Propagation of Rayleigh Wave in a Two-Temperature Generalized Thermoelastic Solid Half-Space

Abstract

The Rayleigh surface wave is studied at a stress-free thermally insulated surface of an isotropic, linear, and homogeneous two-temperature thermoelastic solid half-space in the context of Lord and Shulman theory of generalized thermoelasticity. The governing equations of a two-temperature generalized thermoelastic medium 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 speed of Rayleigh wave 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 thermoelastic 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 Ostoja-Starzewski [3] explained in detail the above theories in their book Thermoelasticity with Finite Wave Speeds. The representative theories in the range of generalized thermoelasticity are reviewed by Hetnarski and Ignaczak [4]. 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 [5], Sinha and Sinha [6], Sinha and Elsibai [7, 8], Sharma et al. [9], Othman and Song [10], Singh [11, 12], and many more.

Gurtin and Williams [13, 14] proposed 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 law, Chen and Gurtin [15] and Chen et al. [16, 17] formulated a theory of thermoelasticity which depends on two distinct temperatures, the conductive temperature Φ and the thermodynamic temperature 𝑇. The two-temperature theory involves a material parameter 𝑎^∗>0. The limit 𝑎^∗→0 implies that Φ→𝑇, 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. According to Warren and Chen [18], these two temperatures can be equal in time-dependent problems under certain conditions, whereas Φ and 𝑇 are generally different in particular problems involving wave propagation. Following Boley and Tolins [19], they studied the wave propagation in the two-temperature theory of coupled thermoelasticity. They showed that the two temperatures, 𝑇 and Φ, and the strain are represented in the form of a traveling wave plus a response, which occurs instantaneously throughout the body. Puri and Jordan [20] studied the propagation of harmonic plane waves in two-temperature theory. Quintanilla and Jordan [21] derived exact solutions of two initial-boundary value problems in the two-temperature theory with dual-phase-lag delay. Youssef [22] developed a theory of two-temperature generalized thermoelasticity. Kumar and Mukhopadhyay [23] extended the work of Puri and Jordan [20] in the context of the linear theory of two-temperature generalized thermoelasticity developed by Youssef [22]. Magaña and Quintanilla [24] showed the uniqueness and growth of solutions in two-temperature generalized thermoelastic theories. Recently, Youssef [25] formulated a theory of two-temperature thermoelasticity without energy dissipation.

In this paper, the Youssef [22] theory is followed for the theoretical study of the Rayleigh wave at the thermally insulated stress-free surface of an isotropic two-temperature thermoelastic solid half-space. 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 the 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 an isotropic, linear, and homogeneous two-temperature thermoelastic solid half-space. Following Youssef [22], the governing equations for a two-temperature generalized thermoelastic half-space with one relaxation time are as follows:(i)the stress-strain-temperature relations 𝜎_𝑖𝑗=𝑐_{𝑖𝑗𝑘𝑙}𝑒_𝑘𝑙−𝛾_𝑖𝑗𝑇−Φ₀, (1)(ii)the displacement-strain relation 𝑒_𝑖𝑗=12𝑢_𝑖,𝑗+𝑢_𝑗,𝑖, (2)(iii)the equation of motion 𝜌𝑢̈_𝑖=𝜎_{𝑗𝑖,𝑗}+𝜌𝐹_𝑖, (3)(iv)the energy equation −𝑞_𝑖,𝑖=𝜌𝑇₀𝑆,̇ (4)(v)the modified Fourier’s law −𝐾_𝑖𝑗Φ_,𝑗=𝑞_𝑖+𝜏₀𝑞̇_𝑖, (5)(vi)the entropy-strain-temperature relation 𝜌𝑆=𝜌𝑐_𝐸𝑇₀𝜃+𝛾_𝑖𝑗𝑒_𝑖𝑗. (6)Here, 𝛾_𝑖𝑗 are the coupling parameters, 𝑇 is the mechanical temperature, Φ₀=𝑇₀ is the reference temperature, 𝜃=𝑇−𝑇₀ with |𝜃/𝑇₀|≪1, 𝜎_𝑖𝑗 is the stress tensor, 𝑒_𝑘𝑙 is the strain tensor, 𝑐_{𝑖𝑗𝑘𝑙} is the tensor of elastic constants, 𝜌 is the mass density, 𝑞_𝑖 is the heat conduction vector, 𝐾_𝑖𝑗 is the thermal conductivity tensor, 𝑐_𝐸 is the specific heat at constant strain, 𝑢_𝑖 are the components of the displacement vector, 𝑆 is the entropy per unit mass, 𝜏₀ is the thermal relaxation time, which will ensure that the heat conduction equation will predict finite speeds of heat propagation, and Φ is the conductive temperature and satisfies the relation Φ−𝑇=𝑎^∗Φ_,𝑖𝑖, (7) where 𝑎^∗>0 is the two-temperature parameter.

3. Analytical 2D Solution

We consider a homogeneous and isotropic thermoelastic medium of an infinite extent with Cartesian coordinates system (𝑥,𝑦,𝑧), which is previously at uniform temperature. The origin is taken on the plane surface, and 𝑧-axis is taken normally into the medium (𝑧≥0). The surface 𝑧=0 is assumed to be stress-free and thermally insulated. The present study is restricted to the plane strain parallel to 𝑥-𝑧 plane, with the displacement vector 𝐮=(𝑢₁,0,𝑢₃). With the help of (1)–(3), we obtain the following two components of the equation of motion: 𝜆+2𝜇𝑢_1,11+𝜆+𝜇𝑢_3,13+𝜇𝑢_1,33−𝛾𝜃_,1=𝜌𝑢̈₁,𝜆+2𝜇𝑢_3,33+𝜆+𝜇𝑢_1,13+𝜇𝑢_3,11−𝛾𝜃_,3=𝜌𝑢̈₃. (8) Equations (4)–(6) lead to the following heat conduction equation: 𝐾Φ_,11+Φ_,33=𝜌𝑐_𝐸𝜃̇+𝜏₀𝜃̈+𝛾𝑇₀𝑢̇_1,1+𝜏₀𝑢̈_1,1+𝛾𝑇₀𝑢̇_3,3+𝜏₀𝑢̈_3,3, (9) and (7) becomes Φ−𝑇=𝑎^∗Φ_,11+Φ_,33. (10) The displacement components 𝑢₁ and 𝑢₃ are written in terms of potentials 𝑞 and 𝜓 as 𝑢₁=𝜕𝑞−𝜕𝑥𝜕𝜓𝜕𝑧,𝑢₃=𝜕𝑞+𝜕𝑧𝜕𝜓.𝜕𝑥 (11) Using (11) in (8)–(10), we obtain 𝜕𝜆+2𝜇²𝑞𝜕𝑥²+𝜕²𝑞𝜕𝑧²−𝛾Φ−𝑎^∗𝜕²Φ𝜕𝑥²+𝜕²Φ𝜕𝑧²𝜕=𝜌²𝑞𝜕𝑡²,𝐾Φ_,11+Φ_,33=𝜌𝑐_𝐸𝜕Φ𝜕𝑡+𝜏₀𝜕²Φ𝜕𝑡²−𝑎^∗𝜌𝑐_𝐸1+𝜏₀𝜕𝜕𝜕𝑡𝜕𝜕𝑡²Φ𝜕𝑥²+𝜕²Φ𝜕𝑧²+𝛾𝑇₀1+𝜏₀𝜕𝜕𝜕𝑡𝜕𝜕𝑡²𝑞𝜕𝑥²+𝜕²𝑞𝜕𝑧²,𝜇𝜕²𝜓𝜕𝑥²+𝜕²𝜓𝜕𝑧²𝜕=𝜌²𝜓𝜕𝑡². (12)(13)(14)

For thermoelastic surface waves in the half-space propagating in 𝑥-direction, the potential functions Φ, 𝑞, and 𝜓 are taken in the following form: Φ,𝑞,𝜓=Φ(𝑧),𝑞(𝑧),𝜓(𝑧)exp𝑖𝜂𝑥−𝜒𝑡, (15) where 𝜒²=𝜂²𝑐², 𝜂 is the wave number, and 𝑐 is the phase velocity.

Substituting (15) with (12) and (13) and eliminating 𝑞, Φ, we obtain the following auxiliary equation: 𝐷⁴−𝐴𝐷²+𝐵=0, (16) where 𝐷=𝑑/𝑑𝑧, 𝐴=2𝜂²+𝜂²𝑐²𝑐₁²+𝜖+𝐾−𝑎^∗𝜒²𝑎^∗𝜖𝜒²−𝑐₁²𝐾−𝑎^∗𝜒²,𝐵=𝜂⁴+𝜂⁴𝑐²𝑐₁²+𝜖+𝐾−𝑎^∗𝜒²𝑎^∗𝜖𝜒²−𝑐₁²𝐾−𝑎^∗𝜒²−𝜂⁴𝑐⁴1𝑎^∗𝜖𝜒²−𝑐₁²𝐾−𝑎^∗𝜒²,𝜖=𝛾²𝑇₀𝑐_𝐸𝜏^∗,𝐾𝐾=𝜌𝑐_𝐸𝜏^∗,𝜏^∗=𝜏₀+𝑖𝜒,𝛾𝛾=𝜌,𝑐₁²=𝜆+2𝜇𝜌. (17) With the help of (16) and keeping in mind that 𝑞,Φ→0 as 𝑧→∞ for surface waves, the solutions 𝑞, Φ are written as 𝑞=𝐴exp−𝜂𝛽₁𝑧+𝐵exp−𝜂𝛽₂𝑧,𝜁exp𝜄𝜂𝑥−𝜒𝑡Φ=₁𝐴exp−𝜂𝛽₁𝑧+𝜁₂𝐵exp−𝜂𝛽₂𝑧exp𝜄𝜂𝑥−𝜒𝑡, (18) where 𝛽₁²=12𝜂²𝐴+𝐴²,𝛽−4𝐵₂₂=12𝜂²𝐴−𝐴²,𝜁−4𝐵_𝑖=𝜒²−𝑐₁²1−𝛽_𝑖²𝜂²𝛾1+𝑎^∗𝜂²1−𝛽_𝑖²,(𝑖=1,2). (19)(20) Substituting (15) with (14) and keeping in mind that 𝜓→0 as 𝑧→∞ for surface waves, we obtain the following solution: 𝜓=𝐶exp−𝜂𝛽₃𝑧+𝜄𝜂𝑥−𝜒𝑡, (21) where 𝛽₃²𝑐=1−²𝑐₂²,𝑐₂²=𝜇𝜌. (22)

4. Derivation of Frequency Equation

The mechanical and thermal conditions at the thermally insulated surface 𝑧=0 are as follows:(i)vanishing of the normal stress component 𝜎_𝑧𝑧=0, (23)(ii)vanishing of the tangential stress component 𝜎_𝑧𝑥=0, (24)(iii)vanishing of the normal heat flux component 𝜕Φ𝜕𝑧=0, (25)where 𝜎_𝑧𝑧𝜕=𝜆²𝑞𝜕𝑥²+𝜕²𝑞𝜕𝑧²𝜕+2𝜇²𝜓𝜕𝜕𝑥𝜕𝑧+2𝜇²𝑞𝜕𝑧²−𝛾Φ−𝑎^∗𝜕²Φ𝜕𝑥²+𝜕²Φ𝜕𝑧²,𝜎_𝑧𝑥2𝜕=𝜇²𝑞−𝜕𝜕𝑥𝜕𝑧²𝜓𝜕𝑧²+𝜕²𝜓𝜕𝑥². (26) Making use of solutions (18) and (21) for 𝑞, Φ, and Ψ in (23) to (25) and eliminating 𝐴, 𝐵, and 𝐶, we obtain the following equation: 4𝜇𝜂²𝛽₁𝛽₂𝛽₃𝜁₂−𝜁₁−1+𝛽₃²×𝜂²−𝜆𝜁₂𝛽₂−𝜁₁𝛽₁+𝜆+2𝜇𝛽₁𝛽₂𝜁₂𝛽₁−𝜁₁𝛽₂−𝜌𝑐₁²𝜁₁𝜁₂𝛽₂−𝛽₁1+𝑎^∗𝜂²1+𝛽₁𝛽₂=0, (27) which is the frequency equation of thermoelastic Rayleigh wave in a two-temperature generalized thermoelastic medium in the context of the Lord and Shulman [1] theory.

5. Special Cases

5.1. Small Thermal Coupling (𝜖≪1)

For most of materials, 𝜖 is small at a normal temperature. For 𝜖≪1 and using (19), we obtain the following approximated relations as 𝛽₁²+𝛽₂²≡2−𝑐²𝑐₁²+𝐾−𝑎^∗𝜒²𝑐₁²𝐾−𝑎^∗𝜒²×11+𝜖𝑐₁²+𝐾−𝑎^∗𝜒²+𝑎^∗𝜒²𝑐₁²𝐾−𝑎^∗𝜒²,𝛽₁²𝛽₂²≡1−𝑐²𝑐₁²+𝐾−𝑎^∗𝜒²𝑐₁²𝐾−𝑎^∗𝜒²×11+𝜖𝑐₁²+𝐾−𝑎^∗𝜒²+𝑎^∗𝜒²𝑐₁²𝐾−𝑎^∗𝜒²+𝑐⁴𝑐₁²𝐾−𝑎^∗𝜒²𝑎1+𝜖^∗𝜒²𝑐₁²𝐾−𝑎^∗𝜒². (28) With the help of these approximations, we can approximate 𝛽₁, 𝛽₂, and the coupling coefficients 𝜁₁ and 𝜁₂. Finally, the frequency equation (27) can be approximated. Further, if we consider 𝜖→0, 𝑎^∗→0, we obtain 𝛽₁≡1−𝑐²/𝑐₁² and 𝛽₂≡1−𝑐²/𝐾.

For numerical purpose, we put 𝑐²=𝑐^∗2+𝜖(𝜉₁+𝑖𝜉₂), where 𝑐^∗ is the classical Rayleigh wave velocity, and 𝜉₁ and 𝜉₂ are two reals, then 𝜒𝜂=𝑐^∗1−𝜖𝜉₁2𝑐^∗2−𝑖𝜖𝜉₂2𝑐^∗2. (29) The velocity of propagation is equal to (𝑐^∗+𝜖𝜉₁/2𝑐^∗), and the amplitude attenuation factor is equal to exp[𝜖𝜒𝜉₂𝑥/2𝑐^∗3] with 𝜉₂<0.

5.2. Small Frequency (𝜒≪1)

For 𝜒≪1 and using (19), we obtain the following approximated relations as 𝛽₁²+𝛽₂²≡2−𝑐²𝑐₁²+𝜖+𝐾𝑐₁²𝐾×1+𝜒²(1+𝜖)𝑎^∗𝑐₁²𝐾−𝑎^∗𝑐₁²+𝜖+𝐾,𝛽₁²𝛽₂²≡1−𝑐²𝑐₁²+𝜖+𝐾𝑐₁²𝐾×1+𝜒²(1+𝜖)𝑎^∗𝑐₁²𝐾−𝑎^∗𝑐₁²+𝜖+𝐾+𝑐⁴𝑐₁²𝐾1+𝜒²(1+𝜖)𝑎^∗𝑐₁²𝐾. (30) With the help of these approximations, we can approximate 𝛽₁, 𝛽₂, and the coupling coefficients 𝜁₁ and 𝜁₂. Finally, the frequency equation (27) can be approximated. Also, if we consider 𝜒→0, 𝜖→0, we obtain 𝛽₁≡1−𝑐²/𝑐₁² and 𝛽₂≡1−𝑐²/𝐾.

5.3. Isotropic Elastic Case

If we neglect thermal parameters, then the frequency equation (27) reduces to 𝑐2−²𝑐₂²²=4𝑐1−²𝑐₁²𝑐1−²𝑐₂², (31) which is the frequency equation of Rayleigh wave for an isotropic elastic case.

6. Numerical Example

The speed of propagation is computed for the following material parameters of aluminium metal: 𝜆=5.775×10¹¹ Dyn·cm⁻², 𝜇=2.646×10¹¹ Dyn·cm⁻², 𝜌=2.7 g·cm⁻³, 𝑐_𝐸=0.236 Cal·g⁻¹·°C⁻¹, 𝐾=0.492 Cal·cm⁻¹·s⁻¹·°C⁻¹, 𝛾=0.05(3𝜆+2𝜇), 𝑇₀=27°C, and 𝑐^∗=0.9554.

The nondimensional speed of Rayleigh wave is shown graphically against the frequency (2–20 Hz) in Figure 1, when the two-temperature parameter 𝑎^∗=0.5. With the increase in frequency, it increases very sharply to its maximum value near 𝜒=4.3 Hz, and then it decreases slowly for the higher frequency range. The nondimensional speed of Rayleigh wave is also shown graphically against the two-temperature parameter 𝑎^∗ in Figure 2 for 𝜒=10 Hz. With the increase in the value of two-temperature parameter, it increases very sharply to its maximum value at 𝑎^∗=0.47. Thereafter, it decreases sharply to its minimum value.

Figure 1 

Variations of the nondimensional speed of propagation of Rayleigh wave against the frequency (𝜒) when 𝑎∗=0.5.

Figure 2 

Variations of the nondimensional speed of propagation of Rayleigh wave against the two-temperature parameter (𝑎∗) when 𝜒=10 Hz.

7. Conclusion

The appropriate solutions of all the governing equations of a two-temperature generalized thermoelastic medium are applied at the boundary conditions at a thermally insulated free surface of a half-space to obtain the frequency equation of Rayleigh wave in the context of the Lord and Shulman [1] theory. The frequency equation is approximated for the case of small thermal coupling and small frequency and reduced for isotropic elastic case. From the 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 speed of propagation on the frequency and two-temperature parameter is shown graphically for a particular material representing the model. The present problem is of geophysical interest, particularly in investigations concerned with earthquakes and other phenomena in seismology. The applications range from geophysical problems to quantitative nondestructive evaluation of mechanical structures and acoustic tomography for medical purposes. The problem though is theoretical, but it can provide useful information for experimental researchers working in the field of geophysics and earthquake engineering and seismologist working in the field of mining tremors and drilling into the crust of the earth.