Crack Models For A Transversely Anisotropic Medium

Abstract

A commonly used model for a transversely anisotropic crack rock is that by Hudson (1980, 1981). This model is based on a simplified analysis of a thin circular crack, with displacement and stress conditions specified on the boundary. These papers have a second order correction in addition to the first order term in porosity/crack density. In this paper we compare the results of Hudson with those of Anderson et al. (1974) and Cheng (1978) using the long wavelength static approximation and the ellipsoidal crack model first proposed by Eshelby (1957). We showed that the Hudson model and those based on the complete Eshelby theory agree for small aspect ratio cracks and small crack densities, as expected, provided the weak inclusion version of Hudson's model (1981) is used. For larger crack densities but small aspect ratios, Hudson's first order term agrees with the Eshelby solution. The expansion in the second order term in crack density is an asymptotic series and not a uniformly converging series. Thus there is no general statement one can make about the accuracy of the second order expansion that is valid for a variety of situations. A new expansion based on the Pade approximation is proposed which is identical to Hudson's expansion up to second order in density. This expansion avoids some of the problems associated with Hudson's second order expansion such as increasing moduli with crack density at relatively small crack densities.