International Journal of Engineering Mathematics

Volume 2016, Article ID 4954063, 9 pages

http://dx.doi.org/10.1155/2016/4954063

## Reflection of Plane Waves in Generalized Thermoelastic Half Space under the Action of Uniform Magnetic Field

^{1}Department of Mathematics, Nabagram K. D. Paul Vidyalaya, 27 G. T. Road, Serampore, India^{2}Department of Mathematics, Gobardanga Hindu College, West Bengal, India^{3}Department of Applied Mathematics, University of Calcutta, 92 A. P. C. Road, Kolkata 700 009, India

Received 27 June 2016; Accepted 5 October 2016

Academic Editor: Elio Usai

Copyright © 2016 Narottam Maity et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Reflection of longitudinal displacement waves in a generalized thermoelastic half space under the action of uniform magnetic field has been investigated. The magnetic field is applied in such a direction that the problem can be considered as a two-dimensional one. The discussion is based on the three theories of generalized thermoelasticity: Lord-Shulman (L-S), Green-Lindsay (G-L), and Green-Naghdi (G-N) with energy dissipation. We compute the possible wave velocities for different models. Amplitude ratios have been presented. The effects of magnetic field on various subjects of interest are discussed and shown graphically.

#### 1. Introduction

It is known that mechanical loadings are not the only cause for deformation in elastic solids; thermal loadings also can play a vital role in producing deformations in structures and machines subject to generation and flow of heat. Thermoelasticity takes care of the deformations and stresses produced due to thermal loadings along with the deformation and stresses produced due to mechanical loadings. Clearly the governing equations of motion in thermoelasticity consist of the coupled equations involving mechanical and thermal stresses and the equation governing heat flow in the solid. The parabolic type of heat conduction equation as was adopted initially in the classical theory encountered a serious drawback in the sense that the speed of the heat propagation in the solid could be infinite, which is absolutely unrealistic. Naturally, scientists endeavoured to develop a new theory compatible with the reality. In 1967 Lord and Shulman [1] employed a modified version of the Fourier law of heat conduction and developed a theory known as generalized theory of thermoelasticity. The new thermoelastic equations yielded a finite velocity of heat propagation by introducing relaxation time into the energy equation, rendering the new thermoelastic field equations fully hyperbolic. In 1972 Green and Lindsay [2] proposed a theory of thermoelasticity with certain special features that contrasts with the previous theory having a thermal relaxation time. In this theory Fourier’s law of heat conduction is unchanged whereas the classical energy equation and stress-strain-temperature relations are modified. The two theories have been developed with independent ideas and one cannot be considered as a particular case of the other. Investigation of various problems characterizing the two theories has been discussed by Chandrasekharaiah [3, 4]. Further modifications in the constitutive equations of thermoelasticity were done by Green and Naghdi [5] to accommodate a wider class of heat flow problems.

Another interesting field of recent study is the field of magnetothermoelasticity in which interacting effects of applied magnetic field on elastic and thermal deformations of a solid are studied. Such studies have applications in several areas, particularly in nuclear devices, biomedical engineering, and geomagnetic investigations. Some of the works related to the interaction of the electromagnetic field, the thermal field, and the electric field may be available in literature, namely, Abd-Alla and Al-Dawy [6, 7], Ezzat and Othman [8], Ezzat [9], Wang et al. [10], and Othman [11–13]. A problem of reflection of plane waves in a rotating transversely isotropic magnetothermoelastic half space has been discussed by Singh and Yadav [14]. Roychoudhuri and Banerjee [15] investigated viscoelastic plane waves in a rotating thermoelastic medium following Green-Naghdi model. Thermoelastic plane waves in a rotating isotropic medium have been studied by Ahmad and Khan [16]. The effects of rotation and relaxation times on plane wave propagation in a generalized thermoelastic medium have been investigated by Roychodhuri [17].

The present paper aims at the study of reflection of longitudinal wave from the flat boundary of an elastic half space in generalized thermoelasticity. The half space is under the action of a uniform magnetic field and our objective is to study the behavior of reflection of longitudinal plane wave in presence of magnetic field. It is assumed that the direction of the magnetic field is such that the problem can be considered as two-dimensional. It is observed that there are two distinct types of longitudinal plane waves in the medium. The effects of magnetic field on the velocity of propagation of these types of waves have been shown graphically. Magnetic field effects on amplitude ratios of the waves have also been shown.

#### 2. Field Equations

In our mathematical expressions we shall use bold letters to represent the vector quantities appearing in our discussion. The field equations governing the displacement and the temperature for a homogeneous isotropic magneto elastic material in generalized magnetothermoelasticity in the absence of body force may be presented in vector form as where is the current density, is the magnetic induction vector, , is the total magnetic field vector, is the magnetic permeability of solid, are Lame’s elastic constants, are relaxation times, is additional material constant, is the specific heat, is the temperature, is the absolute temperature, is the coefficient of linear thermal expansion, is the dilatation, , is the coefficient of thermal conductivity, is the mass density, and is the Kronecker delta.

Equation (2) is a single equation representing the generalized heat conduction equation for three different models depending on the values of suffix . For , (2) represents the Lord-Shulman (L-S) model, for it represents the Green-Lindsay (G-L) model, and for it represents Green-Naghdi (G-N) model. The thermal relaxation time satisfies the inequalities [2] in the case of G-L theory.

The above field equations (1) and (2) are supplemented by the following constitutive equations:in which are the components of symmetric stress tensor and are the components of strain tensor related to displacement components , as and the indices take values 1, 2, and 3.

Due to the application of initial magnetic field , an induced magnetic field , an induced electric field , and current density vector are developed, so the total magnetic field . For a slowly moving homogeneous electrically conducting solid, the linear equations of electrodynamics arewhere , is the electrical conductivity, and is the particle velocity. Here we ignore the small effect of temperature gradient on the current density vector . The deformation vector is supposed to be small and the displacement vector is measured from the steady state deformed position.

Equation (3) when expressed in terms of displacement vector becomes Equation (1) may be expressed in vector notation in the formwhereIt is convenient to write the displacement vector in terms of a scalar potential and a vector potential in the formwith .

#### 3. Formulation of the Problem

We take origin of Cartesian coordinate system on the free surface of the thermoelastic solid occupying a half space. -axis is drawn along the normal to the free surface pointing into the medium, while and axes are taken along free surface of the half space, which is assumed to be thermally insulated (Figure 1). The solid is under the action of a uniform magnetic field . The applied magnetic field will develop induced magnetic field , an induced electric field , and current density .