Buckling analysis of coupled DLGSs systems resting on elastic medium using sinusoidal shear deformation orthotropic plate theory

Abstract

This study proposes a solution for the analysis of critical buckling load of a coupled double-layer graphene sheets (DLGSs) system resting on elastic medium. In the upper and lower parts of the system, two graphene layers, which are under van der Waals force and related to each other through an elastic medium on the basis of Pasternak and Winkler models, are utilized. The performed model is considered to be based on Eringen’s nonlocal elasticity theory and sinusoidal nonlocal shear deformation theory. In addition, the properties of orthotropic plate are applied in the model. Constitutive relations and equations together with boundary conditions are derived on the basis of Hamilton’s principle. Furthermore, the effect of surface stress according to Gurtin–Murdoch theory is considered. Moreover, critical buckling load for in-phase, out-of-phase, and one side fixed states subjected to in-plane forces is obtained. The results of the study reveal the minimum value of critical buckling load for in-phase state and the maximum value of that for out-of-phase state, and that the value of one side fixed state critical buckling load exists between these two states.

Appendix

The elements of matrix A are obtained by deriving $u_{i} , v_{i} , w_{i} , \phi_{x i} , \phi_{y i}$ coefficients from the following relations, regarding the states of in-phase, out-of-phase and one side fixed.

$$\delta u_{i} : \left[ {\left[ {h\left( {C_{11} - \frac{{C_{13}^{2} }}{{C_{33} }}} \right) + 2C^{S}_{11} } \right]\left( {\frac{m}{l}} \right)^{2} + \left[ {C_{66} h + 2C^{S}_{66} } \right]\left( {\frac{n}{b}} \right)^{2} } \right]u_{imn} + \left[ {h\left( {C_{12} - \frac{{C_{13} C_{32} }}{{C_{33} }}} \right) + 2C^{S}_{12} + C_{66} h + 2C^{S}_{66} } \right]v_{imn} \left( {\frac{mn}{lb}} \right) = 0$$

(24)

$$\delta v_{i}:\left[ {\left[ {C_{66} h + 2C_{66}^{\text{S}} } \right]\left( {\frac{m}{l}} \right)^{2} + \left[ {h\left( {C_{22} - \frac{{C_{23}^{2} }}{{C_{33} }}} \right) + 2C_{66}^{\text{S}} } \right]\left( {\frac{n}{b}} \right)^{2} } \right]v_{imn} + \left[ {C_{66} h + 2C_{66}^{\text{S}} + h\left( {C_{21} - \frac{{C_{23} C_{31} }}{{C_{33} }}} \right) + 2C_{21}^{\text{S}} } \right]u_{imn} \left( {\frac{mn}{lb}} \right) = 0$$

(25)

$$\delta {w_1}: \left[ { - {{{h^3}} \over {12}}\left( {{C_{11}} - {{{C_{13}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over 6}{{{C_{13}}} \over {{C_{33}}}}{\tau ^{\rm{s}}} - {{{h^2}} \over 2}{C^{\rm{S}}}_{11} - \mu {\gamma _1}{N_{cr}}} \right]{w_{1mn}}{\left( {{{m\pi } \over l}} \right)^4} + \left[ { - {{{h^3}} \over {12}}\left( {{C_{22}} - {{{C_{23}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over 6}{{{C_{23}}} \over {{C_{33}}}}{\tau ^{\rm{s}}} - {{{h^2}} \over 2}C_{22}^{\rm{S}} - \mu {\gamma _2}{N_{cr}}} \right]{w_{1mn}}{\left( {{{n\pi } \over b}} \right)^4} + \cdots \left[ { - {{{h^3}} \over 6}\left( {{C_{12}} - {{{C_{13}}{C_{32}}} \over {{C_{33}}}}} \right) + {{{h^2}} \over 6}\left( {{{{C_{13}}} \over {{C_{33}}}} + {{{C_{23}}} \over {{C_{33}}}}} \right){\tau ^{\rm{s}}} - {h^2}C_{12}^{\rm{S}} - {{{h^3}{C_{66}}} \over 3} - 2{h^2}C_{66}^{\rm{S}} - \mu {\gamma _1}{N_{cr}} - \mu {\gamma _2}{N_{cr}}} \right]{w_{1mn}}\left( {{{{m^2}{n^2}{\pi ^4}} \over {{l^2}{b^2}}}} \right) + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{11}} - {{{C_{13}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{11}^{\rm{S}}} \right]{\phi _{1xmn}}{\left( {{{m\pi } \over l}} \right)^3} + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{22}} - {{{C_{23}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{22}^{\rm{S}}} \right]{\phi _{1ymn}}{\left( {{{n\pi } \over b}} \right)^3} + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{12}} - {{{C_{13}}{C_{32}}} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{12}^{\rm{S}} + {{4{h^3}{C_{66}}} \over {{\pi ^3}}} + {{2{h^2}C_{66}^{\rm{S}}} \over \pi }} \right]{\phi _{1ymn}}\left( {{{{m^2}n{\pi ^3}} \over {{l^2}b}}} \right) + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{21}} - {{{C_{23}}{C_{31}}} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{21}^{\rm{S}} + {{4{h^3}{C_{66}}} \over {{\pi ^3}}} + {{2{h^2}C_{66}^{\rm{S}}} \over \pi }} \right]{\phi _{1xmn}}\left( {{{m{n^2}{\pi ^3}} \over {l{b^2}}}} \right) - \left[ {{\gamma _1}{N_{cr}} - {c_{12}}\mu } \right]{w_{1mn}}{\left( {{{m\pi } \over l}} \right)^2} - \left[ {{\gamma _2}{N_{cr}} - {c_{12}}\mu } \right]{w_{1mn}}{\left( {{{n\pi } \over b}} \right)^2} - {c_{12}}\mu {w_{2mn}}{\left( {{{m\pi } \over l}} \right)^2} - {c_{12}}\mu {w_{2mn}}{\left( {{{n\pi } \over b}} \right)^2} + {c_{12}}\left( {{w_{1mn}} - {w_{2mn}}} \right) = 0$$

(26)

$$\delta w_{2}: \left[ { - \frac{{h^{3} }}{12}\left( {C_{11} - \frac{{C_{13}^{2} }}{{C_{33} }}} \right) + \frac{{h^{2} }}{6} \frac{{C_{13} }}{{C_{33} }} \tau^{\text{s}} - \frac{{h^{2} }}{2}C_{11}^{\text{S}} - G_{23} \mu - \mu \gamma_{1} N_{cr} } \right]w_{2mn} \left( {\frac{m\pi }{l}} \right)^{4} + \left[ { - \frac{{h^{3} }}{12}\left( {C_{22} - \frac{{C_{23}^{2} }}{{C_{33} }}} \right) + \frac{{h^{2} }}{6} \frac{{C_{23} }}{{C_{33} }}\tau^{\text{s}} - \frac{{h^{2} }}{2}C_{22}^{\text{S}} - G_{23} \mu - \mu \gamma_{2} N_{cr} } \right]w_{2mn} \left( {\frac{n\pi }{b}} \right)^{4} + \left[ { - \frac{{h^{3} }}{6}\left( {C_{12} - \frac{{C_{13} C_{32} }}{{C_{33} }}} \right) + \frac{{h^{2} }}{6}\left( { \frac{{C_{13} }}{{C_{33} }} + \frac{{C_{23} }}{{C_{33} }}} \right)\tau^{\text{s}} - h^{2} C_{12}^{\text{S}} - \frac{{h^{3} C_{66} }}{3} - 2h^{2} C_{66}^{\text{S}} } \right] - 2G_{23} \mu - \mu \gamma_{1} N_{cr} - \mu \gamma_{2} N_{cr} w_{2mn} \left( {\frac{{m^{2} n^{2} \pi^{4} }}{{l^{2} b^{2} }}} \right) + \left[ {\frac{{2h^{3} }}{{\pi^{3} }}\left( {C_{11} - \frac{{C_{13}^{2} }}{{C_{33} }}} \right) + \frac{{h^{2} }}{\pi }C_{11}^{\text{S}} } \right]\phi_{2xmn} \left( {\frac{m\pi }{l}} \right)^{3} + \left[ {\frac{{2h^{3} }}{{\pi^{3} }}\left( {C_{22} - \frac{{C_{23}^{2} }}{{C_{33} }}} \right) + \frac{{h^{2} }}{\pi }C_{22}^{\text{S}} } \right]\phi_{2ymn} \left( {\frac{n\pi }{b}} \right)^{3} + \left[ {\frac{{2h^{3} }}{{\pi^{3} }}\left( {C_{12} - \frac{{C_{13} C_{32} }}{{C_{33} }}} \right) + \frac{{h^{2} }}{\pi }C_{12}^{\text{S}} + \frac{{4h^{3} C_{66} }}{{\pi^{3} }} + \frac{{2h^{2} C_{66}^{\text{S}} }}{\pi }} \right]\phi_{2ymn} \left( {\frac{{m^{2} n\pi^{3} }}{{l^{2} b}}} \right) + \left[ {\frac{{2h^{3} }}{{\pi^{3} }}\left( {C_{21} - \frac{{C_{23} C_{31} }}{{C_{33} }}} \right) + \frac{{h^{2} }}{\pi }C_{21}^{\text{S}} + \frac{{4h^{3} C_{66} }}{{\pi^{3} }} + \frac{{2h^{2} C_{66}^{\text{S}} }}{\pi }} \right]\phi_{2xmn} \left( {\frac{{mn^{2} \pi^{3} }}{{lb^{2} }}} \right) - c_{21} \mu w_{1mn} \left( {\frac{m\pi }{l}} \right)^{2} - \left[ {\gamma_{1} N_{cr} + G_{23} - c_{21} \mu + K_{23} \mu } \right]w_{2mn} \left( {\frac{m\pi }{l}} \right)^{2} + \left[ {K_{23} \mu + G_{23} } \right]w_{3mn} \left( {\frac{m\pi }{l}} \right)^{2} - c_{21} \mu w_{1mn} \left( {\frac{n\pi }{b}} \right)^{2} - \left[ {\gamma_{2} N_{cr} + G_{23} - c_{21} \mu + K_{23} \mu } \right]w_{2mn} \left( {\frac{n\pi }{b}} \right)^{2} + \left[ {K_{23} \mu + G_{23} } \right]w_{3mn} \left( {\frac{n\pi }{l}} \right)^{2} + G_{23} \mu w_{3mn} \left( {\frac{m\pi }{l}} \right)^{4} + 2G_{23} \mu w_{3mn} \left( {\frac{{m^{2} n^{2} \pi^{4} }}{{l^{2} b^{2} }}} \right) + G_{23} \mu w_{3mn} \left( {\frac{n\pi }{b}} \right)^{4} + c_{21} \left( {w_{2mn} - w_{1mn} } \right) - K_{23} \left( {w_{2mn} - w_{3mn} } \right) = 0$$

(27)

$$\delta {w_3}: \left[ { - {{{h^3}} \over {12}}\left( {{C_{11}} - {{{C_{13}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over 6}{\rm{}}{{{C_{13}}} \over {{C_{33}}}}{\rm{}}{\tau ^{\rm{s}}} - {{{h^2}} \over 2}C_{11}^{\rm{S}} - {G_{32}}\mu - \mu {\gamma _1}{N_{cr}}} \right]{w_{3mn}}{\left( {{{m\pi } \over l}} \right)^4} + \left[ { - {{{h^3}} \over {12}}\left( {{C_{22}} - {{{C_{23}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over 6}{\rm{}}{{{C_{23}}} \over {{C_{33}}}}{\tau ^{\rm{s}}} - {{{h^2}} \over 2}C_{22}^{\rm{S}} - {G_{32}}\mu - \mu {\gamma _2}{N_{cr}}} \right]{w_{3mn}}{\left( {{{n\pi } \over b}} \right)^4} + \left[ { - {{{h^3}} \over 6}\left( {{C_{12}} - {{{C_{13}}{C_{32}}} \over {{C_{33}}}}} \right) + {{{h^2}} \over 6}\left( {{\rm{}}{{{C_{13}}} \over {{C_{33}}}} + {{{C_{23}}} \over {{C_{33}}}}} \right){\tau ^{\rm{s}}} - {h^2}C_{12}^{\rm{S}} - {{{h^3}{C_{66}}} \over 3} - 2{h^2}C_{66}^{\rm{S}} - 2{G_{32}}\mu - \mu {\gamma _1}{N_{cr}} - \mu {\gamma _2}{N_{cr}}} \right]{w_{3mn}}\left( {{{{m^2}{n^2}{\pi ^4}} \over {{l^2}{b^2}}}} \right) + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{11}} - {{{C_{13}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{11}^{\rm{S}}} \right]{\phi _{3xmn}}{\left( {{{m\pi } \over l}} \right)^3} + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{22}} - {{{C_{23}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{22}^{\rm{S}}} \right]{\phi _{3ymn}}{\left( {{{n\pi } \over b}} \right)^3} + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{12}} - {{{C_{13}}{C_{32}}} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{12}^{\rm{S}} + {{4{h^3}{C_{66}}} \over {{\pi ^3}}} + {{2{h^2}C_{66}^{\rm{S}}} \over \pi }} \right]{\phi _{3ymn}}\left( {{{{m^2}n{\pi ^3}} \over {{l^2}b}}} \right) + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{21}} - {{{C_{23}}{C_{31}}} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{21}^{\rm{S}} + {{4{h^3}{C_{66}}} \over {{\pi ^3}}} + {{2{h^2}C_{66}^{\rm{S}}} \over \pi }} \right]{\phi _{3xmn}}\left( {{{m{n^2}{\pi ^3}} \over {l{b^2}}}} \right) - {c_{32}}\mu {\rm{}}{w_{4mn}}{\left( {{{m\pi } \over l}} \right)^2} - \left[ {{\gamma _1}{N_{cr}} + {G_{32}} - {c_{32}}\mu + {K_{32}}\mu } \right]{w_{3mn}}{\left( {{{m\pi } \over l}} \right)^2} + \left[ {{K_{32}}\mu + {G_{32}}} \right]{w_{2mn}}{\left( {{{m\pi } \over l}} \right)^2} - {c_{32}}\mu {w_{4mn}}{\left( {{{n\pi } \over b}} \right)^2} - \left[ {{\gamma _2}{N_{cr}} + {G_{32}} - {c_{32}}\mu + {K_{32}}\mu } \right]{w_{3mn}}{\left( {{{n\pi } \over b}} \right)^2} + \left[ {{K_{32}}\mu + {G_{32}}} \right]{w_{2mn}}{\left( {{{n\pi } \over b}} \right)^2} + {G_{32}}\mu {w_{2mn}}{\left( {{{m\pi } \over l}} \right)^4} + 2{G_{32}}\mu {w_{2mn}}\left( {{{{m^2}{n^2}{\pi ^4}} \over {{l^2}{b^2}}}} \right) + {G_{32}}\mu {w_{2mn}}{\left( {{{n\pi } \over b}} \right)^4} + {c_{34}}\left( {{w_{3mn}} - {w_{4mn}}} \right){\rm{}} - {K_{32}}\left( {{w_{3mn}} - {w_{2mn}}} \right){\rm{}} = 0$$

(28)

$$\delta {w_4}:\left[ { - {{{h^3}} \over {12}}\left( {{C_{11}} - {{{C_{13}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over 6}{\rm{}}{{{C_{13}}} \over {{C_{33}}}}{\rm{}}{\tau ^{\rm{s}}} - {{{h^2}} \over 2}C_{11}^{\rm{S}} - \mu {\gamma _1}{N_{cr}}} \right]{w_{4mn}}{\left( {{{m\pi } \over l}} \right)^4} + \left[ { - {{{h^3}} \over {12}}\left( {{C_{22}} - {{{C_{23}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over 6}{\rm{}}{{{C_{23}}} \over {{C_{33}}}}{\tau ^{\rm{s}}} - {{{h^2}} \over 2}C_{22}^{\rm{S}} - \mu {\gamma _2}{N_{cr}}} \right]{w_{4mn}}{\left( {{{n\pi } \over b}} \right)^4} + \left[ { - {{{h^3}} \over 6}\left( {{C_{12}} - {{{C_{13}}{C_{32}}} \over {{C_{33}}}}} \right) + {{{h^2}} \over 6}\left( {{\rm{}}{{{C_{13}}} \over {{C_{33}}}} + {{{C_{23}}} \over {{C_{33}}}}} \right){\tau ^{\rm{s}}} - {h^2}C_{12}^{\rm{S}} - {{{h^3}{C_{66}}} \over 3} - 2{h^2}C_{66}^{\rm{S}} - \mu {\gamma _1}{N_{cr}} - \mu {\gamma _2}{N_{cr}}} \right]{w_{4mn}}\left( {{{{m^2}{n^2}{\pi ^4}} \over {{l^2}{b^2}}}} \right) + \cdots \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{11}} - {{{C_{13}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{11}^{\rm{S}}} \right]{\phi _{4xmn}}{\left( {{{m\pi } \over l}} \right)^3} + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{22}} - {{{C_{23}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{22}^{\rm{S}}} \right]{\phi _{4ymn}}{\left( {{{n\pi } \over b}} \right)^3} + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{12}} - {{{C_{13}}{C_{32}}} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{12}^{\rm{S}} + {{4{h^3}{C_{66}}} \over {{\pi ^3}}} + {{2{h^2}C_{66}^{\rm{S}}} \over \pi }} \right]{\phi _{4ymn}}\left( {{{{m^2}n{\pi ^3}} \over {{l^2}b}}} \right) + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{21}} - {{{C_{23}}{C_{31}}} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{21}^{\rm{S}} + {{4{h^3}{C_{66}}} \over {{\pi ^3}}} + {{2{h^2}C_{66}^{\rm{S}}} \over \pi }} \right]{\phi _{4xmn}}\left( {{{m{n^2}{\pi ^3}} \over {l{b^2}}}} \right) - \left[ {{\gamma _1}{N_{cr}} - {c_{43}}\mu } \right]{w_{4mn}}{\left( {{{m\pi } \over l}} \right)^2} - \left[ {{\gamma _2}{N_{cr}} - {c_{43}}\mu } \right]{w_{4mn}}{\left( {{{n\pi } \over b}} \right)^2} - {c_{43}}\mu {w_{3mn}}{\left( {{{m\pi } \over l}} \right)^2} - {c_{43}}\mu {w_{3mn}}{\left( {{{n\pi } \over b}} \right)^2} + {c_{43}}\left( {{w_{4mn}} - {w_{3mn}}} \right) = 0.$$

(29)

$$\delta \phi_{x i}: \left[ {- \frac{{2h^{3} }}{{\pi^{3} }}\left( {C_{11} - \frac{{C_{13}^{2}}}{{C_{33} }}} \right) + \frac{{4h^{2} }}{{\pi^{3} }} \frac{{C_{13}}}{{C_{33} }}\tau^{\text{s}} - \frac{{h^{2} }}{\pi}C_{11}^{\text{S}} } \right]w_{imn} \left( {\frac{m\pi }{l}}\right)^{3} + \left[ { - \frac{{2h^{3} }}{{\pi^{3} }}\left( {C_{12}- \frac{{C_{13} C_{32} }}{{C_{33} }}} \right) + \frac{{4h^{2}}}{{\pi^{3} }} \frac{{C_{13} }}{{C_{33} }} \tau^{\text{s}} -\frac{{h^{2} }}{\pi }C_{12}^{\text{S}} - \frac{{4h^{3} C_{66}}}{{\pi^{3} }}} \right] - \frac{{2h^{2} C_{66}^{\text{S}} }}{\pi}w_{imn} \left( {\frac{{mn^{2} \pi^{3} }}{{lb^{2} }}} \right) +\left[ {\frac{{h^{3} }}{{2\pi^{2} }}\left( {C_{11} -\frac{{C_{13}^{2} }}{{C_{33} }}} \right) + \frac{{2h^{2} }}{{\pi^{2}}}C_{11}^{\text{S}} } \right]\phi_{ixmn} \left( {\frac{m\pi }{l}}\right)^{2} + \left[ {\frac{{h^{3} }}{{2\pi^{2} }}\left( {C_{12} -\frac{{C_{13} C_{32} }}{{C_{33} }}} \right) + \frac{{2h^{2}}}{{\pi^{2} }}C_{12}^{\text{S}} + \frac{{h^{3} C_{66} }}{{2\pi^{2}}} + \frac{{2h^{2} C_{66}^{\text{S}} }}{{\pi^{2} }}}\right]\phi_{iymn} \left( {\frac{{mn\pi^{2} }}{lb}} \right) + \left[{\frac{{h^{3} C_{66} }}{{2\pi^{2} }} + \frac{{2h^{2}C_{66}^{\text{S}} }}{{\pi^{2} }}} \right]\phi_{ixmn} \left({\frac{n\pi }{b}} \right)^{2} + \frac{{C_{55} h}}{2} \phi_{ixmn} = 0$$

(30)

$$\delta \phi_{y i}: \left[ { - \frac{{2h^{3}}}{{\pi^{3} }}\left( {C_{22} - \frac{{C_{23}^{2} }}{{C_{33} }}}\right) + \frac{{4h^{2} }}{{\pi^{3} }} \frac{{C_{23} }}{{C_{33} }}\tau^{\text{s}} - \frac{{h^{2} }}{\pi }C_{22}^{\text{S}} }\right]w_{imn} \left( {\frac{n\pi }{b}} \right)^{3} + \left[ { -\frac{{2h^{3} }}{{\pi^{3} }}\left( {C_{21} - \frac{{C_{23} C_{31}}}{{C_{33} }}} \right) + \frac{{4h^{2} }}{{\pi^{3} }} \frac{{C_{23}}}{{C_{33} }} \tau^{\text{s}} - \frac{{h^{2} }}{\pi}C_{21}^{\text{S}} - \frac{{4h^{3} C_{66} }}{{\pi^{3} }}} \right] -\frac{{2h^{2} C_{66}^{\text{S}} }}{\pi }w_{imn} \left( {\frac{{m^{2}n\pi^{3} }}{{l^{2} b}}} \right) + \left[ {\frac{{h^{3} C_{66}}}{{2\pi^{2} }} + \frac{{2h^{2} C_{66}^{\text{S}} }}{{\pi^{2} }}}\right]\phi_{iymn} \left( {\frac{m\pi }{l}} \right)^{2} + \left[{\frac{{h^{3} }}{{2\pi^{2} }}\left( {C_{21} - \frac{{C_{23} C_{31}}}{{C_{33} }}} \right) + \frac{{2h^{2} }}{{\pi^{2}}}C_{21}^{\text{S}} + \frac{{h^{3} C_{66} }}{{2\pi^{2} }} +\frac{{2h^{2} C_{66}^{\text{S}} }}{{\pi^{2} }}} \right]\phi_{ixmn}\left( {\frac{{mn\pi^{2} }}{lb}} \right) + \left[ {\frac{{h^{3}}}{{2\pi^{2} }}\left( {C_{22} - \frac{{C_{23}^{2} }}{{C_{33} }}}\right) + \frac{{2h^{2} }}{{\pi^{2} }}C_{22}^{\text{S}} }\right]\phi_{iymn} \left( {\frac{n\pi }{b}} \right)^{2} +\frac{{C_{44} h}}{2} \phi_{iymn} = 0.$$

(31)