Applied Mathematics, 2011, 2, 619-624
doi:10.4236/am.2011.25082 Published Online May 2011 (http://www.SciRP.org/journal/am)
A Problem of a Semi-Infinite Medium Subjected to
Exponential Heating Using a Dual-Phase-Lag
Thermoelastic Model
Ahmed Elsayed Abouelregal
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt
Received January 2, 2011; revised March 39, 2011; accepted April 2, 2011
Abstract
The problem of a semi-infinite medium subjected to thermal shock on its plane boundary is solved in the
context of the dual-phase-lag thermoelastic model. The expressions for temperature, displacement and stress
are presented. The governing equations are expressed in Laplace transform domain and solved in that do-
main. The solution of the problem in the physical domain is obtained by using a numerical method for the
inversion of the Laplace transforms based on Fourier series expansions. The numerical estimates of the dis-
placement, temperature, stress and strain are obtained for a hypothetical material. The results obtained are
presented graphically to show the effect phase-lag of the heat flux q

and a phase-lag of temperature gra-
dient


on displacement, temperature, stress.
Keywords: Generalized Thermoelasticity, Dual-Phase-Lag Model, Semi-Infinite Medium, Laplace
Transform
1. Introduction
Biot [1] (1956) introduced the theory of coupled ther-
moelasticity to overcome the first shortcoming in the
classical uncoupled theory of thermoelasticity where it
predicts two phenomena not compatible with physical
observations. First, the equation of heat conduction of
this theory does not contain any elastic terms. Second,
the heat equation is of a parabolic type, predicting infi-
nite speeds of propagation for heat waves. The governing
equations for Biot theory are coupled, eliminating the
first paradox of the classical theory. However, both theo-
ries share the second shortcoming since the heat equation
for the coupled theory is also parabolic.
Thermoelasticity theories that predict a finite speed for
the propagation of thermal signals have aroused much
interest in the last three decades. These theories are
known as generalized therrnoelasticity theories. The first
generalizations of the thermoelasticity theory is due to
Lord and Shulman [2] who introduced the theory of gen-
eralized thermoelasticity with one relaxation time by po-
stulating a new law of heat conduction to replace the
classical Fourier’ law. This law contains the heat flux
vector as well as its time derivative. It contains also a
new constant that acts as a relaxation time. The heat equ-
ation of this theory is of the wave-type, ensuring finite
speeds of propagation for heat and elastic waves. The re-
maining governing equations for this theory, namely, the
equations of motion and the constitutive relations remain
the same as those for the coupled and the uncoupled
theories. This theory was extended by Dhaliwal and She-
rief [3] to general anisotropic media in the presence of
heat sources.
A generalization of this inequality was proposed by
Green and Laws [4] Green and Lindsay obtained another
version of the constitutive equations in [5]. The theory of
thermoelasticity without energy dissipation is another ge-
neralized theory and was formulated by Green and Na-
ghdi [6]. It includes the thermal displacement gradient
among its independent constitutive variables, and differs
from the previous theories in that it does not accommo-
date dissipation of thermal energy.
Tzou [7,8] proposed the dual-phase-lag (DPL) model,
which describes the interactions between phonons and
electrons on the microscopic level as retarding sources
causing a delayed response on the macroscopic scale. For
macroscopic formulation, it would be convenient to use
the DPL mode for investigation of the micro-structural

A. E. ABOUELREGAL 620
effect on the behavior of heat transfer. The physical
meanings and the applicability of the DPL mode have
been supported by the experimental results [9]. The
dual-phase-lag (DPL) proposed by Tzou [9] is such a
modification of the classical thermoelastic model in
which the Fourier law is replaced by an approximation to
a modified Fourier law with tow different time transla-
tions: a phase-lag of the heat flux q

and a phase-lag of
temperature gradient


. A Taylor series approximation
of the modified Fourier law, together with the remaining
field equations leads to a complete system of equations
describing a dual-phase-lag thermoelastic model. The
model transmits thermoelastic disturbance in a wave-like
manner if the approximation is linear with respect to
q

and


, and 0 ≤


< q

; or quadratic in q

and
linear in


, with q

> 0 and


> 0. This theory is
developed in a rational way to produce a fully consistent
theory which is able to incorporate thermal pulse trans-
mission in a very logical manner.
Danilovskaya [10] was the first to solve an actual
problem in the theory of elasticity with nonuniform heat.
The problem was for a half-space subjected to a thermal
shock in the context of what became known as the theory
of uncoupled thermoelasticity. Chandrasekharaiah and
Srinath [11] studied the one dimensional thermal wave
propagation in a half space based on the theory of ther-
moelasticity without energy dissipation due to a constant
step in temperature applied to the boundary. Roychoud-
huri and Dutta [12] studied thermoelastic interactions in
an isotropic homogeneous thermoelastic solid containing
time-dependent distributed heat sources which vary pe-
riodically for a finite time interval in the context of
Green and Naghdi theory. Sherief and Dhaliwal [13]
solved a generalized one-dimensional thermal-shock
problem for small times. Allam et al. [14] discussed
magneto-thermoelasticity for an infinite body with a
spherical cavity and variable material properties without
energy dissipation.
The present paper is organized as follows. Section 2
describes the formulation of the problem and the gov-
erning equations. Section 3 discusses the Laplace trans-
form technique and the solution in the transformed do-
main is obtained using a potential function. Section 4
summarizes the inverse Laplace transforms using a nu-
merical method based on Fourier expansion techniques.
The last section is devoted to the numerical example for
finding the temperature, displacement and the stress.
These distributions are also depicted graphical.
2. Formulation of the Problem
We shall consider a homogeneous, isotropic, thermoelas-
tic solid, occupying the region where the
0x
x
-axis
is taken perpendicular to the bounding plane of the half-
space pointing inwards. The boundary conditions for tem-
perature is in the form of exponential heating, a more
realistic situation. It is assumed that the state of the me-
dium depends only on
x
and t and that the displace-
ment vector has components .
 
,, 0, 0tux
The equation of motion in the absence of body forces
in the one dimensional case has the form

22
2
2
u
2
u T
x
tx
 



 
 (1)
The constitutive equation will take the following form

xx =+2 T
u
x
 

 (2)

The Chandrasekharaiah and Tzou theory is such a mo-
dified of classical thermoelasticity model in which the
Fourier law is replaced by an approximation of the equa-
tion



q
qxtK xt

,,T


   (*)
The model transmits thermoelastic disturbances in a
wave-like manner if Equation (*) is approximated by
11
qqK T
tt

,

 


 



where 0q




.
Hence, we get the heat conduction equation in the
context of dual-phase-lag model in the form
22
0
2
11
qE
TT
KC
ttt x

u
T
xt


 
 
 

 




 
 

(3)
where
x
x

is the stress,

and

are the Lamé con-
stants,

is equal to


32 t

 , t

is the thermal
expansion coefficient,
K
is the thermal conductivity,
E
C is the specific heat per unit mass at constant strain, ρ
is the density of the medium and is the heat flux
vector.
i
q
Moreover, if we put


= 0 and q

= τ (the first re-
laxation time), fundamental equations possible for the
Lord and Shulman's theory.
The initial and boundary conditions are taken as



0
,
,0,0
t
uxt
uxt x
t

0t







0
0
,
,0
t
t
Txt
Txt x
t


,0,



 

00
//
0
0
0
,
,e, 1e
tt
x
x
uxt
Txt Tx
 



 
 (4)
where 0

is constant and the regularity boundary condi-
tions are


,Txt,


t,ux and
x
x

0 as
x
.

A. E. ABOUELREGAL621
1
,
For convenience, we shall use the following non-di-
mensional variables
2
11
',','
x
cxucu tc t

 
22
1101
',','
qq
tc tctc

2
0
,

  
0
',' ,
xx
xx
T
TT

 
where
1
2
,
E
Cc
K






In terms of these variables, Equations (1)-(3) become
(where the primes are suppressed for simplicity)
22
22 .
uu T
a
x
tx
 

  (5)
22
2
11
q
TT
ttg
ttt x

  








u
xt



(6)
2
xx =ubT
x


 (7)
where
2
00
2
1
2
,, ,.
E
TT
ab g
C c




 

 
and initial and boundary conditions will be
 
0
0
,
,0
t
t
uxt
uxt x
t




,0,
 
0
0
,
,0
t
t
Txt
Txt x
t




,0,




0
/
0
0
,
,e, 1e
tt tt
x
x
uxt
Txt x




 

0
/
(8)
3. Solution in the Laplace Transform
Domain
We use the Laplace transform of both sides of the last
equations which is defined in the form
 
0
ed
st .
f
sft

t
Hence, we obtain Equations (5-7) in the form
2
2
2
d,
d
d
T
sua
d
x
x



 (9)
  
2
2
d
111
d
dqq
ts stsTstsg
2
xx
d
=d
ubT
x
  (11)
where the over bar symbol denotes its Laplace transform
and s denotes the Laplace transform parameter.
The boundary conditions (8) in the Laplace transform
can be expressed in the form



0
00
0
0
0
,,
1
d, .
d1
x
x
t
Txt st
uxt t
x
sst


 

(12)
Introducing the thermoelastic potential function

defined by the relation
d
d
u
x

 (13)
Equations (9-10) reduce to
2
2
2
d,
d
s
aT
x





(14)
  
22
22
dd
111
dd
qq
tsstsTstsg .
x
x






(15)
Eliminating from Equations (14) and (15), we ob-
tain
T

42
22
42
dd
10
dd
sP sP
xx


 ,


 



(16)
where

is the mechanical coupling constant defined
by ag


and



1
1
q
s
ts
Pts


.
The solutions of Equation (16) bounded at infinity can
be written in the form:
1
12
ee
m
AA 


2
m
(17)
where 1
A
and 2
A
are parameters depending on s to be
determined from the boundary conditions, 1
mand 2
are the roots with positive real parts of the characteristic
equation
m


42 22
10msP msP 


 
 (18)
1
m and are given by
2
m
 
2
22
1,2
11
2
2
4
s
PsP
m




sP
(19)
du
x
x





(10)
The expression for displacement and temperature can
be written in the forms
1
112 2
e
m
umAmA

 2 e
m
(20)

A. E. ABOUELREGAL 622



12
22 22
12
12
ee
mm
msm s
TA
aa


A (21) (21)
Substituting from Equations (20) and (21) into Equa-
tion (11), we obtain
Substituting from Equations (20) and (21) into Equa-
tion (11), we obtain
1
2
2
22
11
2
22
22
e
e
m
xx
m
bbs
mA
aa
bbs
mA
aa


















(22)
From the boundary conditions (12), it follows that



22
20
132
01 2
1
1
msts
Asstmm

 
2
,



22
10
232
01 2
1
1
msts
Asstmm


2
4. Numerical Inversion of the Laplace
Transform
In order to determine the conductive and thermal tem-
perature, displacement and stress distributions in the time
domain, we adopt a numerical inversion method based
on a Fourier series expansion [15]. In this method, the
inverse g(t) of the Laplace transform g(s) is approxi-
mated by the relation
 
1
/
1
1
11
Re,0 ,
2
ct Nikt t
k
gc
eik
g
tegct
tt





 





t
(23)
where is the real part and Re 1i is imaginary
number unit and is a sufficiently large integer rep-
resenting the number of terms in the truncated infinite
Fourier series. For faster convergence, numerous nu-
merical experiments have shown that the value of c
satisfies the relation Tzou [9].
N
4.7ct 
N must chosen such that
1
/
1
1
eRee
ikt t ct ik
gc t













(24)
where 1

is a persecuted small positive number that
corresponds to the degree of accuracy to be achieved.
The parameter c is a positive free parameter that must be
greater than the real parts of all singularities of


g
s.
The optimal choice of c was obtained according to the
criteria described in [15].
Formula (23) was used to invert the Laplace trans-
forms in Equations (20)-(22), given the temperature,
stress, and displacement distributions.
5. Numerical Results
Now, we will consider a numerical example for which
computational results are given. For this purpose, Copper
is taken as the thermoelastic material for which we take
the following values of the different physical constants
[16]
368K

, ,
5
1.78 10
t


 383.1,
E
C1.61g

,
10 10
= 8954, =7.7610, =3.8610,   
2
0
= 8886.73, =4, T=293, =0.0168   .
The non-dimensional temperature T, displacement
, and stress component
u
x
x

distributions were evalu-
ated on the
x
-axis. Further by setting the phase-lag of
the heat flux t

to zero, the results due to the Lord and
Shulman's theory are obtained. The computations were
carried out for one value of time, namely for 0.1t

.
The graphs of the temperature, displacement and
thermal stress due to phase-lag of the heat flux q tare
exhibited graphically in Figures 1-3. The results carried
0
0. 005
0.01
0. 015
0.02
0. 025
0.03
00.2 0.40.6 0.81
Disp lacement
0.05
q
t
0.1
q
t
0.2
q
t
Figure 1. Displacement vs. distance for different values of
phaselag of the heatflux.
0
0.2
0.4
0.6
0.8
1
00.20.40.60.81
Temperature
0.05
q
t
0.1
q
t
0.2
q
t
x
0.05
q
t
0.1
q
t
0.2
q
t
x
x
Figure 2. Temperature vs. distance for different values of
phaselag of heatflux.

A. E. ABOUELREGAL623
-0.6
-0.4
-0.2
0
00.2 0.40.6 0.81
Ther m al S tress
x
0.05
q
t
0.1
q
t
0.2
q
t
Figure 3. Thermal stress vs. distance for different values of
phaselag of heatflux.
out for three values of , namely =0.05, 0.1 and 0.2.
q
tq t
We can deduce that:
1) The parameter has significant effects on all the
fields.
q
t
2) The wave has a finite speed of propagation. This
result shows that the DPL model agree with the general-
ized thermoelasticity.
3) The temperature and displacement starts with its
maximum value at the origin and decreases until attain-
ing zero beyond a wave front for the generalized theory,
which agree with the boundary conditions.
4) The magnitude of the stress increases rapidly as
x
increases and it attains a peak value at , thereaf-
ter it decreases slowly with increasing
= 0.1x
x
whereas the
magnitude of the peak value is reduced with the increase
of t.
q
Figures 4-6 show the heat, the displacement, and the
stress respectively with distance
x
at the same instance
with different values of the phase-lag of tem-
perature gradient parameter
0.1t


0q tt t


0.05 t


which means
coupled thermoelasticity model of Biot, which
means generalized thermoelasticity model of Lord and Shul-
man and for (t

and ) means
that DPL model and we found that, the parameter
0t


0.080t

 
t

has a significant effects on all the fields.
In all these figures, it is clear that the values of solu-
tions for L-S theory are large in comparison with those
for DPL model. This may be due to the nature of the
boundary conditions, which we take.
6. Conclusions
In the framework of this article, a problem of a half-
space whose surface is rigidly fixed and subjected to the
effects of a thermal shock on the surface within the con-
text of the theory of generalized thermoelasticity pro-
0
0.005
0.01
0.015
0.02
0.025
0.03
00.2 0.40.6 0.81
Displacement
0t


0.05t


0.08t


x
–0.2
–0.4
–0.6
Figure 4. Displacement vs. distance for different values of
phaselag of gradient of temperature.
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
0 0.20.40.60.8 1
Temperature
0t


0.05t


0.08t


x
–0.1
Figure 5. Temperature vs. distance for different values of
phaselag of gradient of temperature.
-0.6
0
00.20.40.60.81
l Stress
0t


0.05t


0.08t


x
Therma
-0.3
–0.3
–0.6
Figure 6. Thermal stress vs. distance for different values of
phaselag of gradient of temperature.
posed by Tzou.
According to the above results, we can conclude that:
1) As the phase-lag of the heat flux qconstant in-
creases the corresponding components of temperature,
t

A. E. ABOUELREGAL 624
displacement and stress decrease.
2) The increases of the phase-lag of gradient of tem-
perature t

decrease the components of temperature,
displacement and stress distributions.
3) We found that, the parameters and
q
tt

have
significant effects on all the fields.
4) The phenomenon of finite speeds of propagation is
manifested in all these figures.
The comparison of different theories of thermoelastic-
ity, i.e. Lord and Shulman theory and Chandrasekharaiah
and Tzou model is carried out.
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