Rayleigh wave in a quadratic nonlinear elastic half-space (Murnaghan model)

Abstract

The nonlinear equations that underlie the analysis of classical Rayleigh waves are derived for the two-dimensional case of nonlinear elastic deformation described by the Murnaghan model. In addition to the case of presence of both geometrical and physical nonlinearities, two special cases are considered where one only type of nonlinearity is taken into account. It is shown that unlike the one-dimensional problems for plane waves where only three types of nonlinear interaction should be allowed for, the two-dimensional problems should include 24 types of nonlinear interaction. In the case of geometrical nonlinearity alone, a preliminary analysis of the nonlinear equations is carried out. Second-order equations are derived. The second approximation includes the second harmonics of the wave itself and its attenuating amplitude and is nonlinearly dependent on the initial amplitude of the Rayleigh wave and linearly increasing with the distance traveled by the wave