Wave Propagation in an Elastic Half-Space with Quadratic Nonlinearity

Abstract

This study investigates wave propagation in an elastic half-space with quadratic nonlinearity due to a line load on the surface. The consideration of this problem is one of the well known Lamb problems. Even since Lamb's original solution, numerous investigators have obtained solutions to many different variants of the Lamb problem. However, most of the solutions existing in the current literature are limited to wave propagation in a linear elastic half-space. In this work, the Lamb problem in an elastic half-space with quadratic nonlinearity is considered. For this, the problem is first formulated as a hyperbolic system of conservation laws, which is then solved numerically using a semi-discrete central scheme. The numerical method is implemented using the package CentPack. The accuracy of the numerical method is first studied by comparing the numerical solution with the analytical solution for a half-space with linear response (the original Lamb's problem). The numerical results for the half-space with quadratic nonlinearity are than studied using signal-processing tools such as the fast Fourier transform (FFT) in order to analyze and interpret any nonlinear effects. This in particular gives the possibility to evaluate the excitation of higher order harmonics whose amplitude is used to infer material properties. To quantify and compare the nonlinearity of different materials, two parameters are introduced; these parameters are similar to the acoustical nonlinearity parameter for plane waves.