Citation: Zenkour, A.M.; Mashat,
D.S.; Allehaibi, A.M. Thermoelastic
Coupling Response of an Unbounded
Solid with a Cylindrical Cavity Due
to a Moving Heat Source.
Mathematics 2022, 10, 9. https://
doi.org/10.3390/math10010009
Academic Editors: Nikos D. Lagaros
and Vagelis Plevris
Received: 27 November 2021
Accepted: 17 December 2021
Published: 21 December 2021
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Article
Thermoelastic Coupling Response of an Unbounded Solid with
a Cylindrical Cavity Due to a Moving Heat Source
Ashraf M. Zenkour
1,2,
*, Daoud S. Mashat
1
and Ashraf M. Allehaibi
1,3
1
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589,
Saudi Arabia; dmashat@kau.edu.sa (D.S.M.); amlehaibi@uqu.edu.sa (A.M.A.)
2
Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh 33516, Egypt
3
Department of Mathematics, Jamoum University College, Umm Al-Qura University, Jamoum, Makkah 21955,
Saudi Arabia
* Correspondence: zenkour@kau.edu.sa or zenkour@sci.kfs.edu.eg
Abstract:
The current article introduces the thermoelastic coupled response of an unbounded solid
with a cylindrical hole under a traveling heat source and harmonically altering heat. A refined
dual-phase-lag thermoelasticity theory is used for this purpose. A generalized thermoelastic coupled
solution is developed by using Laplace’s transforms technique. Field quantities are graphically
displayed and discussed to illustrate the effects of heat source, phase-lag parameters, and the angular
frequency of thermal vibration on the field quantities. Some comparisons are made with and without
the inclusion of a moving heat source. The outcomes described here using the refined dual-phase-lag
thermoelasticity theory are the most accurate and are provided as benchmarks for other researchers.
Keywords: G–N; L–S and CTE theories; cylindrical hole; dual-phase-lag; moving velocity
1. Introduction
The thermoelasticity theory is adopted in various applications to obtain interesting
formulations due to a variety of microphysical processes. The starting point of the clas-
sical coupled thermoelasticity (CTE) model was founded by Duhamel [
1
]. While Biot [
2
]
formulated the CTE theory by considering the second law of thermodynamics. One of
the first generalized theories is established by Lord and Shulman (L–S) [
3
] by including a
thermal relaxation parameter. While Green and Lindsay [
4
] developed another generalized
model by including two thermal relaxation parameters. Such generalized theories with
one or more thermal relaxation parameters are also stated as hyperbolic thermoelasticity
theories [
5
]. Green and Nagdhi (G–N) [
6
–
8
] formulated three various theories of thermoe-
lasticity in an unusual way. In addition, Tzou [
9
,
10
] presented a modern generalized one
which is called a dual-phase-lag (DPL) theory. A lot of research is presented to include and
modify Tzou’s model (see, e.g., [11–15]).
Many problems found in the literature are concerned with the thermoelastic response
of unbounded bodies with cylindrical cavities. Chandrasekharaiah and Srinath [
16
] applied
the G–N II model to analyze axisymmetric thermoelastic communications in an unbounded
solid including a cylindrical hole. Allam et al. [
17
] discussed thermal distribution field
quantities of a half-space containing a circular cylindrical cavity in the framework of a G–N
model. Ezzat and El-Bary [
18
,
19
] used a fractional-order of both thermo-viscoelasticity
and magneto-thermoelasticity theories to deal with an unbounded perfect conducting
media having a cylindrical hole in the existence of an axial uniform magnetic field. Sharma
et al. [
20
] tried to solve the dynamic formulation of an elasto-thermo-diffusion infinite cylin-
drical hole under various boundary conditions. Kumar and Mukhopadhyay [
21
] presented
the impacts of three-phase-lags (TPLs) on thermoelastic communications under step input
in temperature on a cylindrical hole in an infinite body. Mukhopadhyay and Kumar [
22
]
dealt with the thermoelastic communications in an infinite solid with a cylindrical hole
based on a two-temperature L–S model. Kumar et al. [
23
] described the thermoelastic
Mathematics 2022, 10, 9. https://doi.org/10.3390/math10010009 https://www.mdpi.com/journal/mathematics
