Existence results for diffuse interface models describing phase separation and damage

Abstract

In this paper, we analytically investigate multi-component Cahn-Hilliard and Allen-Cahn systems which are coupled with elasticity and uni-directional damage processes. The free energy of the system is of the form Ω 1 2 Γ ∇c : ∇c + 1 2 |∇z| 2 + W ch (c) + W el (e, c, z) dx with a polynomial or logarithmic chemical energy density W ch , an inhomogeneous elastic energy density W el and a quadratic structure of the gradient of damage variable z. For the corresponding elastic Cahn-Hilliard and Allen-Cahn systems coupled with uni-directional damage processes, we present an appropriate notion of weak solutions and prove existence results based on certain regularization methods and a higher integrability result for strain e.

A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains

“…In the recent [HK11], different techniques have been adopted to analyze models coupling damage with phase separation processes in elastic bodies (see also [HK13]). Also in [HK11], a q-Laplacian regularization with q > d (d being the space dimension) is used in order to ensure C 0 (Ω)-regularity for z.…”

Modeling and analysis of a phase field system for damage and phase separation processes in solids

“…The next Proposition (see [28,31]) collects the basic properties of this concept of weak solution. In particular, note that the sole condition (ii) is weaker than the usual variational inequality that characterizes the doubly nonlinear inclusion (i).…”

“…Note that we can choose r = 0 in (9) due to Φ(0) = Φ (0) = 0, see Lemma 3.7 and Remark 3.8 in [31] for details. q = (u, c, w, z) is called a weak solution of the system (S 0 ) with the initial-boundary conditions (IBC) if the following properties are satisfied:…”

“…A local in time existence result for a complete dissipative damage model with the evolving of temperature can be found in [6]. A coupled system describing incomplete damage, linear elasticity and phase separation appeared firstly in [28,31]. There, existence of weak solutions has been proven under mild assumptions, where, for instance, the stiffness tensor may be material-dependent and the chemical free energy may be of polynomial or logarithmic type.…”

“…The main aim of this work is to show existence of weak solutions of a unified model for phase separation and damage processes, where, in contrast to the existing incomplete damage models in the literature [45,34,28,31] or local in time damage evolution [9,10], different stress-strain relations of damaged and undamaged material are regarded. More precisely, we choose an elastic energy density W el of the form…”