Rayleigh surface wave propagation in orthotropic

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Feb 10, 2017 - homogeneous, orthotropic thermoelastic half-space in the context of three- phase-lag ..... (c44 + c13)u1,13 + c44u3,11 + c33u3,33 β3T,3 = ρ.
Journal of Thermal Stresses

ISSN: 0149-5739 (Print) 1521-074X (Online) Journal homepage: http://www.tandfonline.com/loi/uths20

Rayleigh surface wave propagation in orthotropic thermoelastic solids under three-phase-lag model Siddhartha Biswas, Basudeb Mukhopadhyay & Soumen Shaw To cite this article: Siddhartha Biswas, Basudeb Mukhopadhyay & Soumen Shaw (2017) Rayleigh surface wave propagation in orthotropic thermoelastic solids under three-phase-lag model, Journal of Thermal Stresses, 40:4, 403-419, DOI: 10.1080/01495739.2017.1283971 To link to this article: https://doi.org/10.1080/01495739.2017.1283971

Published online: 10 Feb 2017.

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JOURNAL OF THERMAL STRESSES 2017, VOL. 40, NO. 4, 403–419 http://dx.doi.org/10.1080/01495739.2017.1283971

Rayleigh surface wave propagation in orthotropic thermoelastic solids under three-phase-lag model Siddhartha Biswas, Basudeb Mukhopadhyay, and Soumen Shaw Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal, India ABSTRACT

ARTICLE HISTORY

The present article deals with the propagation of Rayleigh surface waves in a homogeneous, orthotropic thermoelastic half-space in the context of threephase-lag model of thermoelasticity. The frequency equations in closed form are derived and the amplitude ratios of surface displacements and temperature change during the Rayleigh wave propagation on the surface of halfspace have been computed analytically. The path of particles during Rayleigh wave propagation is found elliptical and eccentricity of the ellipse is derived. To illustrate the analytical developments, the numerical solution is performed and the computer-simulated results in respect of phase velocity, attenuation coefficient, and specific loss are presented graphically.

Received 21 April 2016 Accepted 16 January 2017 KEYWORDS

Attenuation coefficient; frequency equation; normal mode analysis; orthotropic medium; path of surface particles; Rayleigh waves; three-phase-lag model

Introduction It is well known that for high heat flux and low temperature cases, Fourier’s law of heat conduction is not appropriate one. To overcome these difficulties several generalized heat conduction models were proposed by several authors. In 1967, Lord and Shulman proposed first generalization of heat conduction model with one relaxation time. Green and Lindsay proposed another generalization by incorporating temperature gradient term into the constitutive relations. A detailed information regarding generalized heat conduction model in the various fields can be found in the monographs of Chandrasekharaiah [1, 2] and Ignaczak and Hetnarski [3]. In the advancement of heat transfer technology, high rate heating is a rapidly emerging area. In the nonequilibrium thermodynamic heat transport, shortening the response time and the macroscopic effects in the energy exchange are the two important aspects. In the high-rate laser heating, the temperature of metal lattice may remain undisturbed while the energy exchange between the phonons and free electrons takes place. Relative to time at which electrons start to receive the phonon energy from the laser sources, enhancement of lattice temperature is delayed due to the phonon–electron interactions of the microscopic level. Thus, a microscopic level time lag is possible in between the temperature gradient, heat flux vector, and thermal displacement (for details see Shaw and Mukhopadhyay [4]). In three-phase-lag (TPL) heat conduction equation, the Fourier law of heat conduction is replaced by an approximation of TPLs for the heat flux vector (τq ), temperature gradient (τT ), and thermal displacement gradient (τν ). The previous established models can be obtained as special cases from this more general model. TPL model is very much useful in the problems of nuclear boiling, exothermic catalytic reactions, phonon–electron interactions, phonon scattering, etc. In this model, the delay time τq captures the thermal wave behavior (a small-scale response in time) and the phase lag τT captures the effect of phonon–electron interactions (a microscopic response in space). The other delay time τν is considered as a constitutive variable, known as phase lag for the thermal displacement gradient. CONTACT Siddhartha Biswas [email protected] Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Botanic Garden, Howrah 711103, West Bengal, India. Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/uths. © 2017 Taylor & Francis

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The coupling between the temperature and strain fields influences both the form of surface wave and its velocity of propagation. The wave propagation in thermoelastic media is applicable in various fields such as earthquake engineering, soil dynamics, nuclear reactors, electronic components, resonators, high-energy particle accelerators, driving microfluidic actuation, and a variety of processes or in the non-destructive evaluation in material process control and fabrication. It is well known that the study of surface waves in elastic solids is of geophysical interest and that the investigation of thermal effects on elastic wave propagation possesses a great importance in many seismological and astrophysical problems. In seismology, surface waves traveling along the earth’s surface play an important role, since they can be the most destructive type of seismic wave produced by earthquakes. Ivanov and Savona [5] and Abo-Dahab [6] studied surface wave propagation in thermoelastic half-space. Temperature ratedependent thermoelastic Rayleigh waves is studied by Chandrasekharaiah and Srikantaiah [7]. Dwan and Chakraborty [8] investigated Rayleigh wave propagation in semi-infinite solids with Green– Lindsay’s model of generalized thermoelasticity and obtained approximations to the frequency equations for different ranges of the parameters involved. Wojnar [9] discussed the propagation of Rayleigh waves by taking into account the thermal relaxation times. The influence of heat conduction and thermal relaxation on the propagation of surface waves polarized in the sagittal plane along the heat-insulated surfaces of the thermoelastic bodies of revolution is investigated by Rossikin and Shitikova [10]. In their article, the modified Maxwell law is used as the law of heat conduction, which allows one to take a finite speed of heat propagation into account. The nonstationary surface waves are interpreted as lines (a straight line or a diverging or converging circumference) on which the temperature and components of the stress and strain tensors experience a discontinuity. Chadwick and Windle [11] as well as Sharma et al. [12] studied the effects of heat conduction upon the propagation of Rayleigh surface waves in a semi-infinite elastic solid. Sharma and Kaur [13] considered Rayleigh waves in rotating thermoelastic solids with voids. The effect of micropolarity, microstretch, and relaxation times on Rayleigh surface waves is discussed by Sharma et al. [14]. Shaw and Mukhopadhyay [15] investigated the electromagnetic effects on Rayleigh surface wave propagation in a homogeneous isotropic thermo-microstretch elastic half-space. Over the few decades, anisotropic materials have been increasingly used. The study of wave propagation in anisotropic materials has been a subject of extensive investigation in the literature. It is of great importance in a variety of applications ranging from seismology to nondestructive testing of composite structures used in aircraft, spacecraft, or other engineering industries. Anisotropy creates qualitatively new properties of elastic waves and acoustic phenomena that have not got close analogous in isotropic media. Some of them have already found their practical applications in real devices. A theoretical description of elastic waves in anisotropic material is a very nontrivial problem. However, the study of the influence of anisotropy upon the Rayleigh waves in a thermoelastic half-space can furnish information that may be useful for experimental seismologists in correcting earthquake estimations. Chakraborty and Pal [16] as well as Chadwick and Seet [17] discussed Rayleigh wave propagation in transversely isotropic medium. Abd-Alla et al. [18] studied the propagation of Rayleigh waves in magnetothermoelastic half-space of a homogeneous orthotropic material under the effect of rotation, initial stress, and gravity field. Kumar and Kansal [19] discussed the propagation of Rayleigh waves in a homogeneous, transversely isotropic, thermoelastic diffusive half-space, subject to stress-free, thermally insulated boundary in the context of generalized thermoelastic diffusion theory. Chirita [20] investigated thermoelastic harmonic Rayleigh surface waves propagating in a homogeneous anisotropic half-space along a plane when the direction of propagation coincides with one of the crystallographic axes and showed that the phase velocity of the seismic waves is influenced by the thermal effects, and the surface waves are damped in time. Surface waves are successfully studied by Stroh formalism [21]. The Stroh formalism is a powerful and elegant mathematical method developed for the analysis of equations of anisotropic elasticity. Recently, Destrade [22], Tanuma [23] and Ting [24, 25] have used the Stroh formalism to study the wave propagation in the classical elasticity on an exponentially graded anisotropic half-space. Tanuma [23] and Ting [24, 25] showed that the anisotropy brings out new features in

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the analysis of surface wave propagation. Secular equations of Rayleigh waves in a nonhomogeneous orthotropic elastic medium under the influence of gravity are studied by Vinh and Seriani [26]. AbdAlla and Ahmed [27] and Abd-Alla et al. [28] studied Rayleigh waves in an orthotropic thermoelastic medium under the influence of gravity, magnetic field, and initial stress. The effects of heat conduction upon the propagation of Rayleigh surface waves in a semi-infinite elastic solid is studied for transversely isotropic materials by Sharma et al. [29] and Sharma and Singh [30]. Kumar and Kansal [31] investigated Lamb wave propagation in transversely isotropic medium. Kumar and Gupta [32] investigated the effect of phase lags on Rayleigh wave propagation in thermoelastic medium. Abouelregal [33] studied Rayleigh wave propagation in thermoelastic half-space in the context of dual-phase-lag model. The effect of viscosity in anisotropic thermoelastic medium with TPL model has been studied by Kumar et al. [34]. Kumar and Chawla [35] examined plane wave propagation in anisotropic thermoelastic medium under TPL and two-phase-lag models. Singh et al. [36] considered Rayleigh surface wave propagation in transversely isotropic media under dual-phase-lag model. Singh and Renu [37] examined the effect of initial stress on surface wave propagation. Propagation of waves in transversely isotropic medium under G–N theory is considered by Gupta [38]. Sharma et al. [39] studied the effect of viscosity on wave propagation in anisotropic thermoelastic with Green–Naghdi theory type-II and type-III. Recently, Shaw and Mukhopadhyay [4] analyzed Rayleigh surface wave propagation in isotropic micropolar solid under TPL model of thermoelasticity. Rayleigh surface wave propagation in orthotropic thermoelastic medium under TPL model has not been attempted so far. In the present investigation, the propagation of Rayleigh waves in orthotropic thermoelastic half-space in the context of TPL model has been considered. Different frequency equations are derived and the path of surface particles is found as elliptical. Numerical results for the different characteristics of waves like phase velocity, attenuation coefficient, and specific loss are computed numerically, and the effect of phase lags on them are presented graphically for various thermoelastic models. This study may find its applications in the design of surface acoustic waves devices, structural health monitoring, and damage characterization of materials.

Fundamental equations The basic governing equations of anisotropic TPL thermoelastic model in the absence of body forces and heat sources are the following: (a) The equation of motion: τij,j = ρ

∂ 2 ui ∂t 2

(1)

(b) The stress–strain temperature relation: τij = cijkl ekl − βij T

(2)

(c) The strain–displacement relation: eij = (d) The energy equation:

 1 ui,j + uj,i 2

− qi,i = ρT0

∂S ∂t

(e) The modified Fourier law with TPL: !     τq2 ∂ 2 ∂ ∂ ∂ ∗ − Kij 1 + τT T,ij − Kij 1 + τν ν,ij = 1 + τq + qi,i ∂t ∂t ∂t 2 ∂t 2 where

∂ν,ij ∂t

= T,ij .

(3)

(4)

(5)

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(f) The energy–strain temperature relation: ρCe T + βij eij (6) T0 where T is the temperature above reference temperature, T0 is the reference uniform temperature of T the body chosen such that T0