Mathematical modelling to investigate the behavior of seismic waves propagation in Earth media and piezoelectric plate with regular and irregular boundaries

Abstract

This thesis entitled Mathematical modelling to investigate the behavior of seismic waves propagation in Earth media and piezoelectric plate with regular and irregular boundaries is a study carried out with the objective of understanding the propagation of Love-type waves in composite mediums. The periodic variations of heterogeneity, external forces, irregularity are taken under consideration. The propagation of SH-waves, Love waves, Torsional waves are the surface waves to be analyzed in the stratified media. The main concern is to examine the influence of different parameters on the phase velocity of the travelling waves. In Chapter 2, we have considered sinusoidal diversity in the piezomagnetic materials under the impact of an impulsive point source to elucidate the hidden characteristics of Love wave behaviour. As a two-dimensional problem, the governing equations of motion are laid down for the proposed model and deduced by Fourier transformation. The frequency relation is derived by using prescribed boundary conditions and the well-known properties of Green's function. The obtained frequency relation is deduced into the conventional form of the Love wave dispersion, validating the proposed model. The fundamental mode along with two higher-order modes of Love wave dispersion is investigated analytically to demonstrate the effects of the functional gradient parameters, piezomagnetic constants, magnetic permeabilities, and amplify gradient parameters on the frequency curves. The outcomes of this study may be considered to secure the high performance of the SAW devices. In Chapter 3, Love wave propagation in the conductive polymer structure is investigated. The silicone rubber polymer of finite thickness is used in the guiding layer with conductive and viscous effects. The functionally graded lithium tantalate with piezoelectric effect is taken in the substrate. The propagation behaviour of the Love wave is examined under the influence of the external force, sinusoidal diversity (SSD), conductivity, and viscoelasticity of the polymers. The impulsive force generated by point source is considered at the interface of the layer and substrate. The equations of motion in the presence of impulsive force are solved by Fourier transform and Green's function technique. The dispersion curve of the Love wave propagation is derived by using boundary conditions. The influence of the conductivity, viscosity, SSD parameters, and piezoelectricity on the propagation velocity of the wave is examined for the first three modes of the Love wave. The dispersion equation reduced to the conventional form of the Love wave dispersion in particular cases. The derived results are appropriate for the manufacturing and design of the surface acoustic wave (SAW) device, transducer, and sensors. Moreover, the results are utilized to enhance the performance of the surface acoustic wave devices and sensors. In Chapter 4, we have considered a hybrid layered-half-space schematic of fiber-reinforced and orthotropic materials. The functionally graded orthotropic materials has great importance in engineering applications. The functionally graded orthotropic materials in the half-space and the reinforced materials under high initial stress are utilized in the superficial layer of finite depth. To examine the effect of heterogeneity on wave phase velocity, the elastic characteristics of orthotropic materials are defined in terms of hyperbolic functions. Moreover, a rectangular plate is installed at the interface of the materials to expose the effect of irregularity in the materials. An external force (impulsive force) is applied to the wave at the point source of the disturbance to understand the influence of such a force on the phase velocity of the wave. The non-homogeneous equations of motion are deduced by Fourier transformation and are solved analytically by Green’s function technique along with possible applications of Dirac-delta function. The dispersion relation of the anti-plane wave propagation is obtained analytically. A graphic explanation of the proposed schematic and the stability of the model has been discussed. Chapter 5 deals with the behaviour of surface horizontally polarized shear waves (SH waves) in the composite multi-material structure with a periodic irregular surface and interface. To unravel the enshrouded features of SH wave propagation in multi-layers structure, we consider a model of three distinct composite materials. In the schematic of the problem, the guiding layer (M-I) contains fluid-saturated porous materials of finite thickness, the intermediate layer (M-II) contains fiber-reinforced composites, and the substrate contains the functionally graded orthotropic materials (M-III). The free surface of M-I and the upper interface of the medium are considered to be irregular on a periodic basis, but the interface of M-II and M-III is supposed to be regular. The dispersion relation is obtained analytically. We demonstrated graphically the phase velocity versus the wave number to analyze the propagation behaviour of the SH wave propagation. This provides a comprehensive evaluation of the impact of regular and irregular boundaries of the composite materials on the phase velocity of the SH waves. The behaviour of the reinforced parameters, initial stress, and porosity on the phase velocity are examined in both scenarios. The obtained results are useful to understand the compositions of the materials on the mountain surface. Irregularity may occur on Earth's surface in the form of mountains due to imperfection of the Earth's crust. Chapter 6 is devoted to explore the influence of horizontally polarized shear waves on mountains. We considered the fluid saturated porous medium (superficial layer) over an orthotropic semi-infinite medium with rigid mountain surface (Model-I) and the soft mountain surface (Model-II) as the medium of wave propagation. The mountain surface is defined mathematically as a periodic function of time domain. The physical interpretation of materials structure is explained in rectangular Cartesian coordinate system and the origin of the Cartesian coordinates system is defined at the contact interface of layer and half-space. The displacement of the mountains has been calculated by solving energy equation analytically. The influence of rigid and soft mountain surfaces on the phase velocity of shear waves has been demonstrated graphically (For graphics we have used MATLAB software). Chapter 7 investigates the possibility of the propagation of torsional wave in a layered structure comprised of fiber-reinforced medium followed by a heterogeneous half-space with periodic variations. The influence of rectangular irregularity present at the interface on the wave propagation has been examined. The governing equations have been established. The dispersion relation has been acquired with the assistance of boundary conditions, displacement and stress components. With the particular cases, the classical dispersion relation has been derived. The impact of reinforced parameters, heterogeneity parameters and the presence of irregularity on these parameters have been shown graphically with the help of MATLAB software. The results of the wave propagation have applications in geophysics, geoengineering and prediction of earthquake.