1 Chapter I Introduction Elasticity is the branch of Physics which deals with analysis of stress and strain. A material is said to be elastic, which deforms under stress and returns to its original shape after removal of the stress causing deformation. The amount of deformation is called the strain. The deformation is a change in shape due to an applied force. This can be due to pulling forces, pushing forces, shear, bending or twisting. External forces are the forces acting between the body and the environment. Hooke s law of elasticity is an approximation which states that the amount by which a material body is deformed is linearly related to the force causing the deformation. Materials for which Hooke s law is a useful approximation are called as linearly elastic or Hookean materials. Hooke s law is named after the 17 th century British Physicist Robert Hooke. He first stated this law in 1676 as a Latin anagram, whose solution was published in 1678. The disturbance of particles in a medium where the particles do not actually move but transmit the disturbance is known as a wave. A seismic wave is a wave that travels through the Earth, most often as a result of a tectonic earthquake, sometimes from an explosion. Seismic waves are also continually excited by the pounding of ocean waves and the wind. Seismic waves are studied by the seismologists and measured by a seismograph, which measures the output of a seismometer or geophone. The study of seismic waves gives useful information regarding oil reservoirs and hydrophones. There are two types of seismic waves namely body wave and surface wave. Other modes of
2 wave propagation also exist than those described here, but they are comparatively of minor importance. Body waves travel through the interior of the earth. They follow ray paths bent by the varying density and modulus of the earth s interior. The density and modulus, in turn, vary according to temperature, composition and phase. This effect is similar to the refraction of light waves. Body waves transmit the first-arriving tremors of an earthquake, as well as many later arrivals. There are two main kinds of body waves: Primary (P waves) and Secondary (S waves). P waves are also known as longitudinal or compressional waves, which mean that the ground is alternately compressed and dilated in the direction of propagation. In solids, the P waves generally travel slightly less than twice as fast as S-waves and can travel through any type of material. When generated by an earthquake they are less destructive than S waves and surface waves that follow them, due to their lesser amplitudes. S waves are transverse or shear waves, which mean that the ground is displaced perpendicularly to the direction of propagation. In the case of horizontally polarized S waves, the ground moves alternatively to one side and then the other. S waves can travel only through solids, as fluids (liquids and gases) do not support shear stresses. Their speed is about 60 % of that of P waves in a given material. Surface waves are analogous to water waves and travel just under the Earth s surface. They travel more slowly than body waves. Because of their low frequency, long duration and large amplitude, they can be the most destructive type of seismic wave. There are two main types of surface waves; Rayleigh waves and Love waves. Theoretically, surface waves can be understood as systems of interacting P and S wave. Rayleigh waves also called ground roll and travel as ripples similar to those on the surface of water. The existence of these waves was predicted by Lord Rayleigh in 1885. They are slower than body waves, roughly 70 % of the velocity of S waves and have been assumed to be visible during an earthquake in an open space. For example, the parking lot of the cars moves up and down with the waves. Love waves are the surface waves that cause horizontal shearing of the ground. They are named after A.E.H. Love, a British mathematician who created a mathematical model of the waves in 1911. They usually travel slightly faster than Rayleigh waves, about 90 % of the S wave velocity. Wide literature is available on the wave propagation in reference books like Stress Waves in Solids by Kolsky (1935), Elastic Waves in Layered Media by Ewing et al. (1957), Progress in Sold Mechanics
3 by Chadwick et al. (1960), Acoustic Fields and Waves in Solids by Auld (1974), Wave Motion in Elastic Solids by Achenbach (1973), Elastic Waves in Solids: Applications to Signal Processing by Dieulesaint and Royer (1980), Seismic Waves and Sources by Ben-Menahem and Singh (1981), Elastic Wave Propagation in Transversely Isotropic Medium by Payton (1983), Wave Motion in Elastic Solids by Graff (1991), etc. Many researchers have worked on wave propagation in elastic solids. For example, Rayleigh (1885) studied the waves propagating along with plane surface of an elastic solid. Buchwald (1961) discussed Rayleigh waves in transversely isotropic media. Abd. Alla (1999) studied the propagation of Rayleigh waves in an elastic half space of orthotropic material. A material is said to be anisotropic if the elastic response of a material is not independent of the material s orientation for a given stress configuration. Anderson (1961) studied the elastic wave propagation in layered anisotropic media and derived the periodic equations for Rayleigh, Love and Stoneley waves. The elastic moduli of an anisotropic material are different for different directions in the material. Keith and Crampin (1977) investigated that three types of body waves with mutually orthogonal particle motion can propagate in an anisotropic elastic solid medium. In general, the particle motion is neither purely longitudinal nor purely transverse. Due to this, the three types of body waves in an anisotropic medium are referred to as qp, qsv and qsh, rather than as P, SV and SH, the symbols used for propagation in an isotropic medium. A monoclinic medium has one plane of elastic symmetry. For wave propagation in the plane of symmetry, SH motion is decoupled from the P-SV motion. While the particle motion of SH waves is purely transverse, it is neither purely longitudinal nor purely transverse in the case of P-SV waves. Chattopadhyay and Choudhury (1995) discussed the reflection of qp waves at the plane free boundary of a monoclinic half-space. In a subsequent paper, Chattopadhya et al. (1996) studied the reflection of qsv waves at a plane free boundary of a monoclinic half-space. In above two papers, they assumed that qp waves are purely longitudinal and qsv waves are purely transverse. Singh (1999) pointed out the errors in these papers and Singh and Khurana (2002) restudied the reflection of P and SV waves at the free surface of a monoclinic half-space. Singh and
4 Khurana (2001) studied the reflection and transmission of P and SV waves at the interface between two monoclinic elastic half-spaces. Singh and Singh (2004) studied the propagation of plane waves in fibre-reinforced, anisotropic, elastic media is discussed. They obtained the expressions for the phase velocity of quasi-p and quasi-sv waves propagating in a plane containing the reinforcement direction as functions of the angle between the propagation and reinforcement directions. They also obtained the closed form expressions for the amplitude ratios for qp and qsv waves reflected at the free surface of a fibre-reinforced, anisotropic, homogeneous, elastic half-space. The dynamical theory of thermoelasticity is the study of interaction between thermal and mechanical fields in solid bodies and is of much importance in various engineering fields. Biot (1956a,b,c, 1962) formulated the theory of coupled thermoelasticity to eliminate the paradox inherent in the classical uncoupled theory that elastic changes have no effect on the temperature. The heat equations for both coupled and uncoupled theories of the diffusion type predicting infinite speeds of propagation for heat waves are contrary to physical observations. Hetnarski and Ignaczack (1999) examined five generalizations to the coupled theory and obtained a number of important analytical results. The first generalized theory of thermoelasticity is due to Lord and Shulman (1967) who introduced the theory of generalized thermoelasticity with one relaxation time by postulating a new law of heat conduction to replace the classical Fourier law. This new law contains the heat flux vector as well as its time derivative. It contains also a new constant that acts as a relaxation time. The heat equation of this theory is of the wave-type, ensuring finite speeds of propagation for heat and elastic waves. The remaining governing equations for this theory, namely, the equation of motions and constitutive relations remain the same as those for the coupled and the uncoupled theories. The second generalization to the coupled theory is known as the generalized theory with two relaxation times. Muller (1971) introduced the theory of generalized thermoelasticity with two relaxation times. A more explicit version was then introduced by Green and Laws (1972), Green and Lindsay (1972) and independently by Suhubi (1975). In this theory the temperature rates are considered among the constitutive variables. This theory also predicts finite speeds of propagation for heat and elastic waves
5 similar to the Lord-Shulman theory. It differs from the latter in that Fourier s law of heat conduction is not violated if the body under consideration has a center of symmetry. Dhaliwal and Sherief (1980) extended the Lord and Shulman (L-S) theory for an anisotropic media. Chandrasekharaiah (1986) referred to this wave-like thermal disturbance as "second sound". Green and Naghdi (1991) established a new thermomechanical theory of deformable media that uses a general entropy balance as postulated in Green and Naghdi (1977). The theory is explained in detail in the context of flow of heat in a rigid solid, with particular reference to the propagation of thermal waves at finite speed. A theory of thermoelasticity for nonpolar bodies, based on the new procedure, was discussed by Green and Naghdi (1993). This theory permits the flow of heat as thermal waves at finite speed, and the heat flow does not involve energy dissipation. Dhaliwal and Wang (1993) formulated the heat-flux dependent thermoelasticity theory for an elastic material with voids. This theory includes the heat-flux among the constitutive variables and assumes an evolution equation for the heat-flux. Literature on generalized thermoelasticity is available in the books like Thermoelasticity with Finite Wave Speeds by Ignaczak and Ostoja-Starzewski (2009) Thermoelastic Models of Continua by Iesan (2004), Thermoelastic Deformations by lesan and Scalia (1996), Thermoelastic Solids by Suhubi (1975), etc. The wave propagation in thermoelastic media is of much importance in various fields such as earthquake engineering, soil dynamics, aeronautics, astronautics, nuclear reactors, high energy particle accelerator, etc. Many researchers have worked on various interesting problems in context of coupled and generalized thermoelasticity. For example, Deresiewicz (1957) studied the plane waves in thermoelastic solid. He has shown that in an isotropic thermoelastic solid, there exist one shear wave and two distinct dilatational waves, where shear wave is not altered by thermal effects. Chadwick and Sneddon (1958) studied the propagation of waves in an isotropic thermoelastic solid. It is shown that the shear waves are uninfluenced by thermal effects but that two distinct dilatational waves exist, one being similar in nature to a purely elastic longitudinal wave, but dispersed and attenuated by the medium, and the other similar to a purely thermal wave. Lockett (1958)
6 studied the effect of thermal properties of solid on velocity of Rayleigh wave. Deresiewicz (1960) studied the effects of boundaries on the waves in a thermoelastic solid and reflection of plane waves from a plane boundary. Chadwick (1960) have discussed the propagation of surface waves in homogeneous thermoelastic media. Chadwick and Windle (1964) studied the effects of heat conduction upon the propagation of Rayleigh surface waves in a semi-infinite elastic solid theoretically in two special cases: (i) when the surface of the solid is maintained at constant temperature and (ii) when the surface is thermally insulated. Nowacki (1966a, b, c) studied the couple stresses in the theory of thermoelasticity. Chadwick and Seet (1970) studied the wave propagation in transversely isotropic heat conducting elastic materials. Nayfeh and Nasser (1971) used the Maxwell s modified heat conduction equation to study plane harmonic waves in unbounded media as well as Rayleigh surface waves propagating along a half space consisting of linearly elastic materials that conduct heat. They obtained explicitly the expressions for various parameters that characterize these waves. McCarthy (1972) studied the wave propagation in generalized thermoelasticity. Sinha and Sinha (1974) studied problems on reflection of thermoelastic waves at a solid half-space in context of Lord and Shulman theory. Agarwal (1978) discussed the surface waves in generalized thermoelasticity of Lord and Shulman and of Green and Lindsay. Sharma and Singh (1985) have studied thermoelastic surface waves in a transversely isotropic half-space. Singh and Sharma (1985) studied the propagation of generalized thermoelastic waves in transversely isotropic medial. They solved basic equations by a general method after decoupling the SH-wave, which is not affected by thermal variations and is independent of the rest of the motion. They revealed that in general there are three distinct waves in a transversely isotropic medium. Sharma and Sidhu (1986) studied the propagation of plane harmonic waves in homogeneous anisotropic generalized thermoelastic materials after deriving the secular equation. They found that four dispersive wave modes are possible. Sharma (1988) studied the reflection of thermoelastic waves from the stress-free insulated boundary of an anisotropic half-space. Mayer (1990) discussed thermoelastic attenuation of surface acoustic waves.
7 Kumar and Singh (1996) discussed Rayleigh-Lamb problem in micropolar generalized thermoelastic layer with stretch and Rayleigh wave in a micropolar generalized thermoelastic half space with stretch. They obtained the frequency and wave velocity equations for symmetric and anti-symmetric vibrations for first problem and derived the frequency equation for second problem. They discussed the special cases of above problems of micropolar generalized thermoelasticity with stretch for G-L and L-S theories. Chandrasekharaiah (1996a, b) studied the one dimensional wave in thermoelastic half-space without energy dissipation. Chandersekharaiah (1997a) investigated the Rayleigh waves in thermoelastic medium without energy dissipation. Chandrasekharaiah (1997b) discussed the complete solutions in the theory of thermoelasticity without energy dissipation. Sinha and Elsibai (1996, 1997) investigated the reflection of thermoelastic waves at the interface of two semi-infinite media in welded contact. Semerak (1997) studied the problem of propagation of Rayleigh surface waves in thermoelastic media on the basis of the generalized coupled theory of thermoelasticity which takes account of the phenomenon of thermal relaxation for sharply nonsteady thermal loads. Shuvalov and Chadwick (1997) studied the unusual hierarchy of degeneracies in the linear theory of thermoelasticity. They deducted that in classical elastic wave theory all degeneracies take the form of acoustic axes i.e. directions in which two or all three plane bulk waves have equal speeds. Singh and Kumar (1998a) studied the plane wave propagation in a generalized thermo-microstretch elastic solid. They also discussed the reflection of the plane waves from a free surface of a generalized thermo-microstretch elastic solid. They obtained the reflection coefficients of various reflected waves with the angle of incidence for the Lord-Shulman theory. Singh and Kumar (1998b) studied the reflection of micropolar thermoelastic waves in micropolar generalized thermoelastic waves in micropolar generalized thermoelastic solid half space. They obtained the reflection coefficients of various reflected waves with the angle of incidences as well as with the thermoelastic coupling coefficients for Green Lindsay and Lord Shulman theories. Singh (2000a, 2001) studied the reflection and refraction of plane sound wave at an interface between a liquid half space and micropolar generalized thermoelastic solid half space. He computed the numerical results in terms of amplitude ratios for aluminum epoxy composite model for both L-S and G-L theories and the
8 comparison revealed the effect of second thermal relaxation time considered by Green and Lindsay. Abd-alla and Al-Dawy (2000) studied the reflection of SV waves from free surface of a thermoelastic solid half-space. Singh (2000b) discussed wave propagation in heat flux dependent generalized thermoelasticity. Cetinkaya and Li (2000) developed a transfer matrix formulation including the second sound effect based on the generalized dynamical theory of thermoelasticity for the longitudinal wave component propagation in a thermoelastic layer. Singh (2002) studied some problems on reflection of the generalized magnetothermo-viscoelastic plane waves from stress-free surface. Verma and Hasebe (2002) studied the boundary value problem in generalized thermoelasticity concerning the propagation of plane harmonic waves in a thin, flat, infinite homogeneous, transversely isotropic plate of finite width. They obtained the frequency equations corresponding to the symmetric and anti-symmetric modes of vibrations of the plate. Song and Zhang (2002) employed the reflection of magneto-thermo-elastic waves to discuss the effect of magnetic filed on the plane harmonic waves of a semi infinite elastic solid nearly a vacuum under Green Nagdhi theory. They obtained the expressions for the reflection coefficients which are the ratios of the amplitude of reflected waves to the amplitude of the incident wave. Sharma and Pathania (2003) studied the propagation of thermoelastic waves in a homogenous, isotropic, thermally conducting plate bordered with layers of inviscid liquid on both sides in the light of coupled thermoelasticity. They derived the secular equations for the plate in closed form and isolated mathematical conditions for symmetric and anti-symmetric wave modes. Sharma et al. (2003) investigated the problem of thermoelastic wave reflection from the insulated and isothermal stress-free as well as rigidly fixed boundaries of a solid half-space in the context of different theories of generalized thermoelasticity. Abd-alla et al. (2003) investigated the reflection of generalized magneto-thermo-viscoelastic waves at the boundary of a semi-infinite solid. For an incident rotational wave from within the solid on its boundary, they obtained the expressions for the reflection coefficients (and their approximate expressions) of the waves generated at the boundary. Sharma et al. (2003) studied the reflection from thermally insulated and isothermal surface for coupled and generalized theories of
9 thermoelasticity. Singh (2003a) studied a problem on wave propagation in an anisotropic generalized thermoelastic solid and obtained a cubic equation, which gives the dimensional velocities of various plane waves. Singh (2003b) studied the plane waves from a free surface of a cracked solid half space. Singh (2004) investigated plane waves in a thermally conducting viscous liquid half space with thermal relaxation times. He investigated that there exists three basic waves, i.e. thermal wave, longitudinal wave and transverse wave in a thermally conducting viscous liquid half space. He studied the reflection of plane waves from the free surface of a thermally conducting viscous liquid half space and obtained the results in terms of amplitude ratios and compared with those without viscosity and thermal disturbances. Sharma (2005) studied the thermoelastic interaction in an infinite Kelvin- Voigt type viscoelastic thermally conducting plate in which upper and lower surfaces are subjected to stress free, thermally insulated conditions. He employed coupled dynamical thermoelasticity and derived complex secular equations for the plate and obtained the results for coupled and uncoupled theories of thermoelasticity as particular cases. He obtained complex secular equitation s that led to two real frequency equations which contains information about wave number, phase velocity, group velocity and attention coefficients. Sharma and Pathania (2005) studied the propagation of circularly crested generalized thermoelastic waves in a homogenous isotopic thermally conducting plate, bordered with layers of inviscid liquid on both sides in the light of conventional coupled thermoelasticity, LS and GL theories of thermoelasticity. They derived secular equations for a circular homogeneous isotropic plate and obtained the results for the uncoupled theory of thermoelasticity. Singh (2005) solved the governing equations for generalized thermodiffusion in an elastic solid in context of L-S theory. There exist three kinds of dilatational waves and a Shear Vertical (SV) wave in a two-dimensional model of the solid. The reflection phenomenon of P and SV waves from free surface of an elastic solid with thermodiffusion is considered. The boundary conditions are solved to obtain a system of four non-homogeneous equations for reflection coefficients. These reflection coefficients are found to depend upon the angle of incidence of P and SV waves, thermodiffusion parameters and other material constants.
10 Singh (2006a) solved the governing equations for generalized thermodiffusion in elastic sold in context of L-S and G-L theories and found that there exist three kinds of dilatational waves and a SV wave in two dimensional model of the solid. He considered the reflection phenomena of SV waves from free surface of an elastic solid with thermodiffusion and solved the boundary conditions to obtain a system of four non homogeneous equations for reflection coefficients. Singh (2006b) studied the propagation of plane waves in a fiber-reinforced, anisotropic, generalized thermoelastic media. He solved the governing equations in xy-plane to obtain a cubic equation in phase velocity. He has shown that three coupled waves namely quasi-p, quasi-sv and quasithermal waves exist. He investigated the propagation of Rayleigh waves at stress free thermally insulated and transversely isotropic, fiber reinforced thermoelasticity solid half space and obtained the frequency equation. Singh (2006c) derived the dispersion relations in a generalized monoclinic thermoelastic solid half-space. Song et al. (2006) studied the wave propagation at interface between two half-spaces of micropolar viscoelastic media. Singh and Othman (2007) studied the governing equations for two dimensional problems in micropolar thermoelastic medium for a half space whose surface is free and subjected to an instantaneous thermal point source. They considered that entire elastic medium is rotating with uniform angular velocity and they applied the formation under five theories of thermoelasticity. Singh and Tomar (2007) studied the possibility of plane wave propagation in an infinite thermo-elastic medium with voids in context of the theory developed by Iesan (1986). They found that three sets of coupled longitudinal waves and a transverse wave can exist in an infinite thermo-elastic medium with voids. Each set of coupled longitudinal waves consists of displacement, void volume fraction and thermal properties. These coupled longitudinal waves are found to be dispersive in nature. They studied the reflection phenomenon of a set of coupled longitudinal waves from a free plane boundary of a thermo-elastic half-space with voids. Singh (2007a) solved the governing equations for two-dimensional homogeneous, isotropic generalized thermoelastic half space with voids in context of LS theory and has shown the existence of three compressional waves and a SV wave exists. He considered the reflection phenomena of compressional or shear wave from the free surface of a thermoelasticity solid with voids. Singh (2007b) studied the reflection of plane waves from thermally
11 insulated as well as isothermal boundaries of fibre-reinforced thermoelastic composites is studied. He obtained the reflection coefficients of three reflected quasi-waves in closed form. Brock (2007) solved the governing equations to study the Stoneley signals for each of two perfectly bonded, dissimilar thermoelastic half-spaces which include as special cases the Fourier heat conduction model and models with either one or two thermal relaxation times. Kumar and Singh (2008) studied the reflection/ transmission of plane waves at an imperfectly bonded interface of two orthotropic generalized thermoelastic half-spaces. Kumar and Singh (2009) studied the propagation of plane waves at an imperfectly bonded interface of two orthotropic generalized thermoelastic rotating halfspaces with different elastic and thermal properties. Singh (2010a) studied the reflection of plane waves at the free surface of a monoclinic thermoelastic solid half-space. Singh (2010b) discussed the wave propagation in an initially stressed transversely isotropic thermoelastic solid half-space. Singh (2011) solved the linear governing equations of generalized porothermoelasticity for a two-dimensional solution, which results in one shear wave and four kinds of coupled longitudinal waves. The phase velocities and inverse quality factors of these plane waves depend on frequency, porosity, relaxation times, and other material parameters. Initial stresses are developed in the medium due to many reasons, resulting from difference of temperature, process of quenching, shot pinning and cold working, slow process of creep, differential external forces, gravity variations, etc. The Earth is supposed to be under high initial stresses. It is therefore of great interest to study the effect of these stresses on the propagation of stress waves. During the last five decades considerable attention has been directed toward these phenomena. It was the achievement of Biot (1965) to show the acoustic propagation under initial stresses would be fundamentally different from that under stresses of free state. He has obtained the velocities of longitudinal and transverse waves along the coordinate axis only. Flavin and Green (1961) studied the propagation of plane thermoelastic waves of small amplitude in an infinite body having been subjected to large uniform extensions, at constant temperature in three perpendicular directions, two of the extension ratios being equal. Chattopadhyay et al. (1982) studied the reflection of elastic waves under initial stress at a
12 free surface. Sidhu and Singh (1983) presented comments on the above paper. Bouden and Datta (1984) studied the Rayleigh waves in granular medium over an initially stressed elastic half-space. Dey et al. (1985) studied the reflection and refraction of P- waves under initial stresses at an interface. Abd-Alla and Ahmed (1998) studied the Rayleigh waves in an orthotropic thermoelastic medium under gravity and initial stress. Ahmed (1999) studied the influence of gravity on the propagation of waves in granular medium. Ahmed (2000) studied the Rayleigh waves in a thermoelastic granular medium under initial stress. Abd- Alla and Ahmed (2003) discussed the Stoneley and Rayleigh waves in a nonhomogeneous orthotropic elastic medium under influence of gravity. Abd-Alla et al. (2004) studied the Rayleigh waves in magneto-elastic half-space of orthotropic material under influence of initial stress and gravity field. Addy and Chakraborty (2005) studied the Rayleigh waves in a viscoelastic half-space under initial hydrostatic stress in presence of the temperature field. Selim and Ahmed (2006) studied the propagation and attenuation of seismic body waves in dissipative medium under initial and couple stresses. Sharma (2007) studied the effect of initial stress on reflection at the free surfaces of anisotropic elastic medium. Selim (2008) discussed the effect of initial stresses on the reflection coefficients of plane waves in a dissipative medium context of Biot's incremental deformation theory. Acharya et al. (2009) investigated the general theory of transversely isotropic magneto-elastic interface waves in conducting media under initial hydrostatic tension or compression. Singh (2010c) studied the reflection of plane waves at a traction-free and electrically shorted/charge-free surface of a prestressed piezoelectric medium is studied. He derived the reflection coefficients of qp and qsv waves for electrically shorted and charge-free cases. Abd-Alla et al. (2011a) studied the propagation of shear waves in a non-homogeneous anisotropic incompressible medium with gravity field and initial stresses. The velocity of propagation of the shear waves depends upon the direction of propagation the anisotropy, gravity field, non-homogeneity of the medium, and the initial stress. Abd-Alla et al. (2011b) studied the influence of the gravity field, relaxation times and initial stress on propagation of Rayleigh waves in an orthotropic magneto-thermoelastic solid medium has been investigated. Son and Kang
13 (2011) studied the effect of initial stress on the propagation behavior of SH waves in piezoelectric coupled plates. Montanaro (1999) investigated the isotropic linear thermoelasticity with hydrostatic initial stress. Singh et al. (2006) studied the reflection of thermoelastic waves from free surface of thermoelastic solid half-space under hydrostatic initial stresses in context of Lord-Shulman theory. Effect of hydrostatic initial stresses is shown graphically on these coefficients. Othman and Song (2007) discussed the reflection of plane waves from a thermoelastic elastic solid half-space under hydrostatic initial stress without energy dissipation. Singh (2008) studied the reflection from insulated and isothermal stress free surface of a thermoelastic solid half space under hydrostatic initial stress He obtained the reflection coefficients and energy ratios of reflected waves for incident P and SV waves. He has shown graphically the effect of hydrostatic initial stresses on reflection coefficients and energy ratios. Ailawalia and Narah (2009) studied the deformation of a rotating generalized thermoelastic medium with a hydrostatic initial stress. They obtained the components of displacement, force stress, and temperature distribution in the Laplace and Fourier domains by applying integral transforms. Ailawalia et al. (2009) studied the deformation in a generalized thermoelastic medium with hydrostatic initial stress subjected to different sources. Ailawalia and Budhiraja (2011) employed the Green and Naghdi theory of thermoelasticity to study the deformation of thermoelastic solid half-space under hydrostatic initial stress and rotation with two-temperature. The present thesis is organized as follows: In Chapter-1, the research papers on development of generalized thermoelasticity are reviewed. The research papers on wave propagation in coupled and generalized thermoelasticity with various additional parameters are reviewed. In particular, the work on wave propagation in thermoealsticity with hydrostatic initial stresses is reviewed in detail. In Chapter-2, a model of two half-spaces of different thermoelastic solids is considered in welded contact under hydrostatic initial stress. The appropriate boundary
14 conditions are satisfied at the interface to obtain reflection and refraction coefficients of various reflected and refracted waves for the incidence of thermal wave. A particular numerical example is considered to show the effect of hydrostatic initial stress on these coefficients graphically for a certain range of the angle of incidence. In Chapter-3, an interface between two different half-spaces of thermoelastic solids with initial stress is considered. A reflection and refraction phenomena at an interface is studied for obtaining the reflection and refraction coefficients of various reflected and refracted waves, when a SV wave is incident. The numerical computations of these coefficients are carried out for a particular model. The effects of hydrostatic initial stresses are observed on the coefficients of reflected and refracted waves and are shown graphically for certain range of angle of incidence. In Chapter-4, Green and Naghdi theory of thermoelasticity is employed to study the reflection and refraction at an interface between two thermoelastic solid half-spaces with hydrostatic initial stress. The amplitude ratios of the reflected and refracted waves are obtained for the incident plane waves. Numerical computations are carried out for a particular example of the model. The effect of hydrostatic initial stress is observed on the amplitude ratios of reflected and refracted waves. In Chapter-5, an imperfectly bonded interface between two monoclinic thermoelastic half-spaces is chosen to study the reflection and transmission of plane waves in context of gerneralized thermoelasticity. Six relations between amplitudes of incident, reflected and transmitted quasi-p (qp) waves, quasi thermal (qt) waves and quasi-sv (qsv) waves are obtained. Some particular cases are obtained which agree with earlier well established results. A procedure for computing reflected and refracted angles is derived for a given incident wave, where the angles of reflection are found not equal to the angles of incident waves in a monoclinic thermoelastic medium. In the last chapter, the surface wave propagation in isotropic thermoelasticity with hydrostatic initial stress is studied. The governing equations are solved to obtain the general solution in x-y plane. The appropriate boundary conditions at an interface between two dissimilar half-spaces are satisfied by appropriate particular solutions to obtain the frequency equation of the surface wave in the medium. The frequency equation for Rayleigh wave is derived as a limiting case.