Study on Some Elastodynamic Problems in Materials with Microstructure and with Reinforced Fibres

Abstract

Theoretical problems involving analysis of elastic wave propagation in complex media is an integral part of geophysics and Earth sciences. It presents fascinating tool to examine Earth's interior. In this light, the present research contributes to the study of some elastodynamic problems like wave propagation and moving load problems in mediums with various material properties and geometry. Graphical user interface (GUI) software in MATLAB has been developed for several problems. The thesis is structured in six chapters dealing with di erent problems. Major contributions and conclusions of the chapters are as follows: Chapter 1 This chapter contains evolution and historical overview of elasticity, microcontinuum theories and anisotropic materials. Recent developments in the elds have been recorded along with the basic governing equations and constitutive relations. Chapter 2 This chapter consists of two problems highlighting the e ect of moving load on the stresses produced in an irregular half-space. First problem of this chapter studies the stresses produced in an irregular bre-reinforced half-space due to a normal moving load on a free surface. The closed form expression of stresses has been obtained. Three di erent cases of irregularity viz. rectangular, parabolic and iii Abstract iv no irregularity have been discussed and compared. It is distinctly marked out that the stresses produced due to normal moving load are a ected by depth of half-space, depth and type of irregularity. Also, these e ects are highlighted through numerical illustrations. Second problem of this chapter presents a theoretical model to study the response of moving load on an irregular micropolar half-space. The expression of stresses produced due to moving load have been obtained in closed form. The irregularity has been taken in three di erent forms viz. rectangular, parabolic and no irregularity. E ects of frictional coe cient, microstructure and irregularity on stresses have been studied and depicted by means of graphs for various cases. Chapter 3 This chapter includes a comprehensive study of surface wave propagation in an anisotropic medium. Secular equation for the propagation of Rayleigh-type surface waves in self-reinforced half-space under the in uence of gravity and liquid loading has been derived in closed form. The e ect of reinforcement, gravity and liquid loading on the phase velocity of Rayleigh-type waves has been distinctly observed. Graphical demonstration has been carried out to highlight the important peculiarities of the problem. Moreover, the comparative study has been made of reinforced over reinforced-free case to unravel the reinforcement e ect. Also, the e ect of absence of liquid loading on propagation of Rayleigh-type waves is analysed. Chapter 4 This chapter deals with propagation of SH-wave in vertically heterogeneous viscoelastic layer lying over a micropolar elastic half-space. Dispersion equation and damping equation are obtained in closed form and are plotted for di erent variations in relevant parameters of heterogeneity, viscoelasticity and micropolarity. v Abstract The heterogeneity in viscoelastic layer is caused by consideration of exponential variation in rigidity, internal friction and density. The dispersion equation has been matched with classical Love wave equation as a special case of the problem when the isotropic layer is lying over an isotropic half-space. Moreover, a comparative study is made to study the impact of presence and absence of micropolarity in the medium of elastic half-space. Chapter 5 We have examined two problems of shear wave propagation in elastic medium with imperfect bonding between layer and half-space in this chapter. In rst problem of this chapter, propagation behaviour of horizontally polarized shear wave in layered structure consisting of a vertically heterogeneous bre-reinforced layer imperfectly bonded to a micropolar elastic half-space is studied. An analytical expression of dispersion equation has been obtained in closed form. The exponential form of heterogeneity is considered in bre-reinforced layer. The signi cant e ects of imperfectness, heterogeneity, reinforcement, micropolarity and coupling factor have been studied and shown graphically. The second problem in this chapter discuss the propagation of shear wave in micropolar elastic half-space imperfectly bonded with a heterogeneous viscoelastic layer. In said model, dispersion equation and damping equation are obtained in closed form. The e ect of imperfect common interface, heterogeneity present in layer, internal friction associated with viscoelastic layer and micropolarity associated with half-space have been investigated and graphical demonstration has been performed to highlight these e ects. Abstract vi Chapter 6 This chapter emerges with the study of horizontally polarized shear wave propagation in a heterogeneous bre-reinforced layer lying over an initially stressed isotropic elastic half-space. The interface between layer and half-space is considered as corrugated and loosely bonded. The heterogeneity in the layer is caused due to exponential variation of depth. The dispersion relation has been found analytically in closed form. The e ect of presence and absence of the corrugated common surface with loose bonding on the dispersion curves has been meticulously examined. Moreover, the substantial e ect of reinforcement, anisotropy, heterogeneity, initial stress, undulation parameter and position parameter on phase velocity of SH-wave have been remarkably traced out. Comparative study is also performed to compare reinforced (anisotropic) case with reinforced-free (isotropic) case, heterogeneous case with homogeneous case and loosely bonded corrugated interface case with perfectly bonded planar interface case. Numerical computation along with graphical demonstration has been carried out for the problem to unravel the hidden facts.