13 results
Search Results
Now showing 1 - 10 of 13
- 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.
- 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.
- ItemDamage of nonlinearly elastic materials at small strain : existence and regularity results(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Thomas, Marita; Mielke, AlexanderLiteraturverz. 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.
- 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.
- ItemFrom adhesive to brittle delamination in visco-elastodynamics(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Rossi, Riccarda; Thomas, MaritaIn this paper we analyze a system for brittle delamination between two visco-elastic bodies, also subject to inertia, which can be interpreted as a model for dynamic fracture. The rate-independent flow rule for the delamination parameter is coupled with the momentum balance for the displacement, including inertia. This model features a nonsmooth constraint ensuring the continuity of the displacements outside the crack set, which is marked by the support of the delamination parameter. A weak solvability concept, generalizing the notion of energetic solution for rate-independent systems to the present mixed rate-dependent/rate-independent frame, is proposed. Via refined variational convergence techniques, existence of solutions is proved by passing to the limit in approximating systems which regularize the nonsmooth constraint by conditions for adhesive contact. The presence of the inertial term requires the design of suitable recovery spaces small enough to provide compactness but large enough to recover the information on the crack set in the limit.
- 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.
- 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.
- 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.
- 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.
- ItemRate-independent damage in thermo-viscoelastic materials with inertia(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Lazzaroni, Giuliano; Rossi, Riccarda; Thomas, Marita; Toader, RodicaWe present a model for rate-independent, unidirectional, partial damage in visco-elastic materials with inertia and thermal effects. The damage process is modeled by means of an internal variable, governed by a rate-independent flow rule. The heat equation and the momentum balance for the displacements are coupled in a highly nonlinear way. Our assumptions on the corresponding energy functional also comprise the case of the Ambrosio-Tortorelli phase-field model (without passage to the brittle limit). We discuss a suitable weak formulation and prove an existence theorem obtained with the aid of a (partially) decoupled time-discrete scheme and variational convergence methods. We also carry out the asymptotic analysis for vanishing viscosity and inertia and obtain a fully rate-independent limit model for displacements and damage, which is independent of temperature.