The critical voltage of a GPL-reinforced composite microdisk covered with piezoelectric layer

Abstract

In this research, electrically characteristics of a graphene nanoplatelet (GPL)-reinforced composite (GPLRC) microdisk are explored using generalized differential quadrature method. Also, the current microstructure is coupled with a piezoelectric actuator (PIAC). The extended form of Halpin–Tsai micromechanics is used to acquire the elasticity of the structure, whereas the variation of thermal expansion, Poisson’s ratio, and density through the thickness direction is determined by the rule of mixtures. Hamilton’s principle is implemented to establish governing equations and associated boundary conditions of the GPLRC microdisk joint with PIAC. The compatibility conditions are satisfied by taking perfect bonding between the core and PIAC into consideration. Maxwell’s equation is employed to capture the piezoelectricity effects. The numerical results revealed the important role of ratios of length scale and nonlocal to thickness, outer-to-inner ratio of radius ($R_{\text{o}} /R_{\text{i}}$), ratio of piezoelectric to core thickness (h_p/h), and GPL weight fraction ($g_{\text{GPL}}$) on the critical voltage of the system. Another important consequence is that by increasing $R_{\text{o}} /R_{\text{i}}$, the critical voltage of the smart structure increases more intensely in comparison with the $g_{\text{GPL}}$.

Appendix

The governing equations of the structure are presented as follows:

$$\begin{aligned} & \delta u^{i}_{0} :\\ & \quad (1 - l^{2} \nabla^{2} )\left[ {\begin{array}{*{20}l} {\frac{\partial }{\partial R}\left( {\begin{array}{*{20}l} {\left[ {A^{i}_{11} \frac{{\partial u^{i}_{0} }}{{\partial R^{i} }} + B^{i}_{11} \frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} - D^{i}_{11} c_{1} \left( {\frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} + \frac{{\partial^{2} w^{i}_{0} }}{{\partial R^{i2} }}} \right)} \right]} \hfill \\ {\quad + \left[ {A^{i}_{12} \frac{{u^{i}_{0} }}{{R^{i} }} + B^{i}_{12} \frac{{u_{1} }}{R} - D^{i}_{12} c_{1} \left( {\frac{{u^{i}_{1} }}{{R^{i} }} + \frac{{\partial w^{i}_{0} }}{{R^{i} \partial R^{i} }}} \right)} \right] - X_{31} \phi } \hfill \\ \end{array} } \right)} \hfill \\ {\quad - \frac{1}{R}\left( {\begin{array}{*{20}l} {\left[ {A^{i}_{12} \frac{{\partial u^{i}_{0} }}{{\partial R^{i} }} + B^{i}_{12} \frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} - D^{i}_{12} c_{1} \left( {\frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} + \frac{{\partial^{2} w^{i}_{0} }}{{\partial R^{i2} }}} \right)} \right]} \hfill \\ {\quad + Q^{i}_{22} \left[ {A^{i}_{22} \frac{{u^{i}_{0} }}{{R^{i} }} + B^{i}_{22} \frac{{u^{i}_{1} }}{{R^{i} }} - D^{i}_{22} c_{1} \left( {\frac{{u^{i}_{1} }}{{R^{i} }} + \frac{{\partial w^{i}_{0} }}{{R^{i} \partial R^{i} }}} \right)} \right]} \hfill \\ \end{array} } \right)} \hfill \\ \end{array} } \right] = 0;\quad i = c,p \\ \end{aligned}$$

(P-1)

$$\begin{aligned} & \delta w^{i}_{0} :\\ & \quad (1 - l^{2} \nabla^{2} )\left[ {\begin{array}{*{20}l} {c_{1} \frac{{\partial^{2} }}{{\partial R^{i2} }}\left( {\begin{array}{*{20}l} {\left[ {D^{i}_{11} \frac{{\partial u^{i}_{0} }}{{\partial R^{i} }} + E^{i}_{11} \frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} - G^{i}_{11} c_{1} \left( {\frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} + \frac{{\partial^{2} w^{i}_{0} }}{{\partial R^{i2} }}} \right)} \right]} \hfill \\ {\quad + \left[ {D^{i}_{12} \frac{{u^{i}_{0} }}{{R^{i} }} + E^{i}_{12} \frac{{u^{i}_{1} }}{{R^{i} }} - G^{i}_{12} c_{1} \left( {\frac{{u^{i}_{1} }}{{R^{i} }} + \frac{{\partial w^{i}_{0} }}{{R^{i} \partial R^{i} }}} \right)} \right] - X_{33} \phi } \hfill \\ \end{array} } \right)} \hfill \\ {\quad - c_{1} \frac{\partial }{{R^{i} \partial R^{i} }}\left( {\begin{array}{*{20}l} {\left[ {D^{i}_{12} \frac{{\partial u^{i}_{0} }}{{\partial R^{i} }} + E^{i}_{12} \frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} - G^{i}_{12} c_{1} \left( {\frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} + \frac{{\partial^{2} w^{i}_{0} }}{{\partial R^{i2} }}} \right)} \right]} \hfill \\ {\quad + Q^{i}_{22} \left[ {D^{i}_{22} \frac{{u^{i}_{0} }}{{R^{i} }} + E^{i}_{22} \frac{{u^{i}_{1} }}{{R^{i} }} - G^{i}_{22} c_{1} \left( {\frac{{u^{i}_{1} }}{{R^{i} }} + \frac{{\partial w^{i}_{0} }}{{R^{i} \partial R^{i} }}} \right)} \right]} \hfill \\ \end{array} } \right)} \hfill \\ {\quad + \frac{\partial }{{\partial R^{i} }}\left( {(A^{i}_{55} - 3C^{i}_{55} c_{1} )\left( {u^{i}_{1} + \frac{{\partial w^{i}_{0} }}{{\partial R^{i} }}} \right) + X_{11} \partial \phi /\partial R} \right)} \hfill \\ {\quad - 3c_{1} \frac{\partial }{{\partial R^{i} }}\left( {(C^{i}_{55} - 3E^{i}_{55} c_{1} )\left( {u^{i}_{1} + \frac{{\partial w^{i}_{0} }}{{\partial R^{i} }}} \right) + X_{12} \partial \phi /\partial R^{i} } \right) - N_{1}^{p} w^{i}_{{0,x^{2} }} } \hfill \\ \end{array} } \right] = 0;\quad i = c,p \\ \end{aligned}$$

(P-2)

$$\begin{aligned} & \delta u^{i}_{1} :\\ & \quad (1 - l^{2} \nabla^{2} )\left[ {\begin{array}{*{20}l} {\frac{\partial }{{\partial R^{i} }}\left( {\begin{array}{*{20}l} {\left[ {B^{i}_{11} \frac{{\partial u^{i}_{0} }}{{\partial R^{i} }} + C^{i}_{11} \frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} - E^{i}_{11} c_{1} \left( {\frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} + \frac{{\partial^{2} w^{i}_{0} }}{{\partial R^{i2} }}} \right)} \right]} \hfill \\ {\quad + \left[ {B^{i}_{12} \frac{{u^{i}_{0} }}{{R^{i} }} + C^{i}_{12} \frac{{u^{i}_{1} }}{{R^{i} }} - E^{i}_{12} c_{1} \left( {\frac{{u^{i}_{1} }}{{R^{i} }} + \frac{{\partial w^{i}_{0} }}{{R^{i} \partial R^{i} }}} \right)} \right] - X_{32} \phi } \hfill \\ \end{array} } \right)} \hfill \\ {\quad - \frac{{c_{1} \partial }}{{\partial R^{i} }}\left( {\begin{array}{*{20}l} {\left[ {D^{i}_{11} \frac{{\partial u^{i}_{0} }}{{\partial R^{i} }} + E^{i}_{11} \frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} - G^{i}_{11} c_{1} \left( {\frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} + \frac{{\partial^{2} w^{i}_{0} }}{{\partial R^{i2} }}} \right)} \right]} \hfill \\ {\quad + \left[ {D^{i}_{12} \frac{{u^{i}_{0} }}{{R^{i} }} + E^{i}_{12} \frac{{u^{i}_{1} }}{{R^{i} }} - G^{i}_{12} c_{1} \left( {\frac{{u^{i}_{1} }}{{R^{i} }} + \frac{{\partial w^{i}_{0} }}{{R^{i} \partial R^{i} }}} \right)} \right] - X_{33} \phi } \hfill \\ \end{array} } \right)} \hfill \\ {\quad - \frac{1}{{R^{i} }}\left( {\begin{array}{*{20}l} {\left[ {B^{i}_{12} \frac{{\partial u^{i}_{0} }}{{\partial R^{i} }} + C^{i}_{12} \frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} - E^{i}_{12} c_{1} \left( {\frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} + \frac{{\partial^{2} w^{i}_{0} }}{{\partial R^{i2} }}} \right)} \right]} \hfill \\ {\quad + Q^{i}_{22} \left[ {B^{i}_{22} \frac{{u^{i}_{0} }}{{R^{i} }} + C^{i}_{22} \frac{{u^{i}_{1} }}{{R^{i} }} - E^{i}_{22} c_{1} \left( {\frac{{u^{i}_{1} }}{{R^{i} }} + \frac{{\partial w^{i}_{0} }}{{R^{i} \partial R^{i} }}} \right)} \right]} \hfill \\ \end{array} } \right)} \hfill \\ {\quad + \frac{{c_{1} }}{{R^{i} }}\left( {\begin{array}{*{20}l} {\left[ {D^{i}_{12} \frac{{\partial u^{i}_{0} }}{{\partial R^{i} }} + E^{i}_{12} \frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} - G^{i}_{12} c_{1} \left( {\frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} + \frac{{\partial^{2} w^{i}_{0} }}{{\partial R^{i2} }}} \right)} \right]} \hfill \\ {\quad + Q^{i}_{22} \left[ {D^{i}_{22} \frac{{u^{i}_{0} }}{{R^{i} }} + E^{i}_{22} \frac{{u^{i}_{1} }}{{R^{i} }} - G^{i}_{22} c_{1} \left( {\frac{{u^{i}_{1} }}{{R^{i} }} + \frac{{\partial w^{i}_{0} }}{{R^{i} \partial R^{i} }}} \right)} \right]} \hfill \\ \end{array} } \right)} \hfill \\ {\quad - \left[ {(A^{i}_{55} - 3C^{i}_{55} c_{1} )\left( {u^{i}_{1} + \frac{{\partial w^{i}_{0} }}{{\partial R^{i} }}} \right) + X_{11} \partial \phi /\partial R^{i} } \right]} \hfill \\ {\quad + 3c_{1} \left( {S^{i}_{Rz} = (C^{i}_{55} - 3E^{i}_{55} c_{1} )\left( {u^{i}_{1} + \frac{{\partial w^{i}_{0} }}{{\partial R^{i} }}} \right) + X_{12} \partial \phi /\partial R^{i} } \right)} \hfill \\ \end{array} } \right] = 0;\quad i = c,p \\ \end{aligned}$$

(P-3)

$$\begin{aligned} & \delta \phi :\\ & \quad (1 - l^{2} \nabla^{2} )\left( {\begin{array}{*{20}l} { + X_{31} \frac{{\partial u^{p}_{0} }}{{\partial R^{p} }} + (X_{11} - 3X_{12} )\frac{{\partial^{2} w^{p}_{0} }}{{\partial R^{p2} }} - X_{33} \frac{{\partial^{2} w^{p}_{0} }}{{\partial R^{p2} }}} \hfill \\ {\quad - ( - X_{11} + 3X_{12} )\frac{{\partial u^{p}_{1} }}{{\partial R^{p} }} + X_{32} \frac{{\partial u^{p}_{1} }}{{\partial R^{p} }} - X_{33} \frac{{\partial u^{p}_{1} }}{{\partial R^{p} }}} \hfill \\ {\quad - X_{41} \frac{{\partial^{2} \phi }}{{\partial R^{p2} }} + X_{42} \phi } \hfill \\ \end{array} } \right) = 0 \\ \end{aligned}$$

(P-4)

The GDQ form can be given as follows:

$$\begin{aligned} & \delta u^{i}_{0} :\\ & \quad \left( {1 - l^{2} \left( \begin{aligned} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } + \hfill \\ \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{1}{{R^{i} }} \hfill \\ \end{aligned} \right)} \right)\left[ {\begin{array}{*{20}l} {\left( {\begin{array}{*{20}l} {\left[ {\begin{array}{*{20}l} {A^{i}_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } u^{i}_{0} + B^{i}_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } u^{i}_{1} } \hfill \\ {\quad - D^{i}_{11} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } u^{i}_{1} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } w^{i}_{0} } \right)} \hfill \\ \end{array} } \right]} \hfill \\ {\quad + \left[ {\begin{array}{*{20}l} { + B^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} + A^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{0} }}{{R_{n}^{i} }}} \hfill \\ {\quad - D^{i}_{12} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R^{i} }} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{w^{i}_{0} }}{{R_{n}^{i} }}} \right)} \hfill \\ \end{array} } \right] - X_{31} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \phi_{n} } \hfill \\ \end{array} } \right)} \hfill \\ {\quad - \frac{1}{R}\left( {\begin{array}{*{20}l} {\left[ {\begin{array}{*{20}l} {A^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{0} }}{{R_{n}^{i} }} + B^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }}} \hfill \\ {\quad - D^{i}_{12} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{w^{i}_{0} }}{{R_{n}^{i} }}} \right)} \hfill \\ \end{array} } \right]} \hfill \\ {\quad + Q^{i}_{22} \left[ {\begin{array}{*{20}l} {A^{i}_{22} \sum\limits_{v = 1}^{{N_{n} }} {\frac{{u^{i}_{0} }}{{R^{2i} {}_{n}}}} + B^{i}_{22} \sum\limits_{v = 1}^{{N_{n} }} {\frac{{u^{i}_{1} }}{{R^{2i} {}_{n}}}} } \hfill \\ {\quad - D^{i}_{22} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {\frac{{u^{i}_{1} }}{{R^{2i} {}_{n}}}} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{w^{i}_{0} }}{{R^{2i} {}_{n}}}} \right)} \hfill \\ \end{array} } \right]} \hfill \\ \end{array} } \right)} \hfill \\ \end{array} } \right] = 0;\quad i = c,p \\ \end{aligned}$$

(P-5)

$$\begin{aligned} & \delta w^{i}_{0} :\\ & \quad \left( {1 - l^{2} \left( \begin{aligned} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } + \hfill \\ \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{1}{{R_{{}}^{i} }} \hfill \\ \end{aligned} \right)} \right)\left[ {\begin{array}{*{20}l} {c_{1} \left( {\begin{array}{*{20}l} {\left[ {\begin{array}{*{20}l} {D^{i}_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(3)}_{n,v} } u^{i}_{0} + E^{i}_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(3)}_{n,v} } u^{i}_{1} } \hfill \\ {\quad - G^{i}_{11} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(3)}_{n,v} } u^{i}_{1} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(4)}_{n,v} } w^{i}_{0} } \right)} \hfill \\ \end{array} } \right]} \hfill \\ {\quad + \left[ {\begin{array}{*{20}l} {D^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{u^{i}_{0} }}{{R_{n}^{i} }} + E^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }}} \hfill \\ {\quad - G^{i}_{12} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(3)}_{n,v} } \frac{{w^{i}_{0} }}{{R_{n}^{i} }}} \right)} \hfill \\ \end{array} } \right] - X_{33} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \phi } \hfill \\ \end{array} } \right)} \hfill \\ {\quad - c_{1} \left( {\begin{array}{*{20}l} {\left[ {\begin{array}{*{20}l} {D^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{u^{i}_{0} }}{{R_{n}^{i} }} + E^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }}} \hfill \\ {\quad - G^{i}_{12} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(3)}_{n,v} } \frac{{w^{i}_{0} }}{{R_{n}^{i} }}} \right)} \hfill \\ \end{array} } \right]} \hfill \\ {\quad + Q^{i}_{22} \left[ {\begin{array}{*{20}l} {D^{i}_{22} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{0} }}{{R^{2i} {}_{n}}} + E^{i}_{22} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R^{2i} {}_{n}}}} \hfill \\ {\quad - G^{i}_{22} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R^{2i} {}_{n}}} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{w^{i}_{0} }}{{R^{2i} {}_{n}}}} \right)} \hfill \\ \end{array} } \right]} \hfill \\ \end{array} } \right)} \hfill \\ {\quad + \left( {(A^{i}_{55} - 3C^{i}_{55} c_{1} )\left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } u^{i}_{1} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } w^{i}_{0} } \right) + X_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \phi } \right)} \hfill \\ {\quad - 3c_{1} \left( {\begin{array}{*{20}l} {(C^{i}_{55} - 3E^{i}_{55} c_{1} )\left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } u^{i}_{1} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } w^{i}_{0} } \right)} \hfill \\ {\quad + X_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \phi } \hfill \\ \end{array} } \right) - N_{1}^{p} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } w^{i}_{0} } \hfill \\ \end{array} } \right] = 0;\quad i = c,p \\ \end{aligned}$$

(P-6)

$$\begin{aligned} & \delta u^{i}_{1} :\\ & \quad \left( {1 - l^{2} \left( \begin{aligned} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } + \hfill \\ \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{1}{{R_{{}}^{i} }} \hfill \\ \end{aligned} \right)} \right)\left[ {\begin{array}{*{20}l} {\left( \begin{aligned} \left[ {B^{i}_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } u^{i}_{0} + C^{i}_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } u^{i}_{1} - E^{i}_{11} c_{1} \left( {\begin{array}{*{20}l} {\sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } u^{i}_{1} } \hfill \\ {\quad + \sum\limits_{v = 1}^{{N_{n} }} {C^{(3)}_{n,v} } w^{i}_{0} } \hfill \\ \end{array} } \right)} \right] \hfill \\ \quad + \left[ {\begin{array}{*{20}l} {B^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{0} }}{{R_{n}^{i} }} + C^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} - } \hfill \\ {E^{i}_{12} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{w^{i}_{0} }}{{R_{n}^{i} }}} \right)} \hfill \\ \end{array} } \right] - X_{32} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \phi \hfill \\ \end{aligned} \right)} \hfill \\ {\quad - c_{1} \left( {\begin{array}{*{20}l} {\left[ {D^{i}_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } u^{i}_{0} + E^{i}_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } u^{i}_{1} - G^{i}_{11} c_{1} \left( {\begin{array}{*{20}l} {\sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } u^{i}_{1} } \hfill \\ {\quad + \sum\limits_{v = 1}^{{N_{n} }} {C^{(3)}_{n,v} } w^{i}_{0} } \hfill \\ \end{array} } \right)} \right]} \hfill \\ {\quad + \left[ {\begin{array}{*{20}l} {D^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{0} }}{{R_{n}^{i} }} + E^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }}} \hfill \\ {\quad - G^{i}_{12} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{w^{i}_{0} }}{{R_{n}^{i} }}} \right)} \hfill \\ \end{array} } \right] - X_{33} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \phi } \hfill \\ \end{array} } \right)} \hfill \\ {\quad - \left( {\begin{array}{*{20}l} {\left[ {\begin{array}{*{20}l} {B^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{0} }}{{R_{n}^{i} }} + C^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }}} \hfill \\ {\quad - E^{i}_{12} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{w^{i}_{0} }}{{R_{n}^{i} }}} \right)} \hfill \\ \end{array} } \right]} \hfill \\ {\quad \; + Q^{i}_{22} \left[ {B^{i}_{22} \sum\limits_{v = 1}^{{N_{n} }} {\frac{{u^{i}_{0} }}{{R^{2i} {}_{n}}}} + C^{i}_{22} \sum\limits_{v = 1}^{{N_{n} }} {\frac{{u^{i}_{1} }}{{R^{2i} {}_{n}}}} - E^{i}_{22} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {\frac{{u^{i}_{1} }}{{R^{2i} {}_{n}}}} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{w^{i}_{0} }}{{R^{2i} {}_{n}}}} \right)} \right]} \hfill \\ \end{array} } \right)} \hfill \\ {\quad + c_{1} \left( {\begin{array}{*{20}l} {\left[ {\begin{array}{*{20}l} {D^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{0} }}{{R_{n}^{i} }} + E^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }}} \hfill \\ {\quad - G^{i}_{12} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{w^{i}_{1} }}{{R_{n}^{i} }}} \right)} \hfill \\ \end{array} } \right]} \hfill \\ {\quad \; + Q^{i}_{22} \left[ {D^{i}_{22} \sum\limits_{v = 1}^{{N_{n} }} {\frac{{u^{i}_{0} }}{{R^{2i} {}_{n}}}} + E^{i}_{22} \sum\limits_{v = 1}^{{N_{n} }} {\frac{{u^{i}_{1} }}{{R^{2i} {}_{n}}}} - G^{i}_{22} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {\frac{{u^{i}_{1} }}{{R^{2i} {}_{n}}}} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} \frac{{w^{i}_{0} }}{{R^{2i} {}_{n}}}} } \right)} \right]} \hfill \\ \end{array} } \right)} \hfill \\ {\quad - \left[ {(A^{i}_{55} - 3C^{i}_{55} c_{1} )\left( {\sum\limits_{v = 1}^{{N_{n} }} {u^{i}_{0} } I_{v} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} w^{i}_{0} } } \right) + X_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \phi } \right]} \hfill \\ {\quad + 3c_{1} \left( {(C^{i}_{55} - 3E^{i}_{55} c_{1} )\left( {\sum\limits_{v = 1}^{{N_{n} }} {u^{i}_{1} } I_{v} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} w^{i}_{0} } } \right) + X_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \phi } \right)} \hfill \\ \end{array} } \right] = 0;\quad i = c,p \\ \end{aligned}$$

(P-7)

$$\begin{aligned} & \delta \phi :\\ & \quad \left( {1 - l^{2} \left( \begin{aligned} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } + \hfill \\ \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{1}{{R_{{}}^{i} }} \hfill \\ \end{aligned} \right)} \right)\left( {\begin{array}{*{20}l} { + X_{31} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } u^{p}_{0} + (X_{11} - 3X_{12} - X_{33} )\sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } w^{p}_{0} } \hfill \\ {\quad - ( - X_{11} + 3X_{12} - X_{32} + X_{33} )\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } u^{p}_{1} } \hfill \\ {\quad - X_{41} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \phi + X_{42} \sum\limits_{v = 1}^{{N_{n} }} \phi I_{v} } \hfill \\ \end{array} } \right) = 0 \\ \end{aligned}$$

(P-8)