Vibration characteristics and shape optimization of FG-GPLRC cylindrical shell with magneto-electro-elastic face sheets

Abstract

The objective of this study was to evaluate the effects of various geometrical parameters, materials, and boundary conditions on the vibrations of a smart cylindrical sandwich shell. The study also aimed to use the bees algorithm to maximize the natural frequencies based on the geometrical and material parameters of the smart sandwich shell. The structure of the shell consists of two outer layers of magneto-electro-elastic (MEE) and a middle layer of functionally graded graphene platelets reinforced composite (FG-GPLRC). The material properties of the MEE face sheets depend on the volume fraction of the piezoelectric and piezo-magnetic phases. The FG-GPLRC core layer is a multilayer isotropic polymer material reinforced by GPLs that are functionally graded and distributed in each layer. The shell is simultaneously subjected to external pressure and thermo-magneto-electro loads. Reddy's higher-order shear deformation shell theory was used to derive the basic equations, along with the relationships between deflection amplitude and time. The natural frequencies were obtained using Galerkin and Runge–Kutta methods, and the Budiansky–Roth standard was employed to determine the critical dynamic buckling load. The results of the study are discussed through parametric studies.

Appendices

Appendix A

$$ \begin{aligned} \left( {\widetilde{{A_{mn} }},\,\,\widetilde{{B_{mn} }},\,\,\widetilde{{D_{mn} }},\,\,\widetilde{{E_{mn} }},\,\,\widetilde{{F_{mn} }},\,\widetilde{{\,H_{mn} }}} \right) & = \int\limits_{{ - h_{f} - 0.5h_{c} }}^{{ - 0.5h_{c} }} {\overline{{C_{mn} }} B\,{\text{d}}z} + \int\limits_{{ - 0.5h_{c} }}^{{0.5h_{c} }} {Q_{mn}^{c} B\,{\text{d}}z\,\,\,\,} \\ & \quad + \int\limits_{{0.5h_{c} }}^{{0.5h_{c} + h_{f} }} {\overline{{C_{mn} }} B\,{\text{d}}z,\,\,\,\,\,\left( {mn = 11,12,22,66} \right),\,} B = (1,\,\,z,\,\,z^{2} ,\,\,z^{3} ,\,\,z^{4} ,\,\,z^{6} ) \\ (\widetilde{{A_{kl} }},\,\,\widetilde{{D_{kl} }},\,\,\widetilde{{F_{kl} }}) & = \int\limits_{{ - h_{f} - 0.5h_{c} }}^{{ - 0.5h_{c} }} {\overline{{C_{kl} }} D{\text{d}}z\,} + \int\limits_{{ - 0.5h_{c} }}^{{0.5h_{c} }} {Q_{kl}^{c} D\,{\text{d}}z} \,\, + \int\limits_{{0.5h_{c} }}^{{0.5h_{c} + h_{f} }} {\overline{{C_{kl} }} D{\text{d}}z,\,\left( {kl = 44,55} \right),} D = (1,\,\,z^{2} ,\,\,z^{4} ) \\ \Phi_{i} & = \int\limits_{{ - h_{f} - 0.5h_{c} }}^{{ - 0.5h_{c} }} {\overline{{e_{31} }} \,E{\text{d}}z} + \int\limits_{{0.5h_{c} }}^{{h_{f} + 0.5h_{c} }} {\overline{{e_{31} }} \,\,E{\text{d}}z} ,\,\Gamma_{i} = \int\limits_{{ - h_{f} - 0.5h_{c} }}^{{ - 0.5h_{c} }} {\overline{{q_{31} }} E{\text{d}}z} + \int\limits_{{0.5h_{c} }}^{{h_{f} + 0.5h_{c} }} {\overline{{q_{31} }} \,\,E{\text{d}}z} ,\, \\ \Phi_{j} & = \int\limits_{{ - h_{f} - 0.5h_{c} }}^{{ - 0.5h_{c} }} {\overline{{e_{32} }} \,E{\text{d}}z} + \int\limits_{{0.5h_{c} }}^{{h_{f} + 0.5h_{c} }} {\overline{{e_{32} }} \,\,E{\text{d}}z} ,\Gamma_{j} = \int\limits_{{ - h_{f} - 0.5h_{c} }}^{{ - 0.5h_{c} }} {\overline{{q_{32} }} \,E\,{\text{d}}z} + \int\limits_{{0.5h_{c} }}^{{h_{f} + 0.5h_{c} }} {\overline{{q_{32} }} \,E\,{\text{d}}z} ,\,E = \left( {1,z,z^{3} } \right), \\ \alpha_{i} & = \int\limits_{{ - h_{f} - 0.5h_{c} }}^{{ - 0.5h_{c} }} {\left( {\overline{{C_{11} }} + \overline{{C_{12} }} } \right)\overline{{\alpha_{1} }} E\,{\text{d}}z} + \int\limits_{{ - 0.5h_{c} }}^{{0.5h_{c} }} {Q_{11}^{c} \alpha_{11}^{c} \,E{\text{d}}z} + \int\limits_{{ - 0.5h_{c} }}^{{0.5h_{c} }} {Q_{12}^{c} \alpha_{22}^{c} E\,{\text{d}}z} + \\ & \quad + \int\limits_{{0.5h_{c} }}^{{h_{f} + 0.5h_{c} }} {\left( {\overline{{C_{11} }} + \overline{{C_{12} }} } \right)\overline{{\alpha_{1} }} E\,{\text{d}}z} ,\,\,\,\left( {i = 1,3,5} \right),\left( {j = 2,4,6} \right), \\ \end{aligned} $$

$$ \begin{aligned} \alpha_{j} & = \int\limits_{{ - h_{f} - 0.5h_{c} }}^{{ - 0.5h_{c} }} {\left( {\overline{{C_{12} }} + \overline{{C_{22} }} } \right)\overline{{\alpha_{2} }} \,{{K}}\,{\text{d}}z} + \int\limits_{{ - 0.5h_{c} }}^{{0.5h_{c} }} {Q_{12}^{c} \alpha_{11}^{c} \,{{K}}{\text{d}}z} \\ & \,\,\,\,\,\,\,\,\, + \int\limits_{{ - 0.5h_{c} }}^{{0.5h_{c} }} {Q_{22}^{c} \alpha_{22}^{c} \,{{K}}\,{\text{d}}z} + \int\limits_{{0.5h_{c} }}^{{h_{f} + 0.5h_{c} }} {\left( {\overline{{C_{12} }} + \overline{{C_{22} }} } \right)\overline{{\alpha_{2} }} \,{{K}}{\text{d}}z} ,\,\,\,\left( {j = 2,4,6} \right), \\ \Phi_{k} & = \int\limits_{{ - h_{f} - 0.5h_{c} }}^{{ - 0.5h_{c} }} {\overline{{e_{24} }} H\,{\text{d}}z} + \int\limits_{{0.5h_{c} }}^{{h_{f} + 0.5h_{c} }} {\overline{{e_{24} }} H\,{\text{d}}z} ,\,\,\,\,\left( {k = 8,10} \right), \\ \Gamma_{k} & = \int\limits_{{ - h_{f} - 0.5h_{c} }}^{{ - 0.5h_{c} }} {\,\,\overline{{q_{24} }} H\,{\text{d}}z} + \int\limits_{{0.5h_{c} }}^{{h_{f} + 0.5h_{c} }} {\overline{{q_{24} }} H\,\,\,{\text{d}}z} ,\,\,\,\left( {k = 8,10} \right), \\ \Phi_{k} & = \int\limits_{{ - h_{f} - 0.5h_{c} }}^{{ - 0.5h_{c} }} {\overline{{e_{15} }} H\,{\text{d}}z} + \int\limits_{{0.5h_{c} }}^{{h_{f} + 0.5h_{c} }} {\overline{{e_{15} }} H{\text{d}}z} ,\,\,\,\left( {k = 8,10} \right), \\ \Gamma_{l} & = \int\limits_{{ - h_{f} - 0.5h_{c} }}^{{ - 0.5h_{c} }} {\,\,\overline{{q_{15} }} H\,{\text{d}}z} + \int\limits_{{0.5h_{c} }}^{{h_{f} + 0.5h_{c} }} {\,\,\overline{{q_{15} }} H\,{\text{d}}z} ,\,\,\,\left( {l = 7,9} \right),\,H = \left( {1,z,z^{3} } \right),K = \left( {1,z^{2} } \right). \\ \end{aligned} $$

$$ \begin{aligned} \overline{{C_{11} }} & = C_{11}^{f} - \frac{{\left( {C_{13}^{f} } \right)^{2} }}{{C_{33}^{f} }}{,}\overline{{C_{12} }} = C_{12}^{f} - \frac{{C_{13}^{f} C_{23}^{f} }}{{C_{33}^{f} }},\overline{{C_{55} }} = C_{55}^{f} {,}\,\overline{{C_{44} }} = C_{44}^{f} {, }\overline{{C_{22} }} = C_{22}^{f} - \frac{{\left( {C_{23}^{f} } \right)^{2} }}{{C_{33}^{f} }}{,} \\ \overline{{C_{66} }} & = C_{66}^{f} ,\,\overline{{e_{31} }} = e_{31}^{f} - \frac{{C_{13}^{f} e_{33}^{f} }}{{C_{33}^{f} }}{,}\overline{{e_{15} }} = e_{15}^{f} {, }\,\overline{{e_{32} }} = e_{32}^{f} - \frac{{C_{23}^{f} e_{33}^{f} }}{{C_{33}^{f} }},\,\overline{{e_{24} }} = e_{24}^{f} ,\,\overline{{q_{31} }} = q_{31}^{f} - \frac{{C_{13}^{f} q_{33}^{f} }}{{C_{33}^{f} }},\, \\ \overline{{q_{15} }} & = q_{15}^{f} {,}\overline{{q_{32} }} = q_{32}^{f} - \frac{{C_{23}^{f} q_{33}^{f} }}{{C_{33}^{f} }},\,\overline{{q_{24} }} = q_{24}^{f} ,\,\,\overline{{\mu_{33} }} = \mu_{33}^{f} + \frac{{\left( {q_{33}^{f} } \right)^{2} }}{{C_{33}^{f} }},\overline{{\mu_{11} }} = \mu_{11}^{f} {,}\,\overline{{\mu_{22} }} = \mu_{22}^{f} {,} \\ \end{aligned} $$

$$ \overline{{\eta_{11} }} = \eta_{11}^{f} {,}\,\overline{{\eta_{22} }} = \eta_{22}^{f} {,}\overline{{m_{11} }} = m_{11}^{f} {,}\,\overline{{m_{22} }} = m_{22}^{f} {,}\,\overline{{\lambda_{1} }} = \lambda_{1}^{f} {,}\,\,\overline{{\lambda_{2} }} = \lambda_{2}^{f} {,} $$

$$ \overline{{\alpha_{2} }} = \alpha_{2}^{f} - C_{23}^{f} \alpha_{3}^{f} /C_{33}^{f} {,}\overline{{p_{3} }} = p_{3}^{f} + e_{33}^{f} \alpha_{33}^{f} /C_{33}^{f} ,\overline{{\lambda_{3} }} = \lambda_{3}^{f} + q_{33}^{f} \alpha_{33}^{f} /C_{33}^{f} , $$

$$ \overline{{p_{1} }} = p_{1}^{f} {,}\,\overline{{p_{2} }} = p_{2}^{f} {,}\overline{{\eta_{33} }} = \eta_{33}^{f} + \frac{{\left( {e_{33}^{f} } \right)^{2} }}{{C_{33}^{f} }},\,\overline{{m_{33} }} = m_{33}^{f} + \frac{{e_{33}^{f} q_{33}^{f} }}{{C_{33}^{f} }},\overline{{\alpha_{1} }} = \alpha_{1}^{f} - \frac{{C_{13}^{f} \alpha_{3}^{f} }}{{C_{33}^{f} }}. $$

Appendix B

$$ \begin{aligned} \Delta & = \widetilde{{A_{11} }}\widetilde{{A_{22} }} - \widetilde{{A_{12}^{2} }},\,\,\,\,\Im_{11}^{{}} = \frac{{\widetilde{{A_{22} }}}}{\Delta },\,\,\,\,\Im_{12}^{{}} = \frac{{\widetilde{{A_{12} }}}}{\Delta },\,\,\Im_{13}^{{}} = \frac{{\widetilde{{B_{12} }}\widetilde{{A_{12} }} - \widetilde{{B_{11} }}\widetilde{{A_{22} }}}}{\Delta },\,\,\, \\ \Im_{14}^{{}} & = \frac{{\widetilde{{B_{22} }}\widetilde{{A_{12} }} - \widetilde{{B_{12} }}\widetilde{{A_{22} }}}}{\Delta },\Im_{15}^{{}} = \frac{{\widetilde{{E_{12} }}\widetilde{{A_{12} }} - \widetilde{{E_{11} }}\widetilde{{A_{22} }}}}{\Delta },\Im_{24}^{{}} = \frac{{\widetilde{{B_{12} }}\widetilde{{A_{12} }} - \widetilde{{B_{22} }}\widetilde{{A_{11} }}}}{\Delta },\,\, \\ \Im_{25}^{{}} & = \frac{{\widetilde{{E_{11} }}\widetilde{{A_{12} }} - \widetilde{{E_{12} }}\widetilde{{A_{11} }}}}{\Delta }\,,\Im_{26}^{{}} = \frac{{\widetilde{{E_{12} }}\widetilde{{A_{12} }} - \widetilde{{E_{22} }}\widetilde{{A_{11} }}}}{\Delta },\,\Im_{16}^{{}} = \frac{{\widetilde{{E_{22} }}\widetilde{{A_{12} }} - \widetilde{{E_{12} }}\widetilde{{A_{22} }}}}{\Delta },\, \\ \,\Im_{17}^{{}} & = 2\frac{{2e_{32} \widetilde{{A_{12} }} - 2e_{31} \widetilde{{A_{22} }}}}{\Delta },\,\Im_{18}^{{}} = 2\frac{{2q_{32} \widetilde{{A_{12} }} - 2q_{31} \widetilde{{A_{22} }}}}{\Delta },\Im_{27}^{{}} = 2\frac{{2e_{31} \widetilde{{A_{12} }} - 2e_{32} \widetilde{{A_{11} }}}}{\Delta },\, \\ \Im_{28}^{{}} & = 2\frac{{2q_{31} \widetilde{{A_{12} }} - 2q_{32} \widetilde{{A_{11} }}}}{\Delta },\Im_{29}^{{}} = \frac{{ - \alpha_{1} \widetilde{{A_{12} }} + \alpha_{2} \widetilde{{A_{11} }}}}{\Delta },\,\,\Im_{19}^{{}} = \frac{{ - \alpha_{2} \widetilde{{A_{12} }} + \alpha_{1} \widetilde{{A_{22} }}}}{\Delta }, \\ \Im_{21}^{{}} & = \frac{{\widetilde{{A_{11} }}}}{\Delta },\,\,\Im_{23}^{{}} = \frac{{\widetilde{{B_{11} }}\widetilde{{A_{12} }} - \widetilde{{B_{12} }}\widetilde{{A_{11} }}}}{\Delta },\Im_{31}^{{}} = \frac{1}{{\widetilde{{A_{66} }}}},\,\,\Im_{32}^{{}} = - \frac{{\widetilde{{B_{66} }}}}{{\widetilde{{A_{66} }}}},\,\,\Im_{33}^{{}} = - \frac{{\widetilde{{E_{66} }}}}{{\widetilde{{A_{66} }}}},\, \\ \end{aligned} $$

$$ \begin{aligned} \widetilde{{\aleph_{1}^{{}} }} & = \aleph_{1} ,\,\widetilde{{\aleph_{1}^{*} }} = \aleph_{1} + \frac{{2\aleph_{2} }}{R},\,\widetilde{{\aleph_{2}^{{}} }} = \aleph_{2} - c_{1} \aleph_{4} ,\,\widetilde{{\aleph_{2}^{*} }} = \aleph_{2} + \frac{{\aleph_{3} }}{R} - c_{1} \aleph_{4} - \frac{{c_{1} \aleph_{5} }}{R};\, \\ \widetilde{{\aleph_{3}^{{}} }} & = c_{1} \aleph_{4} ,\,\widetilde{{\aleph_{3}^{*} }} = c_{1} \aleph_{4} + \frac{{c_{1} \aleph_{5} }}{R},\widetilde{{\aleph_{4}^{{}} }} = \aleph_{3} - 2c_{1} \aleph_{5} + c_{1}^{2} \aleph_{7} ,\,\widetilde{{\aleph_{4}^{*} }} = \widetilde{{\aleph_{4}^{{}} }}, \\ \,\widetilde{{\aleph_{5}^{{}} }} & = c_{1} \aleph_{5} - c_{1}^{2} \aleph_{7} ,\widetilde{{\aleph_{5}^{*} }} = \widetilde{{\aleph_{5}^{{}} }}, \\ \,\aleph_{i} & = \int\limits_{{ - h_{f} - \frac{{h_{c} }}{2}}}^{{ - \frac{{h_{c} }}{2}}} {\rho_{f} (z)z^{i - 1} {\text{d}}z} + \,\,\,\int\limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} {\rho_{c} (z)z^{i - 1} {\text{d}}z} + \int\limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2} + h_{f} }} {\rho_{f} (z)z^{i - 1} {\text{d}}z} ,\,\,\left( {i = \overline{1,5} \,\,and\,\,7} \right). \\ \end{aligned} $$

$$ \begin{aligned} T_{11} \left( w \right) & = S_{11} w_{{,x^{2} }} + S_{12} w_{{,y^{2} }} + S_{13} w_{{,x^{4} }} + S_{14} w_{{,x^{2} y^{2} }} + S_{15} w_{{,y^{4} }} - k_{1} w + k_{2} \left( {w_{{,x^{2} }} + w_{{,y^{2} }} } \right), \\ T_{12} \left( {\phi_{x} } \right) & = S_{11} \phi_{x,x} + S_{16} \phi_{{x,x^{3} }} + S_{17} \phi_{{x,xy^{2} }} ,\,\,T_{13} \left( {\phi_{y} } \right) = S_{12} \phi_{y,y} + S_{18} \phi_{{y,y^{3} }} + S_{19} \phi_{{y,x^{2} y}} , \\ T_{14} (\wp ) & = S_{110} \frac{{\partial^{4} \wp }}{{\partial x^{4} }} + S_{111} \frac{{\partial^{4} \wp }}{{\partial x^{2} \partial y^{2} }} + S_{112} \frac{{\partial^{4} \wp }}{{\partial y^{4} }}, \\ T_{15} \left( \Phi \right) & = \left( {S_{114} \cos \left( {\eta z} \right) - S_{115} \eta \sin \left( {\eta z} \right)} \right)\frac{{\partial^{2} \Phi }}{{\partial^{2} x}} + \left( {S_{113} \cos \left( {\eta z} \right) - S_{116} \eta \sin \left( {\eta z} \right)} \right)\frac{{\partial^{2} \Phi }}{{\partial^{2} y}}, \\ T_{16} \left( \Psi \right) & = \left( {S_{118} \cos \left( {\eta z} \right) - S_{119} \eta \sin \left( {\eta z} \right)} \right)\frac{{\partial^{2} \Psi }}{{\partial^{2} x}} + \left( {S_{117} \cos \left( {\eta z} \right) - S_{120} \eta \sin \left( {\beta z} \right)} \right)\frac{{\partial^{2} \Psi }}{{\partial^{2} y}}, \\ X\left( {w,\wp } \right) & = \frac{{\partial^{2} \wp }}{{\partial y^{2} }}\frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{{\partial^{2} \wp }}{{\partial x^{2} }}\frac{{\partial^{2} w}}{{\partial y^{2} }} - 2\frac{{\partial^{2} \wp }}{\partial x\partial y}\frac{{\partial^{2} w}}{\partial x\partial y} + \frac{1}{R}\frac{{\partial^{2} \wp }}{{\partial x^{2} }}, \\ \end{aligned} $$

$$ \begin{aligned} T_{21} \left( w \right) & = S_{21} \frac{\partial w}{{\partial x}} + S_{22} \frac{{\partial^{3} w}}{{\partial x^{3} }} + S_{23} \frac{{\partial^{3} w}}{{\partial x\partial y^{2} }},\,\, \\ T_{22} \left( {\phi_{x} } \right) & = S_{21} \phi_{x} + S_{24} \frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }} + S_{25} \frac{{\partial^{2} \phi_{x} }}{{\partial y^{2} }}, \\ T_{23} \left( {\phi_{y} } \right) & = S_{26} \frac{{\partial^{2} \phi_{y} }}{\partial x\partial y},\, \\ \end{aligned} $$

$$ \begin{aligned} T_{25} \left( \Phi \right) & = - \left( {S_{114} \cos \left( {\eta z} \right) + S_{29} \eta \sin \left( {\eta z} \right)} \right)\frac{\partial \Phi }{{\partial x}}, \\ T_{26} \left( \Psi \right) & = - \left( {S_{118} \cos \left( {\eta z} \right) + S_{210} \eta \sin \left( {\eta z} \right)} \right)\frac{\partial \Psi }{{\partial x}}, \\ T_{31} \left( w \right) & = S_{31} \frac{\partial w}{{\partial y}} + S_{32} \frac{{\partial^{3} w}}{{\partial x^{2} \partial y}} + S_{33} \frac{{\partial^{3} w}}{{\partial y^{3} }},\,\,\, \\ \end{aligned} $$

$$ \begin{array}{*{20}c} \begin{gathered} T_{24} (\wp ) = S_{27} \frac{{\partial^{3} \wp }}{{\partial x^{3} }} + S_{28} \frac{{\partial^{3} \wp }}{{\partial x\partial y^{2} }}, \hfill \\ T_{33} \left( {\phi_{y} } \right) = S_{31} \phi_{y} + S_{35} \frac{{\partial^{2} \phi_{y} }}{{\partial x^{2} }} + S_{36} \frac{{\partial^{2} \phi_{y} }}{{\partial y^{2} }},\,\, \hfill \\ T_{34} (\wp ) = S_{37} \frac{{\partial^{3} \wp }}{{\partial x^{2} \partial y}} + S_{38} \frac{{\partial^{3} \wp }}{{\partial y^{3} }}, \hfill \\ \end{gathered} & \begin{gathered} T_{32} \left( {\phi_{x} } \right) = S_{34} \frac{{\partial^{2} \phi_{x} }}{\partial x\partial y}, \hfill \\ T_{35} \left( \Phi \right) = - \left( {S_{113} \cos \left( {\eta z} \right) + S_{39} \eta \sin \left( {\eta z} \right)} \right)\frac{\partial \Phi }{{\partial y}}, \hfill \\ T_{36} \left( \Psi \right) = - \left( {S_{117} \cos \left( {\eta z} \right) + S_{310} \beta \sin \left( {\eta z} \right)} \right)\frac{\partial \Psi }{{\partial y}}. \hfill \\ \end{gathered} \\ \end{array} $$

$$ \begin{aligned} \overline{\overline{{\aleph_{3} }}} & = \overline{{\aleph_{4} }} - \left( {\overline{{\aleph_{2} }} } \right)^{2} /\overline{{\aleph_{1} }} ,\,\overline{\overline{{\aleph_{3}^{*} }}} = \overline{{\aleph_{4}^{*} }} - \left( {\overline{{\aleph_{2}^{*} }} } \right)^{2} /\overline{{\aleph_{1}^{*} }} ,\,\overline{\overline{{\aleph_{5} }}} = \overline{{\aleph_{5} }} - \overline{{\aleph_{2} }} \overline{{\aleph_{3} }} /\overline{{\aleph_{1} }} , \\ \overline{\overline{{\aleph_{5}^{*} }}} & = \overline{{\aleph_{5}^{*} }} - \overline{{\aleph_{2}^{*} }} \overline{{\aleph_{3}^{*} }} /\overline{{\aleph_{1}^{*} }} ,\,\overline{\overline{{\aleph_{7} }}} = \left( {\overline{{\aleph_{3} }} } \right)^{2} /\overline{{\aleph_{1} }} - c_{1}^{2} \aleph_{7} ,\overline{\overline{{\aleph_{7}^{*} }}} = \left( {\overline{{\aleph_{3}^{*} }} } \right)^{2} /\overline{{\aleph_{1}^{*} }} - c_{1}^{2} \aleph_{7} . \\ \end{aligned} $$

Appendix C

$$ \begin{aligned} T_{11}^{*} (w^{*} ) & = S_{11} w_{{,x^{2} }}^{*} { + }S_{12} w_{{,y^{2} }}^{*} ,T_{21}^{*} (w^{*} ) = S_{21} w_{,x}^{*} ,\,\,T_{31}^{*} (w^{*} ) = S_{31} w_{,y}^{*} . \\ X_{{}}^{ * } \left( {w^{*} ,\wp } \right) & = \wp_{{,y^{2} }} w_{{,x^{2} }}^{*} - 2\wp_{,xy} w_{,xy}^{*} + \wp_{{,x^{2} }} w_{{,y^{2} }}^{*} , \\ \end{aligned} $$

$$ \begin{aligned} J_{1} & = \Im_{31}^{{}} - 2\Im_{12}^{{}} ,\,J_{2} = \Im_{23}^{{}} - c_{1} \Im_{25}^{{}} ,\,\,J_{3} = \Im_{13}^{{}} - c_{1} \Im_{15}^{{}} - \Im_{32}^{{}} + c_{1} \Im_{33}^{{}} , \\ J_{4} & = \Im_{14}^{{}} - c_{1} \Im_{16}^{{}} ,\,J_{5} = \Im_{24}^{{}} - c_{1} \Im_{26}^{{}} - \Im_{32}^{{}} + c_{1} \Im_{33}^{{}} ,\,\,J_{6} = - c_{1} \Im_{15}^{{}} - c_{1} \Im_{26}^{{}} + 2c_{1} \Im_{33}^{{}} . \\ \end{aligned} $$

$$ \begin{aligned} H_{1} & = \frac{{\beta^{2} }}{{32\Im_{21}^{{}} \alpha_{{}}^{2} }}W(W + 2\mu^{*} ),\,\,\,\,H_{2} = \frac{{\alpha_{{}}^{2} }}{{32\Im_{11}^{{}} \beta^{2} }}W(W + 2\mu^{*} ),\,\,\,H_{3} = P_{1} W + P_{2} \Phi_{x} + P_{3} \Phi_{y} , \\ P_{1} & = \left( {\alpha^{2} /R + c_{1} \Im_{25}^{{}} \alpha^{4} + c_{1} \Im_{16}^{{}} \beta^{4} - J_{6} \alpha^{2} \beta^{2} } \right)/G\,,P_{2} = - (J_{2} \alpha^{3} + J_{3} \alpha \beta^{2} )/G, \\ P_{3} & = - (J_{4} \beta^{3} + J_{5} \alpha^{2} \beta )/G,\,G = \Im_{21}^{{}} \alpha^{4} + J_{1} \alpha^{2} \beta^{2} + \Im_{11}^{{}} \beta^{4} , \\ \end{aligned} $$

$$ \begin{aligned} t_{11} & = - k_{1} - k_{2} \left( {\alpha^{2} + \beta^{2} } \right) + S_{13} \alpha^{4} + S_{14} \alpha^{2} \beta^{2} + S_{15} \beta^{4} + S_{110} P_{1} \alpha^{4} + S_{111} P_{1} \alpha^{2} \beta^{2} + S_{112} P_{1} \beta^{4} - P_{1} \frac{{\alpha^{2} }}{R}, \\ t_{12} & = - S_{11} \alpha + S_{16} \alpha^{3} + S_{17} \alpha \beta^{2} + S_{110} P_{2} \alpha^{4} + S_{111} P_{2} \alpha^{2} \beta^{2} + S_{112} P_{2} \beta^{4} - P_{2} \frac{{\alpha^{2} }}{R}, \\ l_{13} & = - S_{12} \beta + S_{18} \beta^{3} + S_{19} \alpha^{2} \beta + S_{110} P_{3} \alpha^{4} + S_{111} P_{3} \alpha^{2} \beta^{2} + S_{112} P_{3} \beta^{4} - P_{3} \frac{{\alpha^{2} }}{R}, \\ t_{14} & = \frac{{32P_{2} \alpha \beta }}{3LR\pi },\,\,t_{15} = \frac{{32P_{3} \alpha \beta }}{3LR\pi },\,\,t_{16} = \eta \sin \left( {\eta z} \right)\left( {\alpha^{2} S_{115} + \beta^{2} S_{116} } \right), \\ \end{aligned} $$

$$ l_{2} = \frac{{32P_{1} \beta \alpha }}{3LR\pi },l_{1} = - S_{11} \alpha^{2} - S_{12} \beta^{2} ,\,\,t_{17} = \eta \sin \left( {\eta z} \right)\left( {S_{119} \alpha^{2} + S_{120} \beta^{2} } \right), $$

$$ \begin{aligned} O_{11} & = \int\limits_{{ - h_{f} - 0.5h_{c} }}^{{0.5h_{c} }} {e_{15} T{\text{cos}}\left( {\eta z} \right)} {\text{d}}z + \int\limits_{{ - 0.5h_{c} }}^{{0.5h_{c} + h_{f} }} {e_{15} T{\text{cos}}\left( {\eta z} \right)} {\text{d}}z,\,T = \left( {1 - 3c_{1} z^{2} } \right) \\ \eta_{11}^{*} & = \int\limits_{{ - h_{f} - 0.5h_{c} }}^{{0.5h_{c} }} {e_{15} \eta_{11} {\text{cos}}\left( {\eta z} \right){\text{d}}z} + \int\limits_{{ - 0.5h_{c} }}^{{0.5h_{c} + h_{f} }} {e_{15} \eta_{11} {\text{cos}}\left( {\eta z} \right){\text{d}}z} , \\ m_{11}^{*} & = \int\limits_{{ - h_{f} - 0.5h_{c} }}^{{0.5h_{c} }} {m_{11} {\text{cos}}\left( {\eta z} \right){\text{d}}z} + \int\limits_{{ - 0.5h_{c} }}^{{0.5h_{c} + h_{f} }} {m_{11} {\text{cos}}\left( {\eta z} \right){\text{d}}z} , \\ \end{aligned} $$

Appendix D

$$ t_{14}^{1} = t_{14}^{{}} ,t_{15}^{1} = t_{15}^{{}} ,\,l_{1}^{1} = l_{1} + \alpha^{2} \left( {\frac{{\Im_{17}^{{}} }}{{\Im_{11} }}\phi_{0} + \frac{{\Im_{18}^{{}} }}{{\Im_{11} }}\psi_{0} + \frac{{\Im_{19}^{{}} }}{{\Im_{11} }}\Delta T} \right),\,\,l_{4}^{1} = \left( {l_{4} - \frac{{\alpha^{4} }}{{8\Im_{11} }}} \right), $$

$$ \begin{array}{*{20}c} \begin{gathered} t_{11}^{1} = \left( {t_{11} + t_{16} h_{11} + t_{17} h_{21} } \right), \hfill \\ t_{12}^{1} = \left( {t_{12} + t_{16} h_{12} + t_{17} h_{22} } \right), \hfill \\ t_{13}^{1} = \left( {t_{13} + t_{16} h_{13} + t_{17} h_{23} } \right), \hfill \\ \end{gathered} & \begin{gathered} t_{21}^{1} = \left( {t_{21} { + }t_{24} h_{11} { + }t_{25} h_{21} } \right), \hfill \\ t_{22}^{1} = \left( {t_{22} + t_{24} h_{12} + t_{25} h_{22} } \right), \hfill \\ t_{23}^{1} = \left( {t_{23} + t_{24} h_{13} + t_{25} h_{23} } \right), \hfill \\ \end{gathered} & \begin{gathered} t_{31}^{1} = \left( {t_{31} { + }t_{34} h_{11} { + }t_{35} h_{21} } \right), \hfill \\ t_{32}^{1} = \left( {t_{32} + t_{34} h_{12} + t_{35} h_{22} } \right), \hfill \\ t_{33}^{1} = \left( {t_{33} + t_{34} h_{13} + t_{35} h_{23} } \right), \hfill \\ \end{gathered} \\ \end{array} $$

$$ \begin{array}{*{20}c} \begin{gathered} h_{11} = \frac{{\left( {t_{45} t_{51}^{{}} { - }t_{41}^{{}} t_{55} } \right)}}{{\left( {t_{44} t_{55} - t_{45} t_{54} } \right)}}, \hfill \\ h_{12} = \frac{{\left( {t_{45} t_{52}^{{}} - t_{42}^{{}} t_{55} } \right)}}{{\left( {t_{44} t_{55} - t_{45} t_{54} } \right)}}, \hfill \\ \end{gathered} & \begin{gathered} h_{13} = \frac{{\left( {t_{45} t_{53}^{{}} - t_{43}^{{}} t_{55} } \right)}}{{\left( {t_{44} t_{55} - t_{45} t_{54} } \right)}}, \hfill \\ h_{21} = \frac{{\left( {t_{41}^{{}} t_{54} { - }t_{44} t_{51}^{{}} } \right)}}{{\left( {t_{44} t_{55} - t_{45} t_{54} } \right)}}, \hfill \\ \end{gathered} & \begin{gathered} h_{22} = \frac{{\left( {t_{42}^{{}} t_{54} - t_{44} t_{52}^{{}} } \right)}}{{\left( {t_{44} t_{55} - t_{45} t_{54} } \right)}}, \hfill \\ h_{23} = \frac{{\left( {t_{43}^{{}} t_{54} - t_{44} t_{53}^{{}} } \right)}}{{\left( {t_{44} t_{55} - t_{45} t_{54} } \right)}}. \hfill \\ \end{gathered} \\ \end{array} $$