Applied Mathematics
Vol.4 No.1(2013), Article ID:27209,6 pages DOI:10.4236/am.2013.41019

On Rayleigh Wave in Two-Temperature Generalized Thermoelastic Medium without Energy Dissipation

Baljeet Singh1, Kiran Bala2,3

1Department of Mathematics, Post Graduate Government College, Chandigarh, India

2Department of Mathematics, Government College, Haryana, India

3Research Scholar in Mathematics, Singhania University, Rajasthan, India


Received October 8, 2012; revised November 8, 2012; accepted November 15, 2012

Keywords: Two-Temperature; Generalized Thermoelasticity; Rayleigh Wave; Energy Dissipation


In this paper, Rayleigh surface wave is studied at a stress free thermally insulated surface of a two-temperature thermoelastic solid half-space in absence of energy dissipation. The governing equations of two-temperature generalized thermoelastic medium without energy dissipation are solved for surface wave solutions. The appropriate particular solutions are applied to the required boundary conditions to obtain the frequency equation of the Rayleigh wave. Some special cases are also derived. The non-dimensional speed is computed numerically and shown graphically to show the dependence on the frequency and two-temperature parameter.

1. Introduction

Lord and Shulman [1] and Green and Lindsay [2] extended the classical dynamical coupled theory of thermoelasticity to generalized thermoelasticity theories. These theories treat heat propagation as a wave phenomenon rather than a diffusion phenomenon and predict a finite speed of heat propagation. Ignaczak and OstojaStarzewski [3] explained in detail, the above theories in their book on “Thermoelasticity with Finite Wave Speeds”. The theory of thermoelasticity without energy dissipation is another generalized theory, which was formulated by Green and Naghdi [4]. It includes the isothermal displacement gradients among its independent constitutive variables and differs from the previous theories in that it does not accommodate dissipation of thermal energy. The representative theories in the range of generalized thermoelasticity are reviewed by Hetnarski and Ignaczak [5]. Wave propagation in thermoelasticity has many applications in various engineering fields. Some problems on wave propagation in coupled or generalized thermoelasticity are studied by various researchers, for example, Deresiewicz [6], Sinha and Sinha [7], Sinha and Elsibai [8,9], Sharma, et al. [10], Othman and Song [11], Singh [12,13], and many more.

Gurtin and Williams [14,15] suggested the second law of thermodynamics for continuous bodies in which the entropy due to heat conduction was governed by one temperature, that of the heat supply by another temperature. Based on this suggestion, Chen and Gurtin [16] and Chen et al. [17,18] formulated a theory of thermoelasticity which depends on two distinct temperatures, the conductive temperature and the thermodynamic temperature T. The two-temperature theory involves a material parameter. The limit implies that and the classical theory can be recovered from two-temperature theory. The two-temperature model has been widely used to predict the electron and phonon temperature distributions in ultrashort laser processing of metals. Warren and Chen [19] stated that these two temperatures can be equal in time-dependent problems under certain conditions, whereas and are generally different in particular problems involving wave propagation. Following Boley and Tolins [20], they studied the wave propagation in the two-temperature theory of coupled thermoelasticity. They showed that the two temperatures and, and the strain are represented in the form of a travelling wave plus a response, which occurs instantaneously throughout the body. Puri and Jordan [21] discussed the propagation of harmonic plane waves in two temperature theory. Quintanilla and Jordan [22] presented exact solutions of two initial-boundary value problems in the two temperature theory with dual-phase-lag delay. Youssef [23] formulated a theory of two-temperature generalized thermoelasticity. Kumar and Mukhopadhyay [24] extended the work of Puri and Jordan [21] in the context of the linear theory of two-temperature generalized thermoelasticity formulated by Youssef [23]. Magana and Quintanilla [25] studied the uniqueness and growth of solutions in two-temperature generalized thermoelastic theories. Recently, Youssef [26] presented a theory of two-temperature thermoelasticity without energy dissipation.

In the present paper, Youssef [26] theory is applied to study the Rayleigh wave at the thermally insulated stressfree surface of an isotropic two-temperature thermoelastic solid half-space without energy dissipation. The frequency equation of the Rayleigh wave is obtained. The frequency equation is also approximated by assuming small thermal coupling. The dependence of numerical values of non-dimensional speed of the Rayleigh wave on material parameters, frequency and two-temperature parameters is shown graphically for a particular material of the model.

2. Basic Equations

We consider a two-temperature thermoelastic solid halfspace in absence of energy dissipation. Following Youssef [26], the governing equations for a two-temperature generalized thermoelastic half-space without energy dissipation are i) The heat conduction equation


ii) The displacement-strain relation


iii) The equation of motion


iv) The constitutive equations


where is the coupling parameter and is the thermal expansion coefficient. and are called Lame’s elastic constants. is the Kronecker delta. is material characteristic constant. T is the mechanical temperature, is the reference temperature. with. is the stress tensor. is the strain tensor. is the mass density. is the specific heat at constant strain. are the components of the displacement vector. is the conductive temperature and satisfies the relation


where is the two-temperature parameter. The superposed dots in the above equations denote the time derivatives. The subscripts followed by comma in these equations denote the space derivatives.

3. Analytical 2D Solution

We consider a homogeneous and isotropic two-temperature thermoelastic medium without energy dissipation of an infinite extent with Cartesian coordinates system, which is previously at uniform temperature. The origin is taken on the plane surface and the z-axis is taken normally into the medium. The surface is assumed stress-free and thermally insulated. The present study is restricted to the plane strain parallel to plane, with the displacement vector. Now, Equation (3) has the following two components in plane



The heat conduction Equation (1) is written in x-z plane as


and, Equation (5) becomes,


The displacement components u1 and u3 are written in terms of scalar potentials and as


Using Equations (9) and (10) in Equations (6) to (8), we obtain




where .

Using the following quantities

, , ,

, , ,

where, in Equations (11) to (13) and suppressing the primes, we obtain the Equations (11) to (13) in dimensionless form as




where, and


is the coefficient of thermoelastic coupling.

For thermoelastic surface waves in the half-space propagating in x-direction, the potential functions and are taken in the following form


where, is wave number and is the phase velocity.

Substituting Equation (18) in Equations (14) and (16), we obtain



where and


Eliminating from Equations (19) and (20), we obtain the following auxiliary equation



With the help of Equation (21) and keeping in mind that as for surface waves, the solutions are written as








Substituting Equation (18) in Equation (15) and keeping in mind that as for surface waves, we obtain the following solution




4. Derivation of Frequency Equation

The mechanical and thermal conditions at the thermally insulated surface are i) Vanishing of the normal stress component


ii) Vanishing of the tangential stress component


iii) Vanishing of the normal heat flux component





Equation (29) to (31) are written in non-dimensional form as




Making use of solutions (22), (23) and (27) for in the Equations (34) to (36), we obtain the following homogenous system of three equations in A, B and C




The non-trivial solution of Equations (37) to (39) exists if the determinant of the coefficients of A, B and C vanishes, i.e.,


which is the the frequency equation of thermoelastic Rayleigh wave in a two-temperature generalized thermoelastic medium without energy dissipation.

5. Special Cases

5.1. Small Thermal Coupling

In order to have an idea of the effect of two-temperature parameter on the speed of propagation of Rayleigh wave, we consider the case of small thermoelastic coupling. For most of materials, is small at normal temperature. Hence we can approximate the frequency equation by assuming. For, the Equations (24) and (25) are approximated as




, ,



With the help of these approximations for β1 and β2, the coupling coefficients and are approximated and hence the frequency Equation (40) is approximated.

5.2. Isotropic Elastic Case

If we neglect thermal parameters, then the frequency Equation (40) reduces to


which is the frequency equation of Rayleigh wave for an isotropic elastic case.

6. Numerical Example

If we put, where is the classical Rayleigh wave velocity and and are two reals, then


The velocity of propagation is equal to

and the amplitude-attenuation factor is equal to

with The non-dimensional speed of propagation is computed for the following material parameters, , , . , , x = 1 cm,.

The non-dimensional speed of Rayleigh wave is shown graphically against the range of frequency in Figure 1, when two-temperature is 0.75. With the increase in frequency, it increases very sharply at low frequency range and slowly for higher frequency range. The non-dimensional speed of Rayleigh wave is also shown graphically against the range of two-temperature parameter in Figure 2, when the frequency. With the increase in value of two-temperature parameter, it increases very slowly. It seems almost constant in Figure 2, but it increases for the whole range of the two-temperature parameter.

7. Conclusion

The appropriate solutions of the governing equations of two-temperature generalized thermoelastic medium without energy dissipation are applied at the boundary conditions at a thermally insulated free surface of a halfspace to obtain the frequency equation of Rayleigh wave. The frequency equation is approximated for the case of small thermal coupling and reduced for isotropic elastic case. From frequency equation of Rayleigh wave, it is observed that the phase speed of Rayleigh wave depends on various material parameters including the two-temperature parameter. The dependence of numerical values

Figure 1. Variation of the non-dimensional speed of Rayleigh wave against frequency when a* = 0.75.

Figure 2. Variation of the non-dimensional speed of Rayleigh wave against two-temperature parameter when χ = 0.1.

of non-dimensional speed on the frequency and twotemperature parameter is shown graphically for a particular material representing the model.


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