Nonlinear effects in elastic Rayleigh waves

Abstract

The propagation of nonlinear surface elastic waves, or Rayleigh waves, is studied in the case of a harmonic elastic material. In the linear theory Rayleigh waves are non-dispersive, and linear profiles of any shape are acceptable. When nonlinear effects are taken into consideration, special restrictions on the permissible wave profiles need to be imposed. Parker and Talbot investigated the possibility of steady-shape profiles within the second-order approximation of the Rayleigh problem. They established numerically the existence of several families of periodic Rayleigh waves of a permanent form, which may move at phase speeds that differ from the linear Rayleigh wave velocity or at speeds that equal the Rayleigh wave velocity. The distinguishing feature of the Parker-Talbot numerical solutions is that plots of the surface horizontal displacement have singularities which resemble finite corners. The goal of our work has been to gain an understanding of the Rayleigh wave solution structure, particularly in the context of the second-order theory. We propose that there exist infinite families of steady-profile periodic solutions moving at speeds equal to or differing from the linear Rayleigh wave velocity. We present various examples of such solutions, and discuss a simple numerical procedure for generating any member of the infinite families of solutions. The nature of the singularities of the presented numerical profiles is examined, and empirical evidence is provided that the surface horizontal displacements display fractional singularity behavior rather than the corner-like behaviour of the corresponding Parker-Talbot profiles. The numerical approach is then modified to accommodate the proposed singular behavior. We use a solvability condition of a Fredholm type to perform a local study of the surface displacement singularities of Rayleigh waves on an infinite interval. The hypothesis of the existence of fractional singularities in the surface horizontal displacement is tested using two different approximations of the solvability condition. We show that within the second-order theory the steady-profile Rayleigh waves can have only one type of fractional singularity behavior, whether they move at the linear Rayleigh wave velocity or not.