Steady-state anti-plane shear wave scattering by a crack with friction

The present article describes the steady-state numerical modeling of anti-plane shear wave scattering by a crack with frictional boundary conditions. The system is composed of an unbounded elastic solid that includes a closed crack under static compressive stress. A time-harmonic anti-plane shear wave is incident, and dynamic friction between the crack faces is induced as a nonlinear phenomenon. The anti-plane wave scattering can be described in a retarded potential integral equation by taking the nonlinearity into account. The present article introduces the steady-state expression as an asymptotic vibration of crack faces after a sufficient elapsed time. In order to solve the equations describing nonlinear steady-state vibration, a harmonic balance method is integrated into a boundary element method. Fourier coefficients of crack opening displacement distributed on the crack face are treated as unknown variables. The system of nonlinear equations is solved by means of a numerical continuation method. The present numerical results show almost complete agreement with those obtained by the conventional time-domain analysis after a sufficient elapsed time. Furthermore, the robustness and effectiveness of the proposed method are demonstrated numerically.