Evolving Microstructure and Homogenization

In this article we formulate a mathematical model for the temporally evolving microstructure generated by phase changes and study the homogenization of this model. The investigations are partially formal, since we do not prove existence or convergence of solutions of the microstructure model to solutions of the homogenized problem. To model the microstructure, the sharp interface approach is used. The evolution of the interface is governed by an everywhere defined distribution partial differential equation for the characteristic function of one of the phases. This avoids the disadvantage commonly associated with this approach of an evolution equation only defined on the interface. To derive the homogenized problem, a family of solutions of the microstructure problem depending on the fast variable is introduced. The homogenized problem obtained contains a history functional, which is defined by the solution of an initial-boundary value problem in the representative volume element. In the special case of a temporally fixed microstructure the homogenized problem is reduced to an evolution equation to a monotone operator. Received March 15, 2000