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- ItemQuasistatic damage evolution with spatial BV-regularization(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Thomas, MaritaAn 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.
- ItemNumerical 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, MaritaWe 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.
- ItemApproximation schemes for materials with discontinuities(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Bartels, Sören; Milicevic, Marijo; Thomas, Marita; Tornquist, Sven; Weber, NicoDamage 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.