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Has files: Yes × Subject: Generalized gradient systems × Subject: energy-dissipation principle ×

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    Balanced-Viscosity solutions for multi-rate systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Mielke, Alexander; Rossi, Riccarda; Savaré, Giuseppe
    Several mechanical systems are modeled by the static momentum balance for the displacement u coupled with a rate-independent flow rule for some internal variable z. We consider a class of abstract systems of ODEs which have the same structure, albeit in a finite-dimensional setting, and regularize both the static equation and the rate-independent flow rule by adding viscous dissipation terms with coefficients α and , where 0<<1 and α>0 is a fixed parameter. Therefore for α different from 1 the variables u and z have different relaxation rates. We address the vanishing-viscosity analysis as tends to 0 in the viscous system. We prove that, up to a subsequence, (reparameterized) viscous solutions converge to a parameterized curve yielding a Balanced Viscosity solution to the original rate-independent system and providing an accurate description of the system behavior at jumps. We also give a reformulation of the notion of Balanced Viscosity solution in terms of a system of subdifferential inclusions, showing that the viscosity in u and the one in z are involved in the jump dynamics in different ways, according to whether α >1, α=1, or 0<α<1.
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    Variational approaches and methods for dissipative material models with multiple scales
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Mielke, Alexander
    In a first part we consider evolutionary systems given as generalized gradient systems and discuss various variational principles that can be used to construct solutions for a given system or to derive the limit dynamics for multiscale problems. These multiscale limits are formulated in the theory of evolutionary Gamma-convergence. On the one hand we consider the a family of viscous gradient system with quadratic dissipation potentials and a wiggly energy landscape that converge to a rate-independent system. On the other hand we show how the concept of Balanced-Viscosity solution arise as in the vanishing-viscosity limit. As applications we discuss, first, the evolution of laminate microstructures in finite-strain elastoplasticity and, second, a two-phase model for shape-memory materials, where H-measures are used to construct the mutual recovery sequences needed in the existence theory.