Group Properties of Defective Crystal Structures

Abstract

The continuum mechanics of crystals draws on the atomistic view of crystal structure by assuming that the symmetries of perfect (atomic) crystal lattices define the material symmetry groups of associated continua. Thus, for example, the discrete (point) group which gives the rotational symmetries of a perfect cubic lattice is employed, commonly, in defining the material symmetry group of the strain energy function of a linearly elastic continuum with cubic symmetry. In fact, the general procedure has also been used successfully in the context of nonlinear elasticity theory; it amounts to a constitutive assumption which transfers some aspect of a relevant discrete structure to the continuum model. Here I begin to extend this procedure so as to encompass crystals with uniform distribution of defects. The extension is of interest because the behavior of defects is reckoned to amount for significant variations in the mechanical strength of crystalline materials. So an understanding of the geometry of configurations of points that represent uniform distributions of defects is a prerequisite for constructing energy functions that model continua where relevant defect densities are non-zero, if we accept that the symmetries of such configurations define the material symmetry group of the corresponding continuum. The paper investigates the structure of these configurations.