Marita Thomas

Rate-Independent Damage in Thermo-Viscoelastic Materials with Inertia

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.

Coupling Rate-Independent and Rate-Dependent Processes: Existence Results

We address the analysis of an abstract system coupling a rate-independent process with a rate-dependent 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 viscoelastic solids with inertia, and to a recently proposed model for damage with plasticity.

Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics

This note deals with the analysis of a model for partial damage, where the rate- independent, unidirectional flow 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 Roubicek [1, 2] with the methods from Lazzaroni/Rossi/Thomas/Toader [3]. The present analysis encompasses, differently from [2], the monotonicity in time of damage and the dependence of the viscous tensor on damage and temperature, and, unlike [3], a nonconstant heat capacity and a time-dependent Dirichlet loading.

Griffith Formula for Mode-III-Interface-Cracks in Strain-Hardening Compounds.

This paper focusses on the numerical computation of the energy release rate for a quasistatic mode-III-interface-crack in 2-D-compounds of strain-hardening alloys. The investigations are made within the framework of deformation theory of plasticity under the assumption of small strains. The material behavior in the plastic region is described by a special power-law, which leads to the p-Laplacian operator, with jumping coefficients on the interface. Existence and uniqueness results for a minimizer of the potential energy referring to the compound are stated. Furthermore, a so called Griffith formula for the calculation of the energy release rate corresponding to the cracked compound is presented and applied to some examples in order to determine this quantity numerically.

From Nonlinear to Linear Elasticity in a Coupled Rate-Dependent/Independent System for Brittle Delamination

We revisit the weak, energetic-type existence results obtained in (Rossi and Thomas, ESAIM Control Optim. Calc. Var. 21, 1–59, (2015)) for a system for rate-independent, brittle delamination between two visco-elastic, physically nonlinear bulk materials and explain how to rigorously extend such results to the case of visco-elastic, linearly elastic bulk materials. Our approximation result is essentially based on deducing the Mosco-convergence of the functionals involved in the energetic formulation of the system. We apply this approximation result in two different situations at small strains: Firstly, to pass from a nonlinearly elastic to a linearly elastic, brittle model on the time-continuous level, and secondly, to pass from a time-discrete to a time-continuous model using an adhesive contact approximation of the brittle model, in combination with a vanishing, super-quadratic regularization of the bulk energy. The latter approach is beneficial if the model also accounts for the evolution of temperature.

On Some Extension of Energy-Drift-Diffusion Models: Gradient Structure for Optoelectronic Models of Semiconductors

We derive an optoelectronic model based on a gradient formulation for the relaxation of electron-, hole- and photon-densities to their equilibrium state. This leads to a coupled system of partial and ordinary differential equations, for which we discuss the isothermal and the non-isothermal scenario separately.