RESEARCH ARTICLE Copyright © 2016 American Scientiﬁc Publishers All rights reserved Printed in the United States of America Journal of Computational and Theoretical Nanoscience Vol. 13, 7433–7447, 2016 Interaction Between Mechanical Forces and Waves Propagation in Magneto-Thermo-Elastic for Fiber-Reinforced Wafer Kh. Lotfy 1 3 ∗ and W. Hassan 2 3 1 Mathamatical Department, Faculty of Science, Zagazig University, Zagazig 44519, Egypt 2 Mathamatical and Physics Department, Faculty of Engineering, Port Said University, Egypt 3 Mathamatical Department, Faculty of Science and Arts, Al-Mithnab, Qassim University 931, Buridah 51931, Al-Mithnab, Saudi Arabia The objective of this paper is to investigate the surface waves in ﬁbre-reinforced anisotropic elastic medium subjected to magnetic and thermal ﬁelds. We introduce the coupled theory (CD), Lord- Shulman (LS) theory and Green-Lindsay (GL) theory to study the inﬂuence of magnetic ﬁeld on 2D problem of a ﬁbre-reinforced thermoelastic. The analytical expressions for displacement com- ponents and force stress are obtained in the physical domain by using the harmonic vibrations. The wave velocity equations have been obtained in different cases. Numerical results for the tem- perature, displacement, and thermal stress components are given and illustrated graphically in the presence and absence of the magnetic ﬁeld of the material medium. A comparison also is made between the three theories in the presence and absence of ﬁber-reinforced. Keywords: Lord-Shulman, Green-Lindsay, Fibre-Reinforced, Surface Waves, Thermoelastic, Magnetic Field. 1. INTRODUCTION In the postwar years, we have seen a rapid development of themoelasticity stimulated by various engineering sci- ences. Most of investigations were done under the assump- tion of the temperature-independent material properties, which limited the applicability of the solutions obtained to certain ranges of temperature. At high temperature, the material characteristics, such as the modulus of elasticity, the Poisson’s ratio, and the coefﬁcient of thermal conduc- tivity, are no longer constants. In recent years, due to the progress in various ﬁelds of science and technology, tak- ing into consideration the real behaviour of the material characteristics becomes an actual necessity. In some inves- tigations, they have been taken as functions of coordinates. A reinforced concrete member should be designed for all conditions of stresses that may occur and in accordance with principles of mechanics. The characteristic property of a reinforced concrete member is that its components, namely concrete and steel, act together as a single unit as long as they remain in the elastic condition i.e., the two components are bounded together so that there can be ∗ Author to whom correspondence should be addressed. no relative displacement between them. In the case of an elastic solid reinforced by a series of parallel ﬁbers, it is usual to assume trans-verse isotropy. In the linear case, the associated constitutive relations, relating inﬁnitesimal stress and strain components have ﬁve material constants. In the last three decades, the analysis of stress and defor- mation of ﬁber-reinforced composite materials has been an important research area of solid mechanics. The wave’s propagation in a reinforced media plays a very interest- ing role in civil engineering and geophysics. The studies of propagation, reﬂection and transmission of waves are of great interest to seismologists. Such studies help them to obtain knowledge about the rock structures as well as their elastic properties and at the same time information regarding minerals and ﬂuids present in side the earth. The idea of introducing a continuous self reinforcement at every point of an elastic solid was given by Belﬁed et al. 1 The model was later applied to the rotation of a tube by Verma and Rana. 2 Verma 3 has also discussed the magneto elastic shear waves in self-reinforced bodies. Sengupta and Nath 4 discussed the problem of the surface waves in ﬁbre- reinforced anisotropic elastic media. Singh 5 showed that, for wave propagation in ﬁbre-reinforced anisotropic media, J. Comput. Theor. Nanosci. 2016, Vol. 13, No. 10 1546-1955/2016/13/7433/015 doi:10.1166/jctn.2016.5737 7433