How tensile fractures form and then propagate in response to the crack-parallel compression has been puzzling the rock mechanics community for decades. The direct application of engineering fracture mechanics to the problem appears to be invalid, because its zero-width mathematical crack model is unresponsive to the normal stress which is coaxial with the crack direction. A Finite-Width Elliptical Crack (FIWEC) model, which retains the crack width as an additional parameter, is proposed to make crack propagation sensitive to the compressive stress acting along the crack path.
ASYMPTOTIC STRESS FIELD AROUND THE FIWEC CRACK
The analysis starts with the stress functions for an elliptical hole . The elliptical coordinates are replaced with their polar equivalents using the focal point as origin. The terms that include the radial distance from the focal point are expanded into power series. The asymptotic variation of the stresses at the crack tip are then obtained by neglecting the higher order terms. The elastic asymptotic stress field around the FIWEC crack tip consists of two singular terms including the 3/2 and the 1/2 powers of the distance from the focal point (r):
[Equation available in full paper] (1)
Here the external loading, P, acts at an angle â to the crack axis; e is the crack aspect ratio of minor axis (b) to major axis (a) and c is the focal length of the ellipse (Figure 1). The mathematical expressions for the coefficients of fij3, fij1, and fij0 are quite complex.