1. INTRODUCTION:
Linear poroelasticity has important
applications in soils and rock materials saturated by groundwater.
Fluid-saturated porous materials permeated by groundwater or oil are found on
and below the surface of the earth. Water saturated ocean sediments are
considered as fluid-saturated porous materials. Biot (1956a) gave the theory of
fluid-saturated porous materials. Biot (1956b, 1962) also formulated a
theoretical framework for isothermal wave propagation in fluid-saturated
elastic porous media for high and low frequency ranges. Many authors followed
the Biot's theory and studied various problems on propagation of plane and
surface waves in fluid-saturated porous materials. Some prominent contributors
are Jones (1961), Deresiewicz and Rice (1962), Hajra and Mukhopadhyay (1982),
Tajuddin (1984), Sharma and Gogna (1991), Carcione (1996), Khalili et al.
(1999), Berrymann (2005), Tajuddin and Hussaini (2005), Lin et al. (2005),
Albers and Wilmanski (2006), Wang et al. (2006), Sharma (2007), Li, et al.
(2007), Zyserman and Santos (2007) and Nakagawa and Schoenberg (2007), Lo
(2008), Sharma (2012), and many others.
Thermal and mechanical coupling
between the phases made the thermo-mechanical coupling in the poroelastic
medium more complex as compared to classical case. There are various areas like
petroleum engineering, chemical engineering, pavement engineering and nuclear
waste management, where the problems on wave propagation in saturated
thermoelastic porous medium may find applications. Many researchers including
Schiffman, 1971; Pecker and Dereziewicz, 1973; Mc Tigue, 1986; Coussy, 1989;
Kurashige, 1989; Bear et al., 1992; Zhou et. al., 1998; Ghassemi and Diek,
2002; Shrefler, 2002; Abousleiman and Ekbote, 2005; Youssef, 2007, Singh, 2011,
and Singh, 2013 have followed Biot (1956c) theory and contributed significant
problems in porothermoelastic materials.
Following Biot (1956a, 1962) and
Lord and Shulman (1967), Youssef (2007) developed a theory of generalized
porothermoelasticity, in which he considered a homogeneous, isotropic, elastic
matrix whose interstices are filled with a compressible ideal liquid, where
both the solid and liquid form continuous and interacting regions and viscous
stresses are neglected in the liquid. He also assumed that the liquid is
capable of exerting a velocity-dependent friction force on the skeleton. This
theory of generalized porothermoelasticity was applied by Singh (2011, 2013) to
show the existence of one shear and four kinds of coupled longitudinal waves
and to study the reflection phenomena in a generalized porothermoelastic solid
half-space.
In this paper, the governing
equations of generalized porothermoelasticity given in Youssef (2007) are
simplified in context of Green and Naghdi (1993) theory of thermoelastcity
without energy dissipation. A problem on Rayleigh type surface wave is
considered in a generalized porothermoelastic solid half-space. A secular
equation in Rayleigh wave speed is derived and solved numerically for a
particular material. The wave speed is plotted to observe the effects of porosity
and various thermal coefficients.
2. GOVERNING EQUATIONS:
According to the theories of
Youssef (2007)and Green and Naghdi (1993), the linear governing equations of
isotropic and homogeneous generalized porothermoelasticity in absence of body
forces and heat sources, are
(a) Constitutive equations
(1)
, (2)
, (3)
, (4)
, (5)
(b) Equations of motion
(6)
(7)
(c) Heat Equations
(8)
(9)
where the meanings of all
symbols are given in Appendix I.
Using the following Helmholtz's
representations
(10)
(11)
the equations (6) to (9) reduce
for 
plane
as
(12)
(13)
(14)
(15)
(16)
(17)
3. SURFACE WAVE
SOLUTIONS:
We now consider a
porothermoelastic half-space occupying the region 
in
the reference configuration with boundary 
and
Rayleigh surface waves propagating along the direction 
.
The solutions of the equations (12) to (15) are now sought in the form
(18)
in which are functions of 
,
, v is the phase speed and k is the wave number.
With the help of equation (18),
the equations (12) to (15) lead to the following equation
(19)
where, and the expressions for and
are given in Appendix II.
We require the following
radiation conditions on for
(20)
The general solutions, which
satisfy the radiation conditions (20) are
(21)
(22)
The expressions for and
relations between, are given in Appendix III.
Similarly, the general solutions of
equations (16) and (17), which satisfy radiation conditions as are
(23)
where and.
4. BOUNDARY CONDITIONS:
The suitable boundary conditions
at free surface are vanishing of normal stress of solid, tangential stress of
solid, liquid stress per unit area, solid heat flux and liquid heat flux
(24)
where
s
The solutions given by (21) to (23)
satisfy the boundary conditions (24) and we obtain the following secular
equation for Rayleigh wave
(25)
where
5. NUMERICAL RESULTS AND DISCUSSION:
The real part of wave speed of
Rayleigh wave is computed for following physical constants at (Yew and Jogi,
1976)
dyne.cm
,
dyne.cm,
dyne.cm,
dyne.cm,
= 0.82 gm.cm
,
= 2.6 gm.cm,
= 0.002137 gm.cm
= 2.1 cal.gm
.C,
= 1.9 cal.gm.
C.
The variations of the wave speed
of Rayleigh wave are shown graphically in figure 1 against the range of
porosity when, = 0.5 cal.cm.s.
C,
and = 0.4 cal.cm.s.C
for porothermoelastic and poroelastic cases by solid and dashed lines,
respectively. In porothermoelastic case, the value of wave speed is cm. s
when. It increases very sharply and attains its maximum value cm. s
at. Thereafter, it decreases very sharply to a value cm. s
at. In absence of thermal effects, this variation reduces to a similar
variation shown by dashed line in figure 1. The comparison of solid and dashed
lines in figure 1 shows the effect of presence of porosity. This effect becomes
more prominent in the range of porosity between 
and.
For and, the variations of the
wave speed of Rayleigh wave are shown graphically in figure 2 against the range
of
coefficient of thermal expansion in solid phase by solid, small dashed and long
dashed lines, respectively when, 
=
0.5 cal.cm.s.
C,
and 
=
0.4 cal.cm.s.C.
For 
,
it has its minimum value 
cm.
s
at 
.
It increases to its maximum value cm. s
at 
.
By comparing the solid and dashed variations in figure 2, the effects of the
presence porosity and coefficient of thermal expansion in solid phase are
observed on wave speed of the Rayleigh wave.
For 
and
,
the variations of the wave speed of Rayleigh wave are shown graphically in
figure 3 against the range 
of
coefficient of thermal expansion in fluid phase by solid, small dashed and long
dashed lines, respectively when 
,
=
0.5 cal.cm.s.
C,
and =
0.4 cal.cm.s.C.
For ,
it has its maximum value 
cm.
s
at 
.
It decreases to its minimum value 
cm.
s
at 
.
The comparison of solid and dashed variations in figure 3 shows the effects of
the presence porosity and coefficient of thermal expansion in fluid phase are
observed on wave speed of the Rayleigh wave.
For and
,
the variations of the wave speed of Rayleigh wave are shown graphically in
figure 4 against the range 
of
thermoelastic coupling in solid phase by solid, small dashed and long dashed
lines, respectively when 
,
=
0.5 cal.cm.s.
C,
and =
0.4 cal.cm.s.C.
For ,
it has a value 
cm.
s
at 
.
It increases sharply to its maximum value 
cm.
s
at 
and
then decreases slowly to a value cm.
s
at 
.
By comparing the solid and dashed variations in figure 4, the effects of the
presence porosity and thermoelastic coupling in solid phase are observed on
wave speed of the Rayleigh wave.
For and
,
the variations of the wave speed of Rayleigh wave are shown graphically in
figure 5 against the range 
of
thermoelastic coupling in fluid phase by solid, small dashed and long dashed
lines, respectively when 
,
=
0.5 cal.cm.s.
C,
and =
0.4 cal.cm.s.C.
For ,
it has a value 
cm.
s
at 
.
It increases very slowly to its maximum value cm.
s
at 
.
The comparison of solid and dashed variations in figure 5 shows the effects of
the presence porosity and thermoelastic coupling in fluid phase are observed on
wave speed of Rayleigh wave.
For and
,
the variations of the wave speed of Rayleigh wave are shown graphically in
figure 6 against the range 
of
the characteristic of solid phase by solid, small dashed and long dashed lines,
respectively when 
and
=
0.4 cal.cm.s.C.
For ,
it has a value 
cm.
s
at 
.
It increases sharply to its maximum value 
cm.
s
at 
and
then decreases to a value 
cm.
s
at 
.
By comparing the solid and dashed variations in figure 6, the effects of the
presence porosity and characteristic of solid phase are observed on wave speed
of the Rayleigh wave.
For and
,
the variations of the wave speed of Rayleigh wave are shown graphically in
figure 7 against the range 
of
the characteristic of fluid phase by solid, small dashed and long dashed lines,
respectively when and
=
0.5 cal.cm.s.C.
For ,
it has a value 
cm.
s
at 
.
It increases sharply to its maximum value 
cm.
s
at 
and
then decreases very slowly to a value 
cm.
s
at 
.
The comparison of solid and dashed variations in figure 7 shows the effects of
the presence porosity and characteristic of fluid phase are observed on wave
speed of the Rayleigh wave.
6. CONCLUSIONS:
The theory of Green and Naghdi
generalized thermoelasticity is employed to study the Rayleigh wave on a
thermally insulated stress free surface of a porothermoelastic solid
half-space. With the help of particular surface wave solutions in half-space
and relevant boundary conditions at free surface, a secular equation in wave
speed of Rayleigh wave is obtained. The secular equation is solved numerically
by a fortran program of iteration method. The real wave speed of Rayleigh wave
is computed for relevant physical constants of porothermoelastic material (Yew
and Jogi, 1976). To observe the dependence of wave speed of various material
parameters, the wave speed of the Rayleigh wave is plotted against porosity,
coefficients of thermal expansion in solid and fluid phases, thermoelastic
coupling coefficients in solid and fluid phases and characteristics of solid
and fluid phases. Some concluding remarks obtained from the numerical
discussion of wave speed in these plots are given as
(i) The
wave speed of Rayleigh wave in Green-Naghdi porothermoelastic solid depends on
porosity of the material. In absence of thermal parameters, the wave speed
increases at each value of porosity in considered range. The thermal effect on
wave speed becomes more prominent beyond 
.
(ii) The
effect of porosity on wave speed increases with the increase in value of
coefficient of thermal expansion in solid phase.
(iii) The
effect of porosity on wave speed decreases with the increase in value of
coefficient of thermal expansion in fluid phase.
(iv) The
effect of porosity on wave speed increases with the increase in values of
coefficients of thermoelastic coupling for both solid to fluid and fluid to
solid.
(v) The
wave speed of Rayleigh wave is affected significantly by characteristics of
solid and fluid phases for different values of porosity.
Seismological observations
provide the evidences of thermal-mechanical processes which control the
formation and evolution of thermoelastic lithosphere below oceans. The present
study may be useful in studying underground layers of porous solids saturated
with oil or groundwater.
7. ACKNOWLEDGEMENT:
Author is highly thankful to
University Grants Commission, New Delhi for granting a Major Research Project
(MRP-MAJOR-MATH-2013-2149).
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APPENDIX I:
are
the density of the fluid and solid phases, 
is
the porosity of the material,
are
the density of the fluid phase per unit volume of the bulk,
are
the density of the solid phase per unit volume of the bulk,
are
the mass coefficient of solid phase,
are
the mass coefficient of fluid phase,
is
the dynamic coupling coefficient,
are
the displacements of the skeleton and fluid phases,
are
the characteristics of the solid and fluid phases,
are
the entropy for the solid and fluid phases,
are
the coefficients of thermal expansion of solid and fluid phases,
are
the thermoelastic coupling between the phases, 
are
the poroelastic
coefficients,
are
the mixed and thermal coefficients,
are
the specific heat at constant strain of the phases,
where, in the
reference state,.
APPENDIX II:
The expressions for and 
are
given as
,
APPENDIX III:
The expressions for are given as
,
where
, , ,
|
Received on 19.01.2018
Accepted on 19.04.2018
Research J. Engineering and
Tech. 2018;9(2): 179-188.
DOI: 10.5958/2321-581X.2018.00025.9
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