Representation of Microstructure and Mechanical Behavior of Polycrystals

Abstract

We review the group theoretical basis of the result that the observed mechanical behavior of a material can be represented by constitutive (differential) equations which govern the evolution of state variables and that these variables are even rank irreducible tensors. On the other hand microscopic observations of internal structure of a polycrystal produce functions that are defined on “curved” objects such as the unit sphere of directions or the set of distinct orientations of a cube, etc. We show, in terms of an example (the crystallite orientation distribution function for a macroscopically homogeneous polycrystal composed of grains of a cubic crystalline solid) that representations of such functions give rise to Fourier coefficients that are also irreducible tensors. The tensorial state variables will be related to these tensorial Fourier coefficients. A major problem of mechanics of materials is to develop methods that enable one, for a given material and for a given purpose, to extract tensorial state variables and the laws for their evolution from the knowledge obtained from the studies of the microstructure and behavior of the material.