Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 671783, 15 pages

http://dx.doi.org/10.1155/2015/671783

## Rotation and Magnetic Field Effect on Surface Waves Propagation in an Elastic Layer Lying over a Generalized Thermoelastic Diffusive Half-Space with Imperfect Boundary

^{1}Mathematics Department, Faculty of Science, Taif University, Taif 888, Saudi Arabia^{2}Mathematics Department, Faculty of Science, SVU, Qena 83523, Egypt^{3}Mathematics Department, Faculty of Science, Zagazig University, Zagazig 44519, Egypt

Received 7 April 2014; Accepted 14 October 2014

Academic Editor: Gongnan Xie

Copyright © 2015 S. M. Abo-Dahab 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 aim of the present investigation is to study the effects of magnetic field, relaxation times, and rotation on the propagation of surface waves with imperfect boundary. The propagation between an isotropic elastic layer of finite thickness and a homogenous isotropic thermodiffusive elastic half-space with rotation in the context of Green-Lindsay (GL) model is studied. The secular equation for surface waves in compact form is derived after developing the mathematical model. The phase velocity and attenuation coefficient are obtained for stiffness, and then deduced for normal stiffness, tangential stiffness and welded contact. The amplitudes of displacements, temperature, and concentration are computed analytically at the free plane boundary. Some special cases are illustrated and compared with previous results obtained by other authors. The effects of rotation, magnetic field, and relaxation times on the speed, attenuation coefficient, and the amplitudes of displacements, temperature, and concentration are displayed graphically.

#### 1. Introduction

The foundations of magnetoelasticity were presented by Knopoff [1] and Chadwick [2] and developed by Kaliski and Petykiewicz [3]. An increasing attention is devoted to the interaction between magnetic field and strain field in a thermoelastic solid due to its many applications in the fields of geophysics, plasma physics, and related topics. All papers quoted above assumed that the interactions between the two fields take place by means of the Lorentz forces appearing in the equations of motion and by means of a term entering Ohm’s law and describing the electric field produced by the velocity of a material particle, moving in a magnetic field. The most ideal interface model, as it is known, is called perfect bond interface where the displacement and traction are continuous across the interface. However, interfaces are seldom perfect. Therefore, various imperfect models such as three-phase and linear models like spring models have been introduced by Yu et al. [4], Yu [5], and Benveniste [6]. Perhaps the most frequently studied imperfect interface model is the smooth bond interface, where the normal components of the displacements and traction are continuous across the interface, while the shear traction components are zero on the interface. Lord and Shulman [7] formulated a generalized theory of thermoelasticity with one thermal relaxation time, who obtained a wave equation by postulating a new law of heat conduction instead of classical Fourier’s law. Green and Lindsay [8] developed a temperature rate-dependent thermoelasticity that includes two thermal relaxation times and does not violate the classical Fourier’s law of heat conduction, when the body under consideration has a center of symmetry. Hetnarski and Ignaczak [9] introduced a review and presentation of generalized theories of thermoelasticity. Diffusion can be defined as the random walk of an assemble of particles from regions of high concentration to that of low concentration. Nowadays, there is a great deal of interest in the study of phenomena due to its applications in geophysics and electronic industry. In integrated circuit fabrication, diffusion is used to introduce “depants” in controlled amounts into semiconductor substance. In particular, diffusion is used to form the base and emitter in bipolar transistors, integrated resistors, and the source/drain in metal oxide semiconductor (MOS) transistors and polysilicon gates in MOS transistors. In most of the applications, the concentration is calculated using Fick’s law. This is simple law which does not take into consideration the mutual interaction between the introduced substance and the medium into which introduced. Study of the diffusion phenomenon is used to improve the conditions of oil extractions. These days’ oil companies are interested in the process of thermoelastic diffusion for more efficient extraction of oil from oil deposits. Until recently, thermodiffusion in solids, especially in metals, was considered as a quantity that is independent of body deformation. Practice, however, indicates that the process of thermodiffusion could have a very considerable influence on the deformation of the body. Thermodiffusion in elastic solid is due to the coupling of temperature, mass diffusion, and strain in addition to the exchange of heat and mass with the environment. Nowacki [10–13] developed the theory of thermoelastic diffusion by using coupled thermoelastic model. This implies infinite speed of propagation of thermoelastic waves. Olesiak and Pyryev [14] investigated the theory of thermoelastic diffusion and coupled quasistationary problems of thermal diffusion for an elastic layer. They studied the influence of cross effects arising from the coupling of the fields of temperature, mass diffusion, and strain due to which the thermal excitation results in additional mass concentration and generates additional fields of temperature. Sherief et al. [15] developed the generalized theory of thermoelastic diffusion with one relaxation time which allows finite speeds of propagation of waves.

Recently, Sherief and Saleh [16] investigated the problem of a thermoelastic half-space in the context of the theory of generalized thermoelastic diffusion with one relaxation time. Singh [17] discussed the reflection phenomena of waves from free surface of a thermoelastic diffusion with one relaxation time and with two relaxation times in [18]. Aouadi [19–21] investigated different problems in thermoelastic diffusion. Sharma and Walia [22, 23] discussed the effect of rotation on Rayleigh waves in the piezothermoelastic half-space. Kumar and Kansal [24] discussed the propagation of Rayleigh waves on free surface in transversely isotropic thermoelastic diffusion. Kumar and Kansal [25] derived the basic equations for generalized thermoelastic diffusion and discussed the Lamb waves. Dawn and Chakraborty [26] studied Rayleigh waves in Green-Lindsay’s model of generalized thermoelastic media. Kumar and Chawla [27] investigated the effect of rotation and stiffness on surface waves propagation in an elastic layer lying over a generalized thermodiffusive elastic half-space with imperfect boundary. New contributions on waves propagation in thermoelastic media have been discussed [28–32].

In this paper, linear model is adopted to represent the imperfectly bonded interface conditions. The linear model is simplified and idealized situation of imperfectly bonded interface, where the discontinuities in displacements at interfaces have a linear relationship with the interface stresses. Taking these applications into account, the surface waves propagation at imperfect boundary between an isotropic elastic layer and isotropic thermodiffusive elastic half-space with rotation in the context of Green-Lindsay theory is investigated. The phase velocity and attenuation coefficients of wave propagation have been computed from the secular equations. The amplitudes of displacements, temperature, concentration, and specific loss are computed and depicted graphically to make clear the influence of magnetic field, rotation, stiffness, relaxation times, and diffusion on the phenomena and compare with the practical results.

#### 2. Basic Equations

The basic governing equations for homogenous generalized thermodiffusive solid in the absence of heat and mass diffusion sources are as follows (Singh [17]).(i)Constitutive relations are(ii)Equation of motion in the rotating frame of reference is where is the centripetal acceleration due to the time varying motion only and is the Coriolis acceleration:whereConsider that the medium is a perfect electric conductor; we take the linearized Maxwell equations governing the electromagnetic field, taking into account absence of the displacement current (SI): where where we have usedthen(iii)Equation of heat conduction is(iv)Equation of mass diffusion is

Here, the medium is rotating with angular velocity , where is the unit vector along the axis of rotation and this equation of motion includes two additional terms, namely,(i)the centripetal acceleration due to time-varying motion,(ii)the Carioles acceleration ,

where and , and are Lame’s constants, is the coefficient of linear thermal expansion, and are, respectively, the density and specific heat at constant strain, , are, respectively, coefficient describing the measure of thermoelastic diffusion effects and of diffusion effects, is the reference temperature assumed to be such that , , are thermal relaxation times with and , are diffusion relaxation times with , and are components of displacement vector. is the temperature change and is the concentration; , are, respectively, the components of stress and strain tensor.

The symbols correspond to partial derivative and time derivative, respectively.

Following Bullen [33], the equations of motion and constitutive relations in isotropic elastic medium are given bywhereand is the displacement vector, is the density of the isotropic medium and are Lame’s constants, are components of stress tensor, and is the Kronecker delta.

#### 3. Formulation of the Problem

As shown in Figure 1, we consider an isotropic elastic layer (Medium ) of thickness overlaying a homogeneous, isotropic, generalized thermodiffusive elastic half-space in rotating frame of reference (Medium ). The origin of the coordinate system is taken at any point on the horizontal surface and -axis in the direction of wave propagation and -axis taking vertically downward into half-space, so that all particles on a line parallel to -axis are equally displaced. Therefore, all the field quantities will be independent of -axis coordinate. The interface between isotropic elastic layer and thermodiffusive elastic half-space with rotation has been taken at an imperfect boundary. The displacement vector , temperature , concentration , and rotation for medium are taken asand displacement vector for the layer (Medium ) is taken asWe define the dimensionless quantities