  Citation: Hobiny, A.; Abbas, I.; Alshehri, H.; Marin, M. Analytical Solutions of Nonlocal Thermoelastic Interaction on Semi-Infinite Mediums Induced by Ramp-Type Heating. Symmetry 2022, 14, 864. https:// doi.org/10.3390/sym14050864 Academic Editor: Danny Arrigo Received: 1 April 2022 Accepted: 12 April 2022 Published: 22 April 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 Analytical Solutions of Nonlocal Thermoelastic Interaction on Semi-Infinite Mediums Induced by Ramp-Type Heating Aatef Hobiny 1 , Ibrahim Abbas 1,2, * , Hashim Alshehri 1 and Marin Marin 3, * 1 Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Mathematics Department, King Abdulaziz University, Jeddah 21577, Saudi Arabia; ahobany@kau.edu.sa (A.H.); hmalshehri@kau.edu.sa (H.A.) 2 Mathematics Department, Faculty of Science, Sohag University, Sohag 82511, Egypt 3 Department of Mathematics and Computer Science, Transilvania University of Brasov, 500093 Brasov, Romania * Correspondence: ibrabbas7@science.sohag.edu.eg (I.A.); m.marin@unitbv.ro (M.M.) Abstract: A novel nonlocal model with one thermal relaxation time is presented to investigate the propagation of waves in a thermoelastic semi-infinite medium. We used Eringen’s theory of the nonlocal continuum to develop these models. Analytical solutions in all physical quantities are provided by using Laplace transforms and eigenvalue techniques. All physical quantities are presented as symmetric and asymmetric tensors. The temperature, the displacement, and the stress variations in semi-infinite materials have been calculated. The effects of nonlocal parameters, ramp type heating, and the thermal relaxation times on the wave propagation distribution of physical fields for mediums are graphically displayed and analyzed. Keywords: eigenvalue approach; Laplace transforms; nonlocal thermoelastic model; thermal relaxation time 1. Introduction The nonlocal elastic theory was first advocated by Eringen [1]. After a period of 2 years, the theory of nonlocal thermoelasticity was explored by Eringen [2]. In continuum mechanics, he addressed constitutive relations, governing equations, laws of equilibrium, and displacement equations/temperature under a nonlocal elastic model. According to the nonlocal elastic model, strain depends on the applied stress of continuous bodies at a place x that is impacted not only by the strain point but also by the strains of the bodies in every other region near this point x. In the uniqueness reference, Altan [3] studied the nonlocal linear elastic theory in depth. Wang and Dhaliwal [4] explained the uniqueness of the theory of nonlocal thermoelasticity. Eringen [5] studied nonlocal electromagnetic solids and superconductivity under the theory of elasticity. Povstenko [6] recommended the nonlocal elastic model to take into account the forces of actions between atoms. Nonlocal theories of field elasticity have been explained in detail by Eringen [7] concerning continuum mechanics. Narendar and Gopalakrishnan [8] studied the effects of nonlocal scale on the ultrasonic wave properties of nanorods. Yu, Tian, and Liu [9] studied Eringen’s nonlocal theory of thermoelasticity with a size-dependent model. Zenkour and Abouelregal [10] studied the vibrations of thermal conductivity under the nonlocal thermoelastic theory due to harmonically variable heat sources. When motivated by the rule of Fourier heat conduction, Biot [11] established the cou- pled thermoelasticity theory (CD theory), which became acceptable for modern engineering applications, particularly in high-temperature cases. On the other hand, the thermoelastic models are physically undesirable at low temperatures and cannot achieve equilibrium. Lord and Shulman [12] (LS) added one relaxation period to the heat conduction equation to resolve the conflict. Symmetry 2022, 14, 864. https://doi.org/10.3390/sym14050864 https://www.mdpi.com/journal/symmetry