Abstract

This article is focused on developing a new mathematical model on the temperature-rate-dependent thermoelasticity theory (Green–Lindsay), using the methodology of memory-dependent derivative (MDD). First, the energy theorem of this model associated with two relaxation times in the context of MDD is derived for homogeneous, isotropic thermoelastic medium. Second, a uniqueness theorem for this model is proved using the Laplace transform technique. A variational principle for this model is also established. Finally, the results for Green–Lindsay model without MDD and coupled theory are obtained from the considered model.

References

1.
Biot
,
M.
,
1956
, “
Thermoelasticity and Irreversible Thermodynamics
,”
J. Appl. Phys.
,
27
(
3
), pp.
240
253
.10.1063/1.1722351
2.
Lord
,
H.
, and
Shulman
,
Y.
,
1967
, “
A Generalized Dynamic Theory of Thermoelasticity
,”
J. Mech. Phys. Solids
,
15
(
5
), pp.
299
309
.10.1016/0022-5096(67)90024-5
3.
Green
,
A. E.
, and
Lindsay
,
K. A.
,
1972
, “
Thermoelasticity
,”
J. Elasticity
,
2
(
1
), pp.
1
7
.10.1007/BF00045689
4.
Chandrasekharaiah
,
D. S.
,
1986
, “
Thermoelasticity With Second Sound: A Review
,”
Appl. Mech. Rev.
,
39
(
3
), pp.
355
376
.10.1115/1.3143705
5.
Chandrasekharaiah
,
D. S.
,
1998
, “
Hyperbolic Thermoelasticity: A Review of Recent Literature
,”
Appl. Mech. Rev.
,
51
(
12
), pp.
705
729
.10.1115/1.3098984
6.
Ignaczak
,
J..
, and
Ostoja-Starzewski
,
M.
,
2009
,
Thermoelasticity With Finite Wave Speeds
,
Oxford University Press
,
New York
.
7.
Green
,
A. E.
, and
Naghdi
,
P. M.
,
1991
, “
A Re-Examination of the Basic Postulates of Thermomechanics
,”
Proc. R. Soc. London, Ser. A
,
432
(
1885
), pp.
171
194
.10.1098/rspa.1991.0012
8.
Green
,
A. E.
, and
Naghdi
,
P. M.
,
1992
, “
On Undamped Heat Waves in an Elastic Solid
,”
J. Therm. Stress.
,
15
(
2
), pp.
253
264
.10.1080/01495739208946136
9.
Green
,
A. E.
, and
Naghdi
,
P. M.
,
1993
, “
Thermoelasticity Without Energy Dissipation
,”
J. Elasticity
,
31
(
3
), pp.
189
208
.10.1007/BF00044969
10.
Green
,
A. E.
, and
Naghdi
,
P. M.
,
1995
, “
A Unified Procedure for Construction of Theories of Deformable Media—I: Classical Continuum Physics
,”
Proc. R. Soc. London, Ser. A
,
448
(
1934
), pp.
335
356
.10.1098/rspa.1995.0020
11.
Green
,
A. E.
, and
Naghdi
,
P. M.
,
1995
, “
A Unified Procedure for Construction of Theories of Deformable Media—II: Generalized Continua
,”
Proc. R. Soc. London, Ser. A
,
448
(
1934
), pp.
357
377
.10.1098/rspa.1995.0021
12.
Green
,
A. E.
, and
Naghdi
,
P. M.
,
1995
, “
A Unified Procedure for Construction of Theories of Deformable Media—III: Mixture of Interacting Continua
,”
Proc. R. Soc. London, Ser. A
,
448
(
1934
), pp.
379
388
.10.1098/rspa.1995.0022
13.
Sarkar
,
N.
, and
Mondal
,
S.
,
2019
, “
Thermoelastic Interactions in a Slim Strip Due to a Moving Heat Source Under Dual-Phase-Lag Heat Transfer
,”
ASME J. Heat Transfer
,
141
(
12
), p.
124501
.10.1115/1.4044920
14.
Singh
,
B.
, and
Pal (Sarkar)
,
S.
,
2019
, “
Thermal Shock Behaviour on Generalized Thermoelastic Semi-Infinite Medium With Moving Heat Source Under Green Naghdi-III Model
,”
Math. Mod. Eng.
,
5
(
3
), pp.
79
89
.10.21595/mme.2019.20904
15.
Shivay
,
O. N.
, and
Mukhopadhyay
,
S.
,
2020
, “
On the Temperature-Rate Dependent Two-Temperature Thermoelasticity Theory
,”
ASME J. Heat Transfer
,
142
(
2
), p.
022102
.10.1115/1.4045241
16.
Sarkar
,
N.
,
De
,
S.
,
Das
,
N.
, and
Sarkar
,
N.
,
2020
, “
Reflection of Thermoelastic Waves From the Insulated Surface of a Solid Half-Space With Time-Delay
,”
ASME J. Heat Transfer
, 142(
9
), p.
092101
.10.1115/1.4046924
17.
Ghosh
,
D.
, and
Lahiri
,
A.
,
2018
, “
A Study on the Generalized Thermoelastic Problem for an Anisotropic Medium
,”
ASME J. Heat Transfer
,
140
(
9
), p.
094501
.10.1115/1.4039554
18.
Caputo
,
M.
, and
Mainardi
,
F.
,
1971
, “
A New Dissipation Model Based on Memory Mechanism
,”
Pure. Appl. Geophys.
,
91
(
1
), pp.
134
147
.10.1007/BF00879562
19.
Caputo
,
M.
, and
Mainardi
,
F.
,
1971
, “
Linear Models of Dissipation in Elastic Solids
,”
L. Riv. Nuovo. Cimento
,
1
(
2
), pp.
161
198
.10.1007/BF02820620
20.
Ezzat
,
M. A.
,
El-Karamany
,
A. S.
, and
Fayik
,
M. A.
,
2012
, “
Fractional Order Theory in Thermoelastic Solid With Three Phase Lag Heat Transfer
,”
Arch. Appl. Mech.
,
82
(
4
), pp.
557
572
.10.1007/s00419-011-0572-6
21.
Sherief
,
H. H.
,
El-Sayed
,
A. M. A.
, and
Abd El-Latief
,
A. M.
,
2010
, “
Fractional Order Theory of Thermoelasticity
,”
Int. J. Solids Struct.
,
47
(
2
), pp.
269
275
.10.1016/j.ijsolstr.2009.09.034
22.
Youssef
,
H. H.
,
2010
, “
Theory of Fractional Order Generalized Thermoelasticity
,”
ASME J. Heat Transfer
,
132
(
6
), p.
061301
.10.1115/1.4000705
23.
Abbas
,
I. A.
,
2014
, “
Functional Graded Material Under Fractional Order Theory of Thermoelasticity
,”
Theor. Appl. Frac. Mech.
,
74
, pp.
18
22
.10.1016/j.tafmec.2014.05.005
24.
Abbas
,
I. A.
,
2015
, “
Eigenvalue Approach to Fractional Order Generalized Magneto Thermoelastic Medium Subjected to Moving Heat Source
,”
J. Mag. Mag. Mater.
,
377
, pp.
452
459
.10.1016/j.jmmm.2014.10.159
25.
Wang
,
J. L.
, and
Li
,
H. F.
,
2011
, “
Surpassing the Fractional  Derivative: Concept of the Memory Dependent Derivative
,”
Comput. Math. Appl.
,
62
(
3
), pp.
562
1567
.10.1016/j.camwa.2011.04.028
26.
Yu
,
Y. J.
,
Hu
,
W.
, and
Tian
,
X. G.
,
2014
, “
A Novel Generalized Thermoelasticity,” Model Based on Memory Dependent Derivatives
,”
Int. J. Eng. Sci.
,
81
, pp.
123
134
.10.1016/j.ijengsci.2014.04.014
27.
Singh
,
B.
,
Pal (Sarkar)
,
S.
, and
Barman
,
K.
,
2019
, “
Thermoelastic Interaction in the Semi-Infinite Solid Medium Due to Three-Phase-Lag Effect Involving Memory Dependent Derivative
,”
J. Therm. Stress.
,
42
(
7
), pp.
874
889
.10.1080/01495739.2019.1602015
28.
Sarkar
,
I.
, and
Mukhopadhyay
,
B.
,
2019
, “
A Domain of Influence Theorem for Generalized Thermoelasticity With Memory-Dependent Derivative
,”
J. Therm. Stress.
,
42
(
11
), pp.
1447
1457
.10.1080/01495739.2019.1642169
29.
Xu
,
Y.
,
Xu
,
Z.
,
Guo
,
Y.
,
Dong
,
Y.
, and
Huang
,
X.
,
2019
, “
A Generalized Magneto-Thermoviscoelastic Problem of a Single-Layer Plate for Vibration Control Considering Memory-Dependent Heat Transfer and Nonlocal Effect
,”
ASME. J. Heat Transfer
,
141
(
8
), p.
082002
.10.1115/1.4044009
30.
Singh
,
B.
,
Pal, (Sarkar)
,
S.
, and
Barman
,
K.
,
2020
, “
Eigenfunction Approach to Generalized Thermo-Viscoelasticity With Memory Dependent Derivative Due to Three Phase Lag Heat Transfer
,”
J. Therm. Stresses
, epub.10.1080/01495739.2020.1770642
31.
Sarkar
,
I.
, and
Mukhopadhyay
,
B.
,
2020
, “
On the Spatial Behavior of Thermal Signals in Generalized Thermoelasticity With Memory Dependent Derivative
,”
Act. Mech.
,
231
, pp.
2989
3001
.10.1007/s00707-020-02687-7
32.
Sarkar
,
I.
, and
Mukhopadhyay
,
B.
,
2020
, “
On Energy, Uniqueness Theorems and Variational Principle for Generalized Thermoelasticity With Memory Dependent Derivative
,”
Int. J. Heat Mass Transfer
,
149
, p.
119112
.10.1016/j.ijheatmasstransfer.2019.119112
33.
Singh
,
B.
,
Pal (Sarkar)
,
S.
, and
Barman
,
K.
,
2020
, “
Memory Dependent Derivative Under Generalized Three-Phase-Lag Thermoelasticity Model With a Heat Source
,”
Multidisc. Mod. Mater. Struct.
, epub.10.1108/MMMS-10-2019-0182
34.
Sarkar
,
I.
, and
Mukhopadhyay
,
B.
,
2020
, “
Thermo-Viscoelastic Interaction Under Dual-Phase-Lag Model With Memory Dependent Derivative
,”
Wav. Rand. Comp. Med.
, epub.10.1080/17455030.2020.1736733
35.
El-Karamany
,
A. S.
, and
Ezzat
,
M. A.
,
2016
, “
Thermoelastic Diffusion With Memory-Dependent Derivative
,”
J. Therm. Stress.
,
39
(
9
), pp.
1035
1050
.10.1080/01495739.2016.1192847
36.
Nowacki
,
W.
,
1975
,
Dynamic Problems of Thermoelasticity
,
Springer
,
Leiden, The Netherlands
37.
Predeleanu
,
M.
,
1959
, “
On Thermal Stresses in Visco-Elastic Bodies
,”
Bull. Math. Soc. Sci. Math. Phys. Repub. Pop. Roum.
,
3
(
2
), pp.
223
228
.
38.
Ionescu-Cazimir
,
V.
,
1964
, “
Problem of Linear Thermoelasticity: Theorems on Reciprocity
,”
Bull. Acad. Polon. Sci. Tech.
,
12
, pp.
473
480
.
39.
Othman
,
M.
,
2004
, “
The Uniqueness and Reciprocity Theorems for Generalized Thermo-Viscoelasticity With Thermal Relaxation Times
,”
Mech. Mech. Eng.
,
7
(
2
), pp.
77
87
.https://www.researchgate.net/publication/286941696_The_uniqueness_and_reciprocity_theorems_for_generalized_thermo-viscoelasticity_with_thermal_relaxation_times
40.
Ezzat
,
M. A.
, and
El-Karamany
,
A. S.
,
2003
, “
On Uniqueness and Reciprocity Theorems for Generalized Thermo-Viscoelasticity With Thermal Relaxation
,”
Can. J. Phys.
,
81
(
6
), pp.
823
833
.10.1139/p03-070
41.
Ezzat
,
M. A.
, and
El-Karamany
,
A. S.
,
2002
, “
The Uniqueness and Reciprocity Theorems for Generalized Thermo-Viscoelasticity for Anisotropic Media
,”
J. Therm. Stress.
,
25
(
6
), pp.
507
522
.10.1080/01495730290074261
42.
Ezzat
,
M. A.
, and
El-Karamany
,
A. S.
,
2015
, “
Two-Temperature Green-Naghdin Theory of Type III in Linear Thermoviscoelastic Anisotropic Solid
,”
Appl. Math. Model
,
39
(
8
), pp.
2155
2171
.10.1016/j.apm.2014.10.031
43.
Ezzat
,
M. A.
, and
El-Karamany
,
A. S.
,
2002
, “
The Uniqueness and Reciprocity Theorems for Generalized Thermo-Viscoelasticity With Two Relaxation Times
,”
Int. J. Eng. Sci.
,
40
(
11
), pp.
1275
1284
.10.1016/S0020-7225(01)00099-4
44.
Love
,
A. E. H.
,
1963
,
The Mathematical Theory of Elasticity
,
Dover Publications
,
New York
.
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