2016, Thomas, Marita, Zanini, Chiara

We study a model for the rate-independent evolution of cohesive zone delamination in a viscoelastic solid, also exposed to dynamics effects. The main feature of this model, inspired by [OP99], is that the surface energy related to the crack opening depends on the history of the crack separation between the two sides of the crack path, and allows for different responses upon loading and unloading. Due to the presence of multivalued and unbounded operators featuring non-penetration and the memory-constraint in the strong formulation of the problem, we prove existence of a weaker notion of solution, known as semistable energetic solution, pioneered in [Rou09] and refined in [RT15a].

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.

2017, Rossi, Riccarda, Thomas, Marita

We revisit the weak, energetic-type existence results obtained in [RT15] for a system for rateindependent, 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.

2020, Bartels, Sören, Milicevic, Marijo, Thomas, Marita, Tornquist, Sven, Weber, Nico

Damage and fracture phenomena are related to the evolution of discontinuities both in space and in time. This contribution deals with methods from mathematical and numerical analysis to handle these: Suitable mathematical formulations and time-discrete schemes for problems with discontinuities in time are presented. For the treatment of problems with discontinuities in space, the focus lies on FE-methods for minimization problems in the space of functions of bounded variation. The developed methods are used to introduce fully discrete schemes for a rate-independent damage model and for the viscous approximation of a model for dynamic phase-field fracture. Convergence of the schemes is discussed.

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.