Crack emergence trajectories at a free surface

Abstract

In the Institute of Mining, Siberian Branch of the Academy of Sciences of the USSR, under the leadership of Chernov [i-3] technology is being developed for rock breaking using normal separation cracks.With this aim, a borehole is drilled at some distance from the free surface and a small nucleating slot is created with a special tool.A working liquid is fed into the borehole under pressure.On reaching a critical pressure determined by the properties of the material being broken and the size and position of the concentrator, the initial crack increases in size. By pumping the liquid and maintaining the pressure required for further crack growth, pieces of rock separate from the rock mass.We must evaluate the size and shape of the separated fragment in relation to the main parameters, i.e., initial size of the nucleating slot, curvature in its shape, distance from the free surface and degree of penetration of the working liquid into a growing crack.In order to explain the qualitative features of the process under plane strain conditions a study has been made of quasistatic hydraulic-rupture crack propagation close to the free surface. It is assumed that the hydraulic-rupture crack is characterized by zero tangential stresses along its length and normal stresses that are constant along its length.In order to find the shape of the crack development trajectory we use the o~-criterion of failure [4,5] ..whose application for an elastically brittle material gives good agreement with the descriPtion for experimental data, e.g., [6,7]. This means that in each step of crack advance only the change in the local stress field is considered in the vicinity of the crack lip caused by its advance and bending, and the effect of dynamic factors is ignored. This last assumption is correct because the hydraulic-rupture crack growth rate is much less than the velocity of Rayleigh surface waves.In solving the problem, we use the method of integral singular equations similar to that described in monographs [5,[8][9][10], and in the last of these this method is generalized for the case of smooth curvilinear slits. Statement of the Problem.Let an isotropic elastic body occupy a lower half-plane Im z ~ 0, z = x + iy, and it contains N smooth curvilinear slits Lk, k = i, N. With each slit L k we connect its local coordinate system Xk0kYk, k = I--?--N. The connection between coordinates of the points of a plane in the main and local coordinate systems is given by the expressionwhere o k is the angle between axes 0kX k and 0x, and Xk ~ yk ~ are coordinates of the origin of the k-th local system in the main system. We assume that the shape of each slit L k in its local coordinate system is known and described by a parametric equationThen solution of the problem in elasticity theory when stresses are specified along the peripheries of the slits as:(a positive sign relates to the upper edge, and a minus sign to the lower edge) is reduced to finding the solution of g'k(~), k = i, N of a set of N singular equations in the form [9, ii]:Institute ...