Journal of Mathematics

Volume 2019, Article ID 1016981, 16 pages

https://doi.org/10.1155/2019/1016981

## Influence of Rotation and Magnetic Fields on a Functionally Graded Thermoelastic Solid Subjected to a Mechanical Load

^{1}Department of Mathematics, Guru Jambheshwar University of Science and Technology, Hisar 125001, Haryana, India^{2}Department of Applied Sciences, School of Engineering and Technology, The NorthCap University, Gurgaon 122017, Haryana, India

Correspondence should be addressed to Kapil Kumar Kalkal; moc.liamffider@ujg_laklaklipak

Received 26 February 2019; Accepted 22 April 2019; Published 10 June 2019

Academic Editor: Viliam Makis

Copyright © 2019 Ankush Gunghas 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

The current manuscript is presented to study two-dimensional deformations in a nonhomogeneous, isotropic, rotating, magneto-thermoelastic medium in the context of Green-Naghdi model III. It is assumed that the functionally graded material has nonhomogeneous mechanical and thermal properties in the -direction. The exact expressions for the displacement components, temperature field, and stresses are obtained in the physical domain by using normal mode technique. These are also computed numerically for a copper-like material and presented graphically to observe the variations of the considered physical variables. Comparisons of the physical quantities are shown in figures to depict the effects of angular velocity, nonhomogeneity parameter, and magnetic field.

#### 1. Introduction

In the classical dynamical coupled theory of thermoelasticity formulated by Biot [1], thermal signals are propagating with infinite speed, which is not physically acceptable. To remove this drawback of the classical coupled dynamical theory of thermoelasticity, several theories of generalized thermoelasticity were developed. The first two generalized thermoelastic theories are Lord-Shulman (L-S) [2] theory and Green-Lindsay (G-L) [3] theory. In L-S theory, one thermal relaxation time parameter is introduced in the classical Fourier’s law of heat conduction, whereas in the G-L theory, two thermal relaxation times are introduced in the constitutive relations for force stress tensor and entropy equation. Green and Naghdi [4–6] speculated three new thermoelasticity theories that permit treatment of a much wider class of heat flow problems, labeled as G-N models I, II, and III. The nature of the constitutive equations in these three models is such that when the respective theories are linearized, the model I is same as the classical heat conduction theory, model II predicts the finite speed of heat propagation involving no energy dissipation, and model III indicates the propagation of thermal signals with finite speed.

The theory of magneto-thermoelasticity is concerned with the effect of magnetic field on elastic and thermoelastic deformations of a solid body and has received the attention of many researchers due to its extensive use in various fields like optics, geophysics, and acoustics. The problem of distribution of thermal stresses and temperature in a perfectly conducting half-space, in contact with a vacuum, permeated by an initial magnetic field was studied by Ezzat [7]. A model of two-dimensional equations of generalized magneto-thermoelasticity with one relaxation time in a perfectly conducting medium was established by Ezzat and Othman [8]. The impacts of fractional order parameter, hydrostatic initial stress, and gravity field on the plane waves in a fiber-reinforced isotropic thermoelastic medium were investigated by Othman* et al.* [9]. Deswal and Kalkal [10] employed Laplace and Fourier transforms technique to study the phenomenon of wave propagation in a fractional order micropolar magneto-thermoelastic half-space. In the frame of fractional order theory of generalized thermoelasticity, Deswal* et al.* [11] studied magneto-thermoelastic interactions in an initially stressed, isotropic, homogeneous half-space. The effects of initial stress and magnetic field on thermoelastic interactions in an isotropic, thermally and electrically conducting half-space, whose surface is subjected to mechanical and thermal loads, were explored by Othman and Eraki [12]. Xiong and Guo [13] investigated the electro-magneto-thermoelastic diffusive plane waves in a half-space with variable material properties under fractional order thermoelastic theory.

Since the large bodies like the earth, the moon, and other planets have an angular velocity, it appears more realistic to study the thermoelastic problems in a rotating medium. The propagation of elastic waves in a rotating, homogeneous, and isotropic medium was investigated by Schoenberg and Censor [14]. Some results in thermoelastic rotating medium are due to Roy Choudhuri and Debnath [15] and Roy Choudhuri and Mukhopadhyay [16]. The effect of rotation in generalized thermoelastic solid under the influence of gravity with an overlying infinite thermoelastic fluid was analyzed by Ailawalia and Narah [17]. Abouelregal and Zenkour [18] used fractional order theory of thermoelasticity to scrutinize the effect of angular velocity on fiber-reinforced generalized thermoelastic medium whose surface is subjected to a Mode-I crack problem. Kumar* et al.* [19] discussed the propagation of plane waves at the free surface of thermally conducting micropolar elastic half-space with two temperatures. Othman* et al.* [20] considered the dual-phase lag model to study the influence of the rotation on a two-dimensional problem of micropolar thermoelastic isotropic medium with two temperatures. Said* et al.* [21] used normal mode technique to study thermodynamical interactions in a micropolar magneto-elastic medium with rotation and two-temperature. Abouelregal and Abo-Dahab [22] investigated a two-dimensional problem in the context of dual-phase-lag model with fiber-reinforcement and rotation using normal mode analysis. The effect of angular velocity on Rayleigh wave propagation in a fiber-reinforced, anisotropic magneto-thermo-viscoelastic media was discussed by Hussien and Bayones [23].

Over the last few decades, some structural materials such as functionally graded materials have been rapidly developed and used in many engineering applications. In functionally graded materials (FGMs), material properties vary gradually with a location within the body. FGMs are usually designed to be used under high-temperature environments. So, FGMs can easily eliminate or control thermal stresses, when sudden heating or cooling happens. These types of material are broadly used in important structures such as body materials in the aerospace field and nuclear reactors. A thermoinelastic response of functionally graded composites was studied by Aboudi* et al.* [24]. Abd-Alla* et al.* [25] analyzed radial vibrations in a functionally graded orthotropic elastic half-space subjected to rotation and gravity field. The electro-magneto-thermoelastic response of an infinite functionally graded cylinder was studied by Abbas and Zenkour [26], by using finite element method. The problem of generalized thermoelasticity in a thick-walled functionally graded cylinder with one relaxation time was considered by Abbas [27]. In this problem, the effects of temperature-dependent properties, volume fraction parameter, and thermal relaxation time on thermophysical quantities are estimated.

The aim of the present contribution is to consider two-dimensional disturbances in an infinite, isotropic, nonhomogeneous, rotating, magneto-thermoelastic medium in the context of G-N model III. All the mechanical and thermal properties of the FGM under consideration are supposed to vary as an exponential power of the space-coordinate. The numerical results for the physical quantities have been obtained for a copper-like material and presented graphically to estimate and highlight the effects of different parameters considered in this problem.

#### 2. Basic Equations

Following Green-Naghdi [5] and Roy Choudhuri and Debnath [15], the field equations and stress-strain-temperature relations in a rotating thermoelastic medium in the presence of body forces are

*Constitutive Law*where

*Stress Equation of Motion*

*Equation of Heat Conduction*where , are Lame’s elastic constants, , is the coefficient of linear thermal expansion, are the components of stress, are the components of strain, is the reference mass density, is the specific heat at constant strain, is the thermal conductivity, is the material constant characteristic for this theory, is the displacement vector, , is the absolute temperature, is the reference temperature of the medium in its natural state assumed to be , is the cubical dilatation, is the rotation vector, and is the Kronecker delta.

For a nonhomogeneous medium, the parameters , , , , , and are no longer constant but become space-dependent. Hence we replace , , , , , and by , , , , , and , respectively, where , , , , , and are supposed to be constants and is a given nondimensional function of the space variable . Using these values of parameters, (1), (3), and (4) take the following form:whereHere, the superposed dot denotes derivative with respect to time and the comma denotes derivative with respect to space variable.

#### 3. Mathematical Model

Consider a nonhomogeneous, isotropic, magneto-thermoelastic half-space under the purview of G-N model III. Rectangular Cartesian coordinates are introduced having the surface of the half-space as the plane , with -axis pointing vertically downwards into the medium. The medium is rotating with an angular velocity . Thus the displacement equation of motion in the rotating plane has two extra terms: , which is the centripetal acceleration due to time varying motion only and , which is the Coriolis acceleration. The present formulation is restricted to -plane and thus all the field variables are independent of the space variable . So the displacement vector and angular velocity will have the components:It is also assumed that material properties are graded only in -direction. So we take as . By virtue of (9), the stresses arising from (5) can be expressed asDue to the application of an initial magnetic field , an induced magnetic field , an induced electric field , and a current density are developed in the considered medium. The simplified linear equations of electrodynamics of a slowly moving medium for a nonhomogeneous, isotropic and thermally conducting elastic solid are given by, neglecting Thomson’s effect [28]:where is the magnetic permeability of the medium.

From the above expressions, one can obtainwhereBy virtue of above expressions and replacing by , the components of the Lorentz force are given byUtilizing the components of Lorentz force into stress equation of motion along with the consideration of two-dimensional problem, the field equations (6) and (7) yieldwhere

#### 4. Exponential Variation of Nonhomogeneity

By assuming , where is a dimensionless parameter, one can conclude that the mechanical and thermal properties of the material vary exponentially along the -direction. The governing equations can be recast in the dimensionless form by introducing the following dimensionless parameters:whereNow, in terms of the dimensionless parameters given in (22), (10)-(12) and (18)-(20) transform towhere

#### 5. Solution Methodology

In this section, the normal mode method is employed, which has the advantage of finding the exact solutions without any assumed constraints on the field variables. In this approach, the solution of the physical variables is decomposed in terms of normal modes and one gets exact solution without any assumed restrictions on the actual physical quantities that appear in the governing equations of the problem considered. Normal mode analysis is, in fact, to look for the solution in Fourier transform domain. It is assumed that all the functions are sufficiently smooth on the real line such that the normal mode analysis of these functions exists. So, the solution for the considered physical variables can be decomposed in terms of normal modes in the following form:where , , , and are the amplitudes of the functions, is the angular frequency, is the imaginary unit, and is the wave number in -direction.

Introducing expression (31) in (27)-(29), we getwhereThe condition for the existence of a nonzero solution of the system of (32)-(34) provides uswhereThe general solution of (36) which is bounded as is given bywhere are parameters, depending upon and , andIn view of solution (38), stress components (24)-(26) take the formwhere

#### 6. Application: Mechanical Load on the Surface of the Half-Space

A nonhomogeneous, rotating, magneto-thermoelastic medium, occupying the half-space , has been considered. The surface of the half-space is acted upon by a mechanical load as shown in Figure 1. So, the boundary conditions are given by the following.