Application of the Dugdale crack model to the stress-strain relations and bulk moduli of a granite

Abstract

Variations of dynamic and static hulk moduli and compressional strain of a rock sample of granite in response to hydrostatic pressure at room temperature are summarized as follows: 1. (1) “ nonlinear” increase of compressional strain becomes “linear” at σ 0 (border pressure, for this rock, σ 0 = 1.36 kbar and is the pressure which divides the compression curve in two parts: the crack-active part and the crack-closed part). 2. (2) static bulk modulus obtained from compressional strain data ϵ drastically increases and levels off at pressures higher than the border pressure σ 0, 3. (3) pressure dependence of dynamic bulk modulus obtained from acoustic measurements has a trend similar to that of the static one, and 4. (4) border pressure σ 0 from (3) is just equal to that from (1) and (2). To interpret these phenomena and understand the underlying reason for these variations, two crack models were examined, i.e., an elliptical crack model with constant aspect ratio and the Dugdale crack model. The difference in closing processes for the two models results in different changes in effective bulk modulus and compressional strain due to the applied pressure. If the total number of cracks cannot change with increasing pressure, the drastic increase in bulk modulus and nonlinear part of the stress-strain curve at low pressure can be attributed to reduction of crack width or crack length. It is shown that the Dugdale crack model explains better the observed results. Theoretical calculations show that the crack density of the rock sample is 0.42 (dimensionless) and that the yield stress of 2.09 kbar acts in the proximity of tips of Dugdale cracks. These values seem reasonable when we consider the values of the porosity of 0.16% and the border pressure of 1.36 kbar.