2010, Heinemann, Christian, Kraus, Christiane

The Cahn-Hilliard model is a typical phase field approach for describing phase separation and coarsening phenomena in alloys. This model has been generalized to the so-called Cahn-Larché system by combining it with elasticity to capture non-neglecting deformation phenomena, which occurs during phase separation in the material. Evolving microstructures such as phase separation and coarsening processes have a strong influence on damage initiation and propagation in alloys. We develop the existing framework of Cahn-Hilliard and Cahn-Larché systems by coupling these systems with a unidirectional evolution inclusion for an internal variable, describing damage processes. After establishing a weak notion of the corresponding evolutionary system, we prove existence of weak solutions for rate-dependent damage processes under certain growth conditions of the energy functional

2012, Heinemann, Christian, Kraus, Christiane

We introduce a complete damage model with a time-depending domain for linear-elastically stressed solids 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 quasi-static balance equation for the displacement field. For the introduced complete damage model, we propose a classical formulation and a corresponding suitable weak formulation in an S BV-framework. We show that the classical differential inclusion can be regained from the notion of weak solutions under certain regularity assumptions. The main aim of this work is to prove local-in-time existence and global-in-time existence in some weaker sense for the introduced model. In contrast to incomplete damage theories, the material can be exposed to damage in the proposed model in such a way that the elastic behavior may break down on the damaged parts of the material, i.e. we loose coercivity properties of the free energy. This leads to several mathematical difficulties. For instance, 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.

2013, Bonetti, Elena, Heinemann, Christian, Kraus, Christiane, Segatti, Antonio

In this work, we analytically investigate a multi-component system for describing phase separation and damage processes in solids. The model consists of a parabolic diffusion equation of fourth order for the concentration coupled with an elliptic system with material dependent coefficients for the strain tensor and a doubly nonlinear differential inclusion for the damage function. The main aim of this paper is to show existence of weak solutions for the introduced model, where, in contrast to existing damage models in the literature, different elastic properties of damaged and undamaged material are regarded. To prove existence of weak solutions for the introduced model, we start with an approximation system. Then, by passing to the limit, existence results of weak solutions for the proposed model are obtained via suitable variational techniques.