Gravitational Effect on Piezo-Thermoelasiticity in the Context of Three Phase Lag Model with Two Temperature

Abstract

The main aim of this article is to study the effect of gravity on piezo-thermoelasticity in the context of three phase lag (TPL) model with two-temperature. The governing equations of piezo-thermoelasticity in TPL model with two-temperature in the presence of gravity are solved by Normal mode analysis in half space and we calculate the stress, strain and displacement components. A cadmium selenide material has been considered for the numerical simulation and the result has been plotted graphically. The results obtained for both Lord Shulman (L–S) and TPL thermoelastic models are almost coincident to each other in case of strain, stress tensors and components of displacement but results differ in the case of one components of electric potentials due to the presence of gravity.

Appendices

Appendix A

${\text{A}}_{1} = \frac{{\left( {{\rho c}^{2} - {\text{C}}_{11} } \right){\upalpha }^{2} }}{{{\text{C}}_{44} }}$	${\text{A}}_{2} = \frac{{\left( {{\text{C}}_{13} + {\text{C}}_{14} } \right){\text{i}}\upalpha }}{{{\text{C}}_{44} }}$	${\text{A}}_{3} = \frac{{\uprho {\text{gi}}\upalpha }}{{{\text{C}}_{44} }}$	${\text{ A}}_{4} = \frac{{\left( {e_{31} + e_{15} } \right)i\alpha }}{{C_{44} }}$
${\text{A}}_{5} = \frac{{\upbeta _{1} \left( {1 + {\text{a}}^{*} \upalpha ^{2} } \right){\text{i}}\upalpha }}{{{\text{C}}_{44} }}$	${\text{A}}_{6} = \frac{{{\text{a}}^{*}\upbeta _{1} {\text{i}}\upalpha }}{{{\text{C}}_{44} }}$	${\text{A}}_{7} = \left( {{\text{C}}_{44} + {\text{C}}_{13} } \right){\text{i}}\upalpha$	$A_{8} = \rho gi\alpha$
$A_{9} = C_{33}$	$A_{10} = \left( {\rho c^{2} - C_{44} } \right)\alpha^{2}$	${\text{A}}_{11} = {\text{e}}_{33}$	$A_{12} = e_{15} \alpha^{2}$
$A_{13} = \beta_{3} \left( {1 + a^{*} \alpha^{2} } \right)$	${\text{A}}_{14} = {\upbeta }_{3} {\text{a}}^{*}$	${\text{A}}_{15} = \left( {{\text{e}}_{15} + {\text{e}}_{31} } \right){\text{i}}\upalpha$	$A_{16} = \epsilon_{11} \alpha^{2}$
$A_{17} = \epsilon_{33}$	$A_{18} = P_{3} (1 + a^{*} \alpha^{2}$)	$A_{19} = P_{3} a^{*}$	$A_{20} = i\alpha^{3} cK_{1} l_{1} - \alpha^{2} l_{2} K_{1}^{*}$
${\text{A}}_{21} =\uprho {\text{c}}_{{\text{e}}} \left( {1 + {\text{a}}^{*} \upalpha ^{2} } \right)\upalpha ^{2} {\text{c}}^{2 }$	$A_{22} = \rho c_{e} a^{*} \alpha^{2} c^{2}$	${\text{A}}_{23} = {\text{K}}_{3}^{*} {\text{l}}_{2} - {\text{iK}}_{3} \upalpha {\text{cl}}_{1}$	${\text{A}}_{24} = {\text{T}}_{0}\upbeta _{1} {\text{i}}\upalpha ^{3} {\text{c}}^{2}$
$A_{25} = T_{0} \beta_{3} \alpha^{2} c^{2}$	$A_{26} = P_{3} \alpha^{2} c^{2}$

Here

$$ {\text{l}}_{1} = \frac{{{\text{l}}_{3} }}{{{\text{l}}_{5} }}\quad {\text{l}}_{2} = \frac{{{\text{l}}_{4} }}{{{\text{l}}_{5} }}\quad {\text{l}}_{3} = 1 - {\text{i}}\upalpha {\text{c}}\uptau _{{\text{T}}} \quad {\text{l}}_{4} = 1 - {\text{i}}\upalpha {\text{c}}\uptau _{{\text{v}}} \quad {\text{l}}_{5} = 1 - {\text{i}}\upalpha {\text{c}}\uptau _{{\text{q}}} - \frac{{\upalpha ^{2} {\text{c}}^{2} }}{2}\uptau _{{\text{q}}}^{2} $$

Appendix B

$$ \begin{aligned} {\text{K}} = & \frac{{{\text{A}}_{9} \left[ {{\text{A}}_{16} \left( {{\text{A}}_{23} - {\text{A}}_{22} } \right) - {\text{A}}_{17} \left( {{\text{A}}_{20} + {\text{A}}_{21} } \right) + {\text{A}}_{18} {\text{A}}_{26} } \right] + {\text{A}}_{17} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{19} {\text{A}}_{26} {\text{A}}_{10} + {\text{A}}_{11} \left[ {{\text{A}}_{13} {\text{A}}_{26} + {\text{A}}_{18} {\text{A}}_{25} - {\text{A}}_{11} \left( {{\text{A}}_{20} + {\text{A}}_{21} } \right)} \right]}}{{{\text{A}}_{9} \left[ {{\text{A}}_{17} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{11} {\text{A}}_{26} } \right] + {\text{A}}_{11} \left[ {{\text{A}}_{11} {\text{A}}_{12} - {\text{A}}_{23} - {\text{A}}_{19} {\text{A}}_{25} } \right] + {\text{A}}_{14} \left[ {{\text{A}}_{17} {\text{A}}_{25} - {\text{A}}_{14} {\text{A}}_{26} } \right]}} \\ & + \frac{{{\text{A}}_{12} \left[ {{\text{A}}_{19} {\text{A}}_{25} + {\text{A}}_{14} {\text{A}}_{26} } \right] - {\text{A}}_{25} \left[ {{\text{A}}_{13} {\text{A}}_{17} + {\text{A}}_{14} {\text{A}}_{16} } \right] - {\text{A}}_{1} {\text{A}}_{11} {\text{A}}_{23} + {\text{A}}_{1} {\text{A}}_{9} \left[ {{\text{A}}_{17} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{19} {\text{A}}_{26} } \right] + {\text{A}}_{1} {\text{A}}_{11}^{2} {\text{A}}_{12} + {\text{A}}_{19} {\text{A}}_{25} }}{{{\text{A}}_{9} \left[ {{\text{A}}_{17} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{11} {\text{A}}_{26} } \right] + {\text{A}}_{11} \left[ {{\text{A}}_{11} {\text{A}}_{12} - {\text{A}}_{23} - {\text{A}}_{19} {\text{A}}_{25} } \right] + {\text{A}}_{14} \left[ {{\text{A}}_{17} {\text{A}}_{25} - {\text{A}}_{14} {\text{A}}_{26} } \right]}} \\ & + \frac{{{\text{A}}_{1} {\text{A}}_{14} {\text{A}}_{26} + {\text{A}}_{1} {\text{A}}_{14} {\text{A}}_{17} {\text{A}}_{25} + {\text{A}}_{2} \left[ {{\text{A}}_{17} \left( {{\text{A}}_{23} - {\text{A}}_{22} } \right) + {\text{A}}_{19} {\text{A}}_{26} } \right] + {\text{A}}_{2} {\text{A}}_{15} \left[ {{\text{A}}_{11} \left( {{\text{A}}_{23} - {\text{A}}_{22} } \right) + {\text{A}}_{14} {\text{A}}_{26} } \right] + {\text{A}}_{2} {\text{A}}_{24} \left[ {{\text{A}}_{11} {\text{A}}_{19} - {\text{A}}_{14} {\text{A}}_{17} } \right]}}{{{\text{A}}_{9} \left[ {{\text{A}}_{17} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{11} {\text{A}}_{26} } \right] + {\text{A}}_{11} \left[ {{\text{A}}_{11} {\text{A}}_{12} - {\text{A}}_{23} - {\text{A}}_{19} {\text{A}}_{25} } \right] + {\text{A}}_{14} \left[ {{\text{A}}_{17} {\text{A}}_{25} - {\text{A}}_{14} {\text{A}}_{26} } \right]}} \\ & + \frac{{{\text{A}}_{4} {\text{A}}_{7} \left[ {{\text{A}}_{11} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{19} {\text{A}}_{25} } \right] + {\text{A}}_{4} {\text{A}}_{9} \left[ {{\text{A}}_{15} \left( {{\text{A}}_{23} - {\text{A}}_{22} } \right)} \right] + {\text{A}}_{4} {\text{A}}_{9} {\text{A}}_{19} {\text{A}}_{24} + {\text{A}}_{4} {\text{A}}_{14} \left[ {{\text{A}}_{25} \left( {{\text{A}}_{11} - {\text{A}}_{15} } \right) + {\text{A}}_{6} \left( {{\text{A}}_{11} {\text{A}}_{26} - {\text{A}}_{17} {\text{A}}_{25} } \right)} \right]}}{{{\text{A}}_{9} \left[ {{\text{A}}_{17} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{11} {\text{A}}_{26} } \right] + {\text{A}}_{11} \left[ {{\text{A}}_{11} {\text{A}}_{12} - {\text{A}}_{23} - {\text{A}}_{19} {\text{A}}_{25} } \right] + {\text{A}}_{14} \left[ {{\text{A}}_{17} {\text{A}}_{25} - {\text{A}}_{14} {\text{A}}_{26} } \right]}} \\ & - \frac{{{\text{A}}_{4} {\text{A}}_{14} \left[ {{\text{A}}_{6} {\text{A}}_{15} {\text{A}}_{9} {\text{A}}_{26} + {\text{A}}_{6} {\text{A}}_{15} {\text{A}}_{11} {\text{A}}_{25} - {\text{A}}_{6} {\text{A}}_{24} \left( {{\text{A}}_{9} {\text{A}}_{17} + {\text{A}}_{11}^{2} } \right)} \right]}}{{{\text{A}}_{9} \left[ {{\text{A}}_{17} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{11} {\text{A}}_{26} } \right] + {\text{A}}_{11} \left[ {{\text{A}}_{11} {\text{A}}_{12} - {\text{A}}_{23} - {\text{A}}_{19} {\text{A}}_{25} } \right] + {\text{A}}_{14} \left[ {{\text{A}}_{17} {\text{A}}_{25} - {\text{A}}_{14} {\text{A}}_{26} } \right]}} \\ \end{aligned} $$

$$ \begin{aligned} {\text{L}} =& \frac{{{\text{A}}_{9} {\text{A}}_{16} \left( {{\text{A}}_{20} + {\text{A}}_{21} } \right) + {\text{A}}_{10} \left[ {{\text{A}}_{16} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{17} \left( {{\text{A}}_{20} + {\text{A}}_{21} } \right) + {\text{A}}_{18} {\text{A}}_{26} } \right] + \left[ {{\text{A}}_{12} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{12} {\text{A}}_{18} {\text{A}}_{25} - {\text{A}}_{13} \left( {{\text{A}}_{20} + {\text{A}}_{21} } \right)} \right]}}{{{\text{A}}_{9} \left[ {{\text{A}}_{17} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{11} {\text{A}}_{26} } \right] + {\text{A}}_{11} \left[ {{\text{A}}_{11} {\text{A}}_{12} - {\text{A}}_{23} - {\text{A}}_{19} {\text{A}}_{25} } \right] + {\text{A}}_{14} \left[ {{\text{A}}_{17} {\text{A}}_{25} - {\text{A}}_{14} {\text{A}}_{26} } \right]}} \\ &+ \frac{{{\text{A}}_{1} {\text{A}}_{9} \left[ {{\text{A}}_{16} \left( {{\text{A}}_{23} - {\text{A}}_{22} } \right) + {\text{A}}_{18} {\text{A}}_{26} - {\text{A}}_{17} \left( {{\text{A}}_{20} + {\text{A}}_{21} } \right)} \right] + {\text{A}}_{1} {\text{A}}_{10} \left[ {{\text{A}}_{17} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{19} {\text{A}}_{16} } \right] + {\text{A}}_{1} {\text{A}}_{11} \left[ {{\text{A}}_{18} {\text{A}}_{25} - \left( {{\text{A}}_{20} + {\text{A}}_{21} } \right) + {\text{A}}_{13} } \right]}}{{{\text{A}}_{9} \left[ {{\text{A}}_{17} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{11} {\text{A}}_{26} } \right] + {\text{A}}_{11} \left[ {{\text{A}}_{11} {\text{A}}_{12} - {\text{A}}_{23} - {\text{A}}_{19} {\text{A}}_{25} } \right] + {\text{A}}_{14} \left[ {{\text{A}}_{17} {\text{A}}_{25} - {\text{A}}_{14} {\text{A}}_{26} } \right]}} \\ & + \frac{{{\text{A}}_{1} {\text{A}}_{12} \left[ {{\text{A}}_{19} {\text{A}}_{25} + {\text{A}}_{14} {\text{A}}_{26} } \right] - {\text{A}}_{1} \left[ {{\text{A}}_{25} \left( {{\text{A}}_{13} {\text{A}}_{17} + {\text{A}}_{14} {\text{A}}_{16} } \right)} \right] + {\text{A}}_{2} \left[ {{\text{A}}_{16} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) + {\text{A}}_{17} \left( {{\text{A}}_{20} + {\text{A}}_{31} } \right) - {\text{A}}_{2} {\text{A}}_{7} {\text{A}}_{26} - {\text{A}}_{2} {\text{A}}_{12} {\text{A}}_{15} {\text{A}}_{22} } \right]}}{{{\text{A}}_{9} \left[ {{\text{A}}_{17} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{11} {\text{A}}_{26} } \right] + {\text{A}}_{11} \left[ {{\text{A}}_{11} {\text{A}}_{12} - {\text{A}}_{23} - {\text{A}}_{19} {\text{A}}_{25} } \right] + {\text{A}}_{14} \left[ {{\text{A}}_{17} {\text{A}}_{25} - {\text{A}}_{14} {\text{A}}_{26} } \right]}} \\& + \frac{{{\text{A}}_{2} {\text{A}}_{12} \left[ {{\text{A}}_{15} {\text{A}}_{23} - {\text{A}}_{19} {\text{A}}_{24} } \right] - \left[ {2{\text{A}}_{15} {\text{A}}_{20} - {\text{A}}_{18} {\text{A}}_{24} } \right] + {\text{A}}_{2} {\text{A}}_{15} {\text{A}}_{26} + {\text{A}}_{2} {\text{A}}_{24} \left[ {{\text{A}}_{14} {\text{A}}_{16} + {\text{A}}_{13} {\text{A}}_{17} } \right] + {\text{A}}_{3} {\text{A}}_{8} \left[ {{\text{A}}_{17} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{19} {\text{A}}_{26} } \right]}}{{{\text{A}}_{9} \left[ {{\text{A}}_{17} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{11} {\text{A}}_{26} } \right] + {\text{A}}_{11} \left[ {{\text{A}}_{11} {\text{A}}_{12} - {\text{A}}_{23} - {\text{A}}_{19} {\text{A}}_{25} } \right] + {\text{A}}_{14} \left[ {{\text{A}}_{17} {\text{A}}_{25} - {\text{A}}_{14} {\text{A}}_{26} } \right]}} \\ & + \frac{{{\text{A}}_{4} {\text{A}}_{7} \left[ {{\text{A}}_{12} \left( {{\text{A}}_{23} - {\text{A}}_{22} } \right)} \right] + {\text{A}}_{4} \left[ {{\text{A}}_{11} \left( {{\text{A}}_{20} - {\text{A}}_{21} } \right) + {\text{A}}_{18} {\text{A}}_{25} } \right] + \left[ {{\text{A}}_{15} \left( {{\text{A}}_{20} + {\text{A}}_{21} - {\text{A}}_{22} } \right) + {\text{A}}_{4} {\text{A}}_{15} \left( {{\text{A}}_{10} {\text{A}}_{23} + {\text{A}}_{13} {\text{A}}_{25} } \right) + {\text{A}}_{4} {\text{A}}_{24} {\text{A}}_{10} {\text{A}}_{19} } \right]}}{{{\text{A}}_{9} \left[ {{\text{A}}_{17} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{11} {\text{A}}_{26} } \right] + {\text{A}}_{11} \left[ {{\text{A}}_{11} {\text{A}}_{12} - {\text{A}}_{23} - {\text{A}}_{19} {\text{A}}_{25} } \right] + {\text{A}}_{14} \left[ {{\text{A}}_{17} {\text{A}}_{25} - {\text{A}}_{14} {\text{A}}_{26} } \right]}} \\ & - \frac{{{\text{A}}_{4} {\text{A}}_{24} {\text{A}}_{9} - {\text{A}}_{4} {\text{A}}_{24} \left( {{\text{A}}_{11} {\text{A}}_{13} + {\text{A}}_{12} {\text{A}}_{14} } \right) + {\text{A}}_{5} \left( {{\text{A}}_{17} {\text{A}}_{25} - {\text{A}}_{11} {\text{A}}_{26} } \right) + {\text{A}}_{5} {\text{A}}_{15} \left( {{\text{A}}_{9} {\text{A}}_{26} + {\text{A}}_{11} {\text{A}}_{25} } \right) - {\text{A}}_{5} {\text{A}}_{24} {\text{A}}_{22} {\text{A}}_{11} {\text{A}}_{24} \left( {{\text{A}}_{9} {\text{A}}_{17} + {\text{A}}_{11}^{2} } \right)}}{{{\text{A}}_{9} \left[ {{\text{A}}_{17} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{11} {\text{A}}_{26} } \right] + {\text{A}}_{11} \left[ {{\text{A}}_{11} {\text{A}}_{12} - {\text{A}}_{23} - {\text{A}}_{19} {\text{A}}_{25} } \right] + {\text{A}}_{14} \left[ {{\text{A}}_{17} {\text{A}}_{25} - {\text{A}}_{14} {\text{A}}_{26} } \right]}} \\ & + \frac{{{\text{A}}_{6} {\text{A}}_{7} \left( {{\text{A}}_{16} {\text{A}}_{25} - {\text{A}}_{12} {\text{A}}_{26} } \right) + {\text{A}}_{6} {\text{A}}_{15} {\text{A}}_{12} {\text{A}}_{25} - {\text{A}}_{6} {\text{A}}_{15} {\text{A}}_{10} {\text{A}}_{26} - {\text{A}}_{6} {\text{A}}_{24} \left( {{\text{A}}_{10} {\text{A}}_{17} - {\text{A}}_{9} {\text{A}}_{16} - 2{\text{A}}_{11} {\text{A}}_{12} } \right)}}{{{\text{A}}_{9} \left[ {{\text{A}}_{17} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{11} {\text{A}}_{26} } \right] + {\text{A}}_{11} \left[ {{\text{A}}_{11} {\text{A}}_{12} - {\text{A}}_{23} - {\text{A}}_{19} {\text{A}}_{25} } \right] + {\text{A}}_{14} \left[ {{\text{A}}_{17} {\text{A}}_{25} - {\text{A}}_{14} {\text{A}}_{26} } \right]}} \\\end{aligned} $$

$$ \begin{aligned} {\text{M}} = & \user2{ }\frac{{\left( {{\text{A}}_{10} {\text{A}}_{16} - {\text{A}}_{12}^{2} } \right)\left( {{\text{A}}_{20} + {\text{A}}_{21} } \right) + {\text{A}}_{1} \left[ {{\text{A}}_{9} \left( {{\text{A}}_{20} + {\text{A}}_{21} } \right) + {\text{A}}_{10} \left( {{\text{A}}_{23} - {\text{A}}_{22} } \right)} \right] - {\text{A}}_{1} {\text{A}}_{10} \left[ {{\text{A}}_{17} \left( {{\text{A}}_{20} + {\text{A}}_{21} } \right)} \right] + {\text{A}}_{1} {\text{A}}_{18} {\text{A}}_{26} + 2{\text{A}}_{1} {\text{A}}_{11} {\text{A}}_{12} {\text{A}}_{20} }}{{{\text{A}}_{9} \left[ {{\text{A}}_{17} \left( {{\text{A}}_{22} - {\text{A}}_{23} } \right) - {\text{A}}_{11} {\text{A}}_{26} } \right] + {\text{A}}_{11} \left[ {{\text{A}}_{11} {\text{A}}_{12} - {\text{A}}_{23} - {\text{A}}_{19} {\text{A}}_{25} } \right] + {\text{A}}_{14} \left[ {{\text{A}}_{17} {\text{A}}_{25} - {\text{A}}_{14} {\text{A}}_{26} } \right]}} \\ & + \frac{{A_{1} A_{12} \left[ {A_{12} \left( {A_{22} - A_{23} } \right) - A_{18} A_{25} - A_{13} A_{26} } \right] + A_{1} A_{13} A_{16} A_{25} - \left( {A_{20} + A_{21} } \right)A_{2} \left( {A_{7} A_{16} + A_{12} A_{15} } \right) - A_{2} A_{24} \left( {A_{12} A_{18} - A_{13} A_{26} } \right)}}{{A_{9} \left[ {A_{17} \left( {A_{22} - A_{23} } \right) - A_{11} A_{26} } \right] + A_{11} \left[ {A_{11} A_{12} - A_{23} - A_{19} A_{25} } \right] + A_{14} \left[ {A_{17} A_{25} - A_{14} A_{26} } \right]}} \\ & - \frac{{A_{3} A_{8} \left[ {A_{16} \left( {A_{23} - A_{22} } \right)} \right] + A_{3} A_{8} \left( {A_{18} A_{26} - A_{17} A_{20} } \right) + \left( {A_{20} + A_{21} } \right)A_{4} \left( {A_{7} A_{12} + A_{10} A_{15} } \right) + A_{4} A_{24} \left( {A_{12} A_{13} - A_{10} A_{18} } \right) + A_{5} A_{7} A_{12} A_{26} }}{{A_{9} \left[ {A_{17} \left( {A_{22} - A_{23} } \right) - A_{11} A_{26} } \right] + A_{11} \left[ {A_{11} A_{12} - A_{23} - A_{19} A_{25} } \right] + A_{14} \left[ {A_{17} A_{25} - A_{14} A_{26} } \right]}} \\ & + \frac{{A_{5} A_{7} A_{16} A_{25} + A_{5} A_{10} \left( {A_{15} A_{26} - A_{17} A_{24} } \right) + A_{5} A_{12} \left( {2A_{11} A_{24} - A_{15} A_{25} } \right) + A_{5} A_{9} A_{16} A_{24} + A_{6} A_{24} \left( {A_{12}^{2} - A_{10} A_{16} } \right)}}{{A_{9} \left[ {A_{17} \left( {A_{22} - A_{23} } \right) - A_{11} A_{26} } \right] + A_{11} \left[ {A_{11} A_{12} - A_{23} - A_{19} A_{25} } \right] + A_{14} \left[ {A_{17} A_{25} - A_{14} A_{26} } \right]}} \\ \end{aligned} $$

$$ {\text{N}} = \frac{{\left( {A_{10} A_{16} - A_{12}^{2} } \right)\left[ {A_{1} \left( {A_{20} + A_{21} } \right) + A_{5} A_{24} } \right] + A_{3} A_{8} A_{16} \left( {A_{20} + A_{21} } \right)}}{{ A_{9} \left[ {A_{17} \left( {A_{22} - A_{23} } \right) - A_{11} A_{26} } \right] + A_{11} \left[ {A_{11} A_{12} - A_{23} - A_{19} A_{25} } \right] + A_{14} \left[ {A_{17} A_{25} - A_{14} A_{26} } \right]}} $$

Appendix C

$$ \begin{array}{*{20}l} {{\text{f}} = 2\left[ {\frac{{3{\text{K}}^{2} }}{8} - \frac{{3{\text{K}}^{2} }}{4} + {\text{L}}} \right]} \hfill & {u = \sqrt {f^{2} - 3g } } \hfill & {{\text{g}} = \left[ {\frac{{3{\text{K}}^{2} }}{8} - \frac{{3{\text{K}}^{2} }}{4} + {\text{L}}} \right] - 4\left[ {\frac{{{\text{K}}^{4} }}{256} - \frac{{{\text{K}}^{4} }}{64} + \frac{{{\text{K}}^{2} {\text{L}}}}{16} - \frac{{{\text{KM}}}}{4} + {\text{N}}} \right]} \hfill \\ { h = - \left[ { - \frac{{K^{3} }}{16} + \frac{{3K^{2} }}{16} - \frac{KL}{2} + M} \right] } \hfill & {{\text{v}} = \frac{{{\text{sin}}^{ - 1} {\text{w}}}}{3}} \hfill & {{\text{w}} = \frac{{ 2f^{3} - 9fg + 27h}}{{2u^{3} }}} \hfill \\ {t_{1} = \sqrt {\frac{1}{3}\left[ {2usin\left( v \right) - f} \right]} } \hfill & {t_{2} = \sqrt {\frac{1}{3}\left[ { - f - u\left( {\sqrt 3 cosv + sinv} \right)} \right]} } \hfill & { t_{3} = \sqrt {\frac{1}{3}\left[ { - f + u\left( {\sqrt 3 cosv - sinv} \right)} \right]} } \hfill \\ \end{array} $$

Appendix D

$$ \begin{aligned} T_{1n} = & \frac{1}{{q_{1} }}\left[ {r_{1} - \left\{ {\frac{{q_{3} \left( {q_{1} r_{3} - q_{7} r_{1} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right) - q_{3} \left( {q_{8} q_{1} - q_{7} q_{2} } \right)\left( {q_{1} r_{2} - q_{4} r_{1} } \right)}}{{\left( {q_{9} q_{1} - q_{7} q_{3} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right) - \left( {q_{8} q_{1} - q_{7} q_{2} } \right)\left( {q_{6} q_{1} - q_{4} q_{3} } \right)}}} \right\}} \right. \\ & - \left\{ {\frac{{q_{2} \left( {q_{1} r_{2} - q_{4} r_{1} } \right)\left( {q_{9} q_{1} - q_{7} q_{3} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right) - q_{2} \left( {q_{1} r_{2} - q_{4} r_{1} } \right)\left( {q_{8} q_{1} - q_{7} q_{2} } \right)\left( {q_{6} q_{1} - q_{4} q_{3} } \right)}}{{\left( {q_{9} q_{1} - q_{7} q_{3} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right) - \left( {q_{8} q_{1} - q_{7} q_{2} } \right)\left( {q_{6} q_{1} - q_{4} q_{3} } \right)}}} \right\} \\ & \left. { - \left\{ {\frac{{q_{2} \left( {q_{6} q_{1} - q_{4} q_{3} } \right)\left( {q_{1} r_{3} - q_{7} r_{1} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right) + q_{2} \left( {q_{6} q_{1} - q_{4} q_{3} } \right)\left( {q_{8} q_{1} - q_{7} q_{2} } \right)\left( {q_{6} r_{1} - q_{4} r_{3} } \right)}}{{\left( {q_{9} q_{1} - q_{7} q_{3} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right) - \left( {q_{8} q_{1} - q_{7} q_{2} } \right)\left( {q_{6} q_{1} - q_{4} q_{3} } \right)}}} \right\}} \right] \\ \end{aligned} $$

$$ \begin{aligned} T_{2n} = & \frac{1}{{\left( {q_{5} q_{1} - q_{4} q_{2} } \right)}}\left[ {\left\{ {\frac{{\left( {q_{1} r_{2} - q_{4} r_{1} } \right)\left( {q_{9} q_{1} - q_{7} q_{3} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right)}}{{\left( {q_{9} q_{1} - q_{7} q_{3} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right) - \left( {q_{8} q_{1} - q_{7} q_{2} } \right)\left( {q_{6} q_{1} - q_{4} q_{3} } \right)}}} \right\}} \right. \\ & - \left\{ {\frac{{\left( {q_{1} r_{2} - q_{4} r_{1} } \right)\left( {q_{8} q_{1} - q_{7} q_{3} } \right)\left( {q_{6} q_{1} - q_{4} q_{3} } \right)}}{{\left( {q_{9} q_{1} - q_{7} q_{3} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right) - \left( {q_{8} q_{1} - q_{7} q_{2} } \right)\left( {q_{6} q_{1} - q_{4} q_{3} } \right)}}} \right\} \\ & - \left\{ {\frac{{\left( {q_{6} r_{1} - q_{4} q_{3} } \right)\left( {q_{1} r_{3} - q_{7} r_{1} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right)}}{{\left( {q_{9} q_{1} - q_{7} q_{3} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right) - \left( {q_{8} q_{1} - q_{7} q_{2} } \right)\left( {q_{6} q_{1} - q_{4} q_{3} } \right)}}} \right\} \\ & \left. { - \left\{ {\frac{{\left( {q_{6} q_{1} - q_{4} q_{3} } \right)\left( {q_{1} q_{8} - q_{7} q_{2} } \right)\left( {q_{1} r_{2} - q_{4} r_{1} } \right)}}{{\left( {q_{9} q_{1} - q_{7} q_{3} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right) - \left( {q_{8} q_{1} - q_{7} q_{2} } \right)\left( {q_{6} q_{1} - q_{4} q_{3} } \right)}}} \right\}} \right] \\ \end{aligned} $$

$$ \begin{aligned} T_{2n} = & \frac{1}{{\left( {q_{5} q_{1} - q_{4} q_{2} } \right)}}\left[ {\left\{ {\frac{{\left( {q_{1} r_{2} - q_{4} r_{1} } \right)\left( {q_{9} q_{1} - q_{7} q_{3} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right)}}{{\left( {q_{9} q_{1} - q_{7} q_{3} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right) - \left( {q_{8} q_{1} - q_{7} q_{2} } \right)\left( {q_{6} q_{1} - q_{4} q_{3} } \right)}}} \right\}} \right. \\ & - \left\{ {\frac{{\left( {q_{1} r_{2} - q_{4} r_{1} } \right)\left( {q_{8} q_{1} - q_{7} q_{3} } \right)\left( {q_{6} q_{1} - q_{4} q_{3} } \right)}}{{\left( {q_{9} q_{1} - q_{7} q_{3} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right) - \left( {q_{8} q_{1} - q_{7} q_{2} } \right)\left( {q_{6} q_{1} - q_{4} q_{3} } \right)}}} \right\} \\ & - \left\{ {\frac{{\left( {q_{6} r_{1} - q_{4} q_{3} } \right)\left( {q_{1} r_{3} - q_{7} r_{1} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right)}}{{\left( {q_{9} q_{1} - q_{7} q_{3} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right) - \left( {q_{8} q_{1} - q_{7} q_{2} } \right)\left( {q_{6} q_{1} - q_{4} q_{3} } \right)}}} \right\} \\ & \left. { - \left\{ {\frac{{\left( {q_{6} q_{1} - q_{4} q_{3} } \right)\left( {q_{1} q_{8} - q_{7} q_{2} } \right)\left( {q_{1} r_{2} - q_{4} r_{1} } \right)}}{{\left( {q_{9} q_{1} - q_{7} q_{3} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right) - \left( {q_{8} q_{1} - q_{7} q_{2} } \right)\left( {q_{6} q_{1} - q_{4} q_{3} } \right)}}} \right\}} \right] \\ \end{aligned} $$

$$ T_{3n} = \frac{{\left( {q_{1} r_{3} - q_{7} r_{1} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right) - \left( {q_{8} q_{1} - q_{7} q_{2} } \right)\left( {q_{1} r_{2} - q_{4} r_{1} } \right)}}{{\left( {q_{9} q_{1} - q_{7} q_{3} } \right)\left( {q_{5} q_{1} - q_{4} q_{2} } \right) - \left( {q_{8} q_{1} - q_{7} q_{2} } \right)\left( {q_{6} q_{1} - q_{4} q_{3} } \right)}} $$

Here

$$ \begin{aligned} & q_{1} = - A_{2} \lambda_{n} + A_{3} \quad q_{2} = A_{4} \lambda_{n} \quad q_{3} = - A_{5} + A_{6} \lambda_{n}^{2} \quad q_{4} = A_{10} + A_{9} \lambda_{n}^{2} \quad q_{5} = - A_{12} - A_{11} \lambda_{n}^{2} \quad q_{6} = (A_{13} - A_{14} \lambda_{n}^{2} )\lambda_{n} \\ & q_{7} = - A_{12} + A_{11} \lambda_{n}^{2} \quad q_{8} = ( - A_{18} + A_{19} \lambda_{n}^{2} )\lambda_{n} \\ & r_{1} = - \left( {\lambda_{n}^{2} + A_{1} } \right)\quad r_{2} = \lambda_{n} A_{7} + A_{8} \quad r_{3} = A_{15} \lambda_{n} \\ \end{aligned} $$

Appendix E

$$\begin{aligned} & T_{4n} = i\alpha C_{11} - \lambda_{n} C_{13} T_{1n} + e_{31} \lambda_{n} T_{2n} - \beta_{1} \left( {1 + a^{*} \alpha^{2} - a^{*} \lambda_{n}^{2} } \right)T_{3n} \\ & T_{5n} = i\alpha C_{13} - \lambda_{n} C_{33} T_{1n} + e_{33} \lambda_{n} T_{2n} - \beta_{3} \left( {1 + a^{*} \alpha^{2} - a^{*} \lambda_{n}^{2} } \right)T_{3n} \\ & T_{6n} = C_{44} \left( {i\alpha T_{In} - \lambda_{n} } \right) - e_{15} i\alpha T_{2n} \\ & T_{7n} = e_{15} \left( {i\alpha T_{In} - \lambda_{n} } \right) - \alpha^{2} \\ & T_{8n} = i\alpha e_{31} - \lambda_{n} e_{33} T_{1n} + \lambda_{n}^{2} T_{2n} T_{1n} - P_{3} \left( {1 + a^{*} \alpha^{2} - a^{*} \lambda_{n}^{2} } \right)T_{3n} \\ \end{aligned}$$