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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.

Lord and Shulman [

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 [

In the present paper, Youssef [

We consider a two-temperature thermoelastic solid halfspace in absence of energy dissipation. Following Youssef [

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.

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 u_{1} and u_{3} 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

where

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

(22)

(23)

where

and

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

where

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

where

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

(37)

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

(40)

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

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

where

, ,

,

.

With the help of these approximations for β_{1} and β_{2}, the coupling coefficients and are approximated and hence the frequency Equation (40) is approximated.

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.

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

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

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