Group theory and 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 the 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 the 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.