Free vibration analysis of double-viscoelastic nano-composite micro-plates reinforced by FG-SWCNTs based on the third-order shear deformation theory

Abstract

In this paper, free vibration analysis of a double viscoelastic nano-composite plate system reinforced by functionally graded single-walled carbon nanotubes (FG-SWCNT) embedded in a visco-Pasternak medium based on modified strain gradient theory (MSGT) and third-order shear deformation theory (TSDT) is presented. The material properties of the simply-supported nano-composite plates are approximated by the extended rule of mixture. The nonlocal governing equations of motion are obtained using Hamilton’s principle and solved by the Navier and trigonometric methods to obtain natural frequencies for three cases of vibration (out-of-phase vibration, in-phase vibration, and one plate fixed). A parametric study is performed to investigate the effects of material length scale parameter, aspect ratio, stiffness and damping coefficients of visco-Pasternak foundation, and structural damping coefficient on the natural frequency of double viscoelastic nano-composite plate. Also, the influence of volume fraction and different distributions of CNTs on every aspect of the natural frequency of the system is examined. The results show that the values of the natural frequency given by MSGT are higher than those obtained by modified couple stress theory (MCST) and classical theory (CT).

Appendices

Appendix A

$$\begin{aligned} \delta U & = \int {\int_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} {M_{{11}}^{0} \delta u_{{,x}} + M_{{11}}^{1} \delta \varphi _{{1,x}} - cM_{{11}}^{3} \delta \varphi _{{1,x}} - cM_{{11}}^{3} \delta w_{{,xx}} + M_{{22}}^{0} \delta v_{{,y}} } } \\ & + M_{{22}}^{1} \delta \varphi _{{2,y}} - cM_{{22}}^{3} \delta \varphi _{{2,y}} - cM_{{11}}^{3} \delta w_{{,yy}} + M_{{12}}^{0} \left( {\delta u_{{,y}} + \delta v_{{,x}} } \right) + M_{{12}}^{1} (\delta \varphi _{{1,y}} \\ & + \delta \varphi _{{2,x}} ) - cM_{{12}}^{3} \left( {\delta \varphi _{{1,y}} + \delta \varphi _{{2,x}} + 2\delta w_{{,xy}} } \right) + M_{{13}}^{0} \delta \varphi _{1} + M_{{13}}^{0} \delta w_{{,x}} \\ &- 3cM_{{13}}^{2} \left( {\delta \varphi _{1} + \delta w_{{,x}} } \right) + M_{{23}}^{0} \delta \varphi _{2} + M_{{23}}^{0} \delta w_{{,y}} - 3cM_{{23}}^{2} \left( {\delta \varphi _{2} + \delta w_{{,y}} } \right) \\ & + P_{1}^{1} \left( {\delta \varphi _{{1,xx}} + \delta \varphi _{{2,xy}} } \right) - cP_{1}^{3} \left( {\delta \varphi _{{1,xx}} + \delta \varphi _{{2,xy}} } \right) + P_{2}^{0} \left( {\delta u_{{,xy}} + \delta v_{{,yy}} } \right) \\ &- cP_{2}^{3} \left( {\delta w_{{,yyy}} + \delta w_{{,xxy}} } \right) + P_{2}^{1} \left( {\delta \varphi _{{1,xy}} + \delta \varphi _{{2,yy}} } \right) - cP_{2}^{3} \left( {\delta \varphi _{{1,xy}} + \delta \varphi _{{2,yy}} } \right) \\ & +P_{1}^{0} \left( {\delta u_{{,xx}} + \delta v_{{,xy}} } \right) - cP_{1}^{3} \left( {\delta w_{{,xxx}} + \delta w_{{,xyy}} } \right)P_{3}^{0} \left( {\delta \varphi _{{1,x}} + \delta \varphi _{{2,y}} } \right) - 3cP_{3}^{2} (\delta \varphi _{{1,x}} \\ & + \delta \varphi _{{2,y}} ) - 3cP_{3}^{2} \left( {\delta w_{{,xx}} + \delta w_{{,yy}} } \right) + \frac{1}{2}(Y_{{11}}^{0} \delta w_{{,xy}} + 3cY_{{11}}^{2} \delta w_{{,xy}} - Y_{{11}}^{0} \delta \varphi _{{2,x}} \\ & +3cY_{{11}}^{2} \delta \varphi _{{2,x}} ) + \frac{1}{2}\left( {Y_{{22}}^{0} \delta w_{{,xy}} - 3cY_{{22}}^{2} \delta w_{{,xy}} + Y_{{22}}^{0} \delta \varphi _{{1,y}} - 3cY_{{22}}^{2} \delta \varphi _{{1,y}} } \right) \\ &+ \frac{1}{2}(Y_{{12}}^{0} \delta w_{{,yy}} - Y_{{12}}^{0} \delta w_{{,xx}} 3cY_{{12}}^{2} \delta w_{{,yy}} - 3cY_{{12}}^{2} \delta w_{{,xx}} + Y_{{12}}^{0} \delta \varphi _{{1,x}} - Y_{{12}}^{0} \delta \varphi _{{2,y}} \\ & - 3cY_{{12}}^{2} \delta \varphi _{{1,x}} + 3cY_{{12}}^{2} \delta \varphi _{{2,y}} ) + \frac{1}{2}(Y_{{13}}^{0} \delta v_{{,xx}} - Y_{{13}}^{0} \delta u_{{,xy}} + Y_{{13}}^{1} \delta \varphi _{{2,xx}} \\ & -Y_{{13}}^{1} \delta \varphi _{{1,xy}} - cY_{{13}}^{3} \delta \varphi _{{2,xx}} + cY_{{13}}^{3} \delta \varphi _{{1,xy}} + 6cY_{{13}}^{1} + \left( {\delta \varphi _{2} + \delta w_{{,y}} } \right)) + \frac{1}{2}(Y_{{23}}^{0} \delta v_{{,xy}} \\ & - Y_{{23}}^{0} \delta u_{{,yy}} + Y_{{23}}^{1} \delta \varphi _{{2,xy}} - Y_{{23}}^{1} \delta \varphi _{{1,yy}} - cY_{{23}}^{3} \delta \varphi _{{2,xy}} + + cY_{{23}}^{3} \delta \varphi _{{1,yy}} \\ &- 6cY_{{23}}^{1} \left( {\delta \varphi _{1} + \delta w_{{,x}} } \right)) + \frac{1}{5}(2T_{{111}}^{0} \delta u_{{,xx}} - T_{{111}}^{0} \delta u_{{,yy}} - 2T_{{111}}^{0} \delta v_{{,xy}} \\ &- 2cT_{{111}}^{3} \delta w_{{,xxx}} + 3cT_{{111}}^{3} \delta w_{{,xyy}} + \left( {T_{{111}}^{1} - cT_{{111}}^{3} } \right)\left( {2\delta \varphi _{{1,xx}} - \delta \varphi _{{1,yy}} - 2\delta \varphi _{{2,xy}} } \right) \\ & + 6cT_{{111}}^{1} \left( {\delta \varphi _{1} + \delta w_{{,x}} } \right)) + \frac{1}{5}(2T_{{222}}^{0} \delta v_{{,yy}} - T_{{222}}^{0} \delta v_{{,xx}} - 2T_{{222}}^{0} \delta u_{{,xy}} \\ & -2cT_{{222}}^{3} \delta w_{{,yyy}} + 3cT_{{222}}^{3} \delta w_{{,xxy}} + \left( {T_{{222}}^{1} - cT_{{222}}^{3} } \right)\left( {2\delta \varphi _{{2,yy}} - \delta \varphi _{{2,xx}} - 2\delta \varphi _{{1,xy}} } \right) \\ & + 6cT_{{222}}^{1} \left( {\delta \varphi _{2} + \delta w_{{,y}} } \right)) + \frac{1}{5}(\left( {6cT_{{333}}^{2} - T_{{333}}^{0} } \right)\left( {\delta w_{{,xx}} + \delta w_{{,yy}} + \delta \varphi _{{1,x}} + \delta \varphi _{{2,y}} } \right) \\ & - T_{{333}}^{0} \delta \varphi _{{1,x}} - T_{{333}}^{0} \delta \varphi _{{2,y}} ) + \frac{1}{5}(T_{{112}}^{0} \left( {\frac{8}{3}\delta u_{{,xy}} - \frac{4}{3}\delta v_{{,xx}} - \delta v_{{,yy}} } \right) + cT_{{112}}^{3} (\delta w_{{,yyy}} \\ & - 4\delta w_{{,xxy}} )) + \left( {\frac{1}{3}T_{{112}}^{1} - \frac{c}{3}T_{{112}}^{3} } \right)\left( {8\delta \varphi _{{1,xy}} - \delta \varphi _{{2,yy}} + 4\delta \varphi _{{2,xx}} } \right) + 2cT_{{112}}^{1} \\ & \left( {\delta \varphi _{2} + \delta w_{{,y}} } \right)) + \frac{1}{5}(T_{{221}}^{0} \left( {\frac{4}{3}\delta u_{{,yy}} - \delta u_{{,xx}} + \frac{8}{3}\delta v_{{,xy}} } \right) + cT_{{221}}^{3} (\delta w_{{,xxx}} \\ -& 4\delta w_{{,xyy}} ) + \left( {\frac{1}{3}T_{{221}}^{1} - \frac{c}{3}T_{{221}}^{3} } \right)\left( {8\delta \varphi _{{2,xy}} - \delta \varphi _{{1,xx}} + 4\delta \varphi _{{1,yy}} } \right) + 2cT_{{221}}^{1} (\delta \varphi _{1} \\ &+ \delta w_{{,x}} )) + \frac{1}{5}(T_{{331}}^{0} \left( { - \frac{1}{3}\delta u_{{,yy}} - \delta u_{{,xx}} - \frac{2}{3}\delta v_{{,xy}} } \right) + cT_{{331}}^{3} \left( {\delta w_{{,xxx}} + \delta w_{{,xyy}} } \right) \\ &+ \left( { - \frac{1}{3}T_{{331}}^{1} + \frac{c}{3}T_{{331}}^{3} } \right)\left( {\delta \varphi _{{1,yy}} + 3\delta \varphi _{{1,xx}} + 2\delta \varphi _{{2,xy}} } \right) - 8cT_{{331}}^{1} \left( {\delta \varphi _{1} + \delta w_{{,x}} } \right)) \\ & +\frac{1}{5}(T_{{332}}^{0} \left( { - \frac{1}{3}\delta v_{{,xx}} - \delta v_{{,yy}} - \frac{2}{3}\delta u_{{,xy}} } \right) + cT_{{332}}^{3} \left( {\delta w_{{,yyy}} + \delta w_{{,xxy}} } \right) + ( - \frac{1}{3}T_{{332}}^{1} \\ &+ \frac{c}{3}T_{{332}}^{3} )\left( {\delta \varphi _{{2,xx}} + 3\delta \varphi _{{2,yy}} + 2\delta \varphi _{{1,xy}} } \right) - 8cT_{{332}}^{1} \left( {\delta \varphi _{2} + \delta w_{{,y}} } \right)) + \frac{1}{5}((2cT_{{113}}^{2} \\ & - \frac{1}{3}T_{{113}}^{0} )\left( {\delta w_{{,yy}} - 4\delta w_{{,xx}} } \right) + \left( {2cT_{{113}}^{2} - \frac{2}{3}T_{{113}}^{0} } \right)\left( {\delta \varphi _{{2,y}} - 4\delta \varphi _{{1,x}} } \right)) \\ & +\frac{1}{5}(\left( {2cT_{{223}}^{2} - \frac{1}{3}T_{{223}}^{0} } \right)\left( {\delta w_{{,xx}} - 4\delta w_{{,yy}} + \left( {2cT_{{223}}^{2} - \frac{2}{3}T_{{223}}^{0} } \right)\left( {\delta \varphi _{{1,x}} - 4\delta \varphi _{{2,y}} } \right)} \right) \\ & + \left( {2T_{{123}}^{0} - 6cT_{{123}}^{2} } \right)\left( {\delta w_{{,xy}} + \delta \varphi _{{1,y}} + \delta \varphi _{{2,x}} } \right) - 6cT_{{123}}^{2} \delta w_{{,xy}} dzdA \\ \end{aligned}$$

(A-1)

Appendix B

$$\delta u=0 or \left({M}_{11}^{0}-{P}_{1,x}^{0}-{P}_{2,y}^{0}+\frac{1}{2}{Y}_{13,y}^{0}-\frac{2}{5}{T}_{111,x}^{0}+\frac{2}{5}{T}_{222,y}^{0}-\frac{8}{15}{T}_{112,y}^{0}+\frac{1}{15}{T}_{221,x}^{0}+\frac{1}{5}{T}_{331,x}^{0}+\frac{2}{15}{T}_{332,y}^{0}\right){n}_{x}+\left({M}_{12}^{0}+-{P}_{2,x}^{0}+\frac{1}{2}{Y}_{13,x}^{0}\frac{1}{2}{Y}_{23,y}^{0}+\frac{2}{5}{T}_{222,x}^{0}-\frac{8}{15}{T}_{112,x}^{0}+\frac{1}{5}{T}_{111,y}^{0}-\frac{4}{5}{T}_{221,y}^{0}+\frac{1}{15}{T}_{331,y}^{0}+\frac{2}{15}{T}_{332,x}^{0}\right){n}_{y}=0$$

(B-1)

$$\delta v=0 or \left({M}_{12}^{0}-{P}_{1,y}^{0}-\frac{1}{2}{Y}_{13,x}^{0}-\frac{1}{2}{Y}_{23,y}^{0}+\frac{2}{5}{T}_{111,y}^{0}-\frac{4}{15}{T}_{112,x}^{0}+\frac{1}{5}{T}_{222,x}^{0}-\frac{8}{15}{T}_{221,y}^{0}+\frac{2}{15}{T}_{331,y}^{0}+\frac{1}{15}{T}_{332,x}^{0}\right){n}_{x}+\left({M}_{22}^{0}-{P}_{1,x}^{0}+{P}_{2,y}^{0}-\frac{1}{2}{Y}_{23,x}^{0}+\frac{2}{5}{T}_{111,x}^{0}-\frac{2}{5}c{T}_{222,y}^{0}+\frac{1}{15}{T}_{112,y}^{0}-\frac{8}{15}{T}_{221,x}^{0}+\frac{2}{15}{T}_{331,x}^{0}+\frac{1}{5}{T}_{332,y}^{0}\right){n}_{y}=0$$

(B-2)

$$\delta w=0\mathrm{ or }\left(c{M}_{11,x}^{3}+2c{M}_{12,y}^{3}+{M}_{13}^{0}-3c{M}_{13}^{2}-c{P}_{1,xx}^{3}-c{P}_{2,xy}^{3}-c{P}_{1,yy}^{3}-3c{P}_{3,x}^{2}-\frac{1}{2}{Y}_{11,y}^{0}-\frac{3}{2}c{Y}_{11,y}^{2}+\frac{1}{2}{Y}_{22,y}^{0}+\frac{3}{2}c{Y}_{22,y}^{2}+\frac{1}{2}{Y}_{12,x}^{0}+\frac{3}{2}c{Y}_{12,x}^{2}-3c{Y}_{23}^{1}-\frac{2}{5}c{T}_{111,xx}^{3}+\frac{3}{5}c{T}_{111,yy}^{3}+\frac{6}{5}c{T}_{111}^{1}+\frac{3}{5}c{T}_{222,xy}^{3}-\frac{6}{5}c{T}_{333,x}^{2}+\frac{1}{5}{T}_{333,x}^{0}-2{T}_{123,y}^{0}+12c{T}_{123,y}^{3}-\frac{4}{5}c{T}_{112,xy}^{3}+\frac{c}{5}{T}_{221,xx}^{3}-\frac{4}{5}c{T}_{221,yy}^{3}+\frac{2}{5}c{T}_{221}^{1}+\frac{c}{5}{T}_{331,xx}^{3}+\frac{c}{5}{T}_{331,yy}^{3}+\frac{8}{5}c{T}_{331}^{1}+\frac{c}{5}{T}_{332,xy}^{3}+\frac{8}{5}c{T}_{113,x}^{2}-\frac{4}{15}{T}_{113,x}^{0}-\frac{2}{5}c{T}_{223,x}^{2}+\frac{1}{15}{T}_{223,x}^{0}+{k}_{g}({w}_{,x})\right){n}_{x}+\left(c{M}_{22,y}^{3}+2c{M}_{12,x}^{3}+{M}_{23}^{0}-3c{M}_{23}^{0}-c{P}_{2,yy}^{3}+3c{P}_{3,y}^{2}-c{P}_{1,xy}^{3}-c{P}_{2,xx}^{3}-\frac{1}{2}{Y}_{11,x}^{0}-\frac{3}{2}c{Y}_{11,x}^{2}+\frac{1}{2}{Y}_{22,x}^{0}+\frac{3}{2}c{Y}_{22,x}^{2}-\frac{1}{2}{Y}_{12,y}^{0}-\frac{3}{2}c{Y}_{12,y}^{2}+3c{Y}_{13}^{1}\frac{3}{5}c{T}_{111,xy}^{3}+\frac{3}{5}c{T}_{222,xx}^{3}-\frac{2}{5}c{T}_{222,yy}^{3}+\frac{6}{5}c{T}_{222}^{1}-\frac{6}{5}c{T}_{333,y}^{2}+\frac{1}{5}{T}_{333,y}^{0}-\frac{4}{5}c{T}_{221,xy}^{3}-\frac{4}{5}c{T}_{112,xx}^{3}+\frac{c}{5}{T}_{112,yy}^{3}+\frac{2}{5}c{T}_{112}^{1}+\frac{c}{5}{T}_{331,xy}^{3}+\frac{c}{5}{T}_{332,xx}^{3}+\frac{c}{5}{T}_{332,yy}^{3}+\frac{8}{5}c{T}_{332}^{1}+\frac{2}{5}c{T}_{113}^{2}+\frac{1}{15}{T}_{113}^{0}+\frac{8}{5}c{T}_{223,y}^{2}-\frac{4}{15}{T}_{223,y}^{0}-2{T}_{123,x}^{0}+12c{T}_{123,x}^{3}+{k}_{g}({w}_{,y})\right){n}_{y}=0$$

(B-3)

$$\delta {\varphi }_{1}=0\mathrm{ or }\left({M}_{11}^{1}-c{M}_{11}^{3}-{P}_{1,x}^{1}+c{P}_{1,x}^{3}-{P}_{2,y}^{1}+c{P}_{2,y}^{3}+{P}_{3}^{0}-3c{P}_{3}^{2}+\frac{1}{2}{Y}_{12}^{0}-\frac{3}{2}c{Y}_{12}^{2}+\frac{1}{2}{Y}_{13,y}^{1}-\frac{c}{2}{Y}_{13,y}^{3}-\frac{2}{5}{T}_{111,x}^{1}+\frac{2}{5}c{T}_{111,x}^{3}+\frac{2}{5}{T}_{222,y}^{1}-\frac{2}{5}c{T}_{222,y}^{3}+\frac{6}{5}c{T}_{333}^{2}-\frac{2}{5}{T}_{333}^{0}-\frac{8}{15}{T}_{112,y}^{1}+\frac{8}{15}c{T}_{112,y}^{3}+\frac{1}{15}{T}_{221,x}^{1}-\frac{c}{15}{T}_{221,x}^{3}+\frac{3}{15}{T}_{331}^{1}-\frac{3}{15}c{T}_{331}^{3}+\frac{2}{15}{T}_{322,y}^{1}-\frac{2}{15}c{T}_{332,y}^{3}-\frac{8}{5}c{T}_{113}^{2}+\frac{8}{15}{T}_{113}^{0}+\frac{2}{5}c{T}_{223}^{2}-\frac{2}{15}{T}_{223}^{0}\right){n}_{x}+\left({M}_{12}^{1}-c{M}_{12}^{3}-{P}_{2,x}^{1}+c{P}_{2,x}^{3}+\frac{1}{2}{Y}_{22}^{0}-\frac{3}{2}c{Y}_{22}^{2}+\frac{1}{2}{Y}_{13,x}^{1}-\frac{c}{2}{Y}_{13,x}^{3}+\frac{1}{2}{Y}_{23,y}^{1}-\frac{c}{2}{Y}_{23,y}^{3}+\frac{1}{5}{T}_{111,y}^{1}-\frac{1}{5}c{T}_{111,y}^{3}+\frac{2}{5}{T}_{222,x}^{1}-\frac{2}{5}c{T}_{222,x}^{3}-\frac{8}{15}{T}_{112,x}^{1}+\frac{8}{15}c{T}_{112,x}^{3}-\frac{4}{15}{T}_{221,y}^{1}-\frac{4}{15}c{T}_{221,y}^{3}-\frac{1}{15}{T}_{331}^{1}-\frac{c}{15}{T}_{331}^{3}+\frac{2}{15}{T}_{322,x}^{1}-\frac{2}{15}c{T}_{332,x}^{3}+2{T}_{123}^{0}-6c{T}_{123}^{2}\right){n}_{y}=0$$

(B-4)

$$\delta {\varphi }_{2}=0\mathrm{ or }\left({M}_{12}^{1}-c{M}_{12}^{3}-{P}_{1,y}^{1}+c{P}_{1,y}^{3}-\frac{1}{2}{Y}_{11}^{0}+\frac{3}{2}c{Y}_{11}^{2}-\frac{1}{2}{Y}_{13,x}^{1}+\frac{c}{2}{Y}_{13,x}^{3}-\frac{1}{2}{Y}_{23,y}^{1}+\frac{c}{2}{Y}_{23,y}^{3}+\frac{2}{5}{T}_{111,y}^{1}-\frac{2}{5}c{T}_{111,y}^{3}+\frac{1}{5}{T}_{222,x}^{1}-\frac{1}{5}c{T}_{222,x}^{3}-\frac{4}{15}{T}_{112,x}^{1}+\frac{4}{15}c{T}_{112,x}^{3}-\frac{8}{15}{T}_{221,y}^{1}+\frac{8}{15}c{T}_{221,y}^{3}+\frac{2}{15}{T}_{331,y}^{1}-\frac{2}{15}c{T}_{331,y}^{3}+\frac{1}{15}{T}_{332,x}^{1}-\frac{c}{15}{T}_{332,x}^{3}+2{T}_{123}^{0}-6c{T}_{123}^{2}\right){n}_{x}+\left({M}_{22}^{1}-c{M}_{22}^{3}-{P}_{1,x}^{1}+c{P}_{1,x}^{3}-{P}_{2,y}^{1}+c{P}_{2,y}^{3}+{P}_{3}^{0}-3c{P}_{3}^{2}-\frac{1}{2}{Y}_{12}^{0}+\frac{3}{2}c{Y}_{12}^{2}-\frac{1}{2}{Y}_{23,x}^{1}+\frac{c}{2}{Y}_{23,x}^{3}+\frac{2}{5}{T}_{111,x}^{1}-\frac{2}{5}c{T}_{111,x}^{3}-\frac{2}{5}{T}_{222,y}^{1}+\frac{2}{5}c{T}_{222,y}^{3}+\frac{6}{5}c{T}_{333}^{2}-\frac{2}{5}{T}_{333}^{0}+\frac{1}{15}{T}_{112,y}^{1}-\frac{c}{15}{T}_{112,y}^{3}-\frac{8}{15}{T}_{221,x}^{1}+\frac{8}{15}c{T}_{221,x}^{3}+\frac{2}{15}{T}_{331,x}^{1}-\frac{2}{15}c{T}_{331,x}^{3}+\frac{1}{15}{T}_{332,y}^{1}-\frac{c}{15}{T}_{332,y}^{3}+\frac{2}{5}c{T}_{113}^{2}-\frac{2}{15}{T}_{113}^{0}-\frac{8}{5}c{T}_{223}^{2}+\frac{8}{15}{T}_{223}^{0}\right){n}_{y}=0$$

(B-5)

$$\delta {u}_{,x}=0\mathrm{ or }\left({P}_{1}^{0}+\frac{2}{5}{T}_{111}^{0}-\frac{1}{15}{T}_{221}^{0}-\frac{1}{5}{T}_{331}^{0}\right){n}_{x}+\left({P}_{2}^{0}-\frac{1}{2}{Y}_{13}^{0}-\frac{2}{5}{T}_{222}^{0}+\frac{8}{15}{T}_{112}^{0}-\frac{2}{15}{T}_{332}^{0}\right){n}_{y}=0$$

(B-6)

$$\delta {u}_{,y}=0\mathrm{ or }\left({P}_{2}^{0}-\frac{1}{2}{Y}_{13}^{0}-\frac{2}{5}{T}_{222}^{0}+\frac{8}{15}{T}_{112}^{0}-\frac{2}{15}{T}_{332}^{0}\right){n}_{x}+\left(-\frac{1}{2}{Y}_{23}^{0}-\frac{1}{5}{T}_{111}^{0}+\frac{4}{15}{T}_{221}^{0}-\frac{1}{15}{T}_{331}^{0}\right){n}_{y}=0$$

(B-7)

$$\delta {v}_{,x}=0\mathrm{ or }\left(\frac{1}{2}{Y}_{13}^{0}-\frac{1}{5}{T}_{222}^{0}+\frac{4}{15}{T}_{112}^{0}-\frac{1}{15}{T}_{332}^{0}\right){n}_{x}+\left({P}_{1}^{0}+\frac{1}{2}{Y}_{23}^{0}-\frac{2}{5}{T}_{111}^{0}+\frac{8}{15}{T}_{221}^{0}-\frac{2}{15}{T}_{331}^{0}-\frac{2}{15}{T}_{332}^{0}\right){n}_{y}=0$$

(B-8)

$$\delta {v}_{,y}=0\mathrm{ or }\left({P}_{1}^{0}+\frac{1}{2}{Y}_{23}^{0}-\frac{2}{5}{T}_{111}^{0}+\frac{8}{15}{T}_{221}^{0}-\frac{2}{15}{T}_{331}^{0}\right){n}_{x}+\left(-{P}_{2}^{0}+\frac{2}{5}{T}_{222}^{0}-\frac{1}{15}{T}_{112}^{0}-\frac{1}{5}{T}_{332}^{0}\right){n}_{y}=0$$

(B-9)

$$\delta {w}_{,x}=0\mathrm{ or }\left(-c{M}_{11}^{3}+c{P}_{1,x}^{3}+c{P}_{2,y}^{3}-3c{P}_{3}^{2}-\frac{1}{2}{Y}_{12}^{0}-\frac{3}{2}c{Y}_{12}^{2}+\frac{2}{5}c{T}_{111,x}^{3}-\frac{3}{5}c{T}_{222,y}^{3}+\frac{6}{5}c{T}_{333}^{2}-\frac{1}{5}{T}_{333}^{0}+\frac{4}{5}c{T}_{112,y}^{3}-\frac{c}{5}{T}_{221,x}^{3}-\frac{c}{5}{T}_{331,x}^{3}-\frac{c}{5}{T}_{332,y}^{3}-\frac{8}{5}c{T}_{113}^{2}+\frac{4}{15}{T}_{113}^{0}+\frac{2}{5}c{T}_{223}^{2}-\frac{1}{15}{T}_{223}^{0}\right){n}_{x}+\left(-2c{M}_{12}^{3}+c{P}_{1,y}^{3}+c{P}_{2,x}^{3}+\frac{1}{2}{Y}_{11}^{0}+\frac{3}{2}c{Y}_{11}^{2}-\frac{1}{2}{Y}_{22}^{0}-\frac{3}{2}c{Y}_{22}^{2}-\frac{3}{5}c{T}_{111,y}^{3}-\frac{3}{5}c{T}_{222,x}^{3}+\frac{4}{5}{T}_{221,y}^{3}+\frac{4}{5}c{T}_{112,x}^{3}-\frac{c}{5}{T}_{331,y}^{3}-\frac{c}{5}{T}_{332,x}^{3}+2{T}_{123}^{0}-12c{T}_{123}^{2}\right){n}_{y}=0$$

(B-10)

$$\delta {w}_{,y}=0\mathrm{ or }\left(-2c{M}_{12}^{3}-c{M}_{22}^{3}+c{P}_{1,y}^{3}+c{P}_{2,x}^{3}+\frac{1}{2}{Y}_{11}^{0}+\frac{3}{2}c{Y}_{11}^{2}-\frac{1}{2}{Y}_{22}^{0}-\frac{3}{2}c{Y}_{22}^{2}+2{T}_{123}^{0}-12c{T}_{123}^{2}-\frac{3}{5}c{T}_{111,y}^{3}-\frac{3}{5}c{T}_{222,x}^{3}+\frac{4}{5}c{T}_{112,x}^{3}+\frac{4}{5}{T}_{221,y}^{3}-\frac{c}{5}{T}_{331,y}^{3}-\frac{c}{5}{T}_{332,x}^{3}\right){n}_{x}+\left(c{P}_{1,x}^{3}+c{P}_{2,y}^{3}-3c{P}_{3}^{2}+\frac{1}{2}{Y}_{12}^{0}+\frac{3}{2}c{Y}_{12}^{2}-\frac{3}{5}c{T}_{111,x}^{3}+\frac{4}{5}{T}_{221,x}^{3}-\frac{c}{5}{T}_{331,x}^{3}+\frac{2}{5}c{T}_{222,y}^{3}+\frac{6}{5}c{T}_{333}^{2}-\frac{1}{5}{T}_{333}^{0}-\frac{c}{5}{T}_{112,y}^{3}-\frac{c}{5}{T}_{332,y}^{3}+\frac{2}{5}c{T}_{113}^{2}-\frac{1}{15}{T}_{113}^{0}-\frac{8}{5}c{T}_{223}^{2}+\frac{4}{15}{T}_{223}^{0}\right){n}_{y}=0$$

(B-11)

$$\delta {w}_{,xy}=0\mathrm{ or }\left(c{P}_{3}^{3}-c{P}_{2}^{3}+\frac{3}{5}c{T}_{222}^{3}-\frac{4}{5}c{T}_{112}^{3}+\frac{c}{5}{T}_{332}^{3}\right){n}_{x}+\left(-c{P}_{1}^{3}+\frac{3}{5}c{T}_{111}^{3}-\frac{4}{5}c{T}_{221}^{3}+\frac{c}{5}{T}_{331}^{3}\right){n}_{y}=0$$

(B-12)

$$\delta {w}_{,yy}=0\mathrm{ or }\left(-c{P}_{1}^{3}+\frac{3}{5}c{T}_{111}^{3}-\frac{4}{5}c{T}_{221}^{3}+\frac{c}{5}{T}_{331}^{3}\right){n}_{x}+\left(-c{P}_{2}^{3}-\frac{2}{5}c{T}_{222}^{3}+\frac{c}{5}{T}_{112}^{3}+\frac{c}{5}{T}_{332}^{3}\right){n}_{y}=0$$

(B-13)

$$\delta {w}_{,xx}=0\mathrm{ or }\left(-c{P}_{1}^{3}-\frac{2}{5}c{T}_{111}^{3}+\frac{c}{5}{T}_{221}^{3}+\frac{c}{5}{T}_{331}^{3}\right){n}_{x}+\left(-c{P}_{2}^{3}+\frac{3}{5}c{T}_{222}^{3}-\frac{4}{5}c{T}_{112}^{3}+\frac{c}{5}{T}_{332}^{3}\right){n}_{y}=0$$

(B-14)

$$\delta {\varphi }_{1,x}=0 or \left({P}_{1}^{1}-c{P}_{1}^{3}+\frac{2}{5}{T}_{111}^{1}-\frac{2}{5}c{T}_{111}^{3}-\frac{1}{15}{T}_{221}^{1}+\frac{c}{15}{T}_{221}^{3}-\frac{3}{15}{T}_{331}^{1}+\frac{3}{15}c{T}_{331}^{3}\right){n}_{x}+\left({P}_{2}^{1}-c{P}_{2}^{3}-\frac{1}{2}{Y}_{13}^{1}+\frac{c}{2}{Y}_{13}^{3}-\frac{2}{5}{T}_{222}^{1}+\frac{2}{5}c{T}_{222}^{3}+\frac{8}{15}{T}_{112}^{1}-\frac{8}{15}c{T}_{112}^{3}-\frac{2}{15}{T}_{332}^{1}+\frac{2}{15}c{T}_{332}^{3}\right){n}_{y}=0$$

(B-15)

$$\delta {\varphi }_{1,y}=0\mathrm{ or }\left({P}_{2}^{1}-c{P}_{2}^{3}-\frac{1}{2}{Y}_{13}^{1}+\frac{c}{2}{Y}_{13}^{3}-\frac{2}{5}{T}_{222}^{1}+\frac{2}{5}c{T}_{222}^{3}+\frac{8}{15}{T}_{112}^{1}-\frac{8}{15}c{T}_{112}^{3}-\frac{2}{15}{T}_{332}^{1}+\frac{2}{15}c{T}_{332}^{3}\right){n}_{x}+\left(-\frac{1}{2}{Y}_{23}^{1}+\frac{c}{2}{Y}_{23}^{3}-\frac{1}{5}{T}_{111}^{1}+\frac{1}{5}c{T}_{111}^{3}\right){n}_{y}=0$$

(B-16)

$$\delta {\varphi }_{2,x}=0\mathrm{ or }\left(-\frac{1}{2}{Y}_{13}^{1}+\frac{c}{2}{Y}_{13}^{3}-\frac{1}{5}{T}_{222}^{1}+\frac{1}{5}c{T}_{222}^{3}+\frac{4}{15}{T}_{112}^{1}-\frac{4}{15}c{T}_{112}^{3}-\frac{1}{15}{T}_{332}^{1}+\frac{c}{15}{T}_{332}^{3}\right){n}_{x}+\left({P}_{1}^{1}-c{P}_{1}^{3}+\frac{1}{2}{Y}_{23}^{1}-\frac{c}{2}{Y}_{23}^{3}-\frac{2}{5}{T}_{111}^{1}+\frac{2}{5}c{T}_{111}^{3}+\frac{8}{15}{T}_{221}^{1}-\frac{8}{15}c{T}_{221}^{3}-\frac{2}{15}{T}_{331}^{1}+\frac{2}{15}c{T}_{331}^{3}\right){n}_{y}=0$$

(B-17)

$$\delta {\varphi }_{2,y}=0\mathrm{ or }\left({P}_{1}^{1}-c{P}_{1}^{3}+\frac{1}{2}{Y}_{23}^{1}-\frac{c}{2}{Y}_{23}^{3}-\frac{2}{5}{T}_{111}^{1}+\frac{2}{5}c{T}_{111}^{3}+\frac{8}{15}{T}_{221}^{1}-\frac{8}{15}c{T}_{221}^{3}-\frac{2}{15}{T}_{331}^{1}+\frac{2}{15}c{T}_{331}^{3}){n}_{x}+({P}_{2}^{1}-c{P}_{2}^{3}+\frac{2}{5}{T}_{222}^{1}-\frac{2}{5}c{T}_{222}^{3}-\frac{1}{15}{T}_{112}^{1}+\frac{c}{15}{T}_{112}^{3}-\frac{1}{15}{T}_{332}^{1}+\frac{c}{15}{T}_{332}^{3}+\frac{4}{15}{T}_{221}^{1}-\frac{4}{15}c{T}_{221}^{3}-\frac{1}{15}{T}_{331}^{1}+\frac{c}{15}{T}_{331}^{3}\right){n}_{y}=00$$

(B-18)