Rock Mechanics - Orthotropic Relations for Rock Mechanics

The majority of the work done in the field of rock mechanics uses as a basis for analysis the classical linear theory of elasticity. Most workers in rock mechanics realize that removing the assumptions of homogeneity and isotropicity will bring the analysis of stress-strain relationships for geologic materials closer to reality. However, the algebra involved in reducing these restrictions is usually heavy. It is the purpose of this paper to relax the condition of isotropicity to that of orthotropicity and examine the sensitivity of this change for some simple geometries. In addition some necessary relationships between stress and strain for orthotropic materials will be developed. The recent years have seen an increasing interest in the field of rock mechanics. This is amply demonstrated by the number of articles appearing in the technical literature. The majority of these articles use as a basis for analysis the classical linear theory of elasticity. In addition, the assumptions that the material under consideration is elastically homogeneous and isotropic reduce the number of constants necessary to write an analytical relationship between stress and strain to two, e.g., Young's modulus and Poisson's ratio. Most workers in the area of rock mechanics realize that removing the conditions of homogeneity and isotropicity will bring the analysis of stress-strain relationship for geologic materials closer to reality. However, the algebra involved in analysis without imposing these restrictions is usually heavy. It is the purpose of this paper to relax the condition of isotropicity to that of orthotropicity and examine the sensitivity of this change on stress concentration for some simple geometries. In addition, it is shown that the expression for maximum stress concentration for some simple geometries considered can be further simplified if further assumptions are made. The maximum stress concentration, under these additional simplifying assumptions, is shown graphical1y. In the linear theory of elasticity, generalized Hooke's law assumes the following form