independent evolutionary problems
Authors: Mielke, AlexanderToubíček, TomášStefanelli, Ulisse
Issue Date: 2006
Published in: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik , Volume 1156, ISSN 0946-8633
Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
Abstract: This work uses the energetic formulation of rate-independent systems that is based on the stored-energy functionals ε and the dissipation distance D. For sequences (ε k)k ∈ ℕ and (D k)k ∈ ℕ we address the question under which conditions the limits q∞ of solutions qk: [0,T] → Q satisfy a suitable limit problem with limit functionals ε∞ and D∞, which are the corresponding Γ-limits. We derive a sufficient condition, called emphconditional 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 emphjoint 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 convergece 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.
Keywords: Rate-independent problems; energetic formulation; Gamma convergence; relaxation; time-incremental minimization; joint recovery sequence
DDC: 510
License: This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties.
Dieses Dokument darf im Rahmen von § 53 UrhG zum eigenen Gebrauch kostenfrei heruntergeladen, gelesen, gespeichert und ausgedruckt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden.
Appears in Collections:Mathematik

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