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Author: Mielke, Alexander × DDC: 510 × Subject: energetic solutions ×

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Now showing 1 - 4 of 4
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    Damage of nonlinearly elastic materials at small strain : existence and regularity results
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Thomas, Marita; Mielke, Alexander
    Literaturverz. S. 31 In this paper an existence result for energetic solutions of rate-independent damage processes is established and the temporal regularity of the solution is discussed. We consider a body consisting of a physically nonlinearly elastic material undergoing small deformations and partial damage. The present work is a generalization of [Mielke-Roubicek 2006] concerning the properties of the stored elastic energy density as well as the suitable Sobolev space for the damage variable: While previous work assumes that the damage variable z satisfies z ? W^1,r (Omega) with r>d for Omega ? R^d, we can handle the case r>1 by a new technique for the construction of joint recovery sequences. Moreover, this work generalizes the temporal regularity results to physically nonlinearly elastic materials by analyzing Lipschitz- and Hölder-continuity of solutions with respect to time.
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    Averaging of time-periodic dissipation potentials in rate-independent processes : dedicated to Tomáš Roubícek on the occasion of his sixtieth birthday
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Heida, Martin; Mielke, Alexander
    We study the existence and well-posedness of rate-independent systems (or hysteresis operators) with a dissipation potential that oscillates in time with period. In particular, for the case of quadratic energies in a Hilbert space, we study the averaging limit → 0 and show that the effective dissipation potential is given by the minimum of all friction thresholds in one period, more precisely as the intersection of all the characteristic domains. We show that the rates of the process do not converge weakly, hence our analysis uses the notion of energetic solutions and relies on a detailed estimates to obtain a suitable qui-continuity of the solutions in the limit → 0.
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    Rate-independent elastoplasticity at finite strains and its numerical approximation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Mielke, Alexander; Roubíc̆ek, Tomáš
    Gradient plasticity at large strains with kinematic hardening is analyzed as quasistatic rate-independent evolution. The energy functional with a frame-indifferent polyconvex energy density and the dissipation are approximated numerically by finite elements and implicit time discretization, such that a computationally implementable scheme is obtained. The non-selfpenetration as well as a possible frictionless unilateral contact is considered and approximated numerically by a suitable penalization method which keeps polyconvexity and simultaneously by-passes the Lavrentiev phenomenon. The main result concerns the convergence of the numerical scheme towards energetic solutions. In the case of incompressible plasticity and of nonsimple materials, where the energy depends on the second derivative of the deformation, we derive an explicit stability criterion for convergence relating the spatial discretization and the penalizations.
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    On evolutionary [Gamma]-convergence for gradient systems : in memory of Eduard, Waldemar, and Elli Mielke
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Mielke, Alexander
    In these notes we discuss general approaches for rigorously deriving limits of generalized gradient flows. Our point of view is that a generalized gradient system is defined in terms of two functionals, namely the energy functional E and the dissipation potential R or the associated dissipation distance. We assume that the functionals depend on a small parameter and that the associated gradient systems have solutions u. We investigate the question under which conditions the limits u of (subsequences of) the solutions u are solutions of the gradient system generated by the [Gamma]-limits E0 and R0. Here the choice of the right topology will be crucial awell as additional structural conditions. We cover classical gradient systems, where R is quadratic, and rate-independent systems as well as the passage from classical gradient to rate-independent systems. Various examples, such as periodic homogenization, are used to illustrate the abstract concepts and results.