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- ItemRate-independent evolution of sets(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Rossi, Riccarda; Stefanelli, Ulisse; Thomas, MaritaThe goal of this work is to analyze a model for the rate-independent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of a given time-dependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes. In the mathematical modeling of this process, we distinguish the adhesive case, in which the constraint that the (complement of) the `external load' contains the evolving sets is penalized by a term contributing to the driving energy functional, from the brittle case, enforcing this constraint. The existence of Energetic solutions for the adhesive system is proved by passing to the limit in the associated time-incremental minimization scheme. In the brittle case, this time-discretization procedure gives rise to evolving sets satisfying the stability condition, but it remains an open problem to additionally deduce energy-dissipation balance in the time-continuous limit. This can be obtained under some suitable quantification of data. The properties of the brittle evolution law are illustrated by numerical examples in two space dimensions.
- ItemOn evolutionary [Gamma]-convergence for gradient systems : in memory of Eduard, Waldemar, and Elli Mielke(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Mielke, AlexanderIn 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.
- ItemCoupling rate-independent and rate-dependent processes: Existence results(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Rossi, Riccarda; Thomas, MaritaWe address the analysis of an abstract system coupling a rate-independent process with a second order (in time) nonlinear evolution equation. We propose suitable weak solution concepts and obtain existence results by passing to the limit in carefully devised time-discretization schemes. Our arguments combine techniques from the theory of gradient systems with the toolbox for rate-independent evolution, thus reflecting the mixed character of the problem. Finally, we discuss applications to a class of rate-independent processes in visco-elastic solids with inertia, and to a recently proposed model for damage with plasticity.
- ItemAveraging 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, AlexanderWe 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.
- ItemA rate-independent gradient system in damage coupled with plasticity via structured strains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Bonetti, Elena; Rocca, Elisabetta; Rossi, Riccarda; Thomas, MaritaThis contribution deals with a class of models combining isotropic damage with plasticity. It has been inspired by a work by Freddi and Royer-Carfagni [FRC10], including the case where the inelastic part of the strain only evolves in regions where the material is damaged. The evolution both of the damage and of the plastic variable is assumed to be rate-independent. Existence of solutions is established in the abstract energetic framework elaborated by Mielke and coworkers (cf., e.g., [Mie05, Mie11b]).
- ItemExistence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Heinemann, Christian; Kraus, ChristianeIn this paper, we consider a coupled PDE system describing phase separation and damage phenomena in elastically stressed alloys in the presence of inertial effects. The material is considered on a bounded Lipschitz domain with mixed boundary conditions for the displacement variable. The main aim of this work is to establish existence of weak solutions for the introduced hyperbolic-parabolic system. To this end, we first adopt the notion of weak solutions introduced in [HK11]. Then we prove existence of weak solutions by means of regularization, time-discretization and different variational techniques.
- ItemRate-independent elastoplasticity at finite strains and its numerical approximation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Mielke, Alexander; Roubíc̆ek, TomášGradient 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.
- ItemSome remarks on a model for rate-independent damage in thermo-visco-elastodynamics(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Lazzaroni, Giuliano; Rossi, Riccarda; Thomas, Marita; Toader, RodicaThis note deals with the analysis of a model for partial damage, where the rate-independent, unidirectional ow rule for the damage variable is coupled with the rate-dependent heat equation, and with the momentum balance featuring inertia and viscosity according to Kelvin-Voigt rheology. The results presented here combine the approach from Roubí£ek [Rou09, Rou10] with the methods from Lazzaroni/Rossi/Thomas/Toader [LRTT14]. The present analysis encompasses, dierently from [Rou10], the monotonicity in time of damage and the dependence of the viscous tensor on damage and temperature, and, unlike [LRTT14], a nonconstant heat capacity and a time-dependent Dirichlet loading.
- ItemModeling and analysis of a phase field system for damage and phase separation processes in solids(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bonetti, Elena; Heinemann, Christian; Kraus, Christiane; Segatti, AntonioIn this work, we analytically investigate a multi-component system for describing phase separation and damage processes in solids. The model consists of a parabolic diffusion equation of fourth order for the concentration coupled with an elliptic system with material dependent coefficients for the strain tensor and a doubly nonlinear differential inclusion for the damage function. The main aim of this paper is to show existence of weak solutions for the introduced model, where, in contrast to existing damage models in the literature, different elastic properties of damaged and undamaged material are regarded. To prove existence of weak solutions for the introduced model, we start with an approximation system. Then, by passing to the limit, existence results of weak solutions for the proposed model are obtained via suitable variational techniques.
- ItemExistence of weak solutions for a hyperbolic-parabolic phase field system with mixed boundary conditions on non-smooth domains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Heinemann, Christian; Kraus, ChristianeThe aim of this paper is to prove existence of weak solutions of hyperbolic-parabolic evolution inclusions defined on Lipschitz domains with mixed boundary conditions describing, for instance, damage processes and elasticity with inertial effects. To this end, we first present a suitable weak formulation in order to deal with such evolution inclusions. Then, existence of weak solutions is proven by utilizing time-discretization, H2-regularization and variational techniques.