Mathematical modelling of SH-type wave due to Shear stress discontinuity

Abstract

The aim of this thesis is to solve a geophysical problem with the help of mathematical theory. For instance, the geophysical problem is modeled mathematically and solved further. In this project work, the e↵ect of stress discontinuity and moving load on a layered media has been studied. For cal culating displacement functions analytically, the modified Cagniard-de Hoop process, involving contour integration and deformation of the path in the com plex plane has been used. The results of the present study can be used to study the e↵ect of travelling shear stress discontinuity during natural or man made earthquakes and explosions. These stress discontinuities occur during the earthquake and are associated with the propagation of cracks, faults and fractures. These discontinuities can be observed inside the Earth when water accumulates between two layers or when layers glide over each other. The study may also be useful to the seismologists, geophysicists and civil engineers for the analysis of crack initiation, its propagation for a long period of time and growth rate in anisotropic reinforced medium. The present dissertation is divided into four chapters along with the bibli ography at the end. In chapter-1, a brief introduction to continuum mechanics and seismic wave propagation is discussed. In addition to this, a useful theory of solid me chanics and elastic solids is explained. Further, all the mathematical prereq uisites that are required to solve the desired problem have also been discussed in brief. In chapter-2, SH-type wave motion due to the travelling shear stress discontinuity in a smart material composite structure in conjunction with a transversely isotropic fiber-reinforced stratum and a homogeneous extended stratum. The travelling stress discontinuity has an impulsive behaviour in motion for a short distance and does not rise at a uniform velocity. The gen eralized Cagniard-de Hoop process is applied to invert the integral transform in order to find the accurate form of displacement analytically. The dynamic equations of motion are solved in the transformed domain with the help of the Laplace transform w.r.t. time (t) followed by the application of the Fourier transform w.r.t. spatial variable (x1). The inverse Laplace transform is ob tained analytically, by deforming the path of integration for the inverse Fourier transform. At the stress-free surface of the structure, the displacement com ponent is investigated for the following three di↵erent forms of travelling stress discontinuities: (1) the stress discontinuity is generated suddenly at some point and expand with uniform velocity (V ) in the direction of x1; (2) the discontin uous stress region moves with a uniform velocity V in the x1-direction; (3) the discontinuous region is created within x1 = a to x1 = b and then expands with uniform velocity V in the x1-direction. The numerical integration method is applied to calculate the free surface displacement at any point and numerical simulations of SH-type wave propagation are presented using MATLAB soft ware. The simulation shows the non-uniform and impulsive nature of SH-type wave propagation due to the travelling shear stress discontinuity. Some special cases are derived to validate the model with the revealed literature. In chapter-3, This chapter deals with the analysis of displacement at the free surface due to moving load at x3 = ity at the interface of layered media. The media consists of fiber-reinforced medium overlying an orthotropic homogeneous half-space. The normal line load is moving with a velocity V1. In addition to this, two di↵erent types of discontinuities are considered at the interface: (1) the discontinuous stress region moves with a uniform velocity V in the x1-direction; (2) the discontin uous region is created within x1 = a to x1 = b and then expands with uniform velocity V in the x1-direction. The dynamic equations of motion are solved in the transformed domain with the help of the Laplace transform w.r.t. time (t) followed by the application of the Fourier transform w.r.t. spatial variable (x1). The inverse Laplace transform is obtained analytically, by deforming the path of integration for the inverse Fourier transform. Further, some special cases are discussed to validate the model with the results of chapter-2. Chapter-4 contains final conclusion of the thesis and the future plans.