Mathematical approaches to the study of smart materials

The smartness of a shape-memory material is a consequence of its ability to form a flexible variant structure at one temperature while recognizing only a homogeneous equilibrium at a different temperature. The fine scale morphology or microstructure of this variant structure has a clear role in the macroscopic behavior of the material. To investigate these phenomena, two issues are paramount. First, the presence of several stable variants at a given temperature reflects a complicated potential well structure for the free energy of the material. Second, the presence of spatially oscillatory behavior at the small scale suggests competition between the free energy of the material and loading or other environmental effects. Both of these features represent highly nonlinear processes and thus it is to nonlinear analysis we turn for methods to successfully describe these systems. In this report we describe in an expository fashion one such technique which has been applied in several instances especially related to certain alloys or other crystalline materials.