Dispersion of Rayleigh Wave in a Shielded Anisotropic Generalized Thermoelastic Layer

Abstract

A study of Rayleigh wave propagation has been conducted in thermally conducting homogeneous anisotropic layer. The theories of generalized thermo-elasticity which are taken into account are classical dynamical coupled theory (CD theory), Lord and Shulman’s theory (LS theory) and Green and Lindsay’ theory (GL theory) with two thermal relaxation times. An analytical solution obtained and procured a dispersion relation subjected to boundaries as rigid and insulated. In order to deal with the numerical results of the problem, a tri-clinic material has been taken into account. The results are interpreted in terms of graphs. The graphs show significant impacts on the phase velocity of Rayleigh wave with wave number for different thicknesses of layer under all theories.

Appendices

Appendix 1

$$ \begin{array}{lll} a_{11} = s^2 - 2is\xi _1 - \xi _2 \eta _1^2 &{} a_{12} = \xi _3 s^2 - i\xi _4 s - \xi _1 &{} a_{13} = - \xi _6 \left( {\frac{i}{k} + \tau _1 c} \right) \\ a_{21} = \xi _3 s^2 - i\xi _4 s - \xi _1 &{} a_{22} = \xi _5 s^2 - 2i\xi _3 s - \eta _2^2 &{} a_{23} = s\xi _7 \left( {\frac{1}{k} + i\tau _1 c} \right) \\ a_{31} = \xi _8 c\left( {1 - ikc\tau _0 \varOmega } \right) &{} a_{32} = \xi _9 c\left( {i + kc\tau _0 \varOmega } \right) s &{} a_{33} = - s^2 + \xi _{10} - \frac{c}{\nu }\left( {\frac{i}{k} + \tau _0 c} \right) \end{array} $$

Appendix 2

$$ \begin{array}{l} a_0 = \xi _3^2 - \xi _5 ,\,\,a_1 = 2i\xi _3^{} - \xi _3 \xi _4 + 2i\xi _1 \xi _5 \\ a_2 = 2\xi _1 \xi _3^{} + \frac{{ic\xi _3^2 }}{{kv}} - \xi _{10} \xi _3^2 - \frac{{ic\xi _5^{} }}{{kv}} + \xi _{10} \xi _5^{} - \frac{{ic\xi _7^{} \xi _9^{} }}{k} + \xi _2 \xi _5^{} \eta _1^2 + \eta _2^2 + \frac{{c^2 \xi _3^2 \tau _0 }}{v} - \frac{{c^2 \xi _5^{} \tau _0 }}{v} \\ \,\,\,\,\,\,\,\,\,\, - \frac{{c^2 k\omega \xi _7^{} \xi _9^{} \tau _0 }}{k} + c^2 \xi _7^{} \xi _9^{} \tau _1 - ic^3 k\omega \xi _7^{} \xi _9^{} \tau _0 \tau _1 \\ a_3 = - \frac{{2c\xi _3 }}{{kv}} - 2i\xi _{10} \xi _3 + 2i\xi _1 \xi _4 + \frac{{2c\xi _3 \xi _4 }}{{kv}} + 2i\xi _{10} \xi _3 \xi _4 - \frac{{2c\xi _1 \xi _5 }}{{kv}} - 2i\xi _1 \xi _{10} \xi _5 + \frac{{c\xi _3 \xi _7 \xi _8 }}{k} - \frac{{c\xi _3 \xi _6 \xi _9 }}{k} \\ \,\,\,\,\,\,\,\,\,\, - \frac{{2c\xi _1 \xi _7 \xi _9 }}{k} - 2i\xi _2 \xi _3 \eta _1^2 - 2i\xi _1 \eta _2^2 + \frac{{2ic^2 \xi _3 \tau _0 }}{v} - \frac{{2ic^2 \xi _3 \xi _4 \tau _0 }}{v} + \frac{{2ic^2 \xi _1 \xi _5 \tau _0 }}{v} - \frac{{ickc\omega \xi _3 \xi _7 \xi _8 \tau _0 }}{k} \\ \,\,\,\,\,\,\,\,\,\, + \frac{{ickm\omega \xi _3 \xi _6 \xi _9 \tau _0 }}{k} + \frac{{2ickc\omega \xi _1 \xi _7 \xi _9 \tau _0 }}{k} + ic^2 \xi _3 \xi _7 \xi _8 \tau _1 + ic^2 \xi _3 \xi _6 \xi _9 \tau _1 - 2ic^2 \xi _1 \xi _7 \xi _9 \tau _1 \\ \,\,\,\,\,\,\,\,\,\, + c^2 kc\omega \xi _3 \xi _7 \xi _8 \tau _0 \tau _1 + c^2 kc\omega \xi _3 \xi _6 \xi _9 \tau _0 \tau _1 - 2c^2 kc\omega \xi _1 \xi _7 \xi _9 \tau _0 \tau _1 \\ a_4 = \xi _1^2 + \frac{{2ic\xi _1 \xi _3 }}{{kv}} - 2i\xi _1 \xi _{10} \xi _3 - \frac{{ic\xi _4^2 }}{{kv}} + \xi _{10} \xi _4^2 - \frac{{ic\xi _5 \xi _6 \xi _8 }}{k} - \frac{{ic\xi _4 \xi _7 \xi _8 }}{k} + \frac{{ic\xi _4 \xi _6 \xi _9 }}{k} + \frac{{ic\xi _2 \xi _5 \eta _1^2 }}{{kv}} - \xi _{10} \xi _2 \xi _5 \eta _1^2 \\ \,\,\,\,\,\,\,\,\,\,\, + \frac{{ic\xi _2 \xi _7 \xi _9 \eta _1^2 }}{k} - \frac{{ic\eta _2^2 }}{{kv}} - \xi _{10} \eta _2^2 - \xi _2 \eta _1^2 \eta _2^2 + \frac{{2c^2 \xi _1 \xi _3 \tau _0 }}{v} - \frac{{c^2 \xi _4^2 \tau _0 }}{v} - \frac{{ckc\omega \xi _5 \xi _6 \xi _8 \tau _0 }}{k} - \frac{{ckc\omega \xi _4 \xi _7 \xi _8 \tau _0 }}{k} \\ \,\,\,\,\,\,\,\,\,\,\, + \frac{{ckc\omega \xi _4 \xi _6 \xi _9 \tau _0 }}{k} + \frac{{c^2 \xi _2 \xi _5 \eta _1^2 \tau _0 }}{v} + \frac{{ckc\omega \xi _2 \xi _7 \xi _9 \eta _1^2 \tau _0 }}{k} + \frac{{c^2 \eta _2^2 \tau _0 }}{v} - c^2 \xi _5 \xi _6 \xi _8 \tau _1 + c^2 \xi _4 \xi _7 \xi _8 \tau _1 + c^2 \xi _4 \xi _6 \xi _9 \tau _1 \\ \,\,\,\,\,\,\,\,\,\, - c^2 \xi _2 \xi _7 \xi _9 \eta _1^2 \tau _1 + ic^2 kc\omega \xi _5 \xi _6 \xi _8 \tau _0 \tau _1 - ic^2 kc\omega \xi _4 \xi _7 \xi _8 \tau _0 \tau _1 - ic^2 kc\omega \xi _4 \xi _6 \xi _9 \tau _0 \tau _1 + ic^2 kc\omega \xi _2 \xi _7 \xi _9 \eta _1^2 \tau _0 \tau _1 \\ a_5 = - \frac{{2ic\xi _1 \xi _4 }}{{kv}} - 2i\xi _1 \xi _{10} \xi _4 - \frac{{2c\xi _3^{} \xi _6^{} \xi _8^{} }}{k} - \frac{{c\xi _1^{} \xi _7^{} \xi _8^{} }}{k} + \frac{{c\xi _1^{} \xi _6^{} \xi _9^{} }}{k} + \frac{{2c\xi _2^{} \xi _3^{} \eta _1^2 }}{k} + 2i\xi _{10}^{} \xi _2^{} \xi _3^{} \eta _1^2 + \frac{{2c\xi _1^{} \eta _2^2 }}{{kv}} \\ \,\,\,\,\,\,\,\,\,\, + 2i\xi _1^{} \xi _{10}^{} \eta _2^2 + \frac{{2ic^2 \xi _1 \xi _4 \tau _0 }}{v} + \frac{{2ic^2 k\omega \xi _1 \xi _7 \xi _8 \tau _0 }}{k} + \frac{{ic^2 k\omega \xi _1 \xi _7 \xi _8 \tau _0 }}{k} - \frac{{2ic^2 k\omega \xi _1 \xi _7 \xi _8 \tau _0 }}{k} - \frac{{ic^2 k\omega \xi _1 \xi _6 \xi _9 \tau _0 }}{k} \\ \,\,\,\,\,\,\,\,\, - \frac{{2ic^2 k\omega \xi _2 \xi _3 \eta _1^2 \tau _0 }}{k} - \frac{{2ic^2 \xi _1 \eta _2^2 \tau _0 }}{k} + 2ic^2 \xi _3 \xi _6 \xi _8 \tau _1 - ic^2 \xi _1 \xi _7 \xi _8 \tau _1 - ic^2 \xi _1 \xi _6 \xi _9 \tau _1 + 2c^3 k\omega \xi _3 \xi _6 \xi _8 \tau _0 \tau _1 \\ \,\,\,\,\,\,\,\,\, - c^3 k\omega \xi _1 \xi _7 \xi _8 \tau _0 \tau _1 - c^3 k\omega \xi _1 \xi _6 \xi _9 \tau _0 \tau _1 \\ a_6 = \frac{{ic\xi _1^2 }}{{kv}} - \xi _1^2 \xi _{10} + \frac{{ic\xi _6^{} \xi _8^{} \eta _2^2 }}{k} + \xi _{10}^{} \xi _2^{} \eta _1^2 \eta _2^2 + \frac{{c^2 \xi _1^2 \tau _0 }}{v} + \frac{{c^2 k\omega \xi _6^{} \xi _8^{} \eta _2^2 \tau _0 }}{k} - \frac{{c^2 \xi _2^{} \eta _1^2 \eta _2^2 \tau _0 }}{v} + c^2 \xi _6^{} \xi _8^{} \eta _2^2 \tau _1 \\ \,\,\,\,\,\,\,\,\,\, - ic^3 k\omega \xi _6^{} \xi _8^{} \eta _2^2 \tau _1 \tau _0 \end{array} $$

Appendix 3

$$ \begin{array}{l} a_{11,j} = s_j^2 - 2is_j \xi _1 - \xi _2 \eta _1^2 ,\,\,a_{12,j} = \xi _3 s_j^2 - i\xi _4 s_j - \xi _1 ,a_{13,j} = - \xi _6 \left( {\frac{i}{k} + \tau _1 c} \right) ,\,\, \\ a_{21,j} = \xi _3 s_j^2 - i\xi _4 s_j - \xi _1 a_{22,j} = \xi _5 s_j^2 - 2i\xi _3 s_j - \eta _2^2 ,\,\,a_{23,j} = s_j \xi _7 \left( {\frac{1}{k} + i\tau _1 c} \right) ,\left( {j = 1,\ldots ,6} \right) \\ \end{array} $$

Appendix 4

$$ \begin{array}{l} M_{1j} = \mathrm{{exp}}\left[ {\frac{{ks_j h}}{2}} \right] ,\,\,M_{2j} = \delta _j \mathrm{{exp}}\left[ {\frac{{ks_j h}}{2}} \right] ,\,\,\,M_{3j} = \gamma _j s_j \mathrm{{exp}}\left[ {\frac{{ks_j h}}{2}} \right] , \\ M_{4j} = \mathrm{{exp}}\left[ { - \frac{{ks_j h}}{2}} \right] ,\,\,\,M_{5j} = \delta _j \mathrm{{exp}}\left[ { - \frac{{ks_j h}}{2}} \right] ,\,\,\,M_{6j} = \gamma _j s_j \mathrm{{exp}}\left[ { - \frac{{ks_j h}}{2}} \right] ,\,\,\,\left( {j = 1,\ldots ,6} \right) \\ \end{array} $$