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Chapter 1
Introduction and review of
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Introduction and review of literature
Every day there are about fifty earthquakes worldwide that are strong enough to be felt
locally, and every few days an earthquake occurs that is capable of damaging structures.
Each event radiates seismic waves that travel throughout Earth, and several earthquakes per
day produce distant ground motions that, although too week to be felt, are readily detected
with modern instruments anywhere on the globe. Seismology is the science that studies
these waves and what they tell us about the structure of Earth and the physics of
earthquakes. It is the primary means by which scientists learn about Earth’s deep interior,
where direct observations are impossible, and has provided many of the most important
discoveries regarding the nature of our planet.
Seismology occupies an interesting position within the more general fields of geophysics
and Earth science. It presents fascinating theoretical problems involving analysis of elastic
wave propagation in complex media, but it can also be applied simply as a tool to examine
different areas of interest. Applications range from studies of Earth’s core, thousands of
kilometers below the surface, to detailed mapping of shallow crustal structure to help locate
petroleum deposits. Much of the underlying physics is no more advanced than Newton’s
law (F=ma), but the complicated mathematical treatments and extensive use of powerful
computers. Seismology is driven by observations, and improvements in instrumentation and
data availability have often led to breakthroughs both in seismology theory and our
understanding of Earth structure.
The information that seismology provides has widely varying degrees of uncertainty. Some
parameters, such as the average compressional wave travel time through the mental, are
known to a fraction of a percent, while others, such as the degree of damping of seismic
energy within the inner core, are known only very approximately. The average radial
velocity structure of Earth has been known fairly well for over fifty years, and of locations
and seismic radiation patterns of earthquakes are now routinely mapped, but many
important aspects of the physics of earthquakes themselves remains a mystery.
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Elasticity Theory: When a force is applied to a material, it deforms. This means that the
particles of the material are displaced from their original positions. Provided the force does
not exceed a critical value, the displacements are reversible; the particles of the material
return to their original positions when the force is removed, and no permanent deformation
results. This is called elastic behaviour.
Stress and Strain: The term stress (s) is used to express the loading in terms of force
applied to a certain cross-sectional area of an object. From the perspective of loading, stress
is the applied force or system of forces that tends to deform a body. From the perspective of
what is happening within a material, stress is the internal distribution of forces within a
body that balance and react to the loads applied to it. The stress distribution may or may not
be uniform, depending on the nature of the loading condition. For example, a bar loaded in
pure tension will essentially have a uniform tensile stress distribution. However, a bar
loaded in bending will have a stress distribution that changes with distance perpendicular to
the normal axis.
Simplifying assumptions are often used to represent stress as a vector quantity for many
engineering calculations and for material property determination. The word "vector"
typically refers to a quantity that has a "magnitude" and a "direction". For example, the
stress in an axially loaded bar is simply equal to the applied force divided by the bar's
cross-sectional area.
Fig.1.1.1
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Strain is the response of a system to an applied stress. When a material is loaded with a
force, it produces a stress, which then causes a material to deform. Engineering strain is
defined as the amount of deformation in the direction of the applied force divided by the
initial length of the material. This results in a unitless number, although it is often left in the
unsimplified form, such as inches per inch or meters per meter. For example, the strain in a
bar that is being stretched in tension is the amount of elongation or change in length divided
by its original length. As in the case of stress, the strain distribution may or may not be
uniform in a complex structural element, depending on the nature of the loading condition.
Fig.1.1.2
If the stress is small, the material may only strain a small amount and the material will
return to its original size after the stress is released. This is called elastic deformation,
because like elastic it returns to its unstressed state. Elastic deformation only occurs in a
material when stresses are lower than a critical stress called the yield strength. If a material
is loaded beyond it elastic limit, the material will remain in a deformed condition after the
load is removed. This is called plastic deformation. In some rocks failure can occur
abruptly within the elastic range; this is called brittle behaviour. An elastic material
deforms immediately upon application of a stress and maintains a constant strain until the
stress is removed, upon which the strain returns to its original state. A strain-time plot has a
box-like shape. However, in some materials the strain does not reach a stable value
immediately after application of a stress, but rises gradually to a stable value. This type of
strain response is characteristic of anelastic materials strain response is characteristics of
anelastic materials. After removal of the stress, the time-dependent strain returns reversibly
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to the original level. In plastic deformation the strain keeps increasing as long as the stress
is applied. When the stress is removed, the strain does not return to the original level; a
permanent strain is left in the material.
Viscoelastic Materials: Viscoelastic materials are those for which the relationship between
stress and strain depends on time or, in frequency domain, on frequency. The slope of a plot
of stress vs. strain depends on strain rate.
Constitutive model of linear visco-elasticity: Viscoelastic materials, such as amorphous
polymers, semi-crystalline polymers, and biopolymers, can be modeled in order to
determine their stress or strain interactions as well as their temporal dependencies. These
models, which include the Maxwell model, the Kelvin-Voigt model and the standard linear
solid model, are used to predict a material’s response under different loading conditions.
Viscoelastic behaviour has elastic and viscous components modeled as linear combinations
of spring and dashpots, respectively. Each model differs in the arrangement of these
elements, and all of these viscoelastic models can be equivalently modeled as electrical
circuits. In an equivalent electrical circuit’s capacitance and viscosity of a dashpot to a
circuit’s resistance.
The elastic components can be modeled as spring of elastic constant E, given by σ=Eε
where, σ is the stress, E is the elastic modulus of the material, and ε is the strain that occurs
under the given stress.
The viscous components can be modeled as dashpots such that the stress-strain rate
relationship can be given as   
d
where, σ is the stress, η is the viscosity of material,
dt
d
is the time derivative of strain.
dt
Maxwell Model: The Maxwell model can be represented by a purely viscous damper and a
purely elastic spring connected in series, as shown in the fig. 1.1.3.
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Fig. 1.1.3: Maxwell Model.
The model can be represented by the following equation
d  Total d  D d  s  1 d


 
dt
dt
dt  E dt
(1.1.1)
Under this model, if the material is put under a constant strain, the stresses gradually relax.
When a material is put under a constant stress, the strain has two components. First, an
elastic component occurs instantaneously, corresponding to the spring, and relaxes
immediately upon release of stress. The second is a viscous component that grows with
time as long as the stress is applied. The Maxwell model predicts that stress decays
exponentially with time, which is accurate for most polymers. One limitation of this model
is that it does not predict creep accurately. The Maxwell model for creep or constant-stress
conditions postulates that strain will increase linearly with time.
Kelvin-Voigt model: The Kelvin-Voigt model, also known as the voigt model, consists of
a Newtonian damper and Hookean elastic spring connected in parallel, as shown in the
fig.1.1.4. It is used to explain the creep behaviour of polymers.
Fig.1.1.4: Kelvin-Voigt model.
The constitutive relation is expressed as a linear first-order differential equation
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 (t )  E (t )  
d (t )
dt
(1.1.2)
This model represents a solid undergoing reversible, viscoelastic strain. Upon application
of a constant stress, the material deforms at a decreasing rate, asymptotically approaching
the steady-state. When the stress is released, the material gradually relaxes to its
undeformed state. At constant stress (creep), the model is quite realistic as it predicts strain
to tend to  E as time continues to infinity. Similar to the Maxwell model, the KelvinVoigt model also has limitations the model is extremely good with modeling creep in
materials but with regard to relaxation the model is much less accurate.
Standard linear solid model: The standard linear solid model effectively combines the
Maxwell model and Hookean spring in parallel. A viscous material is modeled as a spring
and dashpot in series with each other, both of which are in parallel with a long spring.
Fig.1.1.5: Standard linear solid model.
For this model, the governing constitutive relation is
.

d

dt

E2   d
   E1 

  E2 dt

E1  E2
(1.1.3)
Under a constant stress, the modeled material will instaneously deform to some strain,
which is the elastic portion of the strain, and after that it will continue to deform and
asymptotically approach a steady-state strain. This last portion is the viscous part of strain.
Although the Standard Linear Solid Model is more accurate than the Maxwell and Kelvin-
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Voigt models in predicting material responses, mathematically it returns inaccurate results
for strain under specific loading conditions and is rather difficult to calculate.
Hook’s Law: In 1678 the English mathematician Robert Hooke (1635-1703) states that the
stress is linearly related to strain for some materials which is known as Hooke’s law of elasticity.
He discovered that the extension of a string is directly proportional to the load applied to the
spring. This law is expressed mathematically as:
F=kx
(1.1.4)
Where F is force, x is displacement, and k is a constant of proportionality called the spring
constant. Equation (1.1.4) is valid only if the spring can return to its original length after the
force is removed.
There is another form of Hooke’s law for engineering materials which has the same
mathematical form of equation (1.1.4) but is expressed in terms of stress and strain (Equation
1.1.4).
σ= Eε
(1.1.5)
This form of the Hook’s law, Equation (1.1.5), states that the stress σ in a material is
proportional to the strain ε. Here there is the same limitation which says, this law is true only
for the material which returns to its original size after the removal of the force. These materials
that obey Hook’s law are known as linear-elastic or "Hookean" materials. The elastic modulii,
defined for different type of deformation, are Young’s modulus, the rigidity modulus and bulk
modulus. Young’s modulus is defined from the extensional deformations. Each longitudinal
strain is proportional to the corresponding stress component. The rigidity modulus is defined
from the shear deformation. Like the longitudinal strain the total shear strain in each plane is
proportional to the corresponding shear stress component. The bulk modulus is defined from
the dilatational experienced by a body under hydrostatic pressure.
Anisotropy and Isotropy: In a single crystal, the physical and mechanical properties often
differ with orientation. It can be seen from looking at our models of crystalline structure
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that atoms should be able to slip over one another or distort in relation to one another easier
in some directions than others. When the properties of a material vary with different
crystallographic orientations, the material is said to be anisotropic.
Alternately, when the properties of a material are the same in all directions, the material is
said to be isotropic. For many polycrystalline materials the grain orientations are random
before any working (deformation) of the material is done. Therefore, even if the individual
grains are anisotropic, the property differences tend to average out and, overall, the material
is isotropic. When a material is formed, the grains are usually distorted and elongated in
one or more directions which make the material anisotropic.
Orthotropic: Some engineering materials, including certain piezoelectric materials (e.g.
Rochelle salt) and 2-ply fiber-reinforced composites, are orthotropic. By definition, an
orthotropic material has at least 2 orthogonal planes of symmetry, where material properties
are independent of direction within each plane. Such materials require 9 independent
variables (i.e. elastic constants) in their constitutive matrices.
In contrast, a material without any planes of symmetry is fully anisotropic and requires 21
elastic constants, whereas a material with an infinite number of symmetry planes (i.e. every
plane is a plane of symmetry) is isotropic, and requires only 2 elastic constants.
Seismic Deformation: When an earthquake fault ruptures, it causes two types of
deformation: static and dynamic. Static deformation is the permanent displacement of the
ground due to the event. The earthquake cycle progresses from a fault that is not under
stress, to a stressed fault as the plate tectonic motions driving the fault slowly proceed, to
rupture during an earthquake and a newly-relaxed but deformed state.
Typically, someone will build a straight reference line such as a road, railroad, pole line, or
fence line across the fault while it is in the pre-rupture stressed state. After the earthquake,
the formerly straight line is distorted into a shape having increasing displacement near the
fault, a process known as elastic rebound.
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Fig.1.1.6
The second type of deformation, dynamic motions, is essentially sound waves radiated
from the earthquake as it ruptures. While most of the plate-tectonic energy driving fault
ruptures is taken up by static deformation, up to 10% may dissipate immediately in the
form of seismic waves.
Seismic Waves: The propagation of a seismic disturbance through a heterogeneous
medium is extremely complex. In order to derive equations that describe the propagation
adequately, it is necessary to make simplifying assumptions. The heterogeneity of the
medium is often modified by dividing into parallel layers, in each of which homogeneous
conditions are assumed. By suitable choice of the thickness, density and elastic properties
of each layer, the real conditions can be approximated. The most important assumption
about the propagation of a seismic disturbance is that it travels by elastic displacements in
the medium. This condition certainly does not apply close to the seismic source. In or near
an earthquake focus or the shot point of a controlled explosion the medium is destroyed.
Particles of the medium are displaced permanently from their neighbors; the deformation is
anelastic. However, when a seismic disturbance has travelled some distance away from its
source, its amplitude decreases and the medium deforms elastically to permit its passage.
The particles of the medium carry out simple harmonic motions, and the seismic energy is
transmitted as a complex set of wave motions.
When seismic energy is released suddenly at a point near the surface of a homogeneous
medium part of the energy propagates through the body of the medium as seismic body
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waves. The remaining part of the seismic energy spreads out over the surface as a seismic
surface wave, analogous to the ripples on the surface of a pool of water into which a stone
has been thrown. Seismic waves are the waves of energy caused by the sudden breaking of
rock within the earth or an explosion. They are the energy that travels through the earth and
is recorded on seismographs.
Types of Seismic Waves: There are several different kinds of seismic waves, and they all
move in different ways. The two main types of waves are body waves and surface waves.
Body waves can travel through the earth's inner layers, but surface waves can only move
along the surface of the planet like ripples on water. The large strain energy released during
an earthquake causes radial propagation of waves with the earth (as it is an elastic mass) in
all directions. These elastic waves, called seismic waves, transmit energy from one point of
the earth to another through different layers and finally carry the energy to the surface
which causes the destruction.
Body waves: Traveling through the interior of the earth, body waves arrive before the
surface waves emitted by an earthquake. These waves are of a higher frequency than
surface waves.
Primary or compressional (P) waves: The first kind of body wave is the P wave or
primary wave. This is the fastest kind of seismic wave, and, consequently, the first to
'arrive' at a seismic station. The P wave can move through solid rock and fluids, like water
or the liquid layers of the earth. It pushes and pulls the rock it moves through just like
sound waves push and pull the air. P waves are also known as compressional waves,
because of the pushing and pulling they do. Subjected to a P wave, particles move in the
same direction that the wave is moving in, which is the direction that the energy is traveling
in, and is sometimes called the direction of wave propagation.
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Fig.1.1.7: A P-wave travels through a medium by means of compression and dilation.
Particles are represented by cubes in this model.
Secondary or shear (S) waves: The second type of body wave is the S wave or secondary
wave, which is the second wave you feel in an earthquake. An S wave is slower than a P
wave and can only move through solid rock, not through any liquid medium. It is this
property of S waves that led seismologists to conclude that the Earth's outer core is a
liquid. S waves move rock particles up and down, or side-to-side--perpendicular to the
direction that the wave is traveling in (the direction of wave propagation).
Fig.1.1.8: An S-wave travels through a medium. Particles are represented by cubes in this
model.
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Surface waves: Travelling only through the crust, surface waves are of a lower frequency
than body waves, and are easily distinguished on a seismogram as a result. Though they
arrive after body waves, it is surface waves that are almost entirely responsible for the
damage and destruction associated with earthquakes. This damage and the strength of the
surface waves are reduced in deeper earthquakes.
Love wave: The first kind of surface wave is called a Love wave, named after A.E.H.
Love, a British mathematician who worked out the mathematical model for this kind of
wave in 1911. It's the fastest surface wave and moves the ground from side-to-side.
Confined to the surface of the crust, Love waves produce entirely horizontal motion.
Fig.1.1.9: A Love wave travels through a medium. Particles are represented by cubes in
this model.
Rayleigh wave: The other kind of surface wave is the Rayleigh wave, named for John
William Strutt, Lord Rayleigh, who mathematically predicted the existence of this kind of
wave in 1885. A Rayleigh wave rolls along the ground just like a wave rolls across a lake
or an ocean. Because it rolls, it moves the ground up and down and side-to-side in the same
direction that the wave is moving. Most of the shaking felt from an earthquake is due to the
Rayleigh wave, which can be much larger than the other waves.
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Fig.1.1.10: A Rayleigh wave travels through a medium. Particles are represented by cubes
in this model.
Interior of Earth: The earth consists of several layers. The three main layers are the core,
the mantle and the crust. The core is the inner part of the earth, the crust is the outer part
and between them is the mantle. The earth is surrounded by the atmosphere. Till this
moment it hasn't been possible to take a look inside the earth because the current
technology doesn't allow it. Therefore all kinds of research had to be done to find out; out
of which material the earth consists, what different layers there are and which influence
those have (had) on the earth's surface. This research is called seismology (cf. Rickter
(1958)).
The inner part of the earth is the core. This part of the earth is about 1,800 miles
(2,900 km) below the earth's surface. The core is a dense ball of the elements iron and
nickel. It is divided into two layers, the inner core and the outer core. The inner core - the
Center of earth - is solid and about 780 miles (1,250 km) thick. The outer core is so hot that
the metal is always molten, but the inner core pressures are so great that it cannot melt,
even though temperatures there reach 6700ºF (3700ºC). The outer core is about 1370 miles
(2,200 km) thick. Because the earth rotates, the outer core spins around the inner core and
that causes the earth's magnetism. We know that if exists because it infracts seismic waves
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creating a shadow zone at distances between 103 to 143 degree. We also know that the
outer part of the core is liquid, because S-waves do not pass through it.
The layer above the core is the mantle. It begins about 6 miles (10 km) below the
oceanic crust and about 19 miles (30 km) below the continental crust (see The Crust). The
mantle is to divide into the inner mantle and the outer mantle. It is about 1,800 miles (2,900
km) thick and makes up nearly 80 percent of the Earth's total volume. It consists of dense
silicate rocks. Both P and S waves from earthquakes travel through the mantle
demonstrating that it is solid. However, there is separate evidence that parts of the mantle
behaves as a fluid over very long geological times scale with rocks flowing slowly in giant
convection cells (cf. Rickter (1958)).
The crust layers above the mantle and is the earth's hard outer shell, the surface on
which we are living. In relation with the other layers the crust is much thinner. It floats
upon the softer, denser mantle. The crust is made up of solid material but these material is
not everywhere the same. There is an Oceanic crust and a Continental crust. The first one is
about 4-7 miles (6-11 km) thick and consists of heavy rocks, like basalt. The Continental
crust is thicker than the Oceanic crust, about 19 miles (30 km) thick. Continental crust is
quite complex in structure and is made from many different kinds of rock. It is mainly
made up of light material like granite.
It is the outer part of the earth above the Mohorovicic discontinuity which is the
level at which the P-wave velocity increases to 7.8-8.2 km/s, and is found almost
everywhere. Below it is the mantle, and to a first approximation the crust. The Mohorovicic
discontinuity, the second most prominent boundary in the earth’s interior, is itself less than
a kilometre thick in some areas but appears to be a few kilometres thick in many others.
Movements of Seismic waves: An earthquake occurs when rocks in a fault zone suddenly
slip past each other, releasing stress that has built up over time. The slippage releases
seismic energy, which is dissipated through two kinds of waves, P-waves and S-waves. The
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distinction between these two waves is easy to picture with a stretched-out slinky. If you
push on one end of a slinky, a compression wave passes through the slinky parallel to its
length. If instead you move one end of the slinky up and down rapidly, a "ripple" wave
moves through the slinky. The compression waves are P-waves, and the ripple waves are Swaves. Though both kinds of waves refract, or bend, when they cross a boundary into a
different material, these two types of waves behave differently depending on the
composition of the material they are passing through. One of the biggest differences is that
S-waves cannot travel through liquids whereas P-waves can. We feel the arrival of the Pand S-waves at a given location as a ground-shaking earthquake.
If the earth were the same composition all the way through its interior, seismic waves
would radiate outward from their source (an earthquake) and behave exactly as other waves
behave - taking longer to travel further and dying out in velocity and strength with distance,
a process called attenuation. Given Newton's observations, if we assume that earth's density
increases evenly with depth because of the overlying pressure, wave velocity will also
increase with depth and the waves will continuously refract, traveling along curved paths
back towards the surface. Figure 1.1.11 shows the kind of pattern we would expect to see in
this case. By the early 1900s, when seismographs were installed worldwide, it quickly
became clear that the earth could not possibly be so simple.
Fig.1.1.11: Seismic waves in an earth of the same composition.
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As early as 132 BC, the Chinese had built instruments to measure the ground shaking
associated with earthquakes. The first modern seismographs, however, weren’t built until
the 1880s in Japan by British seismologists to record local earthquakes. It wasn’t long
before those seismologists recognized that they were also recording earthquakes occurring
thousands of kilometers away.
One of the first important observations on the earth's structure was made by Andrija
Mohorovicic, a Croatian seismologist. He noticed that P-waves measured more than 200
km away from an earthquake's epicenter arrived with higher velocities than those within a
200 km radius. Although these results ran counter to the concept of attenuation, they could
be explained if the waves that arrived with faster velocities traveled through a medium that
allowed them to speed up. In 1909, Mohorovicic defined the first major boundary within
the earth’s interior - the boundary between the crust, which forms the surface of the earth,
and a denser layer below, called the mantle. Seismic waves travel faster in the mantle than
they do in the crust because it is composed of denser material. Thus, stations further away
from the source of an earthquake received waves that had made part of their journey
through the denser rocks of the mantle. The waves that reached the closer stations stayed
within the crust the entire time. Although the official name of the crust-mantle boundary is
the Mohorovicic discontinuity, in honor of its discoverer, it is usually called the Moho.
Fig.1.1.12: The Moho.
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Introduction and review of literature
Another observation made by seismologists was the fact that P-waves die out about 105
degrees away from an earthquake, and then reappear about 140 degrees away, arriving
much later than expected. This region that lacks P-waves is called the P-wave shadow zone
(Figure 1.1.13). S-waves, on the other hand, die out completely around 105 degrees from
the earthquake (Figure 1.1.13). Remember that S-waves are unable to travel through liquid.
The S-wave shadow zone indicates that there is a liquid layer deep within the earth that
stops all S-waves but not the P-waves. In 1914, Beno Gutenberg, a German seismologist,
used these shadow zones to calculate the size of another layer inside of the earth, called its
core. He defined a sharp core-mantle boundary at a depth of 2,900 km, where P-waves were
refracted and slowed and S-waves were stopped.
Fig.1.1.13: The P-wave and S-wave shadow zones.
Historical Development: In seismology the problem of the source mechanism consists in
relating observed seismic wave to the parameters that describe the source. In earth,
neglecting the forces of gravity, body forces in the equation of motion may be used to
represent the processes that generate earthquakes. In general, these forces are functions of
the spatial coordinate and time, may be different for each earthquake and are define only
inside a certain volume. The time dependence of these forces simplifies the solution of
many problems in Seismology. A type of body force of great importance in the solution of
many problems of elasto-dynamics is that form by a unit impulsive force in space and time
with an arbitrary direction, this force may be represented mathematically by means of
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Dirac’s Delta function. The Dirac’s Delta function is important in the study of events
related to point action or an impulsive force, such as an action which is highly localized in a
space and or time, including the point force in solid mechanic; the impulsive force in rigid
body dynamics; the point mass in gravitational field theory; and the point heat sources in
the theory of heat conduction. The propagation and generation of waves in a visco-elastic
type of material are important development in the theory of elasticity. The visco-elastic
behavior of the material is described by the mechanical behavior of solid materials with
small voids. The type of linear visco-elasticity generally displayed by linear elastic
materials with voids termed as linear solid. The study of SH-wave propagation in layered
visco-elastic and inhomogeneous medium has been considered by many investigators. It
was Jeffreys (1931) who took up the wave propagation problem in visco-elastic medium
and obtained the result for step pulse disturbance in terms of asymptotic disturbance.
Moodie (1973) has considered the propagation, reflection and transmission shear waves in
visco-elastic solids. Chattopadhyay (1978) studied the effect on the propagation of SHwave due abrupt thickening of a visco-elastic crustal layer. The attenuation of seismic
waves with distance and of normal modes with time indicates the anelastic behaviour of the
earth. The general theory of visco-elasticity describe the linear behaviour of both elastic
and anelastic materials and provides the basis for describing the attenuation of seismic
waves due to an elasticity. Notable work on seismic waves propagation in viscoelastic
media are of Cooper (1967), Shaw and Bugl (1969), Schoenberg (1971), Borcherdt (1973,
2009), Kausik and Chopra (1983), Gogna and Chander (1985), Manolis and Shaw (1996),
Romeo (2003), Cerveny (2004) etc. The work on SH-waves in visco-elastic media and
anisotropic have been well tackled by Romes (2003), Daley and Krebes (2004), Chaudhary
et al. (2005), Cerveny and Psencik (2006). Recently Tomar and Kaur (2007) have
considered the shear waves at a corrugated interface between anisotropic elastic and viscoelastic solid half-spaces. Abd-Alla and Mahmoud (2011) have discussed the magnetothermo-elastic equation of spherical cavity is decomposed into non-homogeneous equation
with boundary conditions. The effect of thermal relaxation times on the wave propagation in
magneto thermo viscoelastic using the GL theory will be discussed. As a simple, but reasonably
realistic, model for studying the effect of earth’s structure on seismic wave propagation we
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shall consider stratified media composed of isotropic, nearly elastic material. The weak
attenuation will be included in the frequency domain by working with complex wave
speeds.
The torsional vibration of an elastic half-space due to a surface force which is
periodic in time was first considered by Reissner (1937). He considered the problem of
periodic surface stress prescribed over a circular area with zero surface stresses elsewhere.
Reissner and Sagoci (1944) have improved torsional problem by considering a new
approach and opened new era in this direction. The axisymmetric torsional impulsive
loading of an elastic half-space where surface stress is prescribed over a circular area has
been considered by Eason (1964). In another paper Eason and Wilson (1971) have obtain
the displacement produced in a composite infinite solid by an impulsive torsional body
force. Dhawan (1981) has analyzed in detail the three part mixed boundary value problem
of a transversely isotropic half-space underlain by a flat annular rigid stamp on the basis of
three dimensional theory of elasticity. In another investigation Tanimura (1984), analyzed
the stress wave propagation in elastic visco-elastic circular cylindrical bodies bounded by a
plane surface and a spherical surface, containing a spherical cavity. The paper Durelli and
Lin (1986) deals with stresses and displacements in circular rings of rectangular crosssection loaded in the plane and perpendicular to the boundary. The work by Erguven (1987)
considered the elastic torsional problem for a non-homogeneous and transversely isotropic
elastic half-space. Investigations on wave produced by point line sources acting on the
surface of or buried in a half-space may be seen in the papers by Ghosh (1971), Eason
(1969) and many others.
The problems of rigid and flexible disks with isotropic media have thoroughly been
studied in the literature (Pak and Gobert 1990; Pak and Saphores 1991). For dynamic
response of the disk, one can refer to see the extensive list o refrences cited in Pak and
Gobert (1991) for a finite embedment depth of the disk, Kausel (2009b) when the disk
resting on the surface, and Selvadurai (1990) for an infinite embedment. The static
problems corresponding to a disk resting on the surface of a transversely isotropic half-
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Introduction and review of literature
space have been considered by Fabrikant (1986, 1997a, 1997b). Eskandari-Ghadi and
Behrestaghi (2010) considered the axisymmetric normal vibration of rigid disk in smooth
contact with the surface of a transversely isotropic half-space. Chen and Sun (2003)
developed the axisymmetric static problem related to the normal translation of a rigid disk
embedded at a finite depth of an isotropic half-space considered previously by Selvadurai
(1993). They modeled the disk and the half-space, respectively, as a flexible thin plate and
a transversely isotropic solid. Later, following Pak and Gobert (1990), Katebi et al. (2010)
considered the special case of this problem for rigid disk. Recently Eskandari-Ghadi et al
(2010) extended the time harmonic normal vibration problem presented by Pak and Gobert
(1991) for an isotropic half-space to a transversely isotropic medium. It is worth noting that
in these mentioned problems considered by Pak and Gobert (1990), Pak and Gobert (1991),
Katebi et al. (2010), and Eskandari-Ghadi ey al. (2010) the displacement boundary
conditions along the disk-medium interface are not completely satisfied. Based on the
approach noted by Kausel (2009a), the torsional vibration of a rigid disk in a transversely
isotropic half-space can be obtained from the corresponding problem in an isotropic solid
presented by Pak and Abedzadeh (1993). Moreover, the results of normal and rocking
vibrations of a rigid disk embedded in a transversely isotropic full space can be found as the
limiting cases of the previous works considered by Eskandari-Ghadi and Behrestaghi
(2010) and Eskandari-Ghadi et al. (2011), respectively. It should be noted that the
previously mentioned flaws in the cited references Eskandari-Ghadi and Behrestaghi and
Eskandari-Ghadi et al. (2011) do not exist for the limiting cases corresponding to a
homogeneous full space. Therefore, the lateral vibration of a rigid disk embedded in neither
isotropic nor transversely isotropic full space has been yet considered. In addition, the
results of all previous works concerning the time harmonic vibrations of rigid disks in
transversely isotropic solids have been presented in the form of semi-infinite integrals
including integrands with oscillatory nature and weak decaying behaviour full space.
The dynamical theory of thermo-elasticity is the study of dynamical interaction between
thermal and mechanical field in solid bodies and is of much importance in various
engineering fields such as earthquake engineering, soil dynamics, aeronautics, astronautics,
~ 21 ~
Introduction and review of literature
nuclear reactors, high energy particle accelerators, etc. Few generalized theories of thermoelasticity have been defined with the introduction of relaxation in temperature field. The
propagation of thermoelastic waves in a plate under plane stress by using generalized
theories of thermoelasticity has been studied by Chandrasekheraiah and Srinantiah (1984,
1985), Massalas (1986). Here, we mention that several authors Puri (1973, 1975), Agrawal
(1978, 1979), Tao and Prevost (1984), Massalas and Kalpakidis (1987b), and Daimaruya
and Naitoh (1987) have considered the propagation of generalized thermoelastic waves in
plates of isotropic media. Massalas and Kalpakidis (1987a) used the generalized theory of
Lord and Shulman to study the characteristics of wave motion in a thin plate under plane
stress state with mixed boundary conditions. They used Lames’s potentials to derive the
frequency equation. Verma and Hasabe (1999) studied the propagation of generalized
thermoelastic vibrations in infinite plates in the context of generalized thermo-elasticity.
Banerjee and Pao (1974) extended this theory to anisotropic heat conducting elastic
materials. Dhaliwal and Sherief (1980) treated the problem in more systematic manner.
They derived governing field equations of generalized thermoelastic media and proved that
these equations have a unique solution. Sharma and Sidhu (1986) discussed the propagation
of plane harmonic waves in a homogeneous anisotropic generalized thermoelastic solid.
Chadwick and Seet (1970), and Chadwick (1979) investigated the thermoelastic wave
propagation in transversely isotropic and homogeneous anisotropic heat conducting elastic
materials, respectively. The theory with one relaxation time (Lord and Shulman, 1967) is
termed as LS theory and another with two relaxations time (Green and Lindsay, 1972) is
termed as GL theory. Using these modified theories, a large number of problems have been
studied on the propagation of plane waves in generalized thermoelastic media (EIKaramany et al. 2002; Sharma et al. 2003). Sharma et al. (2000) studied plane harmonic
waves in orthotropic thermoelastic materials. Kumar and Kumar (2011) have discussed the
wave propagation and reflection of plane waves incident at the stress free, thermally
insulated or isothermal surface of a homogeneous, orthotropic generalized thermoelastic
half-space with voids under initial stress. Bayones (2012) has discussed the effects of
viscosity and diffusion on generalized magneto-thermoelastic interaction in an isotropic
with a spherical cavity. Chirta (2013) has discussed the effect of the temperature as well as
~ 22 ~
Introduction and review of literature
strain on the propagation of Rayleigh waves in an anisotropic thermoelastic half-space by
means of the so-called Stroh formulation.
Liquid-saturated porous rocks are often present on and below the surface of Earth.
Sedimentary layers consisting a sandstone or limestone saturated with water are usually
present below oceans. Layer of porous solids such as sandstone or limestone saturated with
ground water or oil present in the Earth’s crust. The theory of Biot (1956a,b) for wave
propagation in an isotropic fluid-saturated porous solid is a complete one, through which
several investigator, namely Deresiewicz (1960, 1961, 1962 1964a,b, 1965), Deresiewicz
and Rice (1962), Jones (1961), Yew and Jogi (1976), Ogushwitz (1985), Degrande et al.
(1998) and Santosh et al. (1993) solved a number of related problems. Biot (1956a,b;
1962a,b) also discussed the propagation of plane harmonic seismic waves in such solids
and found that two dilatational waves along with one shear wave are possible in such
solids. Abubaker and Hudson (1961) studied the dispersive properties of liquid overlying a
semi-infinite, homogeneous, transversely isotropic half-space. Deresiewicz and Skalak
(1963) have discussed the boundary conditions which are appropriate for the boundaries of
liquid-saturated porous solids. The surface waves are subjected, along the direction of
propagation, to the attenuation of horizontal and vertical displacements due to mechanical
and geometrical phenomena of dissipation, and due to effects of dispersion (Richart et al.,
1970). Wadhwa (1971) has studied the disturbance due to a periodic point source in a
homogeneous liquid layer overlying a heterogeneous liquid half-space and obtained the
solution in terms of modified Bessel functions. Gogna (1979) considered surface wave
propagation in a homogeneous anisotropic layer lying over a homogeneous, isotropic,
elastic half-space and under a uniform layer of liquid. Chattopadhyay et. al (1986) studied
the propagation of Love waves in a homogeneous, isotropic porous layer overlying an
inhomogeneous half-space generated by point source at the interface of the layer and halfspace. Chadwick (1989) presented very comprehensive theory for the elastic wave
propagation in a transversely isotropic material and proved that three waves can propagate
in the material and proved that three waves can propagate in the material. He also discussed
the surface waves in that material (1989). Gazetas (1982), Yamamoto (1983), Sharma et al.
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Introduction and review of literature
(1990, 1991), Kumar & Miglani (1996) etc. have studied surface wave propagation in
liquid-saturated porous solids with different models. Kumar and Hundal (2003, 2005, 2007)
have studied the problem of wave propagation in fluid-saturated incompressible porous
media. Liu and Liu (2004) have studied the influence of anisotropy of the solid skeleton on
the propagation of characteristic of Rayleigh waves in orthotropic fluid-saturated porous
media. Pal (2006) has studied on shear wave propagation in a multilayered medium
including a fluid saturated porous solid stratum. Pham and Ogden (2004) found secular
equation for orthotropic material and also imposed a condition on the speed of Rayleigh
waves. Rangelov et al. (2010) have considered a special case of wave propagation problem
in a restricted class of orthotropic inhomogeneous half-space. Sethi et al. (2013) have
discussed the propagation of surface waves in anisotropic, fibre-reinforced solid media
under the influence of gravity. Biot’s theory of incremental deformations is used to obtain
the wave velocity equation for Stonely, Rayleigh and Love waves.
The generation and propagation of waves in layered isotropic or anisotropic elastic
media may be relevant to mention with brief description in order to indicate our motivation.
Anderson (1961) made an interesting study of elastic wave propagation in layered
anisotropic media with applications. He discussed the period equations for waves of
Rayleigh, Stonely and Love types. Elastic properties are generally anisotropic (transversely
anisotropic) in sedimentary layers. Studies of Uhring and Van Melle (1955), and Anderson
and Harkrinder (1962) have indicated that anisotropy is also present in the near surface
layers, and in the crust and upper mantle regions of the Earth. It has been observed that the
transition region between the crust and mantle is not a single layer but is possibly formed
by set of thin layers. The transient displacement of SH-type produced at the surface of a
homogeneous elastic half-space, due to the sudden occurrence of the discontinuity in the
shearing stress within the medium has been obtained by Nag (1962). The discontinuity has
been assumed to be expanding and uniformly moving with a constant velocity. In another
paper Nag (1963) has considered the displacement at the free surface due to the sudden
application of a stress-discontinuity, moving along the interface of a layer over-lying a
semi-infinite medium of different elastic constants. In a subsequent paper Ng and Pal
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Introduction and review of literature
(1977) solved a problem similar to that of Nag with a shearing stress discontinuity at the
interface of two layers of finite thickness overlying a semi-infinite medium of different
elastic constants. Mention may also be made of the work on forced vibrations of an
anisotropic elastic spherical shell due to a uniformly distributed internal and external
pressure by Sheehan and Debnath (1972).
Complete references to earlier work can be had from the book ‘Elastic waves in
layered media’, by Ewing, Jardetzky and Press (1957), while the Cagniard’s book
translated and revised by Flinn and Dix (1962) contains many references to later work on
the application of Cagniard’s method. Bath (1968) and Achenbach (1973) have also
described the Cagniard’s technique in very lucid language in their books ‘Mathematical
aspects of Seismology’ and ‘Wave Propagation in Elastic Solids’ respectively.
Some
recent application of Cagniard’s method based on some special reduction techniques can be
had from the books ‘Elastic Wave Propagation in Transversely Isotropic Elastic Media’ by
Payton (1983) and Quantitative seismology (Second Edition) by Aki and Rechards (1980).
A brief outline of the Cagniard-De Hoop Technique:
Two problem of this thesis have been solved by Cagniard De Hoop technique. This method
uses Laplace transform and leads to solutions directly in the time domain. It entails
integration in the complex ray-parameter plane; ray path in the physical problem
correspond to saddle points in the integrand under consideration; head waves correspond to
branch cuts; interface wave correspond to poles; and leaking modes correspond to poles on
Riemann sheets other than that on which the radiation condition is satisfied. Cagniard
solution must convolve with a source function and with the instrument response. This
method is characterized as an exact method. The success of the Cagniard method depends
on a certain property of the Laplace transform of the required solution-namely, that it can

be written as a function of s times a factor in the form
 g (t )e
0
and differentiable function.
~ 25 ~
 st
dt , g(t) being a continuous
Introduction and review of literature
In summary, it is clear that the Cagniard-De Hoop solution has great advantages for
solving Lamb’s (1904) problem with a point source or a line source. The impulse response
is obtained directly and with a minimum of computational effort.
The broad aim of the present worker sets out before himself an investigation into the
theme of elastic wave generation and propagation in layered anisotropic and
inhomogeneous media.
To sum up it is not out of place to mention that the result of most of the problems of
this thesis is obtained in terms of either algebraic functions (closed form) or their definite
integral or in terms of approximations. Most of the problems are evaluated numerically by
MATLAB and MATHEMATICA-7 computers and are shown graphically.
~ 26 ~