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The Elastic Modulus of a Medium Containing Strongly Interacting Antiplane Cracks

Paul M. Davis, & Leon Knopoff

Published 1995, SCEC Contribution #170

We calculate the elastic modulus for up to 10000 randomly oriented, strongly interacting, nonintersecting, antiplane cracks that have a logarithmic size distribution for a range of concentrations c from 0 to 2. An antiplane boundary integral method is used to compute elemental dislocations on the crack faces. The ratio of slips in the cases of interacting cracks to those for isolated cracks has a nearly unit average. The effective modulus is well fit by a mean-field model in which the cracks do not interact. Prom Gauss's theorem in two dimensions, we can show that the mean-field approximation is appropriate for the problem of the modulus of a high concentration of randomly distributed cracks. The mean total field is the external stress field. The modulus as a function of concentration is then simply <μ > = μ0/(1 + c/2). This expression differs from the self-consistent result in that the modulus does not become zero at finite concentration. It also differs from modifications to the self-consistent method which predict an exponential decay of modulus with c.

Davis, P. M., & Knopoff, L. (1995). The Elastic Modulus of a Medium Containing Strongly Interacting Antiplane Cracks. Journal of Geophysical Research, 100(B9), 18253-18258.