A Problem of a Semi-Infinite Medium Subjected to Exponential Heating Using a Dual-Phase-Lag Thermoelastic Model

doi:10.4236/am.2011.25082

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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

E-mail: [email protected]

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



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

-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

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



 











 (1)

The constitutive equation will take the following form



xx =+2 T

 



 (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







qxtK xt



,,T





  (*)

The model transmits thermoelastic disturbances in a

wave-like manner if Equation (*) is approximated by

qqK T











 







where 0q







.

Hence, we get the heat conduction equation in the

context of dual-phase-lag model in the form

ttt







 

 

 



 













 



(3)

where



is the stress,



and



are the Lamé con-

stants,



is equal to





32 t





, t



is the thermal

expansion coefficient,

is the thermal conductivity,

C is the specific heat per unit mass at constant strain, ρ

is the density of the medium and is the heat flux

vector.

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

uxt

uxt x



0t















Txt

Txt x



,0,







 



,e, 1e

uxt

Txt Tx









 

 (4)

where 0



is constant and the regularity boundary condi-

tions are





,Txt,





t,ux and



0 as

.

A. E. ABOUELREGAL621

For convenience, we shall use the following non-di-

mensional variables

',','

cxucu tc t







1101

',','

tc tctc











',' ,







where













In terms of these variables, Equations (1)-(3) become

(where the primes are suppressed for simplicity)

22 .

uu T

 





 (5)

ttg

ttt





 























(6)

xx =ubT





 (7)

where

,, ,.

ab g















 

and initial and boundary conditions will be

 

uxt

uxt x







,0,

 

Txt

Txt x







,0,









,e, 1e

tt tt

uxt

Txt x







 



/

(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

 

st .

sft



t

Hence, we obtain Equations (5-7) in the form

sua







 (9)



 

111

dqq

ts stsTstsg

ubT



 (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



d, .

Txt st

uxt t

sst



 



(12)

Introducing the thermoelastic potential function



defined by the relation



 (13)

Equations (9-10) reduce to











(14)



 

111

tsstsTstsg .













(15)

Eliminating from Equations (14) and (15), we ob-

tain



sP sP





 ,





 







(16)

where



is the mechanical coupling constant defined

by ag





and







Pts





.

The solutions of Equation (16) bounded at infinity can

be written in the form:







m

(17)

where 1

and 2

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





42 22

10msP msP







 

 (18)

m and are given by

 

1,2

PsP









sP

(19)











(10)

The expression for displacement and temperature can

be written in the forms

112 2

umAmA



 2

m

(20)

A. E. ABOUELREGAL

622







22 22

msm s



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

bbs























(22)

From the boundary conditions (12), it follows that



132

01 2

msts

Asstmm



 



232

01 2

msts

Asstmm





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

 

Re,0 ,

ct Nikt t

eik

tegct









 





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].

4.7ct 

N must chosen such that

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





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



, ,

1.78 10





 383.1,

C1.61g



10 10

= 8954, =7.7610, =3.8610,

 



= 8886.73, =4, T=293, =0.0168

 

The non-dimensional temperature T, displacement

, and stress component



distributions were evalu-

ated on the

-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. 005

0.01

0. 015

0.02

0. 025

0.03

00.2 0.40.6 0.81

Disp lacement

0.05

t

0.1

t

0.2

t

Figure 1. Displacement vs. distance for different values of

phaselag of the heatflux.

0.2

0.4

0.6

0.8

00.20.40.60.81

Temperature

0.05

t

0.1

t

0.2

t

0.05

t

0.1

t

0.2

t

Figure 2. Temperature vs. distance for different values of

phaselag of heatflux.

A. E. ABOUELREGAL623

-0.6

-0.4

-0.2

00.2 0.40.6 0.81

Ther m al S tress

0.05

t

0.1

t

0.2

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.

We can deduce that:

1) The parameter has significant effects on all the

fields.

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

increases and it attains a peak value at , thereaf-

ter it decreases slowly with increasing

= 0.1x

whereas the

magnitude of the peak value is reduced with the increase

of t.

Figures 4-6 show the heat, the displacement, and the

stress respectively with distance

at the same instance

with different values of the phase-lag of tem-

perature gradient parameter

0.1t





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





0.080t



 



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-

0.005

0.01

0.015

0.02

0.025

0.03

00.2 0.40.6 0.81

Displacement





0.05t





0.08t





–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





0.05t





0.08t





–0.1

Figure 5. Temperature vs. distance for different values of

phaselag of gradient of temperature.

-0.6

00.20.40.60.81

l Stress





0.05t





0.08t





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,

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



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.

7. References

[1] M. Biot, “Thermoelasticity and Irreversible Thermody-

Namics,” Journal of Applied Physics, Vol. 27, No. 3, 1956,

pp. 240-253. doi:10.1063/1.1722351

[2] H. Lord and Y. Shulman, “A Generalized Dynamical

Theory of Thermoelasticity,” Journal of the Mechanics

and Physics of Solids, Vol. 15, No. 5, 1967, pp. 299-309.

doi:10.1016/0022-5096(67)90024-5

[3] R. Dhaliwal and H. Sherief, “Generalized Thermoelastic-

ity for Anisotropic Media,” Quarterly of Applied Mathe-

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[4] A. E. Green and N. Laws, “On the Entropy Production

Inequality,” Archive for Rational Mechanics and Analysis,

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[5] A. E. Green and K. A. Lindsay, “Thermoelasticity,”

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[8] D. Y. Tzou, “A Unified Approach for Heat Conduction

From Macro- to Micro- Scales,” Journal of Heat Transfer,

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[9] D. Y. Tzou, “Experimental Support for the Lagging Be-

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doi:10.2514/3.725

[10] V. Danilovskaya, “Thermal Stresses in an Elastic Half-

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Mekh., In Russian, Vol. 14, 1950, pp. 316-324.

[11] D. S. Chandrasekharaiah and K. S. Srinath, “One-Dimen-

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Science, Vol. 34, No. 13, 1996, pp. 1447-1455.

doi:10.1016/0020-7225(96)00034-1

[12] S. K. Roychoudhuri and P. S. Dutta, “Thermoelastic In-

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ternational Journal of Solids Structures, Vol. 42, 2005,

pp. 4192-4203.

[13] H. Sherief, and R. Dhaliwal, “Generalized One-Dimen-

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