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    Some 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, Rodica
    This 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.
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    Quasistatic damage evolution with spatial BV-regularization
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Thomas, Marita
    An existence result for energetic solutions of rate-independent damage processes is established. We consider a body consisting of a physically linearly elastic material undergoing infinitesimally small deformations and partial damage. In [ThomasMielke10DamageZAMM] an existence result in the small strain setting was obtained under the assumption that the damage variable z satisfies z∈ W1,r(Ω) with r∈(1,∞) for Ω⊂Rd. We now cover the case r=1. The lack of compactness in W1,1(Ω) requires to do the analysis in BV(Ω). This setting allows it to consider damage variables with values in 0,1. We show that such a brittle damage model is obtained as the Γ-limit of functionals of Modica-Mortola type.
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    Rate-independent damage in thermo-viscoelastic materials with inertia
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Lazzaroni, Giuliano; Rossi, Riccarda; Thomas, Marita; Toader, Rodica
    We 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.
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    Numerical approach to a model for quasistatic damage with spatial BV-regularization
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Bartels, Sören; Milicevic, Marijo; Thomas, Marita
    We address a model for rate-independent, partial, isotropic damage in quasistatic small strain linear elasticity, featuring a damage variable with spatial BV-regularization. Discrete solutions are obtained using an alternate time-discrete scheme and the Variable-ADMM algorithm to solve the constrained nonsmooth optimization problem that determines the damage variable at each time step. We prove convergence of the method and show that discrete solutions approximate a semistable energetic solution of the rate-independent system. Moreover, we present our numerical results for two benchmark problems.
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    Fully discrete approximation of rate-independent damage models with gradient regularization
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Bartels, Sören; Milicevic, Marijo; Thomas, Marita; Weber, Nico
    This work provides a convergence analysis of a time-discrete scheme coupled with a finite-element approximation in space for a model for partial, rate-independent damage featuring a gradient regularization as well as a non-smooth constraint to account for the unidirectionality of the damage evolution. The numerical algorithm to solve the coupled problem of quasistatic small strain linear elasticity with rate-independent gradient damage is based on a Variable ADMM-method to approximate the nonsmooth contribution. Space-discretization is based on P1 finite elements and the algorithm directly couples the time-step size with the spatial grid size h. For a wide class of gradient regularizations, which allows both for Sobolev functions of integrability exponent r ∈ (1, ∞) and for BV-functions, it is shown that solutions obtained with the algorithm approximate as h → 0 a semistable energetic solution of the original problem. The latter is characterized by a minimality property for the displacements, a semistability inequality for the damage variable and an energy dissipation estimate. Numerical benchmark experiments confirm the stability of the method.