Asymptotic solutions for mode II dynamic rupture with nonlinear damping

Abstract

In this study, the nonlinear Rayleigh damping (A.H. Nayfeh, Introduction to Perturbation Techniques, Wiley, New York, 1981, pp. 147–150, 216–230) is adopted, and the governing equations are two coupled nonlinear PDEs, their unknown functions being the amplitude components in x and y directions, respectively. By assuming the unknown functions to be proper vibrational pattern functions which fulfill all the boundary conditions, transforming the fixed coordinates into the moving coordinates by Galileo transformation and applying the calculation of Fourier series, the two nonlinear PDEs are reduced to four nonlinear ODEs. Among them, a couple are of order 1, the rest are of order 2. In order to be able to solve them analytically, the nonlinear terms in one ODE of order 1 for the secondary amplitude v are neglected in the first iterative calculation and the rupture velocity is assumed to be a constant value, and the coupled term in one of the order two ODEs to be 0. Then the remaining order two ODEs is reduced to be a homogeneous van der Pol equation. Its solution is readily obtained. Substituting this known function into the other order two ODEs for the unknown function u, it is reduced to be a nonhomogeneous van der Pol equation. Its solution is also unknown.