Nonlinear dynamics of FG-GNPRC multiphase composite membranes with internal pores and dielectric properties

Abstract

In this paper, the nonlinear dynamics of the functionally graded graphene nanoplatelet reinforced composite (FG-GNPRC) dielectric and porous membrane subjected to electro-mechanical loading is investigated. The effective material properties of multiphase composites are determined via a two-step hybrid micromechanical model. Based on the hyperelastic membrane theory, Neo-Hookean constitutive model and the couple dielectric theory, the governing equations are obtained using an energy method considering damping and dielectric properties. Taylor series expansion (TSE) and differential quadrature (DQ) methods are utilized to discretize equations, which are then solved numerically by the incremental harmonic balance (IHB) method combined with arc-length continuation technique. The convergence analysis is carried out and the accuracy of the solution method is verified by comparing with the results of previous studies. The influence of the attributes of the internal pore, GNP, the geometric characteristics of membrane, stretching ratio and the applied electric filed on forced vibration and resonance response of the system are analyzed.

Appendices

Appendix 1

$$ \begin{gathered} A_{1} = \left( {\eta \rho + u} \right)^{3} \left[ {\left( {\eta + u_{\rho } } \right)^{2} + \left( {w_{\rho } } \right)^{2} } \right]^{3} \hfill \\ A_{2} \rho ^{2} \left[ {\left( {\eta + u_{\rho } } \right)^{2} + \left( {w_{\rho } } \right)^{2} } \right]^{2} - \left( {\eta \rho + u} \right)\left[ {\left( {\eta + u_{\rho } } \right)^{2} + \left( {w_{\rho } } \right)^{2} } \right]\left[ {3\rho \left( {\eta + u_{\rho } } \right) + \rho ^{2} u_{{\rho \rho }} } \right] \hfill \\ + 2\rho ^{2} \left( {\eta + u_{\rho } } \right)^{2} \left[ {\left( {\eta + u_{\rho } } \right)^{2} + \left( {w_{\rho } } \right)^{2} } \right] + 4\rho ^{2} \left( {\eta + u_{\rho } } \right)\left( {\eta \rho + u} \right)\left[ {\left( {\eta + u_{\rho } } \right)u_{{\rho \rho }} + w_{\rho } w_{{\rho \rho }} } \right] \hfill \\ A_{3} = \left[ {\left( {\eta + u_{\rho } } \right)^{2} + \left( {w_{\rho } } \right)^{2} } \right]\left[ {2\rho ^{2} w_{\rho } \left( {\eta + u_{\rho } } \right) - \left( {3\rho w_{\rho } + \rho ^{2} w_{{\rho \rho }} } \right)\left( {\eta \rho + u} \right)} \right] + 4\rho ^{2} w_{\rho } \left( {\eta \rho + u} \right)\left[ {\left( {\eta + u_{\rho } } \right)u_{{\rho \rho }} + w_{\rho } w_{{\rho \rho }} } \right] \hfill \\ A_{4} = - 4\eta \lambda _{1} u_{\rho } u_{{\rho \rho }} - \frac{{10\lambda _{1} }}{3}\left( {w_{\rho } } \right)^{2} u_{{\rho \rho }} - \frac{{20\lambda _{1} }}{3}u_{\rho } w_{\rho } w_{{\rho \rho }} + \frac{{4\eta \lambda _{1} }}{3}w_{\rho } w_{{\rho \rho }} + 10\lambda _{1} \left( {u_{\rho } } \right)^{2} u_{{\rho \rho }} \hfill \\ A_{5} = \frac{{8\eta \lambda _{1} }}{3}w_{\rho } u_{{\rho \rho }} + \frac{{8\eta \lambda _{1} }}{3}u_{\rho } w_{{\rho \rho }} - \frac{{20\lambda _{1} }}{3}u_{\rho } w_{\rho } u_{{\rho \rho }} - \frac{{10\lambda _{1} }}{3}\left( {u_{\rho } } \right)^{2} w_{{\rho \rho }} + 2\lambda _{1} \left( {w_{\rho } } \right)^{2} w_{{\rho \rho }} \hfill \\ A_{6} = b_{1} \eta ^{2} \left( { - \frac{1}{{\chi ^{2} }}U + \frac{1}{\chi }U_{\chi } + U_{{\chi \chi }} } \right) - b_{1} \kappa \left[ {\frac{\eta }{{\chi ^{3} }}U^{2} - \frac{{2\eta }}{\chi }UU_{{\chi \chi }} } \right. - \frac{\eta }{\chi }\left( {U_{\chi } } \right)^{2} \hfill \\ \left. {\kappa \frac{1}{{\chi ^{3} }}U^{2} U_{\chi } - \kappa \frac{{U^{2} }}{{\chi ^{2} }}U_{{\chi \chi }} - \kappa \frac{1}{{\chi ^{2} }}U\left( {U_{\chi } } \right)^{2} + \frac{\eta }{\chi }\left( {W_{\chi } } \right)^{2} + \kappa \frac{1}{{\chi ^{2} }}U\left( {W_{\chi } } \right)^{2} } \right] \hfill \\ A_{7} = b_{1} \eta ^{2} \left( {\frac{1}{\chi }W_{\chi } + W_{{\chi \chi }} } \right) + b_{1} \kappa \left[ {\frac{{2\eta }}{\chi }U_{\chi } W_{\chi } + \frac{{2\eta }}{\chi }UW_{{\chi \chi }} + \frac{2}{{\chi ^{2} }}\kappa UU_{\chi } W_{\chi } - \frac{{U^{2} }}{{\chi ^{3} }}\kappa W_{\chi } + \frac{{U^{2} }}{{\chi ^{2} }}\kappa W_{{\chi \chi }} } \right] \hfill \\ A_{8} = a_{{55}} \kappa \left[ {\frac{4}{3}\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} W_{m} } \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 2 \right)}} W_{m} } } \right. + \frac{3}{{\chi _{i} ^{3} }}U_{i} ^{2} + \frac{1}{3}\frac{1}{{\chi _{i} }}\left( {\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} W_{m} } } \right)^{2} - \frac{2}{{\chi _{i} }}U_{i} \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 2 \right)}} U_{m} } - 4\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} U_{m} } \sum\limits_{{m = 1}}^{N} {c_{{_{{im}} }}^{{\left( 2 \right)}} U_{m} } \left. { - \frac{3}{{\chi _{i} }}\left( {\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} U_{m} } } \right)^{2} } \right] \hfill \\ A_{9} = a_{{66}} \kappa ^{2} \left[ {\frac{6}{{\chi _{i} }}\left( {\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} U_{m} } } \right)^{3} - \frac{6}{{\chi _{i} ^{4} }}U_{i} ^{3} } \right. - \frac{3}{{\chi _{i} ^{3} }}U_{i} ^{2} \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} U_{m} } + \frac{3}{{\chi _{i} ^{2} }}U_{i} ^{2} \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 2 \right)}} U_{m} } + \frac{3}{{\chi _{i} ^{2} }}U_{i} \left( {\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} U_{m} } } \right)^{2} \hfill \\ + \frac{1}{{\chi _{i} ^{2} }}U_{i} \left( {\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} W_{m} } } \right)^{2} - \frac{8}{3}\frac{1}{{\chi _{i} }}U_{i} \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} W_{m} } \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 2 \right)}} W_{m} } + \frac{8}{{\chi _{i} }}U_{i} \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} U_{m} } \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 2 \right)}} U_{m} } - \frac{{10}}{3}\left( {\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} W_{m} } } \right)^{2} \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 2 \right)}} U_{m} } \hfill \\ - \frac{{10}}{3}\frac{1}{{\chi _{i} }}\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} U_{m} } \left( {\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} W_{m} } } \right)^{2} - \frac{{20}}{3}\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} U_{m} } \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} W_{m} } \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 2 \right)}} W_{m} } + \left. {10\left( {\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} U_{m} } } \right)^{2} \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 2 \right)}} U_{m} } } \right] \hfill \\ A_{{10}} = a_{{55}} \kappa \left[ {\frac{2}{{\chi _{i} }}\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} U_{m} } \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} W_{m} } + \frac{4}{{3\chi _{i} }}U_{i} \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 2 \right)}} W_{m} } } \right. + \frac{8}{3}\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} W_{m} } \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 2 \right)}} U_{m} } \left. { + \frac{8}{3}\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} U_{m} } \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 2 \right)}} W_{m} } } \right] \hfill \\ A_{{11}} = a_{{66}} \kappa ^{2} \left[ {\frac{1}{{\chi _{i} ^{3} }}U_{i} ^{2} \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} W_{m} } } \right. - \frac{2}{{\chi _{i} ^{2} }}U_{i} \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} U_{m} } \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} W_{m} } - \frac{1}{{\chi _{i} ^{2} }}U_{i} ^{2} \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 2 \right)}} W_{m} } + \frac{2}{{3\chi _{i} }}\left( {\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} W_{m} } } \right)^{3} \hfill \\ - \frac{6}{{\chi _{i} }}\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} W_{m} } \left( {\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} U_{m} } } \right)^{2} - \frac{8}{{3\chi _{i} }}U_{i} \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} W_{m} } \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 2 \right)}} U_{m} } - \frac{{20}}{3}\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} U_{m} } \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} W_{m} } \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 2 \right)}} U_{m} } \hfill \\ - \frac{{10}}{3}\left( {\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} U_{m} } } \right)^{2} \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 2 \right)}} W_{m} } - \frac{8}{{3\chi _{i} }}U_{i} \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} U_{m} } \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 2 \right)}} W_{m} } + \left. {2\left( {\sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 1 \right)}} W_{m} } } \right)^{2} \sum\limits_{{m = 1}}^{N} {c_{{im}}^{{\left( 2 \right)}} W_{m} } } \right] \hfill \\ \end{gathered} $$

(57)

where

$$ \left\{ \begin{gathered} b_{1} = \sum\limits_{i = 1}^{{N_{L} }} {\int_{{h_{i} }}^{{h_{i + 1} }} {\left( {\frac{{V_{(i)}^{2} \varepsilon_{0} \varepsilon_{r(i)} }}{{C_{0} h^{3} }}} \right)} {\text{d}}z} \hfill \\ a_{55} = \frac{{6a_{1} }}{{\eta^{7} }} \hfill \\ a_{66} = \frac{{6a_{1} }}{{\eta^{8} }} \hfill \\ \end{gathered} \right. $$

(58)

Appendix 2

$$ {\varvec{K}}_{{{\text{NL}}}} = \left[ {\begin{array}{*{20}c} {{\varvec{K}}_{{{\text{NL}}}} \left( {11} \right)} & {{\varvec{K}}_{{{\text{NL}}}} \left( {12} \right)} \\ {{\varvec{K}}_{{{\text{NL}}}} \left( {21} \right)} & {{\varvec{K}}_{{{\text{NL}}}} \left( {22} \right)} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} A & \ldots & B & E & \cdots & F \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ C & \cdots & D & G & \cdots & H \\ I & \cdots & J & M & \cdots & N \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ K & \cdots & L & O & \cdots & P \\ \end{array} } \right] $$

(59)

where

$$ \begin{aligned} A = & a_{55} \kappa \left( {\frac{3}{{\chi_{i}^{3} }}U_{1} } \right) - \frac{{2a_{55} \kappa }}{{\chi_{i} }}\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} U_{m} - 4a_{55} \kappa \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} U_{m} } } \right)c_{11}^{\left( 1 \right)} } - \frac{{3a_{55} \kappa }}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} U_{m} } } \right)c_{11}^{\left( 1 \right)} - a_{66} \kappa^{2} \left( {\frac{6}{{\chi_{i}^{4} }}U_{1}^{2} } \right) \\ + & \frac{{6a_{66} \kappa^{2} }}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} U_{m} } } \right)^{2} c_{11}^{\left( 1 \right)} - \frac{{3a_{66} \kappa^{2} }}{{\chi_{i}^{3} }}\left( {U_{1}^{2} } \right)c_{11}^{\left( 1 \right)} + \frac{{3a_{66} \kappa^{2} }}{{\chi_{i}^{2} }}U_{1} \sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} U_{m} + \frac{{3a_{66} \kappa^{2} }}{{\chi_{i}^{2} }}U_{1} \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} U_{m} } } \right)} c_{11}^{\left( 1 \right)} \\ + & \frac{{8a_{66} \kappa^{2} }}{{\chi_{i} }}U_{1} \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} U_{m} } } \right)c_{11}^{\left( 1 \right)} - \frac{{20a_{66} \kappa^{2} }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} W_{m} } } \right)c_{11}^{\left( 1 \right)} + 10a_{66} \kappa^{2} \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} U_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} U_{m} } } \right)c_{11}^{\left( 1 \right)} \\ - & a_{33} \left[ {\frac{1}{{\chi_{i}^{3} }}U_{1} - \frac{2}{{\chi_{i} }}\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 2 \right)} U_{m} } - \frac{1}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 1 \right)} U_{m} } } \right)c_{11}^{\left( 1 \right)} } \right] - a_{44} \left[ {\frac{1}{{\chi_{i}^{3} }}U_{1} \sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 1 \right)} U_{m} } - \frac{1}{{\chi_{i}^{2} }}U_{1} \sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 2 \right)} U_{m} } - \frac{1}{{\chi_{i}^{2} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 1 \right)} U_{m} } } \right)^{2} + \frac{1}{{\chi_{i}^{2} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 1 \right)} W_{m} } } \right)^{2} } \right] \\ \end{aligned} $$

(60)

$$ \begin{aligned} B = & a_{55} \kappa \left( {\frac{3}{{\chi_{i}^{3} }}U_{1} } \right) - \frac{{2a_{55} \kappa }}{{\chi_{i} }}\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} U_{m} - 4a_{55} \kappa \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} U_{m} } } \right)c_{1N}^{\left( 1 \right)} } - \frac{{3a_{55} \kappa }}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} U_{m} } } \right)c_{1N}^{\left( 1 \right)} - a_{66} \kappa^{2} \left( {\frac{6}{{\chi_{i}^{4} }}U_{1}^{2} } \right) \\ + & \frac{{6a_{66} \kappa^{2} }}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} U_{m} } } \right)^{2} c_{1N}^{\left( 1 \right)} - \frac{{3a_{66} \kappa^{2} }}{{\chi_{i}^{3} }}\left( {U_{1}^{2} } \right)c_{1N}^{\left( 1 \right)} + \frac{{3a_{66} \kappa^{2} }}{{\chi_{i}^{2} }}U_{1} \sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} U_{m} + \frac{{3a_{66} \kappa^{2} }}{{\chi_{i}^{2} }}U_{1} \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} U_{m} } } \right)} c_{1N}^{\left( 1 \right)} \\ + & \frac{{8a_{66} \kappa^{2} }}{{\chi_{i} }}U_{1} \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} U_{m} } } \right)c_{1N}^{\left( 1 \right)} - \frac{{20a_{66} \kappa^{2} }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} W_{m} } } \right)c_{1N}^{\left( 1 \right)} + 10a_{66} \kappa^{2} \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} U_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} U_{m} } } \right)c_{1N}^{\left( 1 \right)} \\ - & a_{33} \left[ {\frac{1}{{\chi_{i}^{3} }}U_{1} - \frac{2}{{\chi_{i} }}\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 2 \right)} U_{m} } - \frac{1}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 1 \right)} U_{m} } } \right)c_{1N}^{\left( 1 \right)} } \right] - a_{44} \left[ {\frac{1}{{\chi_{i}^{3} }}U_{1} \sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 1 \right)} U_{m} } - \frac{1}{{\chi_{i}^{2} }}U_{1} \sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 2 \right)} U_{m} } - \frac{1}{{\chi_{i}^{2} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 1 \right)} U_{m} } } \right)^{2} + \frac{1}{{\chi_{i}^{2} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 1 \right)} W_{m} } } \right)^{2} } \right] \\ \end{aligned} $$

(61)

$$ \begin{aligned} C = & a_{55} \kappa \left( {\frac{3}{{\chi_{i}^{3} }}U_{N} } \right) - \frac{{2a_{55} \kappa }}{{\chi_{i} }}\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} U_{m} - 4a_{55} \kappa \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} U_{m} } } \right)c_{N1}^{\left( 1 \right)} } - \frac{{3a_{55} \kappa }}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} U_{m} } } \right)c_{N1}^{\left( 1 \right)} - a_{66} \kappa^{2} \left( {\frac{6}{{\chi_{i}^{4} }}U_{N}^{2} } \right) \\ + & \frac{{6a_{66} \kappa^{2} }}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} U_{m} } } \right)^{2} c_{N1}^{\left( 1 \right)} - \frac{{3a_{66} \kappa^{2} }}{{\chi_{i}^{3} }}\left( {U_{N}^{2} } \right)c_{N1}^{\left( 1 \right)} + \frac{{3a_{66} \kappa^{2} }}{{\chi_{i}^{2} }}U_{N} \sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} U_{m} + \frac{{3a_{66} \kappa^{2} }}{{\chi_{i}^{2} }}U_{N} \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} U_{m} } } \right)} c_{N1}^{\left( 1 \right)} \\ + & \frac{{8a_{66} \kappa^{2} }}{{\chi_{i} }}U_{N} \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} U_{m} } } \right)c_{N1}^{\left( 1 \right)} - \frac{{20a_{66} \kappa^{2} }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} W_{m} } } \right)c_{N1}^{\left( 1 \right)} + 10a_{66} \kappa^{2} \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} U_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} U_{m} } } \right)c_{N1}^{\left( 1 \right)} \\ - & a_{33} \left[ {\frac{1}{{\chi_{i}^{3} }}U_{N} - \frac{2}{{\chi_{i} }}\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 2 \right)} U_{m} } - \frac{1}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 1 \right)} U_{m} } } \right)c_{N1}^{\left( 1 \right)} } \right] - a_{44} \left[ {\frac{1}{{\chi_{i}^{3} }}U_{N} \sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 1 \right)} U_{m} } - \frac{1}{{\chi_{i}^{2} }}U_{N} \sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 2 \right)} U_{m} } - \frac{1}{{\chi_{i}^{2} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 1 \right)} U_{m} } } \right)^{2} + \frac{1}{{\chi_{i}^{2} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 1 \right)} W_{m} } } \right)^{2} } \right] \\ \end{aligned} $$

(63)

$$ \begin{aligned} D = & a_{55} \kappa \left( {\frac{3}{{\chi_{i}^{3} }}U_{N} } \right) - \frac{{2a_{55} \kappa }}{{\chi_{i} }}\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} U_{m} - 4a_{55} \kappa \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} U_{m} } } \right)c_{NN}^{\left( 1 \right)} } - \frac{{3a_{55} \kappa }}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} U_{m} } } \right)c_{NN}^{\left( 1 \right)} - a_{66} \kappa^{2} \left( {\frac{6}{{\chi_{i}^{4} }}U_{N}^{2} } \right) \\ + & \frac{{6a_{66} \kappa^{2} }}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} U_{m} } } \right)^{2} c_{NN}^{\left( 1 \right)} - \frac{{3a_{66} \kappa^{2} }}{{\chi_{i}^{3} }}\left( {U_{N}^{2} } \right)c_{NN}^{\left( 1 \right)} + \frac{{3a_{66} \kappa^{2} }}{{\chi_{i}^{2} }}U_{N} \sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} U_{m} + \frac{{3a_{66} \kappa^{2} }}{{\chi_{i}^{2} }}U_{N} \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} U_{m} } } \right)} c_{NN}^{\left( 1 \right)} \\ + & \frac{{8a_{66} \kappa^{2} }}{{\chi_{i} }}U_{N} \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} U_{m} } } \right)c_{NN}^{\left( 1 \right)} - \frac{{20a_{66} \kappa^{2} }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} W_{m} } } \right)c_{NN}^{\left( 1 \right)} + 10a_{66} \kappa^{2} \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} U_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} U_{m} } } \right)c_{NN}^{\left( 1 \right)} \\ - & a_{33} \left[ {\frac{1}{{\chi_{i}^{3} }}U_{N} - \frac{2}{{\chi_{i} }}\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 2 \right)} U_{m} } - \frac{1}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 1 \right)} U_{m} } } \right)c_{NN}^{\left( 1 \right)} } \right] - a_{44} \left[ {\frac{1}{{\chi_{i}^{3} }}U_{N} \sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 1 \right)} U_{m} } - \frac{1}{{\chi_{i}^{2} }}U_{N} \sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 2 \right)} U_{m} } - \frac{1}{{\chi_{i}^{2} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 1 \right)} U_{m} } } \right)^{2} + \frac{1}{{\chi_{i}^{2} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 1 \right)} W_{m} } } \right)^{2} } \right] \\ \end{aligned} $$

(63)

$$ \begin{aligned} E = & \frac{{a_{55} \kappa }}{{3\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)c_{11}^{\left( 1 \right)} + \frac{{a_{66} \kappa^{2} }}{{\chi^{2}_{i} }}U_{1} \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)c_{11}^{\left( 1 \right)} - \frac{{8a_{66} \kappa^{2} }}{{3\chi_{i} }}U_{1} \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} W_{m} } } \right)c_{11}^{\left( 1 \right)} \\ - & \frac{{10a_{66} \kappa^{2} }}{{3\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} U_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)c_{11}^{\left( 1 \right)} - \frac{{10a_{66} \kappa^{2} }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} U_{m} } } \right)c_{11}^{\left( 1 \right)} + \frac{{4a_{55} \kappa }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} W_{m} } } \right)c_{11}^{\left( 1 \right)} - a_{33} \left[ {\frac{1}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 1 \right)} W_{m} } } \right)c_{11}^{\left( 1 \right)} } \right] \\ \end{aligned} $$

(64)

$$ \begin{aligned} F = & \frac{{a_{55} \kappa }}{{3\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)c_{1N}^{\left( 1 \right)} + \frac{{a_{66} \kappa^{2} }}{{\chi^{2}_{i} }}U_{1} \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)c_{1N}^{\left( 1 \right)} - \frac{{8a_{66} \kappa^{2} }}{{3\chi_{i} }}U_{1} \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} W_{m} } } \right)c_{1N}^{\left( 1 \right)} \\ - & \frac{{10a_{66} \kappa^{2} }}{{3\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} U_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)c_{1N}^{\left( 1 \right)} - \frac{{10a_{66} \kappa^{2} }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} U_{m} } } \right)c_{1N}^{\left( 1 \right)} + \frac{{4a_{55} \kappa }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} W_{m} } } \right)c_{1N}^{\left( 1 \right)} - a_{33} \left[ {\frac{1}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 1 \right)} W_{m} } } \right)c_{1N}^{\left( 1 \right)} } \right] \\ \end{aligned} $$

(65)

$$ \begin{aligned} G = & \frac{{a_{55} \kappa }}{{3\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)c_{N1}^{\left( 1 \right)} + \frac{{a_{66} \kappa^{2} }}{{\chi^{2}_{i} }}U_{N} \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)c_{N1}^{\left( 1 \right)} - \frac{{8a_{66} \kappa^{2} }}{{3\chi_{i} }}U_{N} \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} W_{m} } } \right)c_{N1}^{\left( 1 \right)} \\ - & \frac{{10a_{66} \kappa^{2} }}{{3\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} U_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)c_{N1}^{\left( 1 \right)} - \frac{{10a_{66} \kappa^{2} }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} U_{m} } } \right)c_{N1}^{\left( 1 \right)} + \frac{{4a_{55} \kappa }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} W_{m} } } \right)c_{N1}^{\left( 1 \right)} - a_{33} \left[ {\frac{1}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 1 \right)} W_{m} } } \right)c_{N1}^{\left( 1 \right)} } \right] \\ \end{aligned} $$

(66)

$$ \begin{aligned} H = & \frac{{a_{55} \kappa }}{{3\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)c_{NN}^{\left( 1 \right)} + \frac{{a_{66} \kappa^{2} }}{{\chi^{2}_{i} }}U_{N} \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)c_{NN}^{\left( 1 \right)} - \frac{{8a_{66} \kappa^{2} }}{{3\chi_{i} }}U_{N} \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} W_{m} } } \right)c_{NN}^{\left( 1 \right)} \\ - & \frac{{10a_{66} \kappa^{2} }}{{3\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} U_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)c_{NN}^{\left( 1 \right)} - \frac{{10a_{66} \kappa^{2} }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} U_{m} } } \right)c_{NN}^{\left( 1 \right)} + \frac{{4a_{55} \kappa }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} W_{m} } } \right)c_{NN}^{\left( 1 \right)} - a_{33} \left[ {\frac{1}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 1 \right)} W_{m} } } \right)c_{NN}^{\left( 1 \right)} } \right] \\ \end{aligned} $$

(67)

$$ \begin{aligned} I = & \frac{{2a_{55} \kappa }}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)c_{11}^{\left( 1 \right)} + \frac{{4a_{55} \kappa }}{{3\chi_{i} }}\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} W_{m} + \frac{{8a_{55} \kappa }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} W_{m} } } \right)} c_{11}^{\left( 1 \right)} - \frac{{2a_{66} \kappa^{2} }}{{\chi_{i}^{2} }}U_{1} \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)c_{11}^{\left( 1 \right)} \\ - & \frac{{a_{66} \kappa^{2} }}{{\chi_{i}^{2} }}U_{1} \sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} W_{m} - \frac{{20a_{66} \kappa^{2} }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} U_{m} } } \right)} c_{11}^{\left( 1 \right)} - \frac{{10a_{66} \kappa^{2} }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} U_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} W_{m} } } \right)c_{11}^{\left( 1 \right)} - \frac{{8a_{66} \kappa^{2} }}{{3\chi_{i} }}U_{1} \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} W_{m} } } \right)c_{11}^{\left( 1 \right)} \\ + & a_{33} \left[ {\frac{2}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 1 \right)} W_{m} } } \right)c_{11}^{\left( 1 \right)} + \frac{2}{{\chi_{i} }}\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 2 \right)} W_{m} } } \right] + a_{44} \left[ {\frac{2}{{\chi_{i}^{2} }}\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 1 \right)} U_{m} } \sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 1 \right)} W_{m} } - \frac{{U_{1} }}{{\chi_{i}^{3} }}\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 1 \right)} W_{m} } + \frac{{U_{1} }}{{\chi_{i}^{2} }}\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 2 \right)} W_{m} } } \right] \\ \end{aligned} $$

(68)

$$ \begin{aligned} J = & \frac{{2a_{55} \kappa }}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)c_{1N}^{\left( 1 \right)} + \frac{{4a_{55} \kappa }}{{3\chi_{i} }}\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} W_{m} + \frac{{8a_{55} \kappa }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} W_{m} } } \right)} c_{1N}^{\left( 1 \right)} - \frac{{2a_{66} \kappa^{2} }}{{\chi_{i}^{2} }}U_{1} \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)c_{1N}^{\left( 1 \right)} \\ - & \frac{{a_{66} \kappa^{2} }}{{\chi_{i}^{2} }}U_{1} \sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} W_{m} - \frac{{20a_{66} \kappa^{2} }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} U_{m} } } \right)} c_{1N}^{\left( 1 \right)} - \frac{{10a_{66} \kappa^{2} }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} U_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} W_{m} } } \right)c_{1N}^{\left( 1 \right)} - \frac{{8a_{66} \kappa^{2} }}{{3\chi_{i} }}U_{1} \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} W_{m} } } \right)c_{1N}^{\left( 1 \right)} \\ + & a_{33} \left[ {\frac{2}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 1 \right)} W_{m} } } \right)c_{1N}^{\left( 1 \right)} + \frac{2}{{\chi_{i} }}\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 2 \right)} W_{m} } } \right] + a_{44} \left[ {\frac{2}{{\chi_{i}^{2} }}\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 1 \right)} U_{m} } \sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 1 \right)} W_{m} } - \frac{{U_{1} }}{{\chi_{i}^{3} }}\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 1 \right)} W_{m} } + \frac{{U_{1} }}{{\chi_{i}^{2} }}\sum\limits_{m = 1}^{N} {c_{{_{1m} }}^{\left( 2 \right)} W_{m} } } \right] \\ \end{aligned} $$

(69)

$$ \begin{aligned} K = & \frac{{2a_{55} \kappa }}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)c_{N1}^{\left( 1 \right)} + \frac{{4a_{55} \kappa }}{{3\chi_{i} }}\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} W_{m} + \frac{{8a_{55} \kappa }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} W_{m} } } \right)} c_{N1}^{\left( 1 \right)} - \frac{{2a_{66} \kappa^{2} }}{{\chi_{i}^{2} }}U_{N} \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)c_{N1}^{\left( 1 \right)} \\ - & \frac{{a_{66} \kappa^{2} }}{{\chi_{i}^{2} }}U_{N} \sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} W_{m} - \frac{{20a_{66} \kappa^{2} }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} U_{m} } } \right)} c_{N1}^{\left( 1 \right)} - \frac{{10a_{66} \kappa^{2} }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} U_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} W_{m} } } \right)c_{N1}^{\left( 1 \right)} - \frac{{8a_{66} \kappa^{2} }}{{3\chi_{i} }}U_{N} \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} W_{m} } } \right)c_{N1}^{\left( 1 \right)} \\ + & a_{33} \left[ {\frac{2}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 1 \right)} W_{m} } } \right)c_{N1}^{\left( 1 \right)} + \frac{2}{{\chi_{i} }}\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 2 \right)} W_{m} } } \right] + a_{44} \left[ {\frac{2}{{\chi_{i}^{2} }}\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 1 \right)} U_{m} } \sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 1 \right)} W_{m} } - \frac{{U_{N} }}{{\chi_{i}^{3} }}\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 1 \right)} W_{m} } + \frac{{U_{N} }}{{\chi_{i}^{2} }}\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 2 \right)} W_{m} } } \right] \\ \end{aligned} $$

(70)

$$ \begin{aligned} L = & \frac{{2a_{55} \kappa }}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)c_{NN}^{\left( 1 \right)} + \frac{{4a_{55} \kappa }}{{3\chi_{i} }}\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} W_{m} + \frac{{8a_{55} \kappa }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} W_{m} } } \right)} c_{NN}^{\left( 1 \right)} - \frac{{2a_{66} \kappa^{2} }}{{\chi_{i}^{2} }}U_{N} \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)c_{NN}^{\left( 1 \right)} \\ - & \frac{{a_{66} \kappa^{2} }}{{\chi_{i}^{2} }}U_{N} \sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} W_{m} - \frac{{20a_{66} \kappa^{2} }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} U_{m} } } \right)} c_{NN}^{\left( 1 \right)} - \frac{{10a_{66} \kappa^{2} }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} U_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} W_{m} } } \right)c_{NN}^{\left( 1 \right)} - \frac{{8a_{66} \kappa^{2} }}{{3\chi_{i} }}U_{N} \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} W_{m} } } \right)c_{NN}^{\left( 1 \right)} \\ + & a_{33} \left[ {\frac{2}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 1 \right)} W_{m} } } \right)c_{NN}^{\left( 1 \right)} + \frac{2}{{\chi_{i} }}\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 2 \right)} W_{m} } } \right] + a_{44} \left[ {\frac{2}{{\chi_{i}^{2} }}\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 1 \right)} U_{m} } \sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 1 \right)} W_{m} } - \frac{{U_{N} }}{{\chi_{i}^{3} }}\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 1 \right)} W_{m} } + \frac{{U_{N} }}{{\chi_{i}^{2} }}\sum\limits_{m = 1}^{N} {c_{{_{Nm} }}^{\left( 2 \right)} W_{m} } } \right] \\ \end{aligned} $$

(71)

$$ \begin{aligned} M = & \frac{{8a_{55} \kappa }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} U_{m} } } \right)c_{11}^{\left( 1 \right)} + \frac{{a_{66} \kappa^{2} }}{{\chi_{i}^{3} }}U_{1}^{2} c_{11}^{\left( 1 \right)} - \frac{{6a_{66} \kappa^{2} }}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} U_{m} } } \right)^{2} c_{11}^{\left( 1 \right)} - \frac{{8a_{66} \kappa^{2} }}{{3\chi_{i} }}U_{1} \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} U_{m} } } \right)c_{11}^{\left( 1 \right)} \\ + & \frac{{2a_{66} \kappa^{2} }}{{3\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)^{2} c_{11}^{\left( 1 \right)} + 2a_{66} \kappa^{2} \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} W_{m} } } \right)c_{11}^{\left( 1 \right)} \\ \end{aligned} $$

(72)

$$ \begin{aligned} N = & \frac{{8a_{55} \kappa }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} U_{m} } } \right)c_{1N}^{\left( 1 \right)} + \frac{{a_{66} \kappa^{2} }}{{\chi_{i}^{3} }}U_{1}^{2} c_{1N}^{\left( 1 \right)} - \frac{{6a_{66} \kappa^{2} }}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} U_{m} } } \right)^{2} c_{1N}^{\left( 1 \right)} - \frac{{8a_{66} \kappa^{2} }}{{3\chi_{i} }}U_{1} \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} U_{m} } } \right)c_{1N}^{\left( 1 \right)} \\ + & \frac{{2a_{66} \kappa^{2} }}{{3\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)^{2} c_{1N}^{\left( 1 \right)} + 2a_{66} \kappa^{2} \left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 1 \right)} W_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{1m}^{\left( 2 \right)} W_{m} } } \right)c_{1N}^{\left( 1 \right)} \\ \end{aligned} $$

(73)

$$ \begin{aligned} O = & \frac{{8a_{55} \kappa }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} U_{m} } } \right)c_{N1}^{\left( 1 \right)} + \frac{{a_{66} \kappa^{2} }}{{\chi_{i}^{3} }}U_{N}^{2} c_{N1}^{\left( 1 \right)} - \frac{{6a_{66} \kappa^{2} }}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} U_{m} } } \right)^{2} c_{N1}^{\left( 1 \right)} - \frac{{8a_{66} \kappa^{2} }}{{3\chi_{i} }}U_{N} \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} U_{m} } } \right)c_{N1}^{\left( 1 \right)} \\ + & \frac{{2a_{66} \kappa^{2} }}{{3\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)^{2} c_{N1}^{\left( 1 \right)} + 2a_{66} \kappa^{2} \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} W_{m} } } \right)c_{N1}^{\left( 1 \right)} \\ \end{aligned} $$

(74)

$$ \begin{aligned} P = & \frac{{8a_{55} \kappa }}{3}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} U_{m} } } \right)c_{NN}^{\left( 1 \right)} + \frac{{a_{66} \kappa^{2} }}{{\chi_{i}^{3} }}U_{N}^{2} c_{NN}^{\left( 1 \right)} - \frac{{6a_{66} \kappa^{2} }}{{\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} U_{m} } } \right)^{2} c_{NN}^{\left( 1 \right)} - \frac{{8a_{66} \kappa^{2} }}{{3\chi_{i} }}U_{N} \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} U_{m} } } \right)c_{NN}^{\left( 1 \right)} \\ + & \frac{{2a_{66} \kappa^{2} }}{{3\chi_{i} }}\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)^{2} c_{NN}^{\left( 1 \right)} + 2a_{66} \kappa^{2} \left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 1 \right)} W_{m} } } \right)\left( {\sum\limits_{m = 1}^{N} {c_{Nm}^{\left( 2 \right)} W_{m} } } \right)c_{NN}^{\left( 1 \right)} \\ \end{aligned} $$

(75)