Analysis and homogenisation of a linear Cahn–Larché system with phase separation on the microscale

Abstract

AbstractWe consider the process of phase separation of a binary system under the influence of mechanical stress modelled by the Cahn–Larché system, where the mechanical deformation takes place on a macrosopic scale whereas the phase separation happens on a microscopic level. After linearisation, we prove existence and uniqueness of a weak solution by a Galerkin approach. As discretisation in space leads to a linear differential–algebraic system of equations, we adjust known solution theory for such equations to a weak setting. This approach may be of interest more generally for coupled elliptic–parabolic systems. A‐priori estimates enable us to pass to the homogenisation limit of the linear system rigorously using the concept of two‐scale convergence. A comparison with the formally homogenised full nonlinear problem shows that both systems lead to models of distributed‐microstructure type in the limit and that homogenisation and linearisation commutes.