Journal of Applied Mathematics and Physics, 2013, 1, 110-120
Published Online November 2013 (http://www.scirp.org/journal/jamp)
http://dx.doi.org/10.4236/jamp.2013.15017
Open Access JAMP
Fractional Order Two Temperatur e Thermo-Elastic
Behavior of Piezoelectric Materials
Essam Bassiouny1,2, Refaat Sabry1,3,4
1Department of Mathematics, Faculty of Science and Humanitarian Studies,
Salman Bin AbdulAziz University, Al Kharj, KSA
2Department of Mathematics, Faculty of Science, Fayoum University, Fayoum, Egypt
3Theoretical Physics Group, Department of Physics, Faculty of Science, Mansoura University,
Damietta Branch, New Damietta, Egypt
4International Centre for Advanced Studies in Physical Sciences, Faculty of Physics and Astronomy,
Ruhr University Bochum, Bochum, Germany
Email: esambassiouny@yahoo.com, Sabry_nonlinear@yahoo.com
Received July 13, 2013; revised August 21, 2013; accepted August 27, 2013
Copyright © 2013 Essam Bassiouny, Refaat Sabry. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
A new mathematical model of time fractional order heat equation and fractional order boundary condition have been
constructed in the context of the generalized theory of thermo piezoelasticity. The governing equations have been ap-
plied to a semi infinite piezoelectric slab. The Laplace transform technique is used to remove the time-dependent terms
in the governing differential equations and the boundary condition. The solution of the problem is first obtained in the
Laplace transform domain. Furthermore, a complex inversion formula of the transform based on a Fourier expansion is
used to get the numerical solutions of the field equations which are represented graphically.
Keywords: Fractional Order; Two Temperature; Generalized Thermoelectricity; Weak Diffusion; Thermal Loading;
Piezoelectric Materials; Ceramics
1. Introduction
Chen and Gurtin [1-3] have formulated a theory of heat
conduction in deformable bodies, which depends upon
two distinct temperatures, the conductive temperature
and the dynamical temperature T. Regarding time inde-
pendent situations, the difference between these two
temperatures is proportional to the heat supply. However,
in the absence of any heat supply, the two temperatures
are identical [1,2]. On the other hand in time dependent
problems, particularly for wave propagation problems,
the two temperatures are generally different regardless of
the presence of heat supply. The two temperatures T,
and the strain are found to have representations in the
form of a traveling wave plus a response, which occur
instantaneously throughout the body [4]. Warren and
Chen [5] investigated the wave propagation in the two-
temperature theory of thermoelasticity, but Youssef [7]
investigated this theory in the context of generalized
thermoelasticity.
Because the non-local property of fractional order of
differential equations (FODE), FODE becomes prominent
and extensively used in many applications in fluid me-
chanics, physics, engineering, viscoelasticity and many
other fields. The presence of the fractional order operator
in the differential equations affects the history of the
system, which means that the next states of the system
will depend, on the current state and also upon all of its
previous states, making it more realistic: Caputo [7],
Mainardi [8] and Podlubny [9]. FODE has been used
successfully in modeling of various physical phenomena
and in many applications such as chemistry, biology,
electronic, wave propagation and viscoelasticity Hilfer
[10], Caputo and Mainardi [11], Caputo [12], Bagley and
Torvik [13], Koeller [14] and Rossikhin and Shitikova
[15].
In the second half of the 19th century, both of the the-
ory of fractional derivatives and integrals were estab-
lished. The first application of fractional derivatives was
applied by Abel to solve an integral equation that arises
in the formulation of the Tautochrone problem [16,17].
Various definitions and approaches of fractional de-
rivatives have become the main purpose of many studies
E. BASSIOUNY, R. SABRY 111
[18,19].
Kimmich [19] study time—fractional diffusion—wave
equation and use the Riemann-Liouville fractional inte-
gral as follows:
  

1
0
1d,0 2,
,
t
tf
Ift
ft


0
(1)
where

is the Gamma function.
Fujita [20,21] considered a fractional order heat wave
equation for the case 12
 obtained from the non
local constitutive equation for the heat flux components
in the form
i
q
1, 12
ii
qIT

 (2)
Povestenko [22,23] used the heat Equation (2) to study
the theories of thermal stresses based on space-time frac-
tional telegraph equations.
To eliminate the paradox of the instantaneous propa-
gation of heat, Cattaneo [24] introduced a law of heat
conduction to replace the classical Fourier law of heat
conduction. The propagation of discontinuities of solu-
tions in this theory was investigated by Ezzatt and Kara-
many [25].
Ezzatt and Karamany [25-28] established a new model
of fractional heat equation based on a Taylor expansion
of time-fractional order. They studied the non-homoge-
neous anisotropic elastic solid using two new models and
they considered the uniqueness theorem in linear frac-
tional two temperatures thermoelasticity and the theory
of a perfect conducting thermoelastic medium. Also, they
constructed a new model of electro-thermoelasticity in
the context of a new consideration of heat conduction
with fractional order.
Sherief et al. [16] used the following form of the heat
conduction law
, 01
i
io
qT
qt
t

 
1
(3)
and derived the governing equations of the fractional
order theory of thermoelasticity using Caputo [7] defini-
tion of fractional derivatives of order 0

of ab-
solutely continuous function

f
t given by:
 
1
d
d
f
tIft
t
(4)
 
 
1
0
d
tts
I
ftfs s
(5)
where I
is the fractional integral of the function
f
t
of order
defined by [19]. In the limit as 1
,
Equation (3) can be reduced to Cattaneo law [24].
A new formula of heat conduction has been considered
in the context of fractional integral operator defined by
Youssef [29] who introduces the following form of heat
conduction law
1, 02
ioi i
qq IT
 

(6)
Taking into consideration the works of Fujita [20,21]
and Povestenko [22,23], Youssef proved the uniqueness
of the solutions in this case.
In the present work a model for generalized ther-
mopiezoelasticity has been constructed in the context of
the fractional heat equation where 02

0
to describe
different types of diffusion where 1

corresponds
to weak diffusion, 1
corresponds to normal diffu-
sion, 12
corresponds to strong diffusion and
2
corresponds to ballistic diffusion. This is used to
investigate the propagation of thermal wave through a
semi infinite slab subjected to thermal loading of frac-
tional order of exponential type applied for finite period
of time.
2. Governing Equations
In the absence of body force, free charge and inner heat
sources, we consider generalized thermo-piezoelectric
governing differential equations Youssef [29] and Yous-
sef and Bassiouny [30] follows:
Equations of motion:
2
,2
i
ij j
u
t

(7)
Equation of entropy increment (in the absence of inner
heat source):
,,
ii o
qT
t
 (8)
Stress-strain-temperature:
,
ijijkl klkijkij
ce hD


(9)
Gauss equation and electric field relation:
,0
ii
D (10)
,
i
Ev
i
(11)
iikl klikki
Ehe Dd
 (12)
Equation of entropy density:
ij ijii
edDcT

(13)
Strain-displacement relations:
,,
1
2
ijijj i
euu
(14)
The heat conduction

2
1
2
iioE o
I
CT
tt
 


 

 e
(15)
Open Access JAMP
E. BASSIOUNY, R. SABRY
112
where
  

1
0
1d, 02,
, 0
t
tf
Ift
ft


(16)
The thermodynamical temperature
relates with the
conductive temperature
by the relation
,
ii
a

 (17)
in which is the two-temperature parameter.
0a
In the above equations, a comma followed by a suffix
denotes material derivatives and a superposed dot de-
notes the derivatives with respect to time.
3. One Dimension Formulation
Consider a semi-infinite piezoelectric rod occupying the
region . At the near end a uniform flow of heat is
supplied to the rod during a finite period of time. All the
state functions field will depend only on the dimension x
and the time t. We assume the following form for the
displacement component:
0x
,, 0
xy
uuxtuu
z
(18)
We consider the following forms of the linearized ba-
sic equations in one-dimensional formulation:

22
2
2u
x2
u
x
t
 



(19)

2uhD
x

 
(20)

22
1
22
oEo
I
CT
t
xt



 


 e
(21)
2
2
a
x


(22)

,uxt
e
x
(23)
0
D
x
(24)
,
v
E
x

(25)
where
32,
t
 
 t
is the coefficient of the
linear thermal expansion, is the coefficient of ther-
mal conductivity and
x
is the coordinate taken along
the rod.
It is convenient now to introduce the following dimen-
sionless variables:

2
22
2
, , , ,
2
, , , ,
2
, , ,
2
o
oo
o
o
o
o
oo o
o
E
o
oo
TT
ucutct T
T
tctxcx cT
C
qh
qDD c
kc Tk
 


 
 
 



 
(26)
From Gauss’s law, since there is no free charge inside
the piezoelectric rod we have
0,
D
x
(27)
which gives
constD
(28)
Substituting from Equation (26) into Equations (19)-
(25) and dropping the primes for convenience, we obtain
the following set of non-dimensional equations Youssef
[29] and Youssef and Bassiouny [30] follows:
22
22
e
2
2
e
x
xt



(29)
eD

 (30)
22
1
22
o
I
e
t
xt



 



(31)
and the following relation between the conductive tem-
perature and the thermodynamical one:
2
2
x

 (32)
where
2
22
2,,and
o
E
ac
C
x


 
 
(33)
The boundary conditions are:


1
0,
t
o
e
tt



(34)

 
,0,0, ,0,0
,0,0,0,0
tetett
ttt


 

2
(35)
where 0
, while the initial conditions are as-
sumed to be:

,0 0,,0 0,,0 0,0xex xx

 (36)
Applying the Laplace transform defined by:


 
0
d
st
Lftfse ft t

(37)
to both sides of Equations (29)-(32), we obtain:
Open Access JAMP
E. BASSIOUNY, R. SABRY 113
22
2
22
dd
dd
e
s
e
xx
 (38)
D
e
s

 (39)

2
2
12
1d ,
doo
2
s
sss
sx

 e (40)
2
2
d.
d
x

 (41)
Using Equations (29) and (30) with the definitions (23)
and (37) we can obtain
2d
d
us
x
(42)
where s denotes the complex argument related to the
Laplace transform.
The transformed boundary conditions take the forms



1
0,
t
o
e
s
Fs
t



(43)
where
 
1
1
o
Fsss



(44)
while the Equations (35) become
  
 
,0, 0,,0
,0, 0,0,
seses
ss

 
 
,
(45)
and the corresponding transformed initial conditions of
the Equations (36) assume the form:

,0,0,00, 0xex xx

 (46)
Eliminating
between Equations (40) and (41), we get:
2
DLLe

 (47)
where


22
2
2
12
d
and d
o
o
ss
LLs D
x
sss

 

Substituting from Equation (47) into Equation (41) we
obtain

1LLe

 (48)
Using Equation (47) we can easily eliminate
be-
tween Equation (38) and (48) to obtain
2
De MNe

(49)
where

 
2
1
1
1
and 1
LL
MMs L
s
LL
NNs L


 




(50)
Solving Equations (47) and (49) together we get the
following fourth order equation
42 0kakb



(51)
where



21
111
o
o
ss
ass


(52)



22
11
11 1
o
o
s
ss s
bss


 
  (53)
It is worth mentioning here that the roots of equation
(51) are functions of s and assume the forms:
22
12
44
,
22
aa baa b
kk
 
 (54)
Thus the solutions of the Equations (47) and (49) sat-
isfying the boundary conditions at infinity are:
1
12
2
x
k
AeA e


xk
(55)
1
12
2
x
k
eBe Be


xk
(56)
where 121
,,
A
AB and 2 are coefficients depending on
s to be determined using the boundary conditions (43)
and (45).
B


2
12
22
12
1o
Fs
ANks
Ns kk
s

(57)


2
21
22
12
1o
Fs
ANks
Ns kk
s

(58)
While the constants 12
are related to the con-
stants
,BB
12
,
A
A according to the following relations:
 


211,
1
i
iioo
o
A
Bksss
s
 




1,2
i
(59)
Substituting from Equations (57) and (58) into the
Equations (55) and (56) the heat conduction and the
strain field in the Laplace domain take the forms:








12
22
12
22
21
,
11
xk xk
oo
Fs
xs Nskk
ssNkessNke




 
(60)










1
2
2
11
2
22
1
,1
11
11
o
xk
oo
xk
oo
exs s
Ak ssse
Ak ssse



 

(61)
Using the expressions of
and of e from Equa-
tions (60) and (61) to find the thermodynamical function
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E. BASSIOUNY, R. SABRY
114
and the stress in the Laplace transformed domain,
thus Equations (39) and (48) become:

12
12
,xkxk D
xs ee
s



(62)
1
12
,2
x
kxk
x
se e


 (63)

1
12
,2
x
k
uxs ueue


xk
(64)
where



2111
1
i
ii o
o
Aks s
s
 



(65)
2
1
ii i
Ak
 (66)
2
i
usk
i
i
 (67)
Equations (60)-(64) are the complete solutions of the
,, ,e

and u, respectively, in the Laplace trans-
formed domain.
In order to invert the Laplace transform, we adopt a
numerical inversion method based on a Fourier series
expansion Honig [31]. Using this method, the inverse
f
t of the Laplace transform

f
s is approximated
by
 
1
11
1
1ππ
Reexp,
2
02,
ct N
k
eik
ftfcf c
tt
tt









1
ikt
t
(68)
where N is a sufficiently large integer representing the
number of terms in the truncated Fourier series, chosen
such that

1
11
ππ
exp Reexp
iNiN t
ctf ctt








(69)
where 1
is a prescribed small positive number that
corresponds to the degree of accuracy required and Re is
the real part. The parameter c is a positive free parameter
that must be greater than the real part of all the singulari-
ties of

f
s. The optimal choice of c was obtained ac-
cording to the criteria described in [31].
4. Numerical Results and Discussion
To investigate the role of various physical parameters
involved in the current problem, we have investigated the
role of varying the angular frequency of thermal vibra-
tion on different system parameters, where it is ob-
served that increasing increases the heat conduction,
as depicted in Figure 1(a). Such behavior is in accor-
dance with the fact that increasing the thermal vibrations
will increase the kinetic energy of the ceramic slab
molecules and results in increasing the amount of heat
transferred by conduction mechanism. Variation of the
Figure 1. Variation of (a) Heat conduc tion
(b) Thermo-
dynamical temperature and (c) Displacement against x for
various values of angular frequency of thermal vibration
at 0
=0.3 =0.4=0.30.75t


.
thermodynamical temperature for different , shows a
peculiar behavior, as illustrated in Figure 1(b) and re-
flected through the variation of the displacement distri-
bution shown in Figure 1(c). It is found that increasing
the thermal vibrations will increase the amplitude of the
thermodynamic temperature. The effect of increasing the
thermal vibrations frequency on the stress and strain
shows the same qualitative behavior, as illustrated in
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E. BASSIOUNY, R. SABRY
Open Access JAMP
115
Figure 2. Both of the stress and strain decreases initially
by increasing x, but at a certain critical point further in-
crease in x increases the stress and strain. It is observed
in Figure 2 that the thermal vibration increases the stress
and strain in a symmetrical way with respect to a critical
point.
The effect of increasing time is shown to increase the
heat conduction as well as the thermodynamic tempera-
ture, as reflected in Figures 3(a) and (b), respectively. In
fact it shows the same qualitative behavior of increasing
Figure 2. The role of varying angular frequency
on the (a) Stress and (b) Strain at 0
=0.3 =0.4=0.30.75t

.
Figure 3. Variation of (a) Heat conduction
(b) Thermodynamical temperature and (c) Displacement against x for differ-
nt value of time at 0
=0.2 =0.30.=0.175

. e
E. BASSIOUNY, R. SABRY
116
the frequency of the thermal vibrations on the heat con-
duction, illustrated in Figure 1(a). Such behavior can be
strain will be an increasing function in x, as illustrated in
Figure 4. The amount of energy delivered to the ceramic
explained on the basis that increasing the time of heating
the slab will increase the amount of energy delivered to
the slab. The amount of energy delivered to the ceramic
slab increases the entropy of the thermodynamic tem-
perature. The minimum points of the stress and strain
curves are shown to be an increasing function in x as
time increases. Far from the near end of the slab the ef-
fect of time damped as x increases and the stress and
slab is a factor of heating time, which is the key answer
to such behavior. An inverse proportion is noticed be-
tween the value of the fractional order (i.e., the measure
of the system memory) and the heat conduction at the
near end of the slab, whereas a slight change in the dis-
tribution curves is noticed for large values of x, as de-
picted in Figure 5(a). The thermodynamic temperature is
a decreasing function in the fractional order as shown in
Figure 4. The role of varying time on the (a) Stress and (b) Strain at
0
=0.2 =0.30.75 =0.1
 
.
Figure 5. Variation of (a) Heat c onduction
, (b) Thermodynamical temperature and (c) Displacement against x for differ-
ent fractional order parameter at 0
0.40.2=0.3= 0.1t =
.
Open Access JAMP
E. BASSIOUNY, R. SABRY 117
Figure 5(b). It is apparent that increasing the system
ractional order leads to reverse t
displacement distribution. The amount of energy deliv-
er
he effect of heating as in ed to the ceramic slab is affected by the weak conduc-
f
Figure 5. The qualitative behavior of the displacement
distribution is shown to resemble the same behavior as in
the case of the effects of thermal angular vibration and
time but here; the fractional has a slight effect on the
tivity imposed on the material by the system memory
retained through the fractional parameter. Such behavior
is confirmed through the role of the fractional parameter
on the stress and strain curves as displayed in Figure 6.
0
0.4= 0.2= 0.3= 0.1t

.Figure 6. The role of varying fractional order parameter on the (a) Stress and (b) Strain at
Figure 7. Variation of (a) Heat c onduction
; (b) Thermodynamic temperature
; (c) Displacement U against x for differ-
ent value of the two-temperature parameter at 0
0.25= 0.75= 0.3= 0.1t

.
Open Access JAMP
E. BASSIOUNY, R. SABRY
118
Figure 8. The role of varying the two-temperature parameter
on the (a) Stress and (b) Strain at 0.25t
0
= 0.75= 0.3= 0.1
.
pThe two temperaturearameter ω, which depends on
o distinct temperatures, the conductive temperature
an
tw
d the thermodynamic temperature where the difference
between these two temperatures is proportional to the
heat supply, is found to increase the heat conduction as
shown in Figure 7(a). The thermodynamic temperature
increases by increasing ω as illustrated in Figure 7(b).
The displacement is found to have a critical behavior at
0.4x, as depicted in Figure 7(c) where the displace-
ment distribution curves for all values of ω pass by this
point and the distribution becomes positive after
it. The strain distribution confirms similar qualitative be-
havior as the stress distribution as illustrated in Figure 8.
Both distributions of the stress and strain are decreased
by increasing ω for 0.4
critical
, as shown in Figure 8.
However, for 0.4
the stress and strain distributions
remain decreasing funcn the two temperature pa-
rameter ω.
tions i
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Open Access JAMP
E. BASSIOUNY, R. SABRY
120
Nomenclatures
: The components of relaxation time.
ture parameter.
: Coefficient of linear thermal expansion.
ij
A
a: The two-tempera
E
C: Specific heat at constant strain.
ijkl : The elastic constants.
c
2
o
c
: Longitudinal wave sp
eed.
: The components of electric displacement.
: The pyroe
nsor. conductivity.
vector.
ment vector.
i
D
i
dlectric constants.
i
E: The components of electric field vector.
ijk
el: The components of strain te
ijk
h: The piezoelectric coefficients.
ij
k: The components of thermal.
i
q: The components of the heat flux
T: Absolute temperature.
o
T: Reference temperature.
Time.
t:
i
u: Components of displace
ectric potential function.
i: The elv
2
o
T
: Dimensionless thermoelasti
T
: The components of dielectric tensor.
ik
ij
: The thermal modulus.
32 T

 .
c coupling con-
stant.
: The angular frequency of thermal vibration.
: Kronecker delta function.
ij
E
C
nstant
: Dimensionless mechanical coupling co
: The entropy density.
E
C
k
: The thermal viscosity.
E
o
C
CT
o
TT

that
: The dynamical temperature increment such
1
o
o
TT
T
.
,

: Lamé’s constants.
: Mass density.
ij
: Components of stress tensor.
x
x
: The principal stress component.
o
: One relaxation time parameter.
Open Access JAMP