TY - GEN A1 - Heinemann, Christian A1 - Kraus, Christiane T1 - Complete damage in linear elastic materials — Modeling, weak formulation and existence results N2 - In this work, we introduce a degenerating PDE system with a time-depending domain for complete damage processes under time-varying Dirichlet boundary conditions. The evolution of the system is described by a doubly nonlinear differential inclusion for the damage process and a degenerating quasi-static balance equation for the displacement field which are strongly nonlinearly coupled. In our proposed model, the material may completely disintegrate which is indispensable for a realistic modeling of damage processes in elastic materials. Complete damage theories lead to several mathematical problems since, for instance, coercivity properties of the free energy are lost and, therefore, several difficulties arise. For the introduced complete damage model, we propose a classical formulation and a corresponding suitable weak formulation in an $SBV$-framework. The main aim is to prove existence of weak solutions for the introduced degenerating model. In addition, we show that the classical differential inclusion can be regained from the notion of weak solutions under certain regularity assumptions which is a novelty in the theory of complete damage models of this type. For the existence results, we had to handle the following problem: During the damage process it might occur that not completely damaged material regions are isolated from the Dirichlet boundary. In this case, the deformation field cannot be controlled in the transition from incomplete to complete damage. To tackle this problem, we consider the evolution process on a time-depending domain. In this context, two major challenges arise: Firstly, the time-dependent domain approach leads to jumps in the energy which have to be accounted for in the energy inequality of the notion of weak solutions. To handle this problem, several energy estimates are established by $\Gamma$-convergence techniques. Secondly, the time-depending domain might have bad smoothness properties such that Korn's inequality cannot be applied. To this end, a covering result for such sets with smooth compactly embedded domains has been shown. KW - complete damage KW - doubly nonlinear differential inclusions KW - existence results KW - elliptic-parabolic systems KW - rate-dependent systems KW - energetic solution KW - weak solution KW - linear elasticity Y1 - 2013 UR - https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1193 UR - https://nbn-resolving.org/urn:nbn:de:0296-matheon-11933 ER -