State Space Approach to Thermoelastic Problem with Three-Phase-Lag Model
A two-dimensional problem of generalized thermoelasticity is formulated using state space approach. In this formulation, the governing equations are transformed into a matrix differential equation whose solution enables to write the solution of two-dimensional problem in terms of the boundary condit...
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irk-123456789-1882342023-02-18T01:26:32Z State Space Approach to Thermoelastic Problem with Three-Phase-Lag Model Biswas, S. A two-dimensional problem of generalized thermoelasticity is formulated using state space approach. In this formulation, the governing equations are transformed into a matrix differential equation whose solution enables to write the solution of two-dimensional problem in terms of the boundary conditions. The resulting formulation is applied to an isotropic half-space problem within three-phase-lag model of thermoelasticity. The bounding surface is traction free and subjected to a time dependent thermal shock. The solution for temperature distribution, displacements and stress components are obtained and presented graphically as well as a comparison with other thermoelastic models is made. Двовимірна задача узагальненої термопружності сформульована з використанням підходу простору станів. У цій постановці основні рівняння перетворюються на матричне диференціальне рівняння, розв'язування якого дає змогу записати розв'язок двовимірної задачі через граничні умови. Остаточне формулювання застосовується до задачі про ізотропний напівпростір в рамках моделі запізнювання трьох фаз термопружності. Гранична поверхня є вільною від розтягу і піддається залежному від часу тепловому удару. Отримані та представлені графічно розв'язки для розподілу температури, переміщень та напружень. Також проведено порівняння з іншими термопружними моделями. 2020 Article State Space Approach to Thermoelastic Problem with Three-Phase-Lag Model / S. Biswas // Прикладная механика. — 2020. — Т. 56, № 2. — С. 130-144. — Бібліогр.: 27 назв. — англ. 0032-8243 http://dspace.nbuv.gov.ua/handle/123456789/188234 en Прикладная механика Інститут механіки ім. С.П. Тимошенка НАН України |
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A two-dimensional problem of generalized thermoelasticity is formulated using state space approach. In this formulation, the governing equations are transformed into a matrix differential equation whose solution enables to write the solution of two-dimensional problem in terms of the boundary conditions. The resulting formulation is applied to an isotropic half-space problem within three-phase-lag model of thermoelasticity. The bounding surface is traction free and subjected to a time dependent thermal shock. The solution for temperature distribution, displacements and stress components are obtained and presented graphically as well as a comparison with other thermoelastic models is made. |
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Biswas, S. |
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Biswas, S. State Space Approach to Thermoelastic Problem with Three-Phase-Lag Model Прикладная механика |
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Biswas, S. |
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Biswas, S. |
| title |
State Space Approach to Thermoelastic Problem with Three-Phase-Lag Model |
| title_short |
State Space Approach to Thermoelastic Problem with Three-Phase-Lag Model |
| title_full |
State Space Approach to Thermoelastic Problem with Three-Phase-Lag Model |
| title_fullStr |
State Space Approach to Thermoelastic Problem with Three-Phase-Lag Model |
| title_full_unstemmed |
State Space Approach to Thermoelastic Problem with Three-Phase-Lag Model |
| title_sort |
state space approach to thermoelastic problem with three-phase-lag model |
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Інститут механіки ім. С.П. Тимошенка НАН України |
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2020 |
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http://dspace.nbuv.gov.ua/handle/123456789/188234 |
| citation_txt |
State Space Approach to Thermoelastic Problem with Three-Phase-Lag Model / S. Biswas // Прикладная механика. — 2020. — Т. 56, № 2. — С. 130-144. — Бібліогр.: 27 назв. — англ. |
| series |
Прикладная механика |
| work_keys_str_mv |
AT biswass statespaceapproachtothermoelasticproblemwiththreephaselagmodel |
| first_indexed |
2025-07-16T10:10:32Z |
| last_indexed |
2025-07-16T10:10:32Z |
| _version_ |
1837797878108520448 |
| fulltext |
2020 П Р И К Л А Д Н А Я М Е Х А Н И К А Том 56, № 2
130 ISSN0032–8243. Прикл. механика, 2020, 56, № 2
S . B i s w a s
STATE SPACE APPROACH TO THERMOELASTIC PROBLEM
WITH THREE-PHASE-LAG MODEL
Department of Mathematics, University of North Bengal,
Darjeeling, India; email: siddharthabsws957@gmail.com
Abstract. A two-dimensional problem of generalized thermoelasticity is formulated us-
ing state space approach. In this formulation, the governing equations are transformed into a
matrix differential equation whose solution enables to write the solution of two-dimensional
problem in terms of the boundary conditions. The resulting formulation is applied to an
isotropic half-space problem within three-phase-lag model of thermoelasticity. The bound-
ing surface is traction free and subjected to a time dependent thermal shock. The solution for
temperature distribution, displacements and stress components are obtained and presented
graphically as well as a comparison with other thermoelastic models is made.
Keywords: state space approach, normal mode analysis, thermal shock, three-phase-lag
model.
1. Introduction.
The classical theory of thermoelasticity [1] suffers from so called ‘paradox of heat con-
duction’ i.e. the heat equations for both theories of a mixed parabolic-hyperbolic type, pre-
dicting infinite speeds of propagation for heat waves contrary to physical observations. To
remove this paradox, the conventional theories of thermoelasticity has been generalized,
where the generalization is in the sense that these theories involve a hyperbolic type heat
transport equation supported by experiments, which exhibit the actual occurrence of wave
type heat transport in solids, called second sound effect. To eliminate the second sound par-
adox of classical thermoelasticity theory, Lord and Shulman [2] established a generalized
thermoelasticity theory which is often referred to as LS model and widely used in the case
of heat flux and low temperature. Green and Lindsay [3] introduced one more theory, called
GL theory, which involves two relaxation times. Later Green and Naghdi [4, 5, 6] developed
three models for generalized thermoelasticity of homogeneous isotropic materials, which are
labeled as G-N models I, II, III. Detailed information regarding these theories can be found
in [7, 8].
The next generalization to the thermoelasticity is known as the dual-phase-lag model
(DPL) developed by Tzou [9]. Tzou [9] considered micro-structural effects into the delayed
response in time in macroscopic formulation by taking into account that increase of the lat-
tice temperature is delayed due to phonon-electron interactions on the macroscopic level.
Tzou [9] introduced two-phase lags to both the heat flux vector and the temperature gradient
and considered as constitutive equation to describe the lagging behavior in the heat conduc-
tion in solids. Recently, Roychoudhuri [10] has established a generalized mathematical
model of a coupled thermoelasticity theory that includes three-phase-lags in the heat flux
vector, the temperature gradient and in the thermal displacement gradient. The more general
model (TPL) established reduces to the previous models as special cases.
In three-phase-lag heat conduction equation the Fourier law of heat conduction is re-
placed by an approximation of three phase lags for the heat flux vector ( )q , the temperature
131
gradient ( )T and the thermal displacement gradient ( ) . The previous established models
can be obtained as special cases from this more general model. Three-phase-lag (TPL)
model is very much useful in the problems of nuclear boiling, exothermic catalytic reac-
tions, phonon-electron interactions, phonon scattering etc., where the delay time q captures
the thermal wave behavior (a small scale response in time), the phase lag T captures the
effect of phonon-electron interactions (a microscopic response in space), the other delay
time is effective since in the three-phase-lag model , the thermal displacement gradient is
considered as a constitutive variable.
State space methods are the cornerstone of modern control theory. The necessity of state
space method is the characterization of the processes of interest by differential equations
instead of transfer functions. In the earlier period the processes were simple enough to be
characterized by a single differential equation of fairly low order. In the modern approach
the processes are characterized by systems of coupled, first order differential equations. In
principle, there is no limit to the order (i.e. the number of independent first order differential
equations) and in practice the only limit to the order is the availability of computer software
capable of performing the required calculations reliably.
A method for solving coupled thermoelastic problems by state space approach has been
developed by Bahar and Hetnarski[11] . The state space formulation for the problems that
do not contain heat sources have been done by Anwar and Sherief [12]. Ezzat et al. [13]
considered state space approach to two-dimensional generalized thermoviscoelasticity with
one relaxation time. Youssef and Al-Lehaibi [14] discussed two temperature generalized
thermoelasticity using state space approach. El-Karamany and Ezzat [15] considered ther-
mal shock problem in generalized thermoelasticity under four theories. Sherief et al. [16]
discussed stochastic thermal shock in generalized thermoelasticity. Ezzat and Youssef [17]
investigated three dimensional thermal shock problem of generalized thermoelastic medium.
Wang et al. [18] considered thermoelastic behavior of elastic media with temperature-
dependent properties under thermal shock. Baksi et al. [19] studied magneto-thermoelastic
problem with thermal relaxation and heat sources in a three dimensional infinite rotating
elastic medium. Biswas and Mukhopadhyay[20] proposed eigenfunction expansion method
to analyze thermal shock behavior in magneto-thermoelastic orthotropic medium with three-
phase-lag model. Biswas and S. M. Abo-Dahab [21] considered the effect of phase-lags on
Rayleigh wave propagation in initially stressed magneto-thermoelastic orthotropic medium.
Biswas [22] proposed modeling of memory-dependent derivatives in orthotropic medium
with three phase lag model under the effect of magnetic field. Said [23] investigated the
influence of gravity on generalized magneto-thermoelastic medium for three-phase -lag
model. Kalkal and Deswal [24] examined the effects of phase lags on three dimensional
wave propagation with temperature dependent properties.
El-Karamany and Ezzat [25] discussed linear micropolar thermoelasticity theory with
three-phase-lag model. Sherief and Al-Sayed [26] proposed state space approach to two-
dimensional generalized micropolar thermoelasticity. Ezzat et al. [27] employed state space
approach to study two-dimensional electro-magnetic thermoelastic problem with two relaxa-
tion times.
In this problem we have considered two dimensional generalized thermoelasticity with
three-phase-lag model of thermoelasticity. Normal mode analysis is employed to the gov-
erning equations and then the problem is solved using state space approach. Stress, dis-
placements and temperature with respect to time and distance for different thermoelastic
models are presented graphically.
2. Formulation of the problem.
We consider a two-dimensional problem and assume that the thermoelastic medium is
governed by the equations of generalized thermoelasticity whose state depends on the space
variables ,x y and the time variable .t
The displacement vector has components [ ( , , ), 0, , , ].u x z t w x z t
The displacement equations are given by
132
, , ,( ) j ij i jj i iu u T u . (1)
The constitutive equation is
2ij kk ij ij ije e T (2)
and the strain-displacement relations are as follows:
, ,
1
( )
2ij i j j ie u u . (3)
In the above equations, a superposed dot denotes differentiation with respect to time,
while a comma denotes spatial derivatives.
The heat conduction equation of three-phase-lag model (Roychoudhuri [10]) is given by
2 2
2 * 2
02
1 1 1
2
q
T q eK T K T C T T e
t t t t
, (4)
where ,q T and are the phase lags for heat flux, temperature gradient and thermal dis-
placement gradient respectively.
We use the following non-dimensional variables:
2
1 0 1 0 2
1
, , , , , , ; , , , , , , ; ; ,ij
q T q T ij
T
x z u w c x z u w t c t T
c
where the dashed quantities denote non-dimensional variables.
In terms of these non-dimensional variables, the equations of motion has the form
(dropping primes).
2 2 2 2
, , , ,( 1)xx zz xz xu u w T u ; (5)
2 2 2 2
, , , ,( 1) xz zz xx zu w w T w (6)
and the components of the stress are:
2 2 2
1 , 1 ,( 2)xx x zc u c w T ; (7a)
, ,xz z xu w ; (7b)
2 2 2
1 , 1 ,( 2) .zz x zc u c w T (7c)
The equation (4) in non-dimensional form is obtained as
2 2
, , , , 2
1 1 1 ( )
2
q
T xx zz xx zz qT T K T T T e
t t t t
, (8)
where
*
2
1 0
.
K
K
Kc
These equations will be supplemented with appropriate boundary conditions.
3. Normal mode analysis.
Generally, Laplace transformation and Fourier transformation are employed to solve a
two- dimensional generalized thermoelastic problem. In the application of this method, the
partial differential equations can be converted into ordinary differential equations. By solv-
ing differential equations in the transformed domain and adopting inverse Fourier transfor-
mation and inverse Laplace transformation in the time domain, the solutions of the problem
can be attained. But this method entails a tiresome process. The key problem is that it intro-
133
duces a discrete error and truncation error in the process of numerical inverse integrated
transformation, so the second sound of heat conduction cannot be fully demonstrated. To
compensate for the defects of the above mentioned method, we solve the problem of gener-
alized thermoelasticity by employing normal mode analysis to the considered equations.
Now we seek the solution of equations (5), (6) and (8) in the following form:
, , , , , , , , expu w e T x z t u w e T z ik x ct , (9)
where k is wave number and c is the phase velocity.
Applying normal mode analysis to both sides of the equations (5), (6) and (8), we obtain
2 2 2 2 2 2 2 2( 1) ;k u D u ik Dw ik T k c u (10)
2 2 2 2 2 2 2 2( 1) ;ik Du D w k w DT k c w (11)
2 2 ( )D k T P T e , (12)
where ;
d
D
dz
2 2
1 2
k c
P
ikc K
with 1 2 2
2
1
;
1
2
T
q q
ikc
k c
ikc
2 2 2
2
1
.
1
2q q
ikc
k c
ikc
4. State-space formulation.
We take the quantities , , ,e T De DT as state variables. Now
.e iku Dw (13)
Eliminating u and w between equations (10), (11) and (12) with the help of equation
(13), we obtain the following equations:
2 2 2 2( ) ;D e k c k P e PT (14)
2 2( ) .D T P e k P T (15)
Equations (14) and (15) can be written in matrix differential equation form as follows:
,
dV z
AV z
dz
(16)
where
2 2 2
2
0 0 1 0
0 0 0 1
;
0 0
0 0
A
k c k P P
P k P
.
e
T
V
De
DT
The formal solution of system (16) can be written in the form
0( ) exp( ) ,V z Az V z (17)
where 0z denotes any arbitrarily chosen initial value for .z
The characteristic equation for the matrix A is
4 2 2 2 2 4 2 2 2 2 2( 2 ) ( ) 0.k c k P P k k k c P P Pk c (18)
The roots of the equation (18) satisfy the relations
134
2 2 2 2 2
1 2 2 ;k k k c k P P (19a)
2 2 4 2 2 2 2 2
1 2 ( ) .k k k k k c P P Pk c (19b)
The Mclaurin series expansion of exp Az is given by
0
.
exp .
!
n
n
A z
Az
n
Using Cayley – Hamilton theorem, the infinite series representing exp Az can be truncat-
ed to the following form:
2 3
0 1 2 3exp( ) ,Az L b I b A b A b A (20)
where I is the unit matrix of order 4 and 0 3, ...,b b are some parameters depending on ,z k
and .t
We shall stress here that the above expressions for the matrix exponential is a formal
one. In the actual physical problem, the space is divided into two regions accordingly as
0z or 0.z
By Cayley – Hamilton theorem, the characteristic roots 1k and 2k of the matrix A
must satisfy the equations
2 3
1 0 1 1 2 1 3 1exp( )k z b b k b k b k ; 2 3
1 0 1 1 2 1 3 1exp( )k z b b k b k b k ;
2 3
2 0 1 2 2 2 3 2exp( )k z b b k b k b k ; 2 3
2 0 1 2 2 2 3 2exp( )k z b b k b k b k .
The solution of the above system is given by
2 2
0 1 2 2 12 2
1 2
1
[ cosh( ) cosh( )];b k k z k k z
k k
2 2
1 2
1 2 12 2
2 11 2
1
sinh( ) sinh( )
k k
b k z k z
k kk k
; (21)
2 1 22 2
1 2
1
[cosh( ) cosh( )];b k z k z
k k
3 1 22 2
1 21 2
1 1 1
sinh( ) sinh( ) .b k z k z
k kk k
Substituting the expressions (21) into (20) and computing 2A and 3A , we obtain after
repeated use of equations (19a) and (19b), the elements ( , 1,2,3,4)ijl i j of the matrix L as
2 2 2 2
11 1 1 2 22 2
1 2
1
( )cosh( ) ( ) cosh( ) ;l k k P k z k k P k z
k k
12 1 22 2
1 2
cosh( ) cosh( )
P
l k z k z
k k
;
2 2 2 2
1 2
13 1 22 2
1 21 2
( ) ( )1
sinh( ) sinh( )
k k P k k P
l k z k z
k kk k
;
135
14 1 22 2
1 21 2
1 1
sinh( ) sinh( ) ;
P
l k z k z
k kk k
21 1 22 2
1 2
cosh( ) cosh( ) ;
P
l k z k z
k k
2 2 2 2
22 1 2 2 12 2
1 2
1
( ) cosh( ) ( ) cosh( ) ;l k k P k z k k P k z
k k
23 1 22 2
1 21 2
1 1
sinh( ) sinh( ) ;
P
l k z k z
k kk k
2 2 2 2
1 2
24 2 12 2
2 11 2
( ) ( )1
sinh( ) sinh( )
k k P k k P
l k z k z
k kk k
;
2 2 2 2
31 1 1 1 2 2 22 2
1 2
1
( ) sinh( ) ( ) sinh( )l k k k P k z k k k P k z
k k
;
32 1 1 2 22 2
1 2
sinh( ) sinh( ) ;
P
l k k z k k z
k k
2 2 2 2
33 1 1 2 22 2
1 2
1
( ) cosh( ) ( ) cosh( )l k k P k z k k P k z
k k
;
34 1 22 2
1 2
cosh( ) cosh( )
P
l k z k z
k k
;
41 1 1 2 22 2
1 2
sinh( ) sinh( )
P
l k k z k k z
k k
;
2 2 2 2
42 2 1 2 1 2 12 2
1 2
1
( ) sinh( ) ( ) sinh( )l k k k P k z k k k P k z
k k
; (22)
43 1 22 2
1 2
[cosh( ) cosh( )]
P
l k z k z
k k
;
2 2 2 2
44 1 1 2 22 2
1 2
1
[( ) cosh( ) ( ) cosh( )]l k k P k z k k P k z
k k
.
It should be noted that we have repeatedly used equations (19a) and (19b) in order to
write (29) in the simplest possible form.
Using equation (24), upon equating Matrices we obtain
11 0 12 0 13 0 14 0e z l e l l e l ; (23)
21 0 22 0 23 0 24 0 ,T z l e l l e l (24)
where 0 0 ,e e z 0 0 ,T z 0 0 0 0,e De z DT z
Using equation (22) into equation (23) and (24) we obtain
2
1
cosh sinhi
i i i
i i
M
T M k z k z
k
; (25)
136
2
1
cosh sinhi
i i i
i i
N
e N k z k z
k
, (26)
where
1
2 2
0 02 2
1 2
1
;
i
i iM Pe k k P
k k
1
2 2
0 02 2
1 2
1
;
i
i iM Pe k k P
k k
1
2 2
0 02 2
1 2
1
;
i
i iN k k P e P
k k
1
2 2
0 02 2
1 2
1
.
i
i iN k k P e P
k k
(27)
Substituting from equation (13) in equation (10), we obtain
2 2
2
2 2 2 2
3
1
1
1 cosh sinh ,
i i
i i i i
i i
N M
D k u ik N M k z k z
k
where 2 2 2 2 2
3k k c k .
Now solving the above equation, we get
2 2 2 22
3 2 2 2 2 2 2
11 2 3 3
1 1
cos cosh sinh
i i i i
i i
i i i i
N M N Mik
u C k z k z k z
k k k k k k k
. (28)
Substituting (28) into (13) and integrating the resulting equation, we get
2 2 2
2
3 2 2 2 2
13 1 2 3
2 2 2
2 2 2 2
3
11
sinh sinh
1
cosh .
i ii
i
i i i i
i ii
i
i i i
k N MNikC
w k z k z
k kk k k k k
k N MN
k z
k k k k
(29)
5. Boundary Conditions.
We consider the case where the surface of the half space is subjected to a time depend-
ent thermal shock and the surface is traction free.
(a) Thermal boundary condition that the surface of the half-space is subjected to a time
dependent thermal shock
( , 0, ) ;T x t F t H a x (30)
(b) Mechanical boundary condition that the surface to the half-space is traction free
( , 0, ) 0; ( , 0, ) 0zz xzx t x t , (31)
where H denotes Heaviside function and a is constant.
6. Application.
We shall apply our results to solve a problem for a half-space 0z . Inside the region
0 z , the positive exponential terms, not bounded at infinity, must be suppressed.
Thus, for 0z we should replace each sinh kz by 0,5exp kz and each cosh kz by
0,5exp .kz
The solution of the problem is given by equation (24) with 0z chosen as zero for con-
venience. Thus, the two components of the initial vectors 0 0V V are known, i.e.,
137
0 0 0e .
Now replacing each sinh kz by 0,5exp kz and each cosh kz by 0,5exp kz , we
obtain
2
1
1
exp ;
2 i i
i
T M k z
2
1
1
exp
2 i i
i
e N k z
;
2 22
3 2 2
1 3
1
exp exp
2 2
i i
i
i i
N MC ik
u k z k z
k k
; (32)
2 2 2
2
3 2 2
13 3
11
exp exp
2 2
i ii
i
i i i i
k N MNikC
w k z k z
k k k k k
.
Using (32) in equations (7a – 7c), the stress components are obtained as
2 22 2 2
1
32 2
1 3
2 2 22 2
1
32 2
1 3
2 2
1
1
exp exp
2
12
exp
2
exp exp ;
2
i i
xx i
i i
i i
i
i i
i i
i
N Mc k
k z ikC k z
k k
k N Mc
N k z
k k
M k z ik x ct
(33)
2 2 2 22
1
2 2
1 3
2 2 22 2
1
2 2
1 3
2 2
3
1
2 1
exp
2
1
exp
2
exp exp exp ;
2
i i
zz i
i i
i i
i i
i i
i i
i
k c N M
k z
k k
k N Mc
N k z
k k
ikC k z M k z ik x ct
(34)
2 2
2
2 2
1 3
2 2 2
2
2 2
1 3
2 2
3 3
3
1
exp
2
1
exp
2
exp exp .
2
i i i
xz i
i i
i ii
i
i i i i
N M kik
k z
k k
k N MNik
k z
k k k k
C
k k k z ik x ct
k
(35)
Now applying boundary conditions (37) and (38), we obtain
138
2
1
1
exp
2 i
i
M F t H a x ik x ct
; (36)
2 2 2 22
1
2 2
1 3
2 2 22 22 2
1
2 2
1 13
2 1
2
1
0;
2 2
i i
i i
i i
i i
i ii
k c N M
k k
k N Mc
N ikC M
k k
(37)
2 2 2 2 2
2 2
2 2
32 2 2 2
1 1 33 3
1 1
0.
2 2 2
i i i i ii
i i ii i i
N M k k N MNik ik C
k k
k kk k k k k
(38)
From (27), we see that iM and iN are expressed in terms of 0e and 0 . By solving the
equations (36), (37) and (38), 0 ,e 0 and C can be obtained.
This completes the solution of the problem.
7. Special cases.
Now we discuss some special cases as follows:
(a) If we take * 0K then equation (4) becomes the heat conduction equation for dual-
phase lag (DPL) model.
(b) If we take 0q T then equation (4) will reduce to Green-Naghdi-III (GN-
III) model.
(c) If we take * 2 0T qK and 0q then equation (4) will reduce to Lord-
Shulman (LS) model.
(d) If we take * 0q T K then equation (4) will reduce to classical thermoelastici-
ty (CT).
8. Numerical discussion.
In order to illustrate the above results graphically the time dependent thermal shock
F t is taken in the following form:
0 exp( ).F t bt
The copper material is chosen for purposes of numerical evaluations.
10 27.76 10 N/m ; 10 23.86 10 N/m ; 5 11.78 10 K ;t
3 28954Kg/m ; 383.1m /KeC , *
0386W/mK; 124 W/mKs; 293K;K K T
72 10 s;q
71 10 s;T
81 10 s.
Further for numerical purpose we take 0 10, 0.1, 1.2.b k
The numerical applications will be carried out for the temperature ,T displacements
,u w and stress xx at 0x . Fig. 1 – 4 represent the graphs for , ,zz u w and T versus t for
fixed values of z for stress free boundary. Keeping t fixed, the graphs for , ,zz u w and T
versus z are presented in fig. 5 – 9.
In fig. 1, a significant difference is noticed for the stress field predicted by different
models. It is observed that the value of stress for LS model is greater than TPL model and
less than CT model.
139
Fig. 1. Comparison of stress xx with respect to time.
In fig. 2 it is found that the value of u for three phase lag model is greater than the val-
ue of u for GN-III model but less than the value of u for LS model. It finally converges to
zero with the increase of time.
Fig. 2. Comparison of u with respect to time.
In fig. 3 it is found that the value of w for three phase lag model is greater than the val-
ue of u for GN-III model but less than the value of w for LS model. It finally converges to
zero with the increase of time.
Fig. 3. Comparison of w with respect to time.
140
It is shown in fig. 4 that temperature for three phase lag model is greater than GN-III
model and less than LS model. The region of influence is much larger in case of LS model
as compared to other models. Temperature is decreasing with the increasing of time and
converging towards zero.
Fig. 4. Comparison of temperature with respect to time.
In fig. 5, a significant difference is observed for the stress field predicted by different
models. The value of stress for three-phase-lag model with and without magnetic field is
greater than GN-III model and lower than LS model. Initially stress has the maximum value
and then with the increase of distance it decreases.
Fig. 5. Comparison of stress xx with respect to distance.
In fig. 6, it is observed that the value of u for three phase lag model is greater than the
value of u for GN-III model but less than the value of u for LS model. The figure indicates
that u has maximum value initially and it decreases with the increase of distance.
In fig. 7, it is observed that the value of w for three phase lag model is greater than the
value of w for GN-III model but less than the value of w for LS model. The figure indicates
that w has maximum value initially and it decreases with the increase of distance.
It is shown in fig. 8 that temperature for three phase lag model is greater than GN-III
model and less than LS model. The region of influence is much larger in case of LS model
as compared to other models. Temperature is decreasing with the increasing of distance and
converging towards zero.
141
Fig. 6. Comparison of u with respect to distance.
Fig. 7. Comparison of w with respect to distance.
Fig. 8. Comparison of temperature with respect to distance.
142
Fig. 9. Comparison of temperature with respect to distance.
It is shown in fig. 9 that temperature for three phase lag model is greater than CT model.
Temperature is decreasing with the increase of distance and converging towards zero. Tempera-
ture for TPL model is showing oscillatory behavior like wave with the increase of distance.
9. Conclusion.
The present article provides a detail analysis of propagation of thermoelastic disturb-
ances in presence of a time dependent thermal shock on traction free half-space. With the
view of theoretical analysis and numerical computation, we can conclude the following phe-
nomena:
(a) The importance of state space analysis is recognized in fields where the time behav-
ior of any physical process is of interest. The state space approach is more general than the
classical Laplace and Fourier transform theory. State space theory is applicable to all sys-
tems that can be analyzed by integral transforms in time and is applicable to many systems
for which transform theory breaks down. Furthermore, state space theory gives a somewhat
different insight into the time behavior of linear systems.
(b) The problem with three-phase-lag model is a more general one as the other thermoe-
lastic models can be obtained as special cases from it.
(c) The thermoelastic disturbances converge towards zero with the distances from the
application of the thermal shock.
(d) The influence of various thermal relaxation times is also observed from this investi-
gation. The Lord-Shulman model shows maximum thermoelastic deformation and GN-III
model of thermoelasticity experiences least amount of disturbances.
The results presented in this article may be useful for researchers who are working on
material science, mathematical physics and thermodynamics with low temperatures as well
as on the development of the hyperbolic thermoelasticity theory.
Disclosure statement: There is no conflict of interests.
РЕЗЮМЕ. Двовимірна задача узагальненої термопружності сформульована з використанням
підходу простору станів. У цій постановці основні рівняння перетворюються на матричне диференці-
альне рівняння, розв’язування якого дає змогу записати розв’язок двовимірної задачі через граничні
умови. Остаточне формулювання застосовується до задачі про ізотропний напівпростір в рамках
моделі запізнювання трьох фаз термопружності. Гранична поверхня є вільною від розтягу і піддаєть-
ся залежному від часу тепловому удару. Отримані та представлені графічно розв’язки для розподілу
температури, переміщень та напружень. Також проведено порівняння з іншими термопружними
моделями.
143
Nomenclature:
, Lame constants;
density
eC specific heat at constant strain
t time
T temperature above reference temperature
0T reference temperature chosen so that 0/ 1T T
ij components of stress tensor
ije components of strain tensor
,u w components of displacement vector
K thermal conductivity
*K material constant characteristic of the theory
2
1c
2
2
2c
2
2
1
2
2
c
c
e , ,x zu w
0 eC
K
t coefficient of linear thermal expansion
(3 2 ) t
2
0 .
( 2 )e
T
C
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From the Editorial Board: The article corresponds completely to submitted manuscript.
Поступила 16.07.2018 Утверждена в печать 05.11.2019
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