Rate‐dependent extensional deformation resulting from crack growth in rock

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Abstract
Quasi‐static propagation and dilation of macroscopic mode I cracks is considered as a source of inelastic strain in crustal rocks undergoing extensional deformations. The approach here is to first characterize the propagation of a single dilating crack and then to consider how a large two‐dimensional array of similar, parallel cracks accomodates an applied deformation. The driving force for propagation of each crack, Gi, is found as a function of the applied strain and the instantaneous crack lengths ci. The rate of crack propagation is taken to be a function of the instantaneous crack extension force ċi = ċ (Gi). A specific propagation rate extension force relation is developed that both fits the available experimental data and has vanishing propagation rate at a finite value of G. From these results a specific internal state variable constitutive law is derived relating uniaxial stress to the instantaneous uniaxial strain and crack density. For strains less than those necessary to cause crack propagation, the rock is predicted to behave like a linear elastic solid. Following the onset of propagation, the average stress may continue to increase if the strain rate accommodated by crack growth is slow in comparison to the applied strain rate. For constant strain rate boundary conditions the solutions exhibit strain softening, leading to peak stress and then a stress decrease. The peak stress increases nearly logarithmically with increasing applied strain rate. The response to unloading depends strongly on whether or not cracks heal. Unloading that follows sealing of the cracks with mineral precipitates causes the rock to undergo a permanent nonrecoverable strain.

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