Effect of initial stresses on the elastic waves in transversely isotropic thermoelastic materials

In the present article, we have studied the reflection/transmission of elastic waves in initially stressed transversely isotropic thermoelastic materials. Three quasi type coupled longitudinal (QL), transverse (QT), and thermal waves are found to propagate in initially stressed transversely isotropic thermoelastic materials. For incident QL and QT‐waves at a plane interface, boundary conditions were implemented for obtaining the coefficients of reflection/transmission, the distribution of energy in the reflected and transmitted waves are also discussed. We have observed that the results vary with direction of incidence as well as the parameters due to elasticity, thermal, and initial stresses. Numerical computations have been performed and analyzed the impact of initial stresses on the results. It has been observed critical angles at θ0=30° and 58° for the reflected and transmitted QL‐waves for incident QT‐wave.


INTRODUCTION
Thermoelasticity discusses heat conduction, strains, and thermal stresses in the materials with the inverse effect of temperature distribution. The study of thermoelastic materials has been implemented in many important fields such as seismology, soil dynamics, physical sciences, aeronautics, atomic smasher, and nuclear reactors. With the help of thermoelastic admittance matrix, 1 the effect of the thermoelastic system to mechanical or thermal is presented. Lord and Shulman 2 considered the relations between temperature and strain to generalize the theory of thermoelastic materials. The theory of thermoelastic materials was linearized by Green and Lindsay, 3 they also proved the uniqueness for the theorem. Dhaliwal and Wang 4 presented the theory of linear generalized dipolar thermoelastic materials having initial stress. The uniqueness theorem and reciprocity relation 5 are proved for the generalized thermoelasticity under initial stresses. Researchers like Dhaliwal and Sherief, 6 Chandrasekharaiah, 7 and Hetnarski 8 also contributed in the development of the theory of thermoelasticity. Two quasiwaves, namely, longitudinal (QL) and transverse (QT) and a purely transverse wave can propagate in the transversely isotropic heat-conducting elastic material. 9 McCarthy 10 discussed the general properties of accelerated waves in generalized thermoelastic materials. Acceleration waves are isothermal or isentropic 11 for the elastic materials with heat conduction. Frequency equation 12 of surface waves in thermoelastic anisotropic materials is obtained. The velocity of propagation of the body waves in a thermally conducting transversely isotropic elastic solid are depicted numerically. 13 Singh 14 obtained the propagation speeds for quasi-P and S waves in the generalized thermoelastic materials considering two relaxation times. He also generalized the theory of thermoelasticity for transversely isotropic heat conducting materials under initial stress to analyzed the reflected waves. 15 For isothermal and thermally insulated, the nature of Rayleigh wave has been investigated in a generalized thermoelastic half-space. 16 In a void containing viscoelastic heat conducting material, Tomar et al 17 found four basic waves having different propagation speeds.
For identification of valuable materials buried inside the earth as well as for inspecting hydrocarbons, the technique of wave propagation has been employed. For incident P and SV-waves, Sinha and Sinha 18 investigated the properties of reflection coefficients in thermoelastic solid half-space. The propagation of P-wave is affected due to presence of thermal coupling in thermoelastic materials with two relaxation times, but SV-wave remains unaffected. 19 Tomar and Singh 20 obtained the reflection/transmission coefficients for incident longitudinal wave and they also investigated for the incident transverse wave. 21 At the plane interface of two different thermoelastic materials containing voids, Singh 22,23 investigated the reflection/transmission phenomena of waves due to incident longitudinal and transverse waves. Some useful results of wave propagation are Achenbach, 24 Sinha and Elsibai, 25 Othman and Song, 26,27 Kumar et al, 28 Othman and Atwa, 29 Sharma and Bhargava, 30 Pal et al, 31 Guo and Wei, 32 Palacidi et al, 33 Singh, 34 Sahu et al, 35 Chatterjee et al, 36 Abbas et al, 37 Zorammuana and Singh, 38 Abdalla et al, 39 Othman et al, 40 Singh and Lianngenga, 41 Barak and Kaliraman, 42 Khurana and Tomar, 43 Lianngenga and Singh, 44 Kumar et al, 45 Singh and Lalawmpuia, 46 and Pal and Kanoria. 47 The fundamental equations for the present problem are inspected in Section 2. The formulation of the problem and wave propagation are explained in Section 3. The stress, heat, and displacements are continuous at the plane interface and this analysis has been performed in Section 4. The coefficients and energy distributions due to reflected and transmitted waves are discussed analytically in Sections 5 and 6, respectively. Some particular cases are obtained in Sections 7. Section 8 is concerned on numerical computation and the final section is the conclusion of the problem.

BASIC EQUATIONS
Following Wang et al, 5 the constitutive relations for prestressed bodies with generalized thermoelasticity are ij = c ijmn e mn + e jk P ki − ij T, where ij , q i , , P ki , and e ij are stress, thermal flux, entropy, prestress, and strain tensors, respectively, c ijmn and ij , K ij , a i , h ijk are elastic and thermal coefficients, respectively, the temperature is change from T 0 to T, u i is the component of displacement vector of the material with density , thermal relaxation time , and specific heat C e . For the generalized thermoelastic materials under initial stresses with body force F i and internal heat source S, the equations of motions are given asü Without body forces as well as heat sources and using Equations (1) to (3) in Equations (5) and (6), we get T 0 where d ijmn = c ijmn + jn e mi and jn is the Kronecker's delta. Note that a i = 0 and h ijk = 0 for the uniform temperature prestressed bodies.

WAVE PROPAGATION
We consider Cartesian coordinates with x and y-axes lying horizontally and z-axis as vertically. Two half-spaces M ∶ 0 ≤ z < ∞ and M ′ ∶ −∞ < z ≤ 0 of transversely isotropic thermoelastic medium under initial stresses are assumed to analyze wave propagation in xz-plane. The diagrammatic structure of the problem is shown in Figure 1.
For half-space M, equations of motions are 15 13 3 , 3 = 33 = 2d 13 1 + d 33 3 , d = C e , 1 and 3 are linear thermal expansion coefficients. where For the incident, reflected, and transmitted waves, we have where A n is the amplitude constant, ⟨d (n) 1 , 0, d (n) 3 ⟩ and ⟨p (n) 1 , 0, p (n) 3 ⟩ are unit displacement and propagation vectors, respectively, k n is wavenumber, v n is phase velocity. 15 Note that n = 0 represents incident a QL or a QT wave, n = 1, n = 2, and F I G U R E 1 Problem figure n = 3 represent for reflected QL, QT, and thermal waves (T-mode), respectively, and n = 4, n = 5, and n = 6 represent for the transmitted QL, QT, and thermal waves, respectively. The coupling constant F (n) is given by Using Snell's law, we can have 22 k 0 k r sin 0 = sin r for r = 1, 2, 3, 4, 5, 6.

BOUNDARY CONDITIONS
The stress tractions, heat flow, and displacement components are continuous at z = 0. We have (i) Continuity of normal traction: (ii) Continuity of shear traction: (iii) Continuity of heat flow: (iv) Continuity of displacement components: These boundary conditions may be reduced to Equations (17) to (20) will help to find the reflection and transmission coefficients of the reflected and transmitted waves.

AMPLITUDE RATIO
The matrix representation of Equations (17) to (20) is given as where A is a 6 × 6 matrix, B and Z are 6 × 1 matrices with the following elements Equation (21) is solved for Z r due to incident QL and QT-waves.

ENERGY RATIO
We have considered partition of energy at z = 0 and the rate of transmission is given by Reference 24 Using Equation (22), the energy ratios waves are Note that E r for r = 1, 2, 3 represent energy ratios of the reflected QL, QT, and T-mode waves, respectively, and r = 4, 5, 6 represent for the transmitted QL, QT, and T-mode waves, respectively.
These ratios exactly match Singh. 15 The distribution of energy E 1 , E 2 , and E 3 of the reflected waves are given by Equation (23).

CASE III:
If M ′ is stress free and P 11 = P 33 = 0. Equation (24) will be modified with the following changes The results are exactly same as Sharma. 13 The energy ratios E 1 , E 2 , and E 3 are also given by Equation (23) with the modified value of

NUMERICAL RESULTS
For evaluating the coefficients and energy distributions due to reflected and transmitted waves for incident QL and QT waves, we have used the relevant parametric values given in Table 1 (see Reference 9). The unit propagation and displacement vectors are (for incident quasilongitudinal wave) (for reflected waves) (for transmitted waves)   Figure 2A increases when 0 is increased. In Figure 2B, all the curves corresponding to |Z 2 | increase initially and decrease when the value of 0 get larger. Thereafter, Curve I, Curve II, and Curve III increase to the maximum values at 0 = 71 • , 0 = 70 • , and 0 = 68 • , respectively, and then decrease again. All the curves in Figure 2C for the amplitude ratio Z 3 increase to the maximum values at 0 = 30 • (Curve I), 0 = 28 • (Curve II), and 0 = 26 • (Curve III), which decrease with the increase of 0 . Note that the minimum and maximum effects of initial stresses on |Z 1 | are near grazing and normal incidence, respectively, while the minimum effect on |Z 3 | is near normal incidence. The variation of the transmission coefficients are depicted in Figure 3. We have observed that |Z 4 | decrease with the increase of 0 , while |Z 5 | and |Z 6 | are similar to those of |Z 2 | and |Z 3 |, respectively. All the curves in Figure 3B meet at a point 0 = 60 • . Herein, also the effect of initial stresses on |Z 4 | is maximum when 0 is close to normal angle of incidence.
The energy distribution on the reflected and transmitted waves are shown in Figures 4 and 5, respectively. In Figure 4A, |E 1 | increases when the value of 0 is getting more. The effects of P and P ′ on |E 1 | are minimum and maximum at the grazing and normal angle of incidence, respectively. All the curves in Figure 4B for |E 2 | increase initially and meet at 0 = 52 • , which then increase to the maximum values at 0 = 77 • (Curve I), 0 = 78 • (Curve II), and 0 = 79 • (curve III).
After these points, all the curves decrease with the rise of 0 . The values of |E 3 | in Figure 4C increase to the maximum values for Curve I, Curve II, and Curve III are observed at 0 = 35 • , 0 = 31 • , and 0 = 28 • , respectively, and all decrease with the higher value of 0 . We have observed that the minimum effect of P and P ′ on |E 3 | is near normal angle of incidence. In Figure 5, the value of |E 4 | falls when the value of 0 is increased, while |E 5 | and |E 6 | show similar pattern with |E 2 | and |E 3 |, respectively. The sum of the energy ratios is close to one.

F I G U R E 9
Energy distribution for transmitted waves with 0 and then increase with the increase of 0 . In Figure 6B, |Z 2 | starts decreasing to the minimum value, which then increase with the higher value of 0 . The values of |Z 3 | in Figure 6C increase initially and decrease slightly, which increase and decrease again when the value of 0 is getting larger. In Figure 7A, |Z 4 | increases to the maximum value and then decreases with the rise in the value of 0 , while |Z 5 | in Figure 7B decreases the value of 0 is increased. It is observed that |Z 6 | has similar pattern with |Z 3 |. In this case, we have observed critical angles 0 = 30 • for |Z 1 | and 0 = 58 • for |Z 4 |. We have observed that the effects of P and P ′ on |Z 3 | and |Z 6 | are minimum near normal as well as grazing angle of incidence. The variation of energy ratios corresponding to reflected and transmitted waves are depicted in Figures 8 and 9, respectively. In Figure 8A, all the curves show that |E 1 | increases to some point and then drops to the minimum value, which then rises with the increase of 0 . The values of |E 2 | in Figure 8B increase with the rise in the value of 0 . We notice that the effect of initial stresses is very small near the normal incidence. It is observed that |E 3 | in Figure 8C increases up to certain value and then decreases with the increase of 0 . In Figure 9, we have observed that the variation of |E 4 | and |E 6 | have similar pattern with |Z 4 | and |Z 6 |, respectively. The values of |E 5 | in Figure 9B decrease initially and then increase up to certain value, which decreases again when the value of 0 is increased. Herein, we have noticed critical angles, 0 = 30 • for |E 1 | and 0 = 58 • for |E 4 |. The law of conservation of energy is also hold for this case. It is observed that the effect of initial stresses are very small near the grazing and normal incidence in most of the amplitude and energy ratios for incident QL and QT waves.

CONCLUSION
For incident QL and QT-waves at the interface between two different half-spaces of initially stressed transversely isotropic thermoelastic materials, the reflected and transmitted waves are analyzed. The formula corresponding to the coefficient of reflection/transmission and energy ratios are obtained with the help of appropriate boundary conditions. These formulas are computed numerically for a particular model. We have the following concluding remarks: (i) The reflection/transmission coefficients and the energy distributions are found to depend on angle of incidence, elastic, thermal and initial stress parameters.