This interesting and well-written article deals with the mathematical formulation of a model describing temporally evolving microstructure generated by phase changes (such as e.g. in single crystal alloys), and its homogenization. For the micromodel, the author uses a sharp interface approach with discontinuous order parameter \(S\) (characterizing the two phases of alloy). The exploitation of the second law of thermodynamics provides a constitutive equation for the normal velocity of phase interfaces which is then transformed into an evolution equation for \(S\). \(S\) itself belongs to the space of functions of bounded variations, its evolution equation (which is of Hamilton-Jacobi type) holds in a distributional sense, and is defined everywhere – not only at the interfaces.
To derive the homogenized problem, different techniques can be applied depending on the equations which are subjected to the homogenization procedure. Here, the homogenized equations for the displacement, stress and internal variables are obtained by using the method of asymptotic series. The homogenization of the evolution equation for the order parameter \(S\) necessitates the introduction of a family of solutions to the initial-boundary value problem (IBVP) of the micromodel depending on a fast variable. Moreover, a number of auxiliary results concerning oscillating functions of bounded variation are needed. The formulation of the resulting homogenized IBVP contains also a history functional which is defined by the solution of the IBVP in the representative volume element and which couples the mean stress to the microstress.
In the special case of temporally fixed microstructure, the author reduces the homogenized system of partial differential equations to an evolution equation of monotone type. Numerical investigations and existence and uniqueness proofs for the model proposed are not presented and are the subject of future work.