The steady motion of a circular crack along the interface of a layer and a half-space

An exact analytical solution is given of the direct axisymmetric steady dynamic problem in the theory of elasticity of the motion, with an arbitrary constant subsonic velocity ν = c along the interface of a rigidly coupled layer 0 ≤ z ≤ H and a half-space z ≤ 0, of a circular transverse shear crack 0 ≤ r ≤ r0 + ct (r0 ≤ 0) with and without a cavity at its tip. Using Hankel transformation in terms of biwave potentials, a general solution of the fundamental equations of motion in the theory of elasticity and the basic solutions of the first fundamental boundary-value problem are separately constructed for the layer and the half-space for the case of arbitrary normal and shear stresses in the plane of separation z = 0 in a moving cylindrical system of coordinates r1 = r + ct, z1 = z. A special regularization of the main solution is carried out which ensures the convergence of the integrals for all stresses and displacements while preserving the high accuracy of the solution to whatever level may be desired [1, 2]. On the basis of the main solutions, a mathematical formulation is given of the mixed problem of the motion of a transverse shear crack with a cavity at the tip and its reduction to a system of three singular integral equations with Cauchy kernels which allows of regularization by the Carleman-Vekua method in terms of the closed solution of the corresponding characteristic system of singular integral equations. When the width of the cavity vanishes, one of the equations of the system solves the problem of a transverse shear crack without a cavity. Criteria are established for the existence of a cavity and its absence as a function of the elastic and velocity characteristics of the layer and half-space and the velocity of motion of the crack c.