Citation: Abbas, I.; Marin, M.; Hobiny, A.; Vlase, S. Thermal Conductivity Study of an Orthotropic Medium Containing a Cylindrical Cavity. Symmetry 2022, 14, 2387. https://doi.org/10.3390/ sym14112387 Academic Editors: Vasilis K. Oikonomou and Victor A. Eremeyev Received: 29 September 2022 Accepted: 8 November 2022 Published: 11 November 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 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/). symmetry S S Article Thermal Conductivity Study of an Orthotropic Medium Containing a Cylindrical Cavity Ibrahim Abbas 1,2 , Marin Marin 3, * , Aatef Hobiny 2 and Sorin Vlase 3,4, * 1 Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt 2 Mathematics Department, Faculty of Science, King Abdulaziz University, Jeddah 21521, Saudi Arabia 3 Department of Mathematics and Computer Science, Transilvania University of Brasov, 500036 Brasov, Romania 4 Romanian Academy of Technical Sciences, B-dul Dacia 26, 030167 Bucharest, Romania * Correspondence: m.marin@unitbv.ro (M.M.); svlase@unitbv.ro (S.V.) Abstract: An interesting feature that appears in the thermoelastic interaction in an orthotropic material containing cylindrical cavities is addressed in this study. For this purpose, the Finite Element Method is applied to analyze a generalized thermoelasticity theory with a relaxation time. For the development of the model, a thermal conductivity that is dependent on the temperature of the orthotropic medium was considered. The boundary condition for the internal surface of a cylindrical hollow is defined by the thermal shocks and the traction on the free surface. The nonlinear formulations of thermoelastic based on thermal relaxation time in orthotropic mediums are abbreviated using the Finite Element Method. The nonlinear equations without Kirchhoff’s transformations are presented. The results are graphically represented to demonstrate how changing thermal conductivity affects all physical values. Keywords: thermal relaxation; orthotropic medium; cylindrical cavity; FEM; constitutive law 1. Introduction In solid mechanics and material science, an orthotropic media possesses material properties at specified points that change along the three perpendicular axes, each having twofold rotational symmetry. Over the last four decades, a number of mathematicians and engineers have exhibited a great deal of interest in generalized thermoelastic theories because of their remarkable realistic implications in a range of domains, such as acoustics, continuum mechanic, nuclear engineering, aeronautic, high-energy particle accelerator, and so on. Biot [1] developed the coupled thermoelastic hypothesis to overcome the inconsis- tency appearing by the uncoupled hypothesis. In this theory, the heat transport and elasticity formulations are coupled. Lord and Shulman [2] proposed many extensions of the thermoelastic theory. In 1980, Dhaliwal and Sherief [3] modified the Lord and Shul- man model to include anisotropic cases. Singh [4] has explored the wave propagation in porous materials using thermoelastic models in general. Alesemi [5] used the LS model under the influence of centrifugal force and Coriolis to investigate the plane waves in magneto-thermoelasticity anisotropic materials. Marin et al. [6] presented some results in the Green and Lindsay model of thermoelastic structures. Aboueregal et al. [7] have studied the effects of varying properties and rotations in visco-thermoelastic orthotropic cylinders. Biswas [8] studied the surface waves in porous nonlocal orthotropic thermoe- lastic materials. Abd-Alla et al. [9–11] have discussed the propagations of Rayleigh wave in a generalized thermo-magneto-elasticity orthotropic medium under gravity field with initial stresses. Biswas and Mukhopadhyay [12] have used the eigenfunction expansion approach to analyze thermal shocks behaviors in thermoelastic orthotropic media by using three thermoelastic models with a magnetic field. Demirdži´ c et al. [13] have applied the Symmetry 2022, 14, 2387. https://doi.org/10.3390/sym14112387 https://www.mdpi.com/journal/symmetry