Energy minimization and nonlinear problems in polycrystalline solids

Abstract

Common engineering structural materials -- metallic alloys and ceramics -- are polycrystalline. They are made up of a very large number of grins which have identical crystal structure, but which are oriented differently. The properties of the material depend critically on the texture, by which one means the size, the shape and the orientation distributions of the different grains. If we can systematically understand this dependence, we can identify textures which provide the best possible properties and then try to design a processing technique which gives rise to the texture. Linear properties -- elastic moduli, conductivity etc. -- have received much attention and there are by now many sophisticated methods to study them (see [1] for a recent example). Some nonlinear properties have also been studied extensively -- for example, there has been much work in polycrystal plasticity beginning with the pioneering work of Taylor [2] -- though in general much less is known here. This paper highlights some recent successes in modelling the behavior of polycrystalline solids using energy minimization. Two examples -- the splitting of ceramics subjected to compressive loads and shape-memory polycrystals -- are described.