The centrally cracked Brazilian disc: implications and solutions in case of closing cracks

Abstract

AbstractA recently presented closed-form analytic solution for the displacement field and stress field in a cracked Brazilian disc, under uniform radial pressure along two symmetric arcs of its periphery, revealed that for a wide range of crack-axis inclinations, the lips of the crack tend to overlap each other, leading to a kind of an “unnatural” geometrical configuration. It is here proven that this behavior is a consequence of the inability of the mathematical model to simulate the change of the boundary conditions that appears (for some special configurations) in the physical problem and render the mathematical problem an “ill-posed” one. Indeed, what happens in praxis is that for a given interval of crack inclination angles, the initially stress-free lips are coming in contact and contact stresses appear violating the boundary conditions initially adopted in the mathematical model. This problem is here solved by superposing to the above-mentioned solution the respective one of an auxiliary mixed fundamental problem solved according to Muskhelishvili’s complex potentials method. In this way, physically acceptable displacement fields and stress fields are obtained all over the cracked disc independently from the crack inclination angle. In addition, the contact stresses developed along the crack lips are determined. Moreover, naturally sound formulae for the corresponding stress intensity factors (in case of cracks with lips in contact to each other) are obtained, which are of crucial engineering importance. The solution obtained enlightens some critical aspects related to the practical application of the cracked Brazilian disc as a tool for the standardized determination of the fracture toughness of brittle rock-like materials and concrete.