Nonlinear Rayleigh and Love Surface Waves in Elastic Materials

Abstract

This chapter is devoted to analysis of nonlinear elastic Rayleigh and Love surface waves, which corresponds to the Murnaghan model. The analysis is divided into two parts. In the first part, the Rayleigh wave is analyzed. In the first subsection, elastic surface waves are described, and then the basic moments in the theory of elastic linear Rayleigh surface waves are shown. Further, nonlinear elastic Rayleigh surface waves are discussed, which includes general information (new variants of quadratically nonlinear equations describing the two-dimensional motion in dependence on two spatial coordinates x 1, x 3 and time t), basic equations, procedures of solving the nonlinear wave equations, and, finally, the first two approximations are obtained and commented on. The main nonlinear effect is that the second harmonic appears in the description of the wave propagation. An analysis of nonlinear boundary conditions is carried out separately. Here the boundary conditions for cases of small and large deformations are given and then an analysis of boundary conditions is carried out. Finally, a new nonlinear Rayleigh equation is derived and discussed. The main nonlinear effect is that the phase velocity depends nonlinearly on the initial amplitude. In the second part, the problem of elastic Love waves is considered in a classical statement with additional assumption of a presence of nonlinearity in the description of deformation. The nonlinear Murnaghan model is used. A new nonlinear wave equation in displacements is derived that includes a linear part and a part with summands of the third and fifth orders of nonlinearity, only. Where the physical nonlinearity is allowed, the solution of a new nonlinear equation with nonlinear boundary conditions is obtained by the method of successive approximations within the framework of the first two approximations. A new nonlinear equation for determining the wave number is derived, which shows a new factor in an initial wave profile distortion—distortion owing to the wavelength changes with unchanging frequency. This information can be found in scientific publications on the elastic Rayleigh and Love waves, list of which is given in the reference list (48 titles) of the chapter [1–48].