CRACK MODELS FOR A TRANSVERSELY ISOTROPIC MEDIUM

A commonly used model for a transversely isotropic cracked 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 show 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 material'' version of Hudson's (1981) model 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 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.