Modelling of subsonic crack propagation in a power-law graded material

Two steady-state problems are solved of an interfacial crack moving along the interface between two power-law graded half-planes under conditions of antiplane and plane strain. Both problems are governed by Navier equations with variable coefficients. The antiplane problem is solved explicitly. The plane problem requires a solution of an auxiliary Carleman problem with two shifts of two meromorphic functions in a strip. It is recast as a system of singular integral equations with a fixed singularity and solved numerically. It is shown that the plane model problem is equivalent to a vector Riemann–Hilbert (RH) problem for two functions with a piece-wise constant matrix coefficient, the entries of which are expressed through the solution of the Carleman problem. An exact solution of the RH problem is determined. Asymptotics of the displacement jumps and the stresses at the crack tip are recovered. For modes I, II and III, a Griffith-type criterion of crack propagation is proposed.