A degenerating Cahn-Hilliard system coupled with complete damage processes

Complete damage in elastic solids appears when the material looses all its integrity due to high exposure. In the case of alloys, the situation is quite involved since spinodal decomposition and coarsening also occur at sufficiently low temperatures which may lead locally to high stress peaks. Experimental observations on solder alloys reveal void and crack growth especially at phase boundaries. In this work, we investigate analytically a degenerating PDE system with a time-depending domain for phase separation and complete damage processes under time-varying Dirichlet boundary conditions. The evolution of the system is described by a degenerating parabolic differential equation of fourth order for the concentration, a doubly nonlinear differential inclusion for the damage process and a degenerating quasi-static balance equation for the displacement field. All these equations are strongly nonlinearly coupled. Because of the doubly degenerating character and the doubly nonlinear differential inclusion we are confronted with introducing a suitable notion of weak solutions. We choose a notion of weak solutions which consists of weak formulations of the diffusion equation and the momentum balance, a one-sided variational inequality for the damage function and an energy estimate. For the introduced degenerating system, we prove existence of weak solutions in an SBV -framework. The existence result is based on an approximation system, where the elliptic degeneracy of the displacement field and the parabolic degeneracy of the concentration are eliminated. In the framework of phase separation and damage, this means that the approximation system allows only for partial damage and a non-degenerating mobility tensor. For the approximation system, existence results are established. Then, a passage to the limit shows existence of weak solutions of the degenerating system. 1 Problem description Phase separation and damage processes occur in many fields, including material sciences, biology and chemical reactions. In particular, for the manufacturing and lifetime prediction of micro-electronic devices it is of great importance to understand the mechanisms and the interplay between phase-separation and damage processes in solder alloys. As soon as elastic alloys are quenched sufficiently, spinodal decomposition leads to a fine-grained structure of different chemical mixtures on a short time-scale (see [DM01] for numerical simulations and experimental observations). The long-term evolution is determined by a chemical diffusion process which tends to minimize the bulk and the surface energy of the chemical substances. J.W. Cahn and J.E. Hilliard developed a phenomenological model for the kinetics of phase-separation in a thermodynamically consistent framework known as the Cahn-Hilliard equation [CH58], for which an extensive mathematical literature exists. An overview of modeling and some analytical aspects of the Cahn-Hilliard equation can be found in [Ell89]. The recent literature is mainly focused on coupled systems. For instance, physical observation and numerical simulations reveal that mechanical stresses influence the developing shapes of the chemical phases. A coupling between Cahn-Hilliard systems and elastic deformations have been analytically studied in [Gar00, BCD+02, CMP00, Gar05a, Gar05b, BP05, PZ08]. For numerical results and simulations we refer [Wei01, Mer05, BB99, GRW01, BM10]. Phase separation of the chemical components may also lead to critical stresses at phase boundaries which result in cracks and formation of voids and are of particular interest to understand the aging process in solder materials, cf. [HCW91, USG07, GUaMM+07, FK09].