Group Theoretic Formulation for Elastic Properties of Media Composed of Anisotropic Layers

Abstract

A matrix formalism allows for the simple evaluation of the anisotropic homogeneous medium equivalent, in the long wavelength limit, to a distribution of fine layers, each layer itself being an elastic anisotropic medium. A given cumulative thickness of fine layers of an anisotropic constituent can be shown to be transformable to an element of a commutative group. Combining group elements G A and G B (corresponding to cumulative thickness H A of thin layers of constituent A and cumulative thickness H B of of constituent B, respectively) gives the group element corresponding to total thickness H A +H B of the homogeneous medium equivalent to the interleaved layers of constituents A and B. The inverse element gives us the notion of ‘uncombining’ layers by addition of the inverse; then, if the remaining layer is a stable anisotropic medium, a valid decomposition of the original anisotropic medium is revealed. Systems of parallel fractures and aligned microcracks (of assumed cumulative thickness zero) also are transformable to group elements which can be manipulated as easily as an element corresponding to any other set of layers. Such systems may be azimuthally anisotropic relative to their normal axis, the anisotropy being due either to fracture or crack geometry or to the anisotropy of the infilling material. Multiple fracture or crack systems are dealt with by successive coordinate system rotations and additions of group elements.