Asymptotic analysis of nonlinear waves in certain layered media

Elastic waves propagating in wave guides such as rods, plates, layered half spaces, etc., become dispersive, i.e. the phase velocities of waves depend on the wave number, due to multiple reflections between the boundaries of wave guides. Horizontally polarized surface shear waves propagate on a layered half space providing that the shear wave velocity of the layer is less than the corresponding velocity of the half space. These waves are discovered by Love considering the dispersive waves with large transverse displacement components in the early part of the surface wave trains recorded by seismographs. They are called Love waves. Later, Stoneley and Tillotson extended the theory of Love waves to a double layered half space in 1928 and Stoneley examined the effect of low velocity internal layer on the propagation of surface shear horizontal (SH) waves in 1950. The propagation of surface SH waves in a double layered half space and also the effect of the low velocity internal layer on the propagation characteristics of surface SH waves have been investigated by other researchers as well due to their important applications in geophysics, civil engineering, electronic signal processing devices, nondestructive testing of materials and prediction of damage that can result from earthquakes etc. Moreover, the study of surface elastic SH wave propagation inside the crustal layer of the Earth gives important information about the layered structure of the Earth for exploration of natural resources buried in the Earth's surface such as oil, petroleum, gas and mineral. In the first part of this work, the slow modulation of a harmonic surface SH wave over the surface of a nonlinear hyperelastic half space covered with two different nonlinear hyperelastic layers of uniform thickness is examined. It is assumed that the linear shear wave velocity of the intermediate layer is slower than those of the top layer and the half space. In the linear limit, the problem reduces to the problem investigated by Stoneley in 1950. The effect of the nonlinear properties of the intermediate layer on the propagation characteristics of surface SH waves has been investigated. Moreover, from the comparison of the solutions characterized the nonlinear SH wave propagation on the single layered half space and on the double layered half space, it is observed that SH wave propagation is affected considerably by the existence of a second layer. In the later part of this study, the SH waves propagating in a nonlinear elastic half space coated with a nonlinear elastic layer having irregular boundary surfaces is examined. It is assumed that the irregularities of the boundary surfaces are the functions of the distance in the direction of propagation of the waves. The effect of the varying boundary surfaces as well as the effect of the nonlinear material parameters on the propagation characteristics of the surface SH waves in a layered half space have been investigated. This thesis consists of four chapters. In the first chapter, it is mentioned about purpose and scope of the studied problems, and literature review is given. In the second chapter, the slow modulation of a harmonic surface SH wave over the surface of a nonlinear hyperelastic half space covered with two different nonlinear hyperelastic layers of uniform thickness is examined. In the analysis, it is assumed that between the linear shear wave velocities of the top layer, c1, the intermediate layer, c2, and the half space, c3, the inequality c2 < c1 < c3 is valid. That is the linear shear wave velocity of the intermediate layer is slower than those of the top layer and the half space. The effect of the nonlinear properties of the slow intermediate layer on the existence of solitary waves has been investigated. This chapter consists of four sections. In the first section, the equations of motion and boundary conditions are produced for the nonlinear surface SH wave propagation on the double layered half space. It is assumed that the constituent materials are nonlinear homogenous, isotropic, incompressible hyperelastic and their strain energy functions are only the functions of the first invariant of the Finger deformation tensor c􀀀1 = [xk;Kxl;K], i.e. S = S(I) where I =trc􀀀1. Namely, we consider the double layered half space made of different generalized neo-Hooken materials. In the second section, the problem of linear SH wave propagation is investigated by linearizing the derived boundary value problem, and then the dispersion relations are obtained for cases where wave propagation exists. It is known that a surface SH wave propagates provided that the phase velocity c of this wave satisfies either the condition c2