Response of Non-local and Heat Source in Moore-Gibson-Thompson
Theory of Thermoelasticity with Hyperbolic Two Temperature
RAJNEESH KUMAR1, SACHIN KAUSHAL2, GULSHAN SHARMA2,3
1Department of Mathematics, Kurukshetra University,
Kurukshetra-136119 Haryana,
INDIA
2Department of Mathematics, School of Chemical Engineering and Physical Sciences,
Lovely Professional University,
144411 Phagwara,
INDIA
3PostGraduate Department of Mathematics,
Doaba College Jalandhar,
INDIA
Abstract: - A new mathematical model of the Moore–Gibson–Thompson (MGT) theory of thermoelasticity under
non-local and hyperbolic two-temperature (HTT) has been developed. The preliminary equations are put in two-
dimensional form and are converted into dimensionless form. The obtained equations are simplified by applying
potential functions. The Laplace transform w.r.t time variable and Fourier transforms w.r.t space variable are
employed in the resulting equations. The assumed model has been used to explore the outcome of heat source in
the form of a laser pulse decaying with time and moving with constant velocity in one direction. The problem is
further examined with normal distributed force and ramp type thermal source. In the transformed domain, the
physical field quantities like displacements, stresses, conductive temperature, and thermodynamic temperature are
obtained. The resulting expressions are obtained numerically with the numerical inversion technique of the
transforms. In simulation, various impacts such as non-local, heat source velocity-time, and HTT are examined and
presented in the form of figures. Unique results are also deduced.
Key-Words: - Moore–Gibson–Thompson (MGT), hyperbolic two temperature (HTT), non-local, moving heat
source, thermoelasticity, ramp type thermal source, normal distributed force.
Received: June 9, 2023. Revised: November 11, 2023. Accepted: December 21, 2023. Published: December 31, 2023.
1 Introduction
The theory of thermoelasticity has received intensive
development from researchers due to its wide
applications in the fields of architecture, structural
analysis, geophysics, aeronautics, etc. The first two
popular generalized theories of thermoelasticity are
suggested by [1], [2], respectively. Later on [3], [4],
[5], [6], proposed three different theories of
generalized thermoelasticity, which are commonly
known as GN-I, GN-II, and GN-III theories of
thermoelasticity. However, the results corresponding
to the GN-III model showed that this theory leads to a
defect similar to the traditional Fourier law and
predicts the instantaneous propagation of thermal
waves.
Two-temperature theory of thermoelasticity was
developed by [7], [8]. [9], introduced a theory of two-
temperature generalized thermoelasticity. This theory
has a drawback in that thermal waves were
propagating at infinite speed. To overcome this
obstacle, [10], modified the two-temperature
generalized theory and furnished the HTT generalized
thermoelasticity theory. [11], investigated the
thermoelastic interactions in an infinite elastic
medium with a cylindrical cavity in the context of the
hyperbolic two-temperature generalized
thermoelasticity theory. [12], obtained the analytical
WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.27
Rajneesh Kumar, Sachin Kaushal, Gulshan Sharma
E-ISSN: 2224-3461
310
Volume 18, 2023
solution by taking into account the photo-
thermoelastic model under the new hyperbolic two-
temperature thermoelasticity.
Recently, interest in non-local elasticity has been
growing rapidly. Consideration of non-local factors,
in heat conduction theory, augments the microscopic
effects at a macroscopic level. It has been
successfully applied to the problems of dislocation,
fracture, mechanics, and dispersion of waves, [13],
[14], developed the concept of the non-local theory of
thermoelasticity. [15], discussed the wave
propagation in a nonlocal thermoelastic medium for
Green and Naghdi theory II (without energy
dissipation) of generalized thermoelasticity. [16],
investigated the impact of a moving heat source on a
magneto-thermoelastic rod in the context of Eringen’s
nonlocal theory under three-phase lag with a
memory-dependent derivative. [17], investigated the
impacts of the nonlocal thermoelastic parameters in
the Green and Naghdi model without energy
dissipation for a nanoscale material by the eigen value
approach.
The Moore-Gibson-Thompson (MGT) equation
has attracted the serious attention of several
researchers. This equation arises in many fields, such
as fluid mechanics, nanostructures, and
thermoelasticity. [18], has established a novel
thermoelastic model subjected to Moore-Gibson-
Thompson’s (MGT) equation which originated from a
third-order differential equation. By incorporating a
relaxation time parameter into the heat equation of
Green-Naghdi theory of type III. The considered
model was used to analyze the well-posedness and the
stability of solutions for one dimension and three-
dimension problem. Later, [19], presented a
generalized thermoelastic model using the MGT heat
conduction equation in the context of two
temperatures and proved the well-posedness and the
exponential decay of the solutions.
[20], employed a modified MGT thermoelastic
heat transfer model to investigate the impact of a
magnetic field on isotropic perfectly conducting half-
space subjected to a periodic heat source. [21],
established a domain of influence theorem in the
MGT thermoelasticity for dipolar bodies and
analyzed wave propagation in an isotropic and
infinite body subjected to a continuous thermal line
source. [22], established results for the domain of
influence theorem for potential–temperature
disturbance in the context of MGT theory of
thermoelasticity. Many authors discussed different
problems in the context of MGT theory of
thermoelasticity, notable of them are [23], [24], [25],
[26].
The present investigation has been performed by
adopting MGT theory of thermoelasticity along with
non-local and HTT models. The problem is examined
under heat source in the form of laser pulse decaying
with time and moving with constant velocity in one
direction along with normal distributed force and
ramp type thermal source. After simplifying the
equation with aid of dimensionless parameters, the
potential functions, and integral transform techniques,
the impact of non-local, HTT, and heat source
velocity for normal distributed force and thermal
source are examined on physical quantities and are
represented in the form of graphs.
2 Basic Equations
Following [10], [13], [18], the field equations and
constitutive relations in the context of MGT theory
with hyperbolic two temperature in the absence of
body forces can be written as:
, (1)
,
(2)
, (3)
, (4)
, (5)
where
- Lame's constants,
- displacement vector,
,- coefficient of linear thermal
expansion, - non-local parameter, - components
of stress tensor, t-time, - density and specific
heat respectively, -heat source, - thermal
conductivity,
- material constant, -
Laplacian operator, - conductive temperature, -
thermodynamic temperature, - two temperature
parameter, - relaxation time, -Kronecker’s delta,
- HTT parameter.
The equations (1) - (5) reduces to the following
cases
WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.27
Rajneesh Kumar, Sachin Kaushal, Gulshan Sharma
E-ISSN: 2224-3461
311
Volume 18, 2023
Coupled theory of thermoelasticity
(1980),
Lord - Shulman (L-S) (1967),
Green-Naghdi–II (GN-II) (1993),
Green-Naghdi–III (GN-III) (1992).
3 Formulation and Solution of the
Problem
A domain of thermoelastic half- space under the
MGT thermoelastic model is taken into account under
non-local and HTT impact is considered as the region
, which is homogenous and isotropic. The two-
dimensional problem is in the plane ( , which
is subjected to normally distributed force and Ramp
type thermal source in addition to the heat source in
the form of laser pulse decaying with time and
moving with constant velocity in one direction is
considered, therefore with these consideration, we
have:
31 ,0, uuu
. (6)
Dimensionless quantities are:
′
, ′
, ′
,
′′
,
iiii ux
c
ux ,',
1
1
,
′′, ′
,
′
, ′
( i =1, 3), (7)
where
and
.
Equations (1) - (5) by taking into account (6) and (7)
determine:
, (8)
, (9)
(10)
, (11)
, (12)
, (13)
, (14)
where
,
,
,
,
,
,
,
.
Equations (8) - (11) are decoupled by using the
dimensionless form of potential functions q and as :
,
. (15)
Laplace and Fourier transforms are taken as:
∞
and (16)
∞
∞
Applying (16) on (12) gives,
, (17)
where
Equation (8) - (10) in the accompany of (15) -
(17) yield the resulting expressions (suppressing the
primes for convenience) after some algebraic
calculation as:
(18)
,, (19)
where
,
,
, ,,
,
,
WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.27
Rajneesh Kumar, Sachin Kaushal, Gulshan Sharma
E-ISSN: 2224-3461
312
Volume 18, 2023
,
,
,
,
,
,
.
The bounded solution of equation (18) and (19) i.e.
as ∞ can be expressed as:
, (20)
, (21)
, (22)
where
,
, where
being the roots of the characteristic
equation
.
3.1 Heat Source
Let the physical entities be affected by a moving heat
source which is assumed to be in non-dimensional
form:
, (23)
where is constant, is the time duration of laser
pulse decaying, is the moving heat source, t is time,
( ) is the Dirac delta function.
4 Boundary Conditions
Here, we explore the impact of normal distributed
force and Ramp type thermal sources as
(iii)
at (24)
where
,
,
. (25)
is the magnitude of force, is the constant
temperature applied on the boundary.
Invoking (16) on (23)-(25), we have
, (iii)
at (26)
where
,
,
(27)
Invoking (16) -(17) (after suppressing the primes)
in (13) -(14) along with (20) -(22), the expressions of
displacements, stresses, conductive temperature, and
thermodynamic temperature are obtained by
considering the boundary conditions defined by (26) -
(27) as:
(28)
(29)
(30)
(31)
(32)
(33)
WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.27
Rajneesh Kumar, Sachin Kaushal, Gulshan Sharma
E-ISSN: 2224-3461
313
Volume 18, 2023
where
,
,,
,
,
,
,
,
,
,
,
,
,
, ,
, ,
,
,
.
5 Special Cases
1. MGT thermoelasticity with HTT: let
in equations (28)-(33), determine the
resulting expressions for MGT with HTT.
2. Non Local Lord Shulman Model (L-S
model) with HTT: Substituting in
equations (28)-(33), gives the expression of
generalized thermoelasticity which involves
one relaxation time under nonlocal and HTT.
3. Green- Naghdi-II model (GN-II model)
with HTT: if in
equations (28)-(33) will explore the resulting
quantities for GN type-II model under non
local theory and HTT.
4. Green- Naghdi-III model (GN-III model)
with HTT: i.f , in
equations (28)-(33), determines expressions
for GN type - III model under the influence
of nonlocal and HTT.
Sub cases:
5. 1 if : We obtain the results for MGT
thermoelasticity with two temperatures along
with non-local effect.
5. 2. if : We attain the results for MGT
thermoelasticity with one temperature along with
non-local effect.
6 Inversion of the Transforms
The components of displacement, stresses,
temperature distribution, and conductive temperature
are the functions of s, and which are parameters
of the Laplace transform and Fourier transform
respectively. To invert these quantities into the
physical domain, we invert the transforms by
applying the method explained by [27].
7 Numerical Result and Discussion
To explore the impact of various parameters under
the assumed model, the numerical calculations are
made for five different cases, (i) nonlocal, (ii)
moving heat source and (iii) hyperbolic two
temperature, (iv) normal distributed forces (v) Ramp
type thermal sources for MGL theory of
thermoelasticity.
Following, [28], we take the case of magnesium
crystal, the physical constants used are:
, ,
,
,,
,
and non- dimensional relaxation times are taken as
.
7.1 Non-Local Effect
In this case, we consider non-local parameters as
, , and , HTT
parameter and for the range
The solid line correspond to (), small
dashed line corresponds to ( ) whereas,
solid line with center symbol correspond to
and small dashed line with center
symbol represents the case of ().
7.1.1 Normal Distributed Force
Figure 1 (Appendix) shows the behavior of vs .
It is noticed that the behavior of for
and , is opposite in nature as that of
and for the range and
respectively, while similar trends are
noticed in the left over interval. It is also seen that
near the boundary surface, attains minima for
at .
Figure 2 (Appendix) is a plot of vs , which
demonstrates that for the values of are
higher in contrast to those obtained for for
WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.27
Rajneesh Kumar, Sachin Kaushal, Gulshan Sharma
E-ISSN: 2224-3461
314
Volume 18, 2023
the entire range. It is also seen that for a higher value
(, ), the values increase in the
entire range with a significant difference in their
magnitude.
Figure 3 (Appendix) predicts T vs . It is seen
that the T for intermediate value of non-local
parameter ( , ) are opposite in
nature to those for the value of ( ,
), it is also noticed that the magnitude of values of
T for is higher as compared to other
considered values of non-local parameter.
The graphical representation of with is
represented in Figure 4 (Appendix). It is noticed that
exhibits sa imilar pattern in the range
and for ( , ),
whereas shows an reverse trend for other
considered v values of , which is accounted as the
impact of non-local parameter.
7.1.2 Ramp Type Thermal Source
Figure 5 (Appendix) represents the variations of
vs . begins with large value in the absence of
non-local parameter. As the value of non-local
parameter increases the value of increases and
attain maxima at for .
The plot of vs is represented by Figure 6
(Appendix). It is seen that the trend of for
and are inverse in nature in the entire
range, whereas for higher values of (,
) are similar in nature in the entire range,
magnitude of values are greater for higher value of
.
Figure 7 (Appendix) shows the variations of T vs
. It is noticed that T exhibits oscillatory behavior
for all values of , the magnitude of oscillation is
greater in the absence of non-local parameter i.e.
while for other values of , T shows small
variations about ' 2 '.
Figure 8 (Appendix) exhibits the plot for vs .
The trend of is similar to that for T with
significant difference in their magnitude.
7.2 Moving Heat Source
In this case, we consider moving heat source
parameter , and , non-local
parameter and hyperbolic two-temperature
parameter for the range The
solid line corresponds to ( ), the small
dashed line corresponds to ( ) and the long
dashed line represents the case of
7.2.1 Normal Distributed Force
Figure 9 (Appendix) demonstrates the variations of
vs . It is noticed that the value of decreases
for the range and and
increases in the rest of interval of for all values of .
It is also noticed that the magnitude of values for
are higher for greater value of ( ).
The variation of vs is represented in
Figure 10 (Appendix). shows an increasing trend
for the entire range for all values of . It is also
revealed that the magnitude of the values of is
greater for larger values of .
Figure 11 (Appendix) depicts the trend of T vs
. The value of temperature T begins with large
value for velocity, for interval
and for left over interval, T follows an oscillatory
behavior. The magnitude of oscillation is higher in
case of than other considered value of v.
The graphical presentation of vs is
represented in Figure 12 (Appendix). follows an
oscillatory behavior for all values of for the entire
range, the magnitude of oscillation is greater for
higher values of .
7.2.2 Ramp Type Thermal Source
Figure 13 (Appendix) depicts the variations of vs
. The values of for and
increases for the range , magnitude of
values for are greater in comparison to
those for and decreases in left over interval,
whereas the smaller value of for (),
shows a steady state about the origin.
Figure 14 (Appendix) is a plot of vs . It is
evident from the plot that follows an oscillatory
behavior for all values of , the magnitude of
oscillation is higher for a greater value of (
).
Figure 15 (Appendix) exhibits the plot for T vs
. The value of T for and decreases
for the range and , and
increases for the left over interval, whereas for
the values of T exhibit small variations about
the value 2.
Figure 16 (Appendix) is a plot of vs . The
behavior of is similar in nature as observed for T
with a significant difference in their magnitude,
which reveals the impact of two temperature
parameters.
WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.27
Rajneesh Kumar, Sachin Kaushal, Gulshan Sharma
E-ISSN: 2224-3461
315
Volume 18, 2023
7.3 Hyperbolic Two Temperature
In this case, we consider hyperbolic two temperature
parameter , and two temperature
parameter and , non-local
parameter and moving heat source
for the range The solid line
corresponds to ( ), small dashed line
corresponds to ( ) whereas, the solid line
with the center symbol correspond to
and small dashed line with center symbol represents
the case of ().
7.3.1 Normal Distributed Force
It is evident from Figure 17 (Appendix), which is a
plot of vs that the trend of are similar in
nature for higher value HTT ( , ) in
the entire interval except in the range
with a magnitude of values is higher for smaller
value of hyperbolic two temperature parameter for
, whereas in case of TT and
trend of are similar in nature in the first
half of the interval, whereas opposite behavior in the
rest of the interval.
Figure 18 (Appendix) is a plot of vs . It is
noticed that the values of for show a decreasing
trend in the entire interval, whereas the values of
for and increases in the range
and decreases in the remaining range. It
also seen that the value of for shows a
small variation about the value '-1' and ultimately
approaches zero.
Figure 19 (Appendix) shows the variations of T
vs . It is noticed that the trend of T for HTT i.e.
is opposite in nature as observed for the
case of TT () in the entire range, which
shows a significant impact of hyperbolic two
temperature parameters. It is also noticed that the
trend of T in the absence of two temperature
parameters are reverse in nature as for the case of
HTT i.e. .
Figure 20 (Appendix) is a plot of vs .
shows an oscillatory trend in the entire range. It is
observed that the magnitude of oscillation is higher
for HTT (.) as compared to other considered
values of HTT parameter.
7.3.2 Ramp Type Thermal Source
Figure 21 (Appendix) is a plot of vs . The
values of increase for the entire range for HTT
parameter ( and ), magnitude of
values for are greater as compared to those
for . Also, the values of for
and decreases in the range and
reverse trend is noticed in the left-over interval.
Figure 22 (Appendix) is a plot of vs .
for shows the opposite trend as compared to
other considered cases for the first half of the
interval, which accounted for as significant effect of
two temperature parameters. In the latter half of the
interval, the values of show an oscillatory behavior.
Figure 23 (Appendix) is a plot of T vs . T
shows opposite behavior as compared to other
considered values of HTT parameters for the entire
range. In the absence of two temperature parameters
T shows a steady state for the interval
and later on the values of T decrease for the interval
, and increase in the left-
over interval.
Figure 24 (Appendix) is a plot of vs . The
variation of shows a similar trend as for T with
significant differences in their magnitude, which
reveals the impact of HTT and two temperature
parameters.
8 Conclusion
In this work, we considered a thermomechanical
problem based on the MGT model having non-local,
Ramp type heat source and hyperbolic two
temperature effects. The paper presents an analytic
solution for the thermomechanical behavior of the
deformation problem in which an infinite body is
subjected to the heat source in the form of a laser
pulse decaying with time and moving with constant
velocity in one direction. The problem is further
examined with normal distributed force and ramp
type thermal source. The results are displayed
graphically to illustrate the effect of nonlocal
parameters, moving heat source parameters and
impact of hyperbolic two temperatures. The
following observations are obtained from numerical
computed results:
when normal distributed force is applied, it is
observed that trends of and for,
are opposite in nature as observed for
and . It is also observed that T
and shows oscillatory behavior, the magnitude
of oscillation is higher for .
WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.27
Rajneesh Kumar, Sachin Kaushal, Gulshan Sharma
E-ISSN: 2224-3461
316
Volume 18, 2023
In case of Ramp type thermal source, the value of
increases in most of the interval for all values
of , while , T and shows oscillatory
behavior in the entire range, the magnitude of
oscillation is higher in the absence of non-local
parameter.
It is observed that a higher value of the moving
heat source parameter enhances the value of ,
T and for normal distributed force as well as for
Ramp type thermal source. It is also observed that
the behavior of variation for , , T and for
different values of are similar in nature with
differences in their magnitude of oscillations.
The hyperbolic two temperature effect enhances
the magnitude of T and in contrast to and
. It is also noticed that in most of the range, the
behavior of all entities for a higher value of
hyperbolic two temperature parameters i.e
are opposite in nature as observed for two
temperature parameter and for
classical one temperature for normally
distributed force as well as for Ramp type thermal
source.
The physical applications of the model can be
found in mechanical engineering and geophysics.
References:
[1] H. W. Lord and Y. Shulman, A
generalizeddynamical theory of Solid, J. Mech.
Phys., Vol.15, No.5, 1967, pp. 299–309,
https://doi.org/10.1016/0022-5096(67)90024-5
[2] A. E.Green and K.A.Lindsay,Thermoelasticity,
J. Elast., Vol.2, No.1, 1972, pp. 1–7,
https://doi.org/10.1007/BF00045689
[3] A. E. Green and P.M. Naghdi,A re-
examination of the basic postulates of
thermomechanics, Proc. R. Soc. Lond.,
Vol.432, No.1885, 1991, pp. 171–194,
https://doi.org/10.1098/rspa.1991.0012
[4] A. E. Green and P.M. Naghdi, On undamped
heat waves in an elastic solid, J. Therm.
Stresses, Vol.15, No.2, 1992, pp. 253-264,
https://doi.org/10.1080/01495739208946136
[5] A. E. Green and P.M. Naghdi,
Thermoelasticity without energy dissipation, J.
Elast., Vol.31, No.3, 1993, pp. 189-208,
https://doi.org/10.1007/BF00044969
[6] A. E. Green and P.M. Naghdi,A re-
examination of the base postulates of
thermomechanics, Proceedings of Royal
Society London A, Vol.432,1985, pp.171 -194,
https://doi.org/10.1098/rspa.1991.0012
[7] P J Chen and M E Gurtin, On a theory of heat
conduction involving two-temperatures,
Journal of Applied Mathematics and Physics
(ZAMP), Vol. No.19, 1968, pp. 614-627,
https://doi.org/10.1007/BF01594969
[8] P J Chen, M E Gurtin, and W O Williams, On
the thermodynamics of non-simple elastic
materials with two temperatures, Journal of
Applied Mathematics and Physics (ZAMP),
Vol.20, 1969, pp. 107-112,
https://doi.org/10.1007/BF01591120
[9] H M Youssef, Theory of two-temperature-
generalised thermoelasticity, IMA Journal of
Applied Mathematics, Vol.71,No.3, 2006,
pp.383 -390,
https://doi.org/10.1093/imamat/hxh101
[10] H M Youssef, A A El-Bary, Theory of
hyperbolic two-temperature generalized
thermoelasticity, Mat. Phy. Mech., Vol.40,
2018, pp. 158-171,
http://dx.doi.org/10.18720/MPM.4022018_4
[11] R Kumar, R. Prasad, and R Kumar,
Thermoelastic interactions on hyperbolic two-
temperature generalized thermoelasticity in an
infinite medium with a cylindrical cavity,
Elsevier European Journal of Mechanics -
A/Solids, Vol. 82, 2020,
https://doi.org/10.1016/j.euromechsol.2020.10
4007
[12] A Hobiny, Ibrahim Abbas and M. Marin, The
influences of the hyperbolic Two-
Temperatures theory on waves propagation in a
semiconductor material containing spherical
cavity, Mathematics, Vol.10, No.121, 2022,
https://doi.org/10.3390/math10010121
[13] A.C.Eringen,Nonlocal polar elastic continua,
Int. J. Eng. Sci., Vol.10, 1972, pp. 1-16,
https://doi.org/10.1016/0020-7225(72)90070-
5
[14] A.C.Eringen,Theory of nonlocal
thermoelasticity, Int. J. Eng. Sci. Vol.12, 1974,
pp.1063 -1077, https://doi.org/10.1016/0020-
7225(74)90033-0
[15] N. De Sarkar, and N.S. Sarkar, Waves in
nonlocal thermoelastic solids of type II,
Journal of Thermal Stresses, Vol.42, 2019,
pp.1153 -1170,
WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.27
Rajneesh Kumar, Sachin Kaushal, Gulshan Sharma
E-ISSN: 2224-3461
317
Volume 18, 2023
https://doi.org/10.1080/01495739.2019.161876
0
[16] F. S. Bayones, S. Mondal,S. M. Abo-Dahab
and A. A. Kilany, Effect of moving heat
source on a magneto-thermoelastic rod in the
context of Eringen’s nonlocal theory under
three-phase lag with a memory dependent
derivative, Mechanics Based Design of
Structures and Machines, Vol. 10., 2021,
https://doi.org/10.1080/15397734.2021.19017
35
[17] T. Saeed and I. Abbas, Effects of the nonlocal
thermoelastic Model in a thermoelastic
nanoscale material, Mathematics, Vol.10,
No.2, 2022,
https://doi.org/10.3390/math10020284.
[18] R.Quintanilla,Moore-Gibson-Thompson
thermoelasticity, Math.Mech. Solids, Vol.24,
2019, pp. 4020 -4031,
https://doi.org/10.1177/1081286519862007.
[19] R. Quintanilla, Moore-Gibson-Thompson
thermoelasticity with two temperatures, Appl.
Eng. Sci., Vol.1, 2020,
https://doi.org/10.1016/j.apples.2020.100006.
[20] A. E. Abouelregal, I. E. Ahmed, M. E. Nasr, K.
M. Khalil, A. Zakria, and F. A. Mohammed,
Thermoelastic processes by a continuous heat
source line in an infinite solid via Moore–
Gibson–Thompson thermoelasticity, Materials
(Basel), Vol.13, No.19, 2020, pp. 1-
17, https://doi.org/10.3390/ma13194463.
[21] M. Marin, M. I. A. Othman, A. R. Seadawy
and C. Carstea, A domain of influence in the
Moore–Gibson–Thompson theory of dipolar
bodies, Journal of Taibah University for
Science, Vol. 14, No.1, 2020, pp.653 -660,
https://doi.org/10.1080/16583655.2020.176366
4.
[22] K. Jangid, A domain of influence theorem
under MGT thermoelasticity theory, Sage
Journals, 2020, pp. 1-11,
https://doi.org/10.1177/1081286520946820
[23] M. Pellicer and R. Quintanilla, On uniqueness
and instability for some thermomechanical
problems involving the Moore–Gibson–
Thompson equation, Journal of Applied
Mathematics and Physics (ZAMP),Vol. 71,
No.3, , 2020, https://doi.org/10.1007/s00033-
020-01307-7.
[24] Abouelregal A. E., Hakan Ersoy, and Omer
Civalek, Solution of Moore–Gibson–
Thompson Equation of an Unbounded
Medium with a Cylindrical Hole,
Mathematics, Vol. 9, 2021, pp.1536,
https://doi.org/10.3390/math9131536.
[25] K. Jangid, M. Gupta, and S. Mukhopadhyay,
On propagation of harmonic plane waves
under the Moore – Gibson–Thompson
thermoelasticity theory, Waves in Random and
Complex Media, 2021,
https://doi.org/10.1080/17455030.2021.19490
71.
[26] A. E. Abouelregal,Fractional derivative
Moore-Gibson-Thompson heat equation
without singular kernel for a thermoelastic
medium with a cylindrical hole and variable
properties, ZAMM, 2022,
https://doi.org/10.1002/zamm.202000327.
[27] R. Kumar, S .Kaushal, L.S Reen, and S.K
Garg, Deformation due to various sources in
transversely isotropic thermoelastic material
without energy dissipation and with two-
temperature, Materials physics and
mechanics, Vol. 27, No.1, 2016, pp.21 -31.
[28] R. S Dhaliwal, and A Singh, Dynamical
Coupled Thermoelasticity, Hindustan
Publishers, Delhi, India, (1980), [Online].
https://cir.nii.ac.jp/crid/113028227172740044
8 (Accessed Date: March 13, 2024).
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Rajneesh Kumar carried out visualization
Conceptualization, Ideas and Supervision
- Sachin Kaushal carried out Conceptualization, and
supervision, editing implemented the software
- Gulshan Sharma organized Writing - review and
Resources and has implemented Program.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.27
Rajneesh Kumar, Sachin Kaushal, Gulshan Sharma
E-ISSN: 2224-3461
318
Volume 18, 2023
https://creativecommons.org/licenses/by/4.0/deed.en_
US
APPENDIX
0246810
Distance x1
-3
-2
-1
0
1
2
Normal Stress t33
Fig. 1: Variation of vs x1
(Non local Parameter-Distributed Force)
0 2 4 6 8 10
Distance x1
-0.6
-0.4
-0.2
0
0.2
0.4
Tangential Stress t31
Fig. 2: Variation of vs x1
(Non local parameter-Distributed Force)
0 2 4 6 8 10
Distance x1
0.5
1
1.5
2
2.5
3
3.5
Temperature T
Fig. 3: Variation of vs x1
(Non-local Parameter-Distributed Force)
Fig. 4: Variation of vs x1
(Non local parameter-Distributed Force)
0246810
Distance x1
0
1
2
3
4
5
Conductive Temperature
WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.27
Rajneesh Kumar, Sachin Kaushal, Gulshan Sharma
E-ISSN: 2224-3461
319
Volume 18, 2023
0246810
Distance x
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Normal Stress t33
Fig. 5: Variation of vs x1 (Non local
parameter - Thermal Source)
0246810
Distance x
-12
-8
-4
0
4
8
Tangential Stress t31
Fig. 6: Variation of vs x1
(Non local parameter-Thermal Source)
0 2 4 6 8 10
Distance x
1.5
2
2.5
Temperature T
Fig. 7: Variation of vs x1
(Non local parameter- Thermal Source)
0246810
Distance x
1.5
2
2.5
3
3.5
4
Conductive Temperature
Fig. 8: Variation of vs x1
(Non local parameter- Thermal Source)
WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.27
Rajneesh Kumar, Sachin Kaushal, Gulshan Sharma
E-ISSN: 2224-3461
320
Volume 18, 2023
0246810
Distance x1
-0.5
0
0.5
1
1.5
2
Normal Stress t33
MGT (
v
= 1.75)
MGT (
v
= 1)
MGT (
v
= 0.25)
Fig. 9: Variation of vs x1
(Heat Source Velocity - Distributed Force)
0 2 4 6 8 10
Distance x1
-0.6
-0.4
-0.2
0
0.2
0.4
Tangential Stress t31
MGT (
v
= 1.75)
MGT (
v
= 1)
MGT (
v
= 0.25)
Fig. 10: Variation of vs x1
(Heat Source Velocity - Distributed Force)
0246810
Distance x1
1.7
1.8
1.9
2
2.1
2.2
Temperature T
MGT (
v
= 1.75)
MGT (
v
= 1)
MGT (
v
= 0.25)
Fig. 11: Variation of vs x1
(Heat Source Velocity - Distributed Force)
0246810
Distance x1
2
2.2
2.4
2.6
2.8
3
Conductive Temperature
MGT (
v
= 1.75)
MGT (
v
= 1)
MGT (
v
= 0.25)
Fig. 12: Variation of vs x1
(Heat Source Velocity - Distributed Force)
WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.27
Rajneesh Kumar, Sachin Kaushal, Gulshan Sharma
E-ISSN: 2224-3461
321
Volume 18, 2023
0246810
Distance x1
-1.2
-0.9
-0.6
-0.3
0
0.3
0.6
Normal Stress t33
MGT (
v
= 1.75)
MGT (
v
= 1)
MGT (
v
= 0.25)
Fig. 13: Variation of vs x1
(Heat Source Velocity - Thermal Source)
0 2 4 6 8 10
Distance x1
-3
-2
-1
0
1
2
Tangential Stress t31
MGT (
v
= 1.75)
MGT (
v
= 1)
MGT (
v
= 0.25)
Fig. 14: Variation of vs x1
(Heat Source Velocity - Thermal Source)
0 2 4 6 8 10
Distance x1
1.7
1.8
1.9
2
2.1
2.2
Temperature T
MGT (
v
= 1.75)
MGT (
v
= 1)
MGT (
v
= 0.25)
Fig. 15: Variation of vs x1
(Heat Source Velocity - Thermal Source)
0246810
Distance x1
1.8
2.1
2.4
2.7
3
Conductive Temperature
MGT (
v
= 1.75)
MGT (
v
= 1)
MGT (
v
= 0.25)
Fig. 16: Variation of vs x1
(Heat Source Velocity - Thermal Source)
WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.27
Rajneesh Kumar, Sachin Kaushal, Gulshan Sharma
E-ISSN: 2224-3461
322
Volume 18, 2023
0246810
Distance x1
-4
-3
-2
-1
0
1
2
3
Normal Stress t33
HTT
HTT
TT a=0.104
AT a=0
Fig. 17: Variation of vs x1
(Hyperbolic two temperature - Distributed Force)
0 2 4 6 8 10
Distance x1
-4.5
-3
-1.5
0
1.5
3
4.5
6
7.5
Tangential Stress t31
HTT
HTT
TT a=0.104
AT a=0
Fig. 18: Variation of vs x1
(Hyperbolic two temperature- Distributed Force)
0 2 4 6 8 10
Distance x1
0.3
0.6
0.9
1.2
1.5
1.8
2.1
Temperature T
HTT
HTT
TT a=0.104
AT a=0
Fig. 19: Variation of vs x1
(Hyperbolic two temperature - Distributed Force)
0 2 4 6 8 10
Distance x1
0
1
2
3
Conductive Temperature
HTT
HTT
TT a=0.104
AT a=0
Fig. 20: Variation of vs x1
(Hyperbolic two temperature - Distributed Force)
WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.27
Rajneesh Kumar, Sachin Kaushal, Gulshan Sharma
E-ISSN: 2224-3461
323
Volume 18, 2023
0246810
Distance x1
-1.5
-1
-0.5
0
0.5
Normal Stress t33
HTT
HTT
TT a=0.104
AT a=0
Fig. 21: Variation of vs x1
(Hyperbolic two temperature - Thermal Source)
0 2 4 6 8 10
Distance x1
-4
-2
0
2
4
6
Tangential Stress t31
HTT
HTT
TT a=0.104
AT a=0
Fig. 22: Variation of vs x1
(Hyperbolic two temperature - Thermal Source)
0 2 4 6 8 10
Distance x
0
0.5
1
1.5
2
2.5
3
Temperature T
HTT
HTT
TT a
AT a=0
Fig. 23: Variation of vs x1
(Hyperbolic two temperature - Thermal Source)
0 2 4 6 8 10
Distance x
0
1
2
3
4
Conductive Temperature
HTT
HTT
TT a=0.104
AT a=0
Fig. 24: Variation of vs x1
(Hyperbolic two temperature - Thermal Source)
WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.27
Rajneesh Kumar, Sachin Kaushal, Gulshan Sharma
E-ISSN: 2224-3461
324
Volume 18, 2023
