1. Introduction: The behavior of earth materials (soil and
rocks) is not purely elastic but it is anelastic in nature. The
anelasticnature of earth material arises because it remember its
past. It means stress-strain relation in case of earth materials
also depend on time. The most widely used rheological model for
anelastic material is a linear viscoelastic body. The behaviour of
viscoelastic body is equivalent to the combined behaviour of both
the linear elastic body and linear viscous body. 2. Rheological
elements: 2.1 Linear Elastic Body: The linear elastic body (Hooke
body) represents the behaviour of a perfectly elastic solid
material. Stress is proportional to strain only in case of linear
elastic body. ( ) ( ) ( )
where ( ) is the stress applied as a function of timet, ( ) is
the strain developed in the linear elastic body and is the time
independent modulus of elasticity. In case of linear elastic
body,
the strain immediately develops in the body when a stress is
applied and comes to its rest position suddenly after removal of
stress. It means linear elastic body does not have a memory.Hooke
Body is shown in figure (1).
Fig.(1) Hooke Body Here we will use * ( )+ () for the direct and
( ) , for the inverse fourier transforms ( ) ( ) .
* ( )+
s the angular frequency. An application of the fourier transform
toequation (1) gives ( ) ( ) ( )
2.2 Linear Viscous Body:The behaviour of linear viscous
body(Stokes body) is that of viscous fluid and stress is
proportional to time derivative of strain.
()
( )
( )
where is the time-independent viscosity. In case of linear
viscous body the strain candoes not immediately develop when stress
is applied but it linearly increase with time and there is
permanent deformation in the body after removal of stress. Stokes
Body has extreme memory. Stokes Body is shown in figure (2).
Fig.(2) Stokes Body An application of the fourier transform to
equation(3) gives ( ) ( ) ( )
3. Stress-Strain Relation in Viscoelastic Medium: The
stress-strain relation for a linear isotropic viscoelastic material
is given by Boltzmann superposition and causality principle. ( )
Where ( ) ( ) ( )
( )is the stress relaxation function, which is defined as a
stress response to Heaviside
unit step function in strain equation (5) reveals that the
entire history of the strain until timet is used to compute the
stress at given timet. The integral in equation (5) depicts a time
convolution of the stress relaxation function and strain rate.
Equation (5) can also be written as using properties of
convolution. () Due to properties of convolution, () ( ) () ( ) ()
( ) ( )
the time derivative of stress relaxation function is the stress
response to the Dirac -function in strain. Therefore equation (7)
can be written as () () () ( )
where,
()
( ).
A comparison of equation (1) and equation (8) reveals that the
form of stress-strain relation for viscoelastic bodyis convoluting
in nature and modulus The fourier transform of equation (8) gives (
) is time dependent.
( )
( )
( )
( )
( ) is complex frequency dependent viscoelastic modulus. An
application of the inverse fourier transform to equation (8) gives
() * ( )+ ( )
And due to properties of the fourier transform, () { ( ) } (
)
Similarly using equation (7) and equation (8), fourier transform
of time derivative of stress can be obtained. ( ) ( ) ( ) ( )
As pointed out earlier viscoelastic materials remember that past
so an instantaneous elastic response of the viscoelastic material
can be defined as unrelaxed modulus ( term equilibirium response
can be defined as relaxed modulus( () In the frequency domain, ( )
( ) ( ) () ). ( ) ) and a long
The modulus defect or relaxation of modulus is given by ( )
Fig.(3) MU is the unrelaxed modulus and MR = MU
is the relaxes modulus. ( ) ( )
The quality factor Q( ) can be obtained using viscoelastic
modulus ( ) ( ) ( )( )
It has been shown based on observations that
are a measure of internal friction in a linear
viscoelastic body and it is nearly constant over the seismic
frequency range. 3.1Conversion of the convolutory stress-strain
relation into a differential form: Consider ( ) as a rational
function ( ) With, ( ) ( ) ( ) ( ) ( ( ) ) ( )
(
)
We have stress-strain relation from equation (8) ( ) Put the
value of ( ) ( )
( ) from equation (17) in above relation ( ) ( ) ( ) ( ) ( )
We know that fourier transform of nth derivative of t is given
as
* And the inverse fourier transform * ( )+( )
( )+
(
)
( )
(
)
( )
(
)
The inverse fourier transform of equation (3) * Now we get, ( )
( ) ( ) ( )+ * ( ) ( )+ ( )
(
)
The nth-order differential equation for more easily than the
convolution integral. 4. Viscoelastic Rheological Models:
( ),which can be eventually numerically solved much
Various rheological models like Maxwell body,Kelvin-vigit body
and Zener body are connected of Hooke and Stokes rheological
elements to approximate the rheological properties and behaviour of
the real earth material.The time-domain and frequency-domain form
of Hooke and Stokes rheological elements are given in equations
(1),(2),(3) and (4). These rheological models can be represented in
time or frequency domain using Hooke and Stokes elements and simple
rules. If these rheological elements are connected in series,
stress is equal and strain becomes additive. On the other hand, if
these rheological elements are connected in parallel, stress
becomes additive and strain is equal for both the elements (Moczo
et al, 2004). 4.1 Maxwell Body: The Maxwell body is obtained by
connecting the Hooke and Stokes elements in series. The Hooke and
Stokes rheological elements in frequency domain form are given
below ( ) ( ) ( )
And, ( ) ( ) ( )
Using the rule for connection in series (Moczo et al, 2004), we
have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) )
( ) ( ) Where ( )
[ ( )
] ( ) ( )
( (
) )
is stress relaxation frequency and the inverse of relaxation
frequency are is stress
relaxation time. ( )
Fig.(4) Rheological Models of viscoelastic materials The relaxed
moduli ( ) and unrelaxed moduli ( ) can be obtained using
frequency-
dependent modulus( ( )). ( ) It means Maxwell Body relaxes from
a stress value of strain is applied. The stress relaxation function
for Maxwell body can be written as using equation (11) and equation
(29). ( ) ( )
to a zero stress when a unit-step
()
2
( )
3
(
)
()
{
}
(
)
()
{
}
(
)
()
(
)
(
)
The integral can be evaluated using contour integral method. We
recall Jordans leema. If ( ) converges unifomly to zero as , then
Where 0 and ( ) ( ) ( )
is the upper half of the circle |z| = R 0 but one does the
contour in the lower half plane.
A similar result applies for
In this example we have to evaluate the integral, from equation
(35) () ( )
According to Jordans leema , we need to endoes the contour with
a semi circle in the upper half plane for 0 and in the lower half
plane for () ( ) 0, as ( )
()
(
)
(
)
()
{
(
)
(
)
(
)
()
(
)
Where H(t) is heaviside unit step function and defind as given
below ( ) 2 ( )
4.2 GMB-EK Model: Rheological models which quite approximate the
rheological properties and behaviour of the real earths material
can be developed by connecting the rheological elements. The
generalisedmaxwell body (GMB) in the literature on rheology is
constructed by connecting various Maxwell bodies in paralled. But,
the model used by Emmerich and Korn (1987) is construted by
connecting a single Hooke element in paralled with various Maxwell
bodies. So the model used by Emmerich and korn (1987) is denoted by
GMB-EK. ( ) ( ) ( )
( )
( )
(
)
( )
[
] ( )
(
)
Fig.(5)Rheological model of generalized Maxwell body (GMB-EK)
defined by Emmerich and Korn (1987). MH and Mw denotes elastic
moduli.l
viscosity
If there are n Maxwell body connected in parallel with a single
Hooke body. So, the complex and frequency dependent visco-elastic
modulus for GMB-EK rheological model is ( )
(
)
With relaxation frequencies ( )
The relaxed and unrelaxed moduli are given by ( ) ( )
(
)
So the total elastic modulus in case of n Maxwell Body is
(
)
If no relaxation in case of lth body is ( Then, we can consider
)
(
)
Now the unrelaxed modulus can be written as ( )
(
)
Finally, complex frequency dependent visco-elastic modulus for
GMB-EK model can be written as ( )
(
)
The relaxation function in case of GMB-EK can be obtained as
()
[
{
}]
(
)
{
}
{
}
()
(
)
()
[
(
)]
()
(
)
Where H(t) is heaviside unit step function and defind as given
below ( ) 2 ( )
This equation (54) was used by Emmerich and Korn (1987). The
complex modulus ( ) for GMB-EK model can also be written as
( )
(
)
( )
[
]
( )
[
]
(
)
So, the another relaxation function for GMB-EK model can be
written as
()
[
{
[
]}]
(
)
()
{
}
{ (
)[
]}
(
)
Using equation (59) the term . /
defined by partial fraction . /( ( ( ( )( ) )( ) )( ) ( ) )
(
)
(
)
Comparing both sides, we get A+B=0 and A = 1, when A = 1 put in
A + B = 0, we get B = -1.
These value put in equation (60), we get . / Using equation (59)
and (62) we get () ( )
{
}
{
(
)}
()
*
(
)+
()
[
*
(
)+]
()
(
)
Where H(t) is heaviside unit step function and defind as given
below ( ) 2 ( )
Equation (63) was used by Kristek and Moczo (2003). 5 .Stress
Strain relation for GMB-EK rheological models: The time derivative
of relaxation function can be written as using equation (63) () ( )
( ) ( ) * ( )+] ()
[
(
)
GMB-KM model can be written as ( ) ( )) ( ) ( ) ( ) ( )
(
{
(
(
))}
(
) ( )
(
)
( )
( )
( )
(
(
))
(
)
5.1 Anelastic function: Kristek and Moczo (2003) defined
anelastic function, independent of material properties, in order to
replace the convolution integral by an addition function. In
contrast of this Day and Minster (1984), Emmerich and Korn (1987)
and Carcione et al.(1988) used an addition function dependent on
material properties. The anelastic function proposed by Kristek and
Moczo (2003) is given by () ( ) ( ( )) ( )
GMB-EK model can be written as using equation (67) and equation
(68). ( ) ( ) ( )
(
)
Time derivative of equation (68), we get ( ) [ ( ) ( ) ( ( ( (
)) )) ( )]
( ) ( )
, ()
()
( )( )
( (
) )
The time derivative of stress, which is required for staggered
grid velocity-stress finite difference scheme can be obtained using
the time derivative of equation (69) and equation(71) by replacing
anelastic function proposed by Kristek and Moczo (2003) is given by
( ) ( ) ( )
(
)
Where, () And, ( ) () ( ) ( ) , ( )( )
5.2 Anelastic coefficient: Kristek and Moczo (2003) anelastic
function as given below ( )
Now, the stress-strain relation for GMB-EK model in case of
staggered grid velocity-stress finite difference scheme can be
written as ( ) ( ) ( )
(
)
Equation (76) reveals that in order to compute the time
derivative of stress, unrelaxed modulus ( ) and anelastic
coefficient ( ) will be required.
5.3 Determination of Anelastic- coefficient : The anelastic
coefficient can be determined using any Q( ) law. The visco-elastic
modulus for GMB-EK model can be written as given below using
equation (57) and equation (75). ( )
[
]
(
)
The quality factor Q( ) can be obtained using the real and
imaginary part of equation (77). Q( ) = From above relation, we
find out Real part of( ) ( )
( ) and Imaginary part of
( )
( )
[
]
[
( ( ( ( ( )( ) ] ) ) ]
)
] )
[
[
[
]
{
}
{
}
(
)
From equation (78), we have ( )
{
}
(
)
And, ( )
{
}
(
)
Using equation (79) and equation (80), we have
( )
[
]
(
)
Equation (81) can also be written as ( ) ( )
(
)
Equation (82) was used to numerically fit any Q( )-law. Q( )
nearly constant.
Emmerich and korn (1987) and Graves and Day (2003) concluded
that an accurate approximation to nearly constant Q( ) is obtained
if the relaxation frequency is logrithmically equidistant and
covers the entire frequency of interest. Equation (58) was solved
using least square method for determination of anelastic
coefficient . ): can be obtained from the field
5.4 Determination of Unrelaxed Modulus ( The phase velocity
( ) at certain reference frequency
measurement. For example SASW method (Kramer, 1996), Moczo et al
(1997) has given the details for computations of phase velocity
using real part of complex and frequency dependent visco-elastic
modulus ( ( )) and density ( ). ( ) } ]
( Using equation (79) Re
)
[{
(
)
( ) is given as ( )
{
}
(
)
Using equation (83) and equation (84), we get
(
)
{
{
} }
(
)
(
){
}
(
)
(
)
(
)
6.FD approximation of Anelastic function : ( ) ( ) can be
approximated using a simple arithmetic averaging in time and
central
difference finite difference operator having second order
accuracy. [ ] ( )
[
]
(
)
Where,
denotes the nth time level. Using equation (88) and equation
(89) anelastic function
can be solved 0 1
\ ( )
Using equation (88) and equation (90), we have
[ 0 0 1 1
]
Now stress-strain can be written as
(
)
[
]
[
]
(
)
Let,
[
]
(
)
Let
[
]
Then, we have
(
)
7. F. D. Approximation of Visco-elastic SH-wave equation:
SH-wave equation is given as ( Using equation (94), we have )
(
)
(
)
(
)
(
)
(
)
(
)
Where superscript (
) and (
) denotes the time level. Other parameters like particle
velocity and stress components are at nth time level. ( ) ( ) (
)
( Where,
)
(
)
(
)
is anelastic coefficient and
is unrelaxed modulus, are defined as ( ( ) )
And,
[
]
(
)
[
]
(
)
Where, And and are obtained harmonic mean of at the node
points.
(
)
Anelastic coefficient
,
have been computed with the help of following equation
using Futtermans equation and least square technique. ( ) (
)
(
)
is relaxation frequency which is logarithmically distributed. Q
( ) values at known frequencies , =1, 22n-1. = . = .
(2,4) finite- difference scheme have been used for the
approximation of the equation (95) to equation (99).
( ( {( Where, = -1/24 and ) = 9/8.
)
[ {( [ {( )
) ) }]
( (
) )
} }
{(
)
(
)
}]
) (
(
)
(
)
(
)
.
/
[ {
}
{
}]
[.
/
(
)
]
(
)
(
)
(
)
.
/
[ {
}
{
}]
[.
/
(
)
]
(
)
(
)
(
)
[ {
}
{
}]
(
)
(
)
(
)
[ {
}
{
}]
(
)
