Comment on “Crack models for a transversely isotropic medium” by C. H. Cheng and comment by C. M. Sayers

Sayers [1993] commented that the expressions of Cheng [1993a] for extending the equations of Hudson [1980, 19811 for the elastic constants of distributions of parallel cracks to large crack densities were nonphysical. He offered the formulations of Sayers and Kachanov [ 19911 as preferable alternatives, and Cheng [ 1993b] accepted these comments. In this further comment, I argue that the large crack densities suggested by both Sayers and Cheng when applied to real rock represent such pronounced fracturing that the rock would be fragmentary, the long-wavelength limit would be exceeded, and effective media concepts would no longer apply. Crampin [1978] and Hudson [1980, 19811 developed expressions for the effective elastic constants, in the longwavelength limit, modeling seismic wave propagation through solids containing weak distributions of parallel noninteracting cracks. The elastic constants specify purely elastic anisotropic solids whose variations of velocity match those of two-phase isotropic solids containing regular distributions of parallel cracks. Such effective medium models are important because they allow the behavior of complicated two-phase solids, which are difficult to calculate, to be simulated by calculations using uniform elastic models. These models are valid when the seismic wavelengths are much greater than the dimensions of the cracks (5 or 10 times greater are usually considered sufficient). The expressions of Hudson are correct to the second order in perturbations from isotropy and it is found that the series approximations begin to diverge for crack densities greater than about E = 0.1 [Crampin, 19841. Hudson : 19861 gives expressions for crack-to-crack interactions .vhich are clearly essential for large crack densities but, as expected, make little difference to the behavior for crack densities less than about E = 0.1. One of the obvious signs of the failure of the Hudson [1980, 19811 expressions for larger crack densities is that for E 2 0.15, some of the elastic constants, and hence the seismic velocities, begin to increase with increasing crack density [Sayers and Kachanov, 19911. This behavior is considered by Sayers ynd Kachanov to be nonphysical. (This may be a plausible nterpretation, but since there is no appeal to the actual physical behavior of cracks, the criticism is essentially not proven.)