Chapter 1

Introduction

Theory of elasticity is concerned with the determination of the stresses and displacements in

a body as a result of applied mechanical or thermal loads, for those cases in which the body

reverts to its original state on the removal of the loads. It is assumed that in case of linear

infinitesimal elasticity, in which the stresses and displacements are linearly proportional to

the applied loads and the displacements are small in comparison with the characteristic

length dimensions of the body. These restrictions ensure that linear superposition can be

used and enable us to employ a wide range of series and transform techniques which are not

available for non-linear problems. Most engineers first encounter problems of this kind in

the context of the subject known as strength of materials, which is an important constituent

of most engineering curricula.

The theory of thermoelasticity deals with the effect of mechanical and thermal distur-

bances on an elastic body. The theory of uncoupled thermoelasticity consists of the heat

equation, which is independent of mechanical effects, and the equation of motion, which

contains the temperature as a known function. There are two defects in this theory. First

is that the mechanical state of the body has no effect on the temperature. Second, the heat

equation, which is parabolic, implies that the speed of propagation of the temperature is

infinite, a physically unrealistic phenomenon.

Duhamel (1837) studied firstly the coupling between the strain and temperature fields.

He gave the derivation of the equations in which distribution of strains in an elastic medium

subjected to a temperature gradient was presented. He introduced the dilatation term in the

equation of thermal conductivity, but his equation was not satisfactory thermodynamically.

Neumann (1855), Voigt (1910) and Jeffreys (1930) made attempts at thermodynamical justi-

fication of equations of Duhamel’s theory (1837) and solved a number of interesting problems.

1

2

Biot (1956) was the first, who obtained satisfactory results, when he derived the basic

relations and equations and formulated the various theorems of thermoelasticity on the basis

of thermodynamics of irreversible processes. In Biot (1956) theory of classical thermoelas-

ticity, the equation of motion is hyperbolic in nature, whereas the heat conduction equation

is parabolic in nature as such classical thermoelasticity predicts a finite speed for predom-

inantly elastic disturbances but an infinite speed for predominantly thermal disturbances,

which are coupled together. Obviously, this result is physically unrealistic since experimental

investigations conducted on various solids have shown that heat pulses do propagate with

finite speed.

To take care of the paradox in Biot’s theory, Kaliski (1965) proposed a physical possible

model involving a finite velocity of heat propagation as actually required in nature. Lord

and Shulman (1967) developed a theory in which they modified the Fourier’s law of heat

conduction with the introduction of a thermal relaxation time parameter. This theory is

known as L-S theory or extended thermoelasticity (ETE) theory. The heat equation in this

theory ensures finite speeds of propagation of heat and elastic waves. The equations of

motion and constitutive relations remain the same as those for the coupled and uncoupled

theories of thermoelasticity.

Gurtin and Pipkin (1968) obtained a general heat conduction equation for non-linear rigid

materials with memory for which thermal disturbances propagate with finite speeds. Fox

(1969) proposed a generalization of thermoelasticity which arises from a physically motivated

modification of Fourier’s law of heat conduction. He postulated the constitutive equations

which are valid for finite deformation and temperature variations and these are reduced to

canonical form by the usual techniques of non-linear continuum mechanics. He gave some

exact solutions to illustrate novel features of the non-linear theory.

Muller (1971) proposed an entropy production inequality in a review of thermodynam-

ics of thermoelastic solids with the help of which, he considered restrictions on a class of

constitution equations. Green and Laws (1972) proposed the generalization of this inequal-

ity. Green and Lindsay (1972) formulated a more explicit and simpler theory than Muller

based upon the inequality proposed by Green and Laws (1972). It is known as G-L theory

of thermoelasticity with two relaxation times or the theory of temperature-rate-dependent

thermoelasticity (TRDTE). This theory does not violate the classical Fourier’s law, if the

material has a centre of symmetry at each point. In this theory, both the equations of motion

and heat conduction are hyperbolic, but the equation of motion is modified and differs from

that of classical coupled thermoelasticity theory.

3

Banerjee and Pao (1974) studied thermoelastic waves in anisotropic solids and found four

characteristic wave speeds. Chadwick (1979) investigated the propagation of plane harmonic

waves of small amplitude in a heat-conducting elastic body of unrestricted symmetry that

is not stress free in its undisturbed state. Sharma (1986) studied the transient behaviour

of thermoelastic waves in transversely isotropic solid half-space with thermal relaxation by

using Fourier and Laplace transform techniques. Li (1992) formulated a generalized theory

of thermoelasticity for an anisotropic medium using a form of the heat transport equation,

which includes the time needed for acceleration of the heat flow. Iesan (1998) presented the

fundamental solution in the theory of thermoelasticity without energy dissipation. Sharma,

Kumar and Chand (2003) investigated reflection at the free surface in generalized thermoe-

lastic half-space.

Svanadze (2004) obtained the fundamental solutions of the equations of the equilib-

rium and steady oscillations in theory of thermoelasticity with microtemperatures. Kumar

and Sarthi (2006) discussed the problem of the reflection and refraction of thermoelastic

plane waves at an imperfect interface between two dissimilar thermoelastic solid half-spaces.

Youssef and AI-Harby (2007) obtained the general solution of infinite body with a spherical

cavity when the bounding plane of cavity is subjected to thermal loading(thermal shock and

ramp-type heating) by using Laplace transform and state space techniques.

Das, Lahiri, Sarkar and Basu (2008) investigated the problem of reflection of P-wave

and SV-wave in a homogeneous, isotropic generalized thermoelastic medium at stress-free

with insulated or isothermal boundary conditions. Kumar and Devi (2010) discussed the

problem of reflection of plane waves in the mixture of generalized thermoelastic solid half-

space. Kothari, Kumar and Mukhopadhyay (2010) obtained the fundamental solution in

generalized thermoelasticity with three phase-lages. Kumar and Devi (2011) studied the

plane wave propagation in anisotropic medium in the context of the different theories of

thermoelasticity. Othman (2011) obtained the displacement component, temperature and

stress component in generalized thermoelastic medium due to internal heat sources by using

state space approach and integral transform technique(Laplace and Fourier transforms).

Youssef and El-Bary (2012) obtained temperature and stress field in a magneto-generalized

thermoelastic medium with variable material properties subjected to ramp-type heating by

using state-space approach.

Diffusion can be defined as the random walk of an assemble of particles from region of

high concentration to that of low concentration. Nowadays, there is a great deal of interest in

the study of this phenomenon due to its application in geophysics and electronic industry. In

4

integrated circuit fabrication, diffusion is used to introduce dopants in controlled amounts

into the semiconductor substance. In particular, diffusion is used to form the base and

emitter in bipolar transistors, integrated resistors, and the source/drain regions in metal

oxide semiconductor (MOS) transistors and dope poly-silicon gates in MOS transistors. In

most of the applications, the concentration is calculated using what is known as Fick’s

law. This is a simple law which does not take into consideration the mutual interaction

between the introduced substance and the medium into which it is introduced or the effect

of temperature on this interaction. Study of phenomenon of diffusion is used to improve the

conditions of oil extraction (seeking ways of more efficiently recovering oil from oil deposits).

These days, oil companies are interested in the process of thermodiffusion for more efficient

extraction of oil from oil deposits.

In the case of the thermodiffusive elasticity of a Hookean body the deformation, which is a

reversible process, occurs coupled with the irreversible processes of heat conduction and diffu-

sion. The work in this area started with three papers by Podstrigach (1961), and Podstrigach

and Pavlina (1961, 1965), who gave the relationships between the deformation, temperature

and concentration based upon the thermodynamics of irreversible processes. Muller (1968)

studied the problem of diffusion and thermal diffusion in a mixture of Maxwellian gases.

Podstrigach and Shvechuk (1969) presented a variational equation equivalent to the system

of governing equations of a model which allows description of the interconnection between

the deformation, heat and mater diffusion processes. The concentration dependent diffusion

phenomena was considered by Shampine (1973). Nowacki (1971, 1972, 1974a, 1974b, 1974c)

in a series of papers presented the theorem of virtual work, fundamental energy theorem,

theorem of reciprocity of works, generalized Maxwell reciprocity relations and theorems of

Somiglina and Maysel type. Thermodiffusion in elastic materials with microstructure, and

the non-linear theory of coupled mechanical and thermodiffusional effects for the elastic ma-

terial were considered by Naerlovic-Veljkovic and Plavsic (1974). Cukic (1974) investigated

a problem of diffusion in thermoelastic plates.

Nowacki (1975) determined the stress functions for thermodiffusion solid bodies. Nowacki

(1976) studied dynamic problems of thermodiffusion in solids. Herrera and Billok (1976)

discussed the dual variational principles for diffusion equations. Wyrwal (1985) described

the variational principle for linear coupled dynamic theory of thermodiffusion in solids.

Maruszewski (1986) discussed the thermodynamic fundamentals of magneto thermodiffusion

and electro thermodiffusion in continuous media. Hoffman (1987) studied the thermodiffu-

sion process in a non-deformable medium. Wrobel (1987a, 1987b) investigated variational

5

theorems for the problems of coupled thermoviscoelastic diffusion with finite velocities of

heat and mass propagation and for the problems of thermodiffusion flows coupled with the

stress field respectively. Wyrwal (1987) proved reciprocity theorem for the linear theory of

thermodiffusion in solids. Nowacki and Olesiak (1991) discussed various problems of ther-

modiffusion in solids. Shvets (1999) developed mathematical models that take account not

only the thermodiffusion process but also the viscosity properties of the material.

Gawinecki, Kacprzyk and Bar-Yoseph (2000) proved a theorem about existence, unique-

ness and regularity of the solution to an initial-boundary value problem for a nonlinear cou-

pled parabolic system appearing in the thermodiffusion in solid body. Sherief, Hamza and

Saleh (2004) proved the uniqueness and reciprocity theorems for the equations of generalized

thermoelastic diffusion problem, in isotropic media, on the basis of the variational princi-

ple equations, under restrictive assumptions on the elastic coefficients. Sherief and Saleh

(2005) investigated the problem of a thermoelastic half-space with a permeating substance

in contact with the bounding plane in the context of the theory of generalized thermoelastic

diffusion with one relaxation time. They took the bounding surface of the half-space to be

traction free and subjected to a time dependent thermal shock. Aouadi (2006a) studied the

generalized thermoelastic diffusion problem with variable electrical and thermal conductiv-

ity. Aouadi (2006b, 2007a) studied the interaction between the processes of elasticity, heat

and diffusion in an infinitely long solid cylinder and in an infinite elastic body with spherical

cavity respectively. Due to the inherit complexity of the derivation of the variational prin-

ciple equations, Aouadi (2007b) proved the theorem as given by Sherief, Hamza and Saleh

(2004). Sharma (2007) studied plane harmonic generalized thermoelastic diffusive waves in

heat conducting solids.

Aouadi (2008a) derived the uniqueness and reciprocity theorems in anisotropic media,

under the restriction that the elastic, thermal conductivity and diffusion tensors are positive

definite. Aouadi (2008b) derived some qualitative results of the coupled theory of thermoelas-

tic diffusion for anisotropic media. Choudhary and Deswal (2008) investigated deformation

due to distributed loads in an elastic solid with generalized thermodiffusion. Sharma, Ku-

mar and Ram (2008a, 2008b), respectively, discussed plane strain deformation in generalized

thermoelastic diffusion and dynamical behavior of generalized thermoelastic diffusion with

two relaxation times in frequency domain due to various sources. Ram, Sharma and Kumar

(2008) investigated thermomechanical response of generalized thermoelastic diffusion with

one relaxation time due to time harmonic sources. Aouadi (2009) derived some spatial sta-

bility results for the quasi-static problem in thermoelastic diffusion theory for anisotropic

6

media. Othman, Atwa and Farouk (2009) investigated the disturbances in a homogeneous,

isotropic elastic medium with thermoelastic diffusion based on the Green and Naghdi theory

type III. Othman, Farouk and El-Hamied (2009) established the model of equations of gener-

alized thermoelastic diffusion for a homogeneous elastic half-space under the dependence of

the modulus of elasticity on the reference temperature. Kumar and Chawla (2009) discussed

wave propagation at the imperfect boundary between transversely isotropic thermoelastic

diffusive half-space and an isotropic elastic layer.

Aouadi (2010a) derived the basic equations of a nonlinear theory of thermoelastic dif-

fusion with mixtures. Aouadi (2010b) investigated a one-dimensional contact problem in

linear thermoelastic diffusion theory. Aouadi and Soufyane (2010) obtained polynomial and

exponential stability for one-dimensional problem in thermoelastic diffusion theory. Kuang

(2010) derived the basic equations and proved variational principles for generalized ther-

modiffusion theory in pyroelectricity. Sharma, Kumar and Ram (2010) investigated a two-

dimensional deformation in a homogeneous, anisotropic generalized thermoelastic diffusion

as a result of an inclined load by applying Laplace and Fourier transforms. Kumar, Kothari

and Mukhopadhyay (2011) established a convolutional type variational principle and a reci-

procity theorem for the linear theory of generalized thermoelastic diffusion for isotropic

elastic solids. Ezzat and Fayik (2011) developed a new theory of thermodiffusion in elastic

solids using the methodology of fractional calculus. Aouadi (2011) proved the exponential

stability of the slightly damped and totally hyperbolic thermoelastic diffusive system. Ku-

mar and Chawla (2011a) discussed propagation of surface waves at the imperfect boundary

between transversely isotropic elastic layer and half-space in the context of generalized the-

ory of thermoelastic diffusion. Kumar and Panchal (2011) discussed the propagation of axial

symmetric cylindrical surface waves in a cylindrical bore through a homogeneous isotropic

thermoelastic diffusive medium of infinite extent.

Bijarnia and Singh (2012) discussed the propagation of plane waves in a transversely

isotropic generalized thermoelastic solid with diffusion in the context of the Lord-Shulman

(1967) theory. Kumar and Chawla (2011b) obtained the fundamental solution for two dimen-

sional problem in orthotropic thermodiffusive elastic medium. Kothari and Mukhopadhyay

(2012) presented the Galerkin-type solution in generalized thermoelastic with mass diffusion

medium. Kumar and Kansal (2012a) discussed the plane wave and fundamental solution in

a generalized thermoelastic diffusion medium.

The theory of linear elastic materials with voids is one of the most important general-

izations of the classical theory of elasticity. This theory has practical use for investigating

7

various types of geological and biological materials for which elastic theory is inadequate.

This theory is concerned with elastic materials consisting of a distribution of small pores

(voids), in which the voids volume is included among the kinematics variables and in the

limiting case of volume tending to zero, the theory reduces to the classical theory of elas-

ticity. Goodman and Cowin (1972) established a continuum theory for granular materials,

whose matrix material (or skeletal) is elastic and interstices are voids. They formulated this

theory from the formal arguments of continuum mechanics and introduced the concept of

distributed body, which represents a continuum model for granular materials (sand, grain,

powder, etc) as well as porous materials (rock, soil, sponge, pressed powder, cork etc). The

basic concept underlying this theory is that the bulk density of the material is written as the

product of two fields, the density field of the matrix material and the volume fraction field

(the ratio of the volume occupied by grains to the bulk volume at a point of the material).

This representation was employed by Nunziato and Cowin (1979) to develop a nonlinear

theory of elastic material with voids. Cowin and Nunziato (1983) presented a linear the-

ory of elastic material with voids for the mathematical study of the mechanical behavior of

porous solids. They considered several applications of the linear theory by investigating the

response of the materials to homogeneous deformations, pure bending of beams and small

amplitudes of acoustic waves. Iesan (1985) proved the uniqueness, reciprocity and variational

theorems for the basic governing equations of elastic materials with voids and also studied

the propagation of acceleration waves in such materials.

Iesan (1986) extended the linear theory of elastic materials with voids to include thermal

effect. The existence and uniqueness of system of governing equations for thermoelastic

material with voids was proved by Rusu (1987). Ciarletta (1991) presented a Galerkin type

solution in the theory thermoelastic materials with voids and used this solution to obtain

harmonic Green’s functions in an infinite thermoelastic body with voids. Dey, Gupta and

Gupta (1993) discussed the propagation of torsional surface waves in an elastic medium

with void pores and concluded that there may be two wavefronts for longitudinal waves

in the medium. Pompei and Scalia (1994) also studied the steady vibrations of a linear

homogeneous and isotropic elastic material with voids.

Dhaliwal and Wang (1995) formulated a thermoelasticity theory for elastic materials with

voids to include a heat flux among the constitutive variables. Luo (1996) studied some basic

principles in the dynamical theory of thermoelastic bodies with voids. Marin (1997a, 1997b,

1998) studied uniqueness and domain of influence results in thermoelastic bodies with voids.

Chirita and Scalia (2001) studied the spatial and temporal behaviour in linear thermoelastic

8

body with voids. Pompei and Scalia (2002) studied the asymptotic spatial behavior in linear

thermoelasticity of materials with voids. Scalia, Pompei and Chirita (2004) considered the

steady time harmonic oscillations within the context of linear thermoelasticity for material

with voids and derived the spatial decay results for the amplitude of harmonic vibrations

in a cylinder subjected to plane boundary data varying harmonically in times on its lateral

surface and on one of its bases, provided the angular frequency is lower than a critical value.

Kumar and Rani (2005a, 2005b) investigated some problems of mechanical and thermal

sources in thermoelastic half-space with voids.

Singh and Tomar (2007) studied the propagation of plane waves in an infinite thermo-

elastic medium with voids using the theory developed by Iesan (1986). They found that

three coupled longitudinal waves and one transverse wave can exist in an infinite thermoe-

lastic medium with voids. Each coupled longitudinal wave consists of displacement, void

volume fraction and thermal properties. The reflection phenomenon of these plane waves

from a free plane boundary of a thermo-elastic half-space with voids has been investigated in

detail. Singh (2007) investigated the propagation of plane waves in a generalized thermoe-

lastic material with voids within the context of Lord-Shulman (1967) theory. Oswald (2007)

investigated the hyperbolic equations for thermoelastic solids with voids.

Sharma, Kaur and Sharma (2008) discussed the problem of three dimensional vibration

analysis of a thermoelastic cylindrical panel with voids. Abo-Dahab (2010) discussed the

problem propagation of longitudinal wave from stress-free surface elastic half-space with

voids under thermal relaxation and magnetic field. Aouadi (2012a) studied the uniqueness

and existence theorems in thermoelasticity with voids and without energy dissipation. Later,

Aouadi, Lazzari and Nibbi (2012) studied the problem of exponential decay in thermoelastic

materials with voids and dissipative boundary without thermal dissipation.

Aouadi (2010c) developed a theory of thermoelastic diffusion materials with voids and

derived the uniqueness, reciprocity, continuous dependence and existence theorems. Singh

(2011a) studied the plane wave propagation in an isotropic, homogeneous thermoelastic

diffusion solid with voids and obtained the relations between reflection coefficients for the

incidence of plane waves. Pal and Singh (2011) discussed the problem of surface wave

propagation in a generalized thermoelastic material with voids and diffusion. Aouadi (2012b)

studied the exponential stability for the one dimensional theory of thermoelastic diffusion

materials with voids. Sharma (2012) studied the reflection of plane waves in thermodiffusive

elastic half-space with voids and obtained amplitude ratios.

Initial stresses are developed in the medium due to many reasons, resulting from difference

9

of temperature, process of quenching, shot pinning and cold working, slow process of creep,

differential external of 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. Biot (1939) developed the non-linear theory of elasticity and

the linearized case for a body under initial stress. Later, in (1940), he studied the problem

of influence of initial stress on elastic waves, and after that in (1965), he depicted that the

acoustic propagation under initial stresses would be fundamentally different from that under

stress free state. He has obtained the velocities of longitudinal and transverse waves along

the coordinate axis only. England and Green (1961), and Green (1962) derived the equations

of small thermoelastic deformations superposed on large deformation and have studied some

applications of the theory to special problems. The case of small thermoelastic deformations

in a body that is under initial stress and initial constant temperature was examined in

detail. Flavin and Green (1961) studied the plane thermo-elastic waves in an initially stressed

thermoelastic medium. Dahlen (1972) investigated the effect of a homogeneous anisotropic

initial stress on the propagation of infinitesimal amplitude elastic body waves in a perfectly

elastic, homogeneous medium.

Iesan (1980) developed the linear theory of thermoelasticity with initial stress and ini-

tial nonconstant temperature. In the first part of this paper, he use the method given by

Green and Rivlin (1964) to obtain the basic equations from the balance of energy and the

invariance requirements under superposed rigid body motions. After that, the linear theory

of thermoelastic materials with initial stresses and initial heat flux is presented. The reci-

procity relations and the variational characterization of solution in the dynamic theory are

also established. In (1987), Iesan developed the theory of initially stressed thermoelastic ma-

terial with voids. Later, Iesan (1988) studied the problem of initially stressed thermoelastic

dielectrics. In this paper he established the equations of infinitesimal deformations and weak

fields superimposed on a finite thermoelastic deformation and strong electromagnetic field,

and shows that initially nonuniform heated rigid dielectrics and thermal field are influenced

by the polarization vector field even in the case of equilibrium.

Montanaro (1997) investigated the problem of wave propagation along axes of symmetry

in linearly elastic media with initial stress. Wang and Slattery (2002) discussed the problem

of thermoelasticity without energy dissipation for initially stressed bodies. Hua, Zhenbang,

Zhenbang, Tiejun and Zikum (2003) investigated the problem of propagation of surface

acoustic waves in prestressed anisotropic layered piezoelectric structures and obtained the

equations of phase velocity for electrically free and shorted cases. Iesan (2008) discussed

10

the theory of prestressed thermoelastic cosserat continua. Vinh (2009) derived the secular

equations of Rayleigh waves in elastic media under the influence of gravity and initial stress.

Akbarov and Ilhan (2010) investigated the problem of time-harmonic dynamical stress

field in a system comprising a pre-stressed orthotropic layer and pre-stressed orthotropic

half-plane. Sharma (2010a) discussed the wave propagation in a pre-stressed anisotropic

generalized thermoelastic medium. Son and Kang (2011) studied the effect of initial stress

on the propagation behavior of SH waves in piezoelectric coupled plates. Abd-Alla, Abo-

Dahab and Hammad (2011) discussed propagation of Rayleigh waves in generalized magneto-

thermoelastic orthotropic material under initial stress and gravity field.

Mahmoud (2012) obtained the frequency equation for Rayleigh waves in a granular

medium under effect of initial stress and gravity field. He used the Lame’s potential tech-

niques to obtain the analytical solution. Abd-Alla, Abo-Dahab and Al-Thamali (2012)

studied the propagation of Rayleigh waves in a rotating orthotropic material elastic half-

space under initial stress and gravity. Abd-Alla and Abo-Dahab (2012) investigated one

dimensional problem in a generalized magneto-thermoelastic diffusion in an infinite rotating

medium under initial stress with a spherical cavity subjected to a time dependent thermal

shock and obtained displacement, temperature, stresses, concentration and chemical poten-

tial.

Viscoelastic behavior manifests itself in creep, or continued deformation of a material

under constant load; and in stress relaxation, or progressive reduction in stress while a ma-

terial is under constant deformation. All materials exhibit some viscoelastic response. In

common metals such as steel or aluminum as well as quartz, at room temperature and at

small strain, the behavior does not deviate much from linear elasticity. Synthetic polymers,

woods, and human tissue as well as metals at high temperature display significant viscoelas-

tic effects. In some application, even a small viscoelastic response can be significant. To

complete , an analysis or design involving such materials must incorporate their viscoelastic

behavior. Viscoelastic materials play an important role in many branches of engineering,

technology and, in recent years, biomechanics. Viscoelastic materials, such as amorphous

polymers, semicrystalline polymers, and biopolymers, can be modeled in order to determine

their stress or strain interactions as well as their temporal dependencies. The investigations

of the solutions of viscoelastic wave equations, velocities of seismic wave propagating and

the attenuation of seismic wave in the viscoelastic media are very important for geophysical

prospecting technology.

Most of the work before (1960), on linear viscoelastic wave propagation for which explicit

11

solutions were obtained, was essentially one dimensional and for specific material. However,

a foundation of three dimensional linear viscoelasticity theory was laid by Bland (1960).

He concluded that similar to a perfectly elastic isotropic medium, under the assumption of

small displacements, two types of waves can propagate in an isotropic viscoelastic medium

with complex velocities when body forces are absent. Coleman (1961) discussed the problem

foundation of linear viscoelasticity. Tsai and Kolsky (1968) investigated experimentally

and theoretically surface wave propagation in elastic and linear viscoelastic blocks when

specimens were subjected to normal surface impacts by steel ball. Maier and Tsai (1974)

studied the problem of wave propagation in linear viscoelastic plates of various thicknesses.

Longman (1980) discussed wave propagation in a viscoelastic solid from a Heaviside step

function plane source and from a Heaviside step function point source. Carcione, Kosloff

and Kosloff (1988) studied the wave propagation simulation in a linear viscoelastic medium.

Iesan and Scalia (1989) given some theorems in the theory of thermoviscoelasticity. Cav-

iglia, Morro and Pagani (1990) investigated the inhomogeneous waves in viscoelastic media.

They also discussed the reflection and refraction phenomena at the common boundary of

different types of media. Ciarletta and Scalia (1991) proved the uniqueness theorem in

the linear theory of viscoelastic materials with voids. Romeo (2001) investigated Rayleigh

waves in a viscoelastic solid half-space. Daley and Krebes (2004) obtained reflection and

transmission coefficients in a viscoelastic media when SH-wave is incident. Cerveny and

Psencik (2005) discussed the plane waves in viscoelastic anisotropic media. Gao, Lin, How-

son and Williams (2006) studied the problem of wave propagation in a viscoelastic layered

half-space and used two methods in combining form which gives very efficient and accurate

solution. Garg (2007) investigated wave propagation in viscoelastic anisotropic media with

initial stress.

Chattopadhyay, Gupta, Sharma and Kumari (2009) studied the shear wave propaga-

tion in a viscoelastic layer over a semi-infinite viscoelastic half-space. Jun and Yun (2010)

discussed the seismic wave propagation in Kelvin viscoelastic media. Dhemaied, Rejiba,

Camerlynck and Bodet (2011) investigated the problem of seismic wave propagation in vis-

coelastic media using the auxiliary differential equation method. Mukhopadhyay (2000)

studied thermoviscoelastic interactions in an unbounded body with the spherical cavity sub-

jected to a periodic loading on the boundary. Cicco and Nappa (2003) discussed the problem

of singular surfaces in thermoviscoelastic materials with voids and presented a linear the-

ory of thermoviscielastic material with voids in which the heat flux is independent on the

present temperature gradient, but depends upon the past history of this gradient. Othman

12

and Song (2007a) studied the reflection and refraction of thermo-viscoelastic waves at the

interface between two micropolar viscoelastic media without energy dissipation.

Ezzat and Atef (2011) investigated the problem of magnetothermo-viscoelastic material

with a spherical cavity. They considered generalized thermoelasticity in the context of one

relaxation time. Fahmy (2011) discussed influence of inhomogeneity and initial stress on the

transient magneto-thermo-visco-elastic stress waves in an anisotropic solid. Iesan and Scalia

(2011) developed the theory of thermoviscoelastic mixtures and presented the uniqueness

result in the linearized theory. Ezzat and Awad (2011) established a linear theory of microp-

olar thermoviscoelasticity with mass diffusion and proved uniqueness theorem. Svanadze

(2012) studied the problem of potential method in the linear theories of viscoelasticity and

thermoviscoelasticity for Kelvin-Voigt materials. He constructed the fundamental solutions

of systems of equations of steady vibrations and established the radiation conditions and

basic properties of fundamental solution. Luppe, Conoir, and Norris (2012) investigated the

problem of effective wave numbers for thermoviscoelastic media containing random configu-

rations of spherical scatterers.

Plan of the thesis is as follows: This thesis consists of five more Chapters and a list

of references given at the end of this thesis. The subject matter is laid out in the following

way:

In chapter 2, the plane waves and fundamental solution in initially stressed isotropic ther-

moelastic media with diffusion and voids has been investigated. The generalized theories of

thermoelasticity developed by Lord and Shulman (1967) and for Green and Lindsay (1972),

have been used to investigate the problem. There exist four kinds of longitudinal waves in

addition to transverse(SV) waves of isotropic generalized thermoelastic medium with diffu-

sion and voids, which get decoupled from rest of motion and are not affected by thermal,

mass diffusion and voids parameters. The graphical representation is given for phase propa-

gation velocity and attenuation quality factor of waves. The fundamental solution of system

of differential equations in the theory of thermoelastic diffusion medium with voids in case of

steady oscillations in terms of elementary functions has also been constructed. Some special

cases are also discussed.

Chapter 3, deals with response of thermal and mass concentration sources for initially

stressed isotropic thermoelastic medium with diffusion and voids for one dimensional model

13

and the state-space approach has been used to solve the problem. The generalized theo-

ries of thermoelasticity developed by Lord and Shulman (1967) and for Green and Lindsay

(1972), have been used to investigate the problem. The variation of volume fractional field,

temperature change, mass concentration and stress component with respect to distance are

computed numerically for a particular model. The obtained numerical results are depicted

graphically to show the effect of initial stress and relaxation times. Some special cases are

also deduced.

Chapter 4, concerns with the wave propagation in initially stressed anisotropic thermoe-

lastic media with diffusion and voids. The generalized theories of thermoelasticity developed

by Lord and Shulman (1967) and for Green and Lindsay (1972), have been used to investigate

the problem. As a special case, the governing equations for homogeneous initially stressed

transversely isotropic generalized thermoelastic media with diffusion and voids are deduced.

When plane waves propagate in a principal plane, purely transverse wave mode gets decou-

pled from rest of the motion and is not affected by the thermal, voids and diffusion vibrations.

The different characteristics of waves like phase velocity and attenuation coefficient, specific

loss and penetration depth are computed numerically and presented graphically. Some par-

ticular cases are also deduced.

Chapter 5, deals with the problem of reflection and refraction due to longitudinal and

transverse waves incident obliquely at the plane interface between uniform elastic half-space

and initially stressed isotropic thermoelastic half-space with diffusion and voids. The gen-

eralized theories of thermoelasticity developed by Lord and Shulman (1967) and for Green

and Lindsay (1972), have been used to investigate the problem. It is noticed that amplitude

ratios of various reflected and refracted waves are functions of angle of incidence, frequency

and are influenced by the elastic properties of media. The variations of amplitude ratios

and energy ratios with angle of incidence are shown graphically to depict the effect of initial

stress and thermal relaxation times. The conservation of energy at the interface is verified.

Some particular cases are also discussed.

In chapter 6, the problem of reflection and transmission at the plane boundary between

elastic half-space and initially stressed isotropic viscothermoelastic half-space with diffusion

and voids has been investigated. The generalized theories of thermoelasticity developed by

Lord and Shulman (1967) and for Green and Lindsay (1972), have been used to investigate

the problem. It is noticed that amplitude ratios of various reflected and refracted waves

are functions of angle of incidence, frequency and are influenced by the elastic, viscoelastic

properties of media. The amplitude ratios and energy ratios have been computed numerically.

14

The variations of amplitude ratios and energy ratios with angle of incidence are shown

graphically. The conservation of energy at the interface is verified. Appreciable effects of

initial stress and thermal relaxation times are observed of the resulting quantities. Some

particular cases are also deduced.