Applied Mathematics, 2010, 1, 411-415
doi:10.4236/am.2010.15054 Published Online November 2010 (http://www.SciRP.org/journal/am)
The Quasi-Static Approximation of Heat Waves in
Anisotropic Thermo-Elastic Media
Shaohua Guo
School of Civil Engineering and Architecture, Zhejiang University of Science and Technology,
Hangzhou, China
Received August 6, 2010; revised September 21, 2010; accepted September 25, 2010
Abstract
The equilibrium equations of anisotropic media, coupled to the heat conduction equations, are studied here
based on the standard spaces of the physical presentation, in which an new thermo-elastic model based on
the second law of thermodynamics is induced. The uncoupled heat wave equation for anisotropic media is
deduced. The results show that the equation of heat wave is of the properties of dissipative waves. In final
part of this paper, we discuss the propagation behaviour of heat waves for transversely isotropic media.
Keywords: Anisotropic Media, Thermo-Elastic Model, Heat Wave, Standard Spaces, Modal Equations
1. Introduction
In recent years, the considerable interests have been
shown in the study of thermo-elastic wave propagations
in anisotropic media. The classical theory of thermo-
elasticity is based on Fourier’s law of heat conduction,
which predicts an infinite speed of propagation of heat.
This is physically absurd and some new theories have
been proposed to eliminate this absurdity. For example,
Lord and Shulman [1] obtained a wave-type heat
equation by modifying the Fourier’s law of heat con-
duction. This new law contains the heat flux vector as
well as its time derivative. It also contains a new constant
that acts as a relaxation time. Since the heat equation of
this theory is of the wave type, it automatically ensures
finite speeds of propagation for heat and elastic waves.
The remaining governing equations for this theory,
namely, the equations of motions and constitutive re-
lations, remain the same as those for the coupled and the
uncoupled theories. This theory was extended by Dhali-
wal and Sherief [2] to general anisotropic media in the
presence of heat sources. Later, Green and Lindsay [3]
deduced another theory, known as temperature rate
dependent theory of thermo-elasticity, including the rate
of temperature in the constitutive equations. This theory
contains two constants that act as relaxation times and
modifies all the equations of the coupled theory, not only
the heat equation. The classical Fourier’s law of heat
conduction is not violated if the medium under consider-
ation has a center of symmetry. Although many works
have been done for heat or elastic waves in anisotropic
media, the explicit uncoupled equations of wave equa-
tions in anisotropic media could not be obtained because
of the limitations of classical elastic theory [4-6]. In this
paper, the idea of standard spaces [7-10] is used to deal
with both the equilibrium equation and the heat con-
duction equation based on the quasi-static approxi-
mation of the propagation of heat wave, and by intro-
ducing an new thermo-elastic model that obeys the
second law of thermo-dynamics, a uncoupled modal
equations of heat wave are obtained, which shows that
heat wave is of the properties of dissipative waves.
Meanwhile the propagation speed, propagation direction
and space pattern of heat wave can be completely
determined by the modal equations.
2. Modal Constitutive Equation of
Anisotropic Thermo-Elastic Media
Hooke’s law and entropy relationship for linear thermo-
elastic materials are the following
ijijkl klij
ca



 (1)
kl kl
a



 (2)
Rewriting them in the Voigt’s notation, we have




cS a (3)
T



aS (4)

S. H. GUO 412
The elastic matrix c can be spectrally decomposed
as follows[7-10]

c  (5)
where


123456
,,,,,diag

   ，, are the matrixes
of eigen elasticity.

123456
,,,,,   is the mo-
dal matrix of elastic media, which is both orthogonal and
positive definite matrixes, and satisfy TI

。
Projecting the elastic physical qualities of the geomet-
ric presentation, such as the stress vector

and strain
vector
S
, into the standard spaces of the physical pres-
entation, we get


*

 (6)


*
S
S  (7)
Rewriting Equations (6) and (7) in the form of scalar, we
have
** 1, ,
ii i= m


   (8)
** 1, ,
ii
Si=m

   S (9)
where

6mis number of the elastic independent
subspaces. Equations (8) and (9) show the elastic physi-
cal qualities under the physical presentation.
Substituting Equations (6) and (7) into Equations (3)
and (4), respectively, and multiplying them with the
transpose of modal matrix in the left, we have

 
  
*
cS a (10)
*T

 aS  (11)
Using Equations (5), (6) and (7), we get
*

 
**
S
a (12)
**T

 aS (13)
Rewriting the above equations in the form of scalar, we
have
*** 1
iiii
Sai m     (14)
** 1
kk
aSk m     sum to k (15)
Equations (14) and (15) are just the modal constitutive
equations for anisotropic thermo-elastic media, in which
*
k
a are the coupled thermo-elastic coefficients.
3. Heat Conduction and Thermo-Dynamic
Equations
The Fourier’s law of heat conduction is the following
'iijj
q

  (16)
where 0
TT

,The second law of thermo-dynamics is
the following
ei
ds dsds (17)
where e
dQ
ds T ,0ds  and 0
i
ds . To general ani-
sotropic media in the absence of heat sources, Equation
(17) can be written in terms of entropy density as follows
'ii i
TqT
 



(18)
Now, we suppose that the irreversible part of entropy
density rate is direct proportion to the negative value of
entropy density acceleration
id






(19)
The reason for this is that 0
i
d

, 0d


, which
are requirements to maintain a stable thermodynamic
process. Then, by Equation (18), we have

'1 ii t
qT

   (20)
Differentiating Equation (16) with respect to i and
substituting from Equation (20), we get

'1 ij jitt
T



  (21)
This is the thermodynamic equation based on the sec-
ond law of thermodynamics, which is same as the L-S
model [1].
4. Eigen Expression of Equilibrium and
Thermodynamic Equation
The eigen form of elastic equilibrium equation can be
written as follows [7,8]
** 0
ii


 (22)
where,



** *
T
ii i

 is the stress operator [7,8],
in which




1131 21
22 3221
33 3231
23 2322332131
131312113332
12 1213232211
00 0
00 0
00 0
0
0
0





 




 


  









 


(23)
We can also rewrite Equation (21) in form of matrix as
follows
 
1
T
tt
BT




 (24)
where


11 22 33233112
T
B,,,2,2,2
 
,


1122 33 23 3112
T,,,,,

 .
Let

*T
B

, Equation (24) becomes

S. H. GUO 413

*1tt
T



  (25)
5. Modal Equation of Heat Wave in
Anisotropic Media
Substituting Equations (14,15) into Equations (22) and
(25), we have
***0
iiii
Sa 



(26)


***
1ttkk TaS

  (27)
According to the principle of operator, Equation (26)
can be written as follows
*
*k
k
k
a
S


 (28)
By using Equation (28), Equation (27) becomes
** **
*0 kk kk
ttt
kk
kk
aa aa
TT  


  



(29)
Rewriting Equation (29) in the form of wave equation,
we get
2*
1
ttt c



 (30)
where
**
1
kk
kk
caa T








(31)
It just the speed of heat wave in anisotropic media.
6. Application
In this section, we discuss the propagation laws of heat
wave in a hexagonal (transversely isotropic) crystal. The
material tensors in Equations (1), (2), (16) are represented
by the bottom matrices (32) under the compact notation
where

6611 12
1
2
ccc
.
There are four independent eigenspaces in a hexagonal
(transversely isotropic) crystal [7-10]
(1) (1)(2)(2)
1122336445
[][ ][,][, ]WWWWW φ    
(33)
where as shown in Equation (34)
where i
ξ
is a vector of order 6 in which ith element is 1
and others are 0.
The eigenelasticity of hexagonal crystal are
2
2
11 1233111233
1,2 13
311124 44
2,
22
,
ccc cccc
cc c



 


 




 
(35)
The structures of four independent eigen-spaces are
the following
*
1, 21, 2
φ
φ
,

*
3
31,1,0, 0, 0,1
3
T
φ,

*
4
20, 0, 0,1,1,0
2
T
φ (36)
*
1 ,21,211221, 233
[]aaaa  ,
The relative quantities and operators can be calculated
as follows:
11 1213
121113
11 11
13 1313
11 11
44
33 33
44
66
000
000
000
,,
00000
00 0
00000
ccc
ccc a
ccc a
ca
c
c









 


 




 


 
 

 




(32)

T
1,2 11 12
13
1,2 22 3
1,2 11 1213
T
3
1,1,, 0, 0, 0
()2
21,1, 0, 0, 0, 0
2
,4,5,6
ii
cc
c
c
ccc
i











 




 

 




(34)

S. H. GUO 414

*
31122
3
3
aaa,
*
40a (37)

*222
11 33
x
yz
   (38)
where
13
1,2 22
1,211 1213
()2
c
cc c
   ，1,211 12
1, 2
13
cc
c




Thus, the equation of heat wave in transversely iso-
tropic media becomes
2222 33
11
11
1
tttx yz
c


 


 


(39)
where
**
** ****
33 11 2244
1234
1
caa
aaaaaaT







(40)
Now, we discuss the propagation properties of heat
wave in transversely isotropic media along z direction.
At this moment, Equation (39) becomes
2
2
33 2
1
ttt cz

 


 (41)
The plane wave solution of Equation (41) is the fol-
lowing


0exp it-kz


 (42)
Substituting the above into Equation (41), we have
222
33
1
ick  

 (43)
where
12
k=k ik

(44)
By using Equation (44), Equation (43) becomes

2222
33121 2
1
ick-k+2ikk
 

 (45)
Comparing the real and imaginary parts of Equation
(45), we have

2222
331 2
ck-k  , 2
12 33
2k kc

  (46)
From the above, we have
2
22
42
22
22
33 33
33
10
2
kk
cc
c




  


(47)
when 11
2

, we get
1
33
kc


, 2
33
2
kc


 (48)
Thus, the plane wave solution of Equation (41) can be
written as

0
33
1
exp exp
2zit-kz c
 


 




(49)
or

0
1
exp exp
2tit-kz
 


 


 (50)
It is seen from the above that heat wave is of dissipa-
tive properties, it will attenuate with the distance or time
of wave propagation. A possible explanation for the re-
sults is that in an idealized solid, the thermal energy can
be transported by quantized electronic excitations and by
the quanta of lattice vibrations, these quanta undergo
collisions of a dissipative nature, causing a thermal re-
sistance in the medium.
Two extreme cases are the following
1)

,

0exp it-kz
 



, 0c
2) 0

, 0

, c
Case 1 shows that no heat wave propagates when the
dissipation coefficients in media is large enough. All ther-
mal disturbances will be totally absorbed by the media.
Case 2 shows that heat wave propagates at an infinite
speed when no dissipation exists in media, which is just
the classical result of thermo-elasticity based on the
conventional heat conduction equation.
7. Conclusions
Based on the quasi-static approximation, the heat wave
in anisotropic elastic media is studied by using the eigen
theory. An new thermo-elastic model based on the
second law of thermodynamics is induced, the uncoupled
heat wave equation is deduced, the propagation speed of
heat wave is obtained. The results show that the equation
of heat wave is of the strongly dissipative properties, and
when the dissipation coefficients is large enough, no heat
wave propagates in media, it means that thermal distur-
bances will be totally absorbed by the media, but when
no dissipation exists, heat wave will propagate at an infi-
nite speed, which is certainly unacceptable in physics.
8. References
[1] H. Lord and Y. Shulman, “A Generalized Dynamical
Theory of Thermo-Elasticity,” Journal of the Mechanics

S. H. GUO 415
and Physics, Vol. 15, No. 5, 1967, pp. 299-309.
[2] R. Dhaliwal and H. Sherief, “Generalized Thermoe-
lasticity for Anisotropic Media,” The Quarterly of Appli-
ed Mathematics, Vol. 33, Vol. 1, 1980, pp. 1-8.
[3] A. E. Green and K. E. Lindsay, “Thermoelasticity,”
Journal of Elasticity, Vol. 2, 1972, pp. 1-7.
[4] X. G. Tian, Y. P. Shen, et al., “A Direct Finite Element
Method Study of Generalized Thermoelastic Problems,”
International Journal of Solids and Structures, Vol. 43,
No. 7-8, 2006, pp. 2050-2063.
[5] T. C. Chen and C. I. Weng, “Generalized Coupled
Transient Thermoelastic Plane Problems by Laplace
Transform/Finite Element Method,” Journal of Applied
Mechanics, Vol. 55, No. 2, 1988, pp. 377-382.
[6] J. H. Prevost and D. Tao, “Finite Element Analysis of
Dynamic Coupled Thermoelasticity Problems with Rela-
xation Times,” Journal of the Mechanics and Physics, Vol.
50, 1983, pp. 817-822.
[7] S. H. Guo, “An Eigen Theory of Rheology for Complex
Media,” Acta Mechanica, Vol. 182, 2007, pp. 985-992.
[8] S. H. Guo, “An Eigen Theory of Electromagnetic Waves
Based on the Standard Spaces,” International Journal of
Engineering Science, Vol. 47, 2009, pp. 405-412.
[9] S. H. Guo, “An Eigen Theory of Waves in Piezoelectric
Solids,” Acta Mechanica Sinica, Vol. 26, No. 2, 2010, pp.
241-246.
[10] S. H. Guo, “An Eigen Theory of Electro-Magnetic
Acoustic Waves in Magnetoelectroelastic Media,” Acta
Mechanica, Vol. 211, No. 1-2, 2010, pp. 173-180.