  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 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics 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