Nonlinear resonance of axially moving graphene platelet-reinforced metal foam cylindrical shells with geometric imperfection

Abstract

The present work pays attention to the primary resonance of axially moving graphene-reinforced mental foam (GPLRMF) cylindrical shells with geometric imperfection. Porosities and graphene platelets (GPLs) are uniformly or non-uniformly distributed along the thickness direction of the cylindrical shell. Considering the influences of initial geometric imperfection and axial velocity, the equivalent elastic modulus is calculated by Halpin–Tsai model, and the equivalent density and Poisson’s ratio are described by the mixture rule. Using the energy principle, the nonlinear equations of motion are derived. Considering two different boundary conditions, the nonlinear primary resonance response is obtained using the modified Lindstedt Poincare (MLP) method. The results indicate that the MLP method can effectively overcome the limitation of traditional perturbation method. In the end, we study the effects of the GPLs distribution patterns, GPLs weight fraction, the porosity coefficient, axial velocity, initial geometric imperfection, and the prestressing force on the resonance problems. It can be found that the presence of initial geometric imperfection can alter the frequency response curve from the characteristics of the hard spring to the soft spring.

Appendices

Appendix A

$$L_{11} = A_{11} \frac{{\partial^{2} u}}{{\partial x^{2} }} + A_{66} \frac{{\partial^{2} u}}{{\partial y^{2} }}$$

(34)

$$L_{12} = \left[ {A_{12} + A_{66} + \frac{1}{R}\left( {B_{12} + 2B_{66} } \right)} \right]\frac{{\partial^{2} v}}{\partial x\partial y}$$

(35)

$$\begin{gathered} L_{13} = \frac{{A_{12} }}{R}\frac{\partial w}{{\partial x}} - B_{11} \frac{{\partial^{3} w}}{{\partial x^{3} }} - \left( {B_{12} + 2B_{66} } \right)\frac{{\partial^{3} w}}{{\partial x\partial y^{2} }} \hfill \\ + A_{11} \left( {\frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{{\partial w^{*} }}{\partial x} + \frac{\partial w}{{\partial x}}\frac{{\partial^{2} w^{*} }}{{\partial x^{2} }}} \right) + A_{12} \left( {\frac{{\partial^{2} w}}{\partial x\partial y}\frac{{\partial w^{*} }}{\partial y} + \frac{\partial w}{{\partial y}}\frac{{\partial^{2} w^{*} }}{\partial x\partial y}} \right) \hfill \\ + A_{66} \left( {\frac{{\partial^{2} w}}{\partial x\partial y}\frac{{\partial w^{*} }}{\partial y} + \frac{\partial w}{{\partial y}}\frac{{\partial^{2} w^{*} }}{\partial x\partial y} + \frac{{\partial^{2} w}}{{\partial y^{2} }}\frac{{\partial w^{*} }}{\partial x} + \frac{{\partial^{2} w^{*} }}{{\partial y^{2} }}\frac{\partial w}{{\partial x}}} \right) \hfill \\ \end{gathered}$$

(36)

$$P_{1} = A_{11} \frac{\partial w}{{\partial x}}\frac{{\partial^{2} w}}{{\partial x^{2} }} + A_{12} \frac{\partial w}{{\partial y}}\frac{{\partial^{2} w}}{\partial x\partial y} + A_{66} \left( {\frac{{\partial^{2} w}}{{\partial y^{2} }}\frac{\partial w}{{\partial x}} + \frac{{\partial^{2} w}}{\partial x\partial y}\frac{\partial w}{{\partial y}}} \right)$$

(37)

$$L_{21} = \left[ {A_{66} + A_{12} + \frac{1}{R}\left( {2B_{66} + B_{12} } \right)} \right]\frac{{\partial^{2} u}}{\partial x\partial y}$$

(38)

$$L_{22} = \left[ {A_{66} + \frac{1}{R}\left( {4B_{66} + \frac{4}{R}D_{66} } \right)} \right]\frac{{\partial^{2} v}}{{\partial x^{2} }} + \left[ {A_{22} + \frac{1}{R}\left( {2B_{22} + \frac{1}{R}D_{22} } \right)} \right]\frac{{\partial^{2} v}}{{\partial y^{2} }}$$

(39)

$$\begin{gathered} L_{23} = - \left[ {2B_{66} + B_{12} + \frac{1}{R}\left( {4D_{66} + D_{12} } \right)} \right]\frac{{\partial^{3} w}}{{\partial x^{2} \partial y}} + \frac{1}{R}\left( {A_{22} + \frac{{B_{22} }}{R}} \right)\frac{\partial w}{{\partial y}} - \left( {B_{22} + \frac{1}{R}D_{22} } \right)\frac{{\partial^{3} w}}{{\partial y^{3} }} \hfill \\ + \left( {A_{66} + \frac{{B_{66} }}{R}} \right)\left( {\frac{{\partial^{2} w^{*} }}{{\partial x^{2} }}\frac{\partial w}{{\partial y}} + \frac{{\partial w^{*} }}{\partial x}\frac{{\partial^{2} w}}{\partial x\partial y} + \frac{\partial w}{{\partial x}}\frac{{\partial^{2} w^{*} }}{\partial x\partial y} + \frac{{\partial w^{*} }}{\partial y}\frac{{\partial^{2} w}}{{\partial x^{2} }}} \right) \hfill \\ + \left( {A_{12} + \frac{{B_{12} }}{R}} \right)\left( {\frac{{\partial^{2} w}}{\partial x\partial y}\frac{{\partial w^{*} }}{\partial x} + \frac{\partial w}{{\partial x}}\frac{{\partial^{2} w^{*} }}{\partial x\partial y}} \right) + \left( {A_{22} + \frac{{B_{22} }}{R}} \right)\left( {\frac{{\partial^{2} w}}{{\partial y^{2} }}\frac{{\partial w^{*} }}{\partial y} + \frac{\partial w}{{\partial y}}\frac{{\partial^{2} w^{*} }}{{\partial y^{2} }}} \right) \hfill \\ \end{gathered}$$

(40)

$$P_{2} = \left( {A_{66} + 2\frac{{B_{66} }}{R}} \right)\left( {\frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{\partial w}{{\partial y}} + \frac{\partial w}{{\partial x}}\frac{{\partial^{2} w}}{\partial x\partial y}} \right) + \left( {A_{12} + \frac{{B_{12} }}{R}} \right)\frac{{\partial^{2} w}}{\partial x\partial y}\frac{\partial w}{{\partial x}} + \left( {A_{22} + \frac{{B_{22} }}{R}} \right)\frac{{\partial^{2} w}}{{\partial y^{2} }}\frac{\partial w}{{\partial y}}$$

(41)

$$L_{31} = B_{11} \frac{{\partial^{3} u}}{{\partial x^{3} }} + \left( {2B_{66} + B_{12} } \right)\frac{{\partial^{3} u}}{{\partial x\partial y^{2} }} - \frac{{A_{12} }}{R}\frac{\partial u}{{\partial x}}$$

(42)

$$L_{32} = \left[ {B_{12} + 2B_{66} + \frac{1}{R}\left( {4D_{66} + D_{12} } \right)} \right]\frac{{\partial^{3} v}}{{\partial x^{2} \partial y}} + \left( {B_{22} + \frac{{D_{22} }}{R}} \right)\frac{{\partial^{3} v}}{{\partial y^{3} }} - \left( {\frac{{A_{22} }}{R} + \frac{{B_{22} }}{{R^{2} }}} \right)\frac{\partial v}{{\partial y}}$$

(43)

$$\begin{gathered} L_{33} = \frac{{2B_{12} }}{R}\frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{{2B_{22} }}{R}\frac{{\partial^{2} w}}{{\partial y^{2} }} - D_{11} \frac{{\partial^{4} w}}{{\partial x^{4} }} - D_{22} \frac{{\partial^{4} w}}{{\partial y^{4} }} - (2D_{12} + 4D_{66} )\frac{{\partial^{4} w}}{{\partial x^{2} \partial y^{2} }} - \frac{{A_{22} }}{{R^{2} }}w \hfill \\ + B_{11} \left( {\frac{{\partial^{3} w}}{{\partial x^{3} }}\frac{{\partial w^{*} }}{\partial x} + 2\frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{{\partial^{2} w^{*} }}{{\partial x^{2} }} + \frac{{\partial^{3} w^{*} }}{{\partial x^{3} }}\frac{\partial w}{{\partial x}}} \right) + B_{12} \left( {\frac{{\partial^{3} w}}{{\partial x^{2} \partial y}}\frac{{\partial w^{*} }}{\partial y} + 2\frac{{\partial^{2} w}}{\partial x\partial y}\frac{{\partial^{2} w^{*} }}{\partial x\partial y} + \frac{\partial w}{{\partial y}}\frac{{\partial^{3} w^{*} }}{{\partial x^{2} \partial y}}} \right) \hfill \\ + 2B_{66} \left( \begin{gathered} \frac{\partial w}{{\partial y}}\frac{{\partial^{3} w^{*} }}{{\partial x^{2} \partial y}} + \frac{{\partial^{2} w^{*} }}{{\partial x^{2} }}\frac{{\partial^{2} w}}{{\partial y^{2} }} + 2\frac{{\partial^{2} w}}{\partial x\partial y}\frac{{\partial^{2} w^{*} }}{\partial x\partial y} \hfill \\ + \frac{{\partial^{3} w}}{{\partial x\partial y^{2} }}\frac{{\partial w^{*} }}{\partial x} + \frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{{\partial^{2} w^{*} }}{{\partial y^{2} }} + \frac{{\partial w^{*} }}{\partial y}\frac{{\partial^{3} w}}{{\partial x^{2} \partial y}} + \frac{{\partial^{3} w^{*} }}{{\partial x\partial y^{2} }}\frac{\partial w}{{\partial x}} \hfill \\ \end{gathered} \right) \hfill \\ + B_{12} \left( {\frac{{\partial^{3} w}}{{\partial x\partial y^{2} }}\frac{\partial w}{{\partial x}} + 2\frac{{\partial^{2} w}}{\partial x\partial y}\frac{{\partial^{2} w^{*} }}{\partial x\partial y} + \frac{\partial w}{{\partial x}}\frac{{\partial^{3} w^{*} }}{{\partial x\partial y^{2} }}} \right) \hfill \\ + B_{22} \left( {\frac{{\partial^{3} w}}{{\partial y^{3} }}\frac{{\partial w^{*} }}{\partial y} + 2\frac{{\partial^{2} w}}{{\partial y^{2} }}\frac{{\partial^{2} w^{*} }}{{\partial y^{2} }} + \frac{{\partial^{3} w^{*} }}{{\partial y^{3} }}\frac{\partial w}{{\partial y}}} \right) - \frac{{A_{12} }}{R}\frac{\partial w}{{\partial x}}\frac{{\partial w^{*} }}{\partial x} - \frac{{A_{22} }}{R}\frac{\partial w}{{\partial y}}\frac{{\partial w^{*} }}{\partial y} \hfill \\ \end{gathered}$$

(44)

$$\begin{gathered} P_{3} = B_{11} \left( {\frac{{\partial^{3} w}}{{\partial x^{3} }}\frac{\partial w}{{\partial x}} + \frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{{\partial^{2} w}}{{\partial x^{2} }}} \right) + B_{12} \left( {\frac{{\partial^{3} w}}{{\partial x^{2} \partial y}}\frac{\partial w}{{\partial y}} + \frac{{\partial^{2} w}}{\partial x\partial y}\frac{{\partial^{2} w}}{\partial x\partial y}} \right) + A_{22} \frac{\partial w}{{\partial y}}\frac{{\partial w^{*} }}{\partial y}\frac{{\partial^{2} w}}{{\partial y^{2} }} \hfill \\ + 2B_{66} \left( {\frac{{\partial^{3} w}}{{\partial x^{2} \partial y}}\frac{\partial w}{{\partial y}} + \frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{{\partial^{2} w}}{{\partial y^{2} }} + \frac{{\partial^{2} w}}{\partial x\partial y}\frac{{\partial^{2} w}}{\partial x\partial y} + \frac{\partial w}{{\partial x}}\frac{{\partial^{3} w}}{{\partial x\partial y^{2} }}} \right) + B_{12} \left( {\frac{{\partial^{3} w}}{{\partial x\partial y^{2} }}\frac{\partial w}{{\partial x}} + \frac{{\partial^{2} w}}{\partial x\partial y}\frac{{\partial^{2} w}}{\partial x\partial y}} \right) \hfill \\ + B_{22} \left( {\frac{{\partial^{3} w}}{{\partial y^{3} }}\frac{\partial w}{{\partial y}} + \frac{{\partial^{2} w}}{{\partial y^{2} }}\frac{{\partial^{2} w}}{{\partial y^{2} }}} \right) - \frac{{A_{12} }}{2R}\left( {\frac{\partial w}{{\partial x}}} \right)^{2} - \frac{{A_{22} }}{2R}\left( {\frac{\partial w}{{\partial y}}} \right)^{2} + \frac{{A_{12} }}{R}w\frac{{\partial^{2} w}}{{\partial x^{2} }} - B_{11} \left( {\frac{{\partial^{2} w}}{{\partial x^{2} }}} \right)^{2} \hfill \\ - 2B_{12} \frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{{\partial^{2} w}}{{\partial y^{2} }} - 4B_{66} \left( {\frac{{\partial^{2} w}}{\partial x\partial y}} \right)^{2} + \frac{{A_{22} }}{R}w\frac{{\partial^{2} w}}{{\partial y^{2} }} - B_{22} \left( {\frac{{\partial^{2} w}}{{\partial y^{2} }}} \right)^{2} + A_{11} \frac{\partial w}{{\partial x}}\frac{{\partial w^{*} }}{\partial x}\frac{{\partial^{2} w}}{{\partial x^{2} }} \hfill \\ + A_{12} \frac{\partial w}{{\partial y}}\frac{{\partial w^{*} }}{\partial y}\frac{{\partial^{2} w}}{{\partial x^{2} }} + 2A_{66} \left( {\frac{{\partial w^{2} }}{\partial x\partial y}\frac{{\partial w^{*} }}{\partial x}\frac{\partial w}{{\partial y}} + \frac{{\partial w^{2} }}{\partial x\partial y}\frac{\partial w}{{\partial x}}\frac{{\partial w^{*} }}{\partial y}} \right) + A_{12} \frac{\partial w}{{\partial x}}\frac{{\partial w^{*} }}{\partial x}\frac{{\partial^{2} w}}{{\partial y^{2} }} \hfill \\ \end{gathered}$$

(45)

$$\begin{gathered} P_{4} = \frac{1}{2}A_{11} \left( {\frac{\partial w}{{\partial x}}} \right)^{2} \frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{1}{2}A_{12} \left( {\frac{\partial w}{{\partial y}}} \right)^{2} \frac{{\partial^{2} w}}{{\partial x^{2} }} + 2A_{66} \frac{{\partial^{2} w}}{\partial x\partial y}\frac{\partial w}{{\partial x}}\frac{\partial w}{{\partial y}} \hfill \\ + \frac{1}{2}A_{12} \left( {\frac{\partial w}{{\partial x}}} \right)^{2} \frac{{\partial^{2} w}}{{\partial y^{2} }} + \frac{1}{2}A_{12} \left( {\frac{\partial w}{{\partial y}}} \right)^{2} \frac{{\partial^{2} w}}{{\partial y^{2} }} \hfill \\ \end{gathered}$$

(46)

$$Q_{3} = A_{11} \frac{\partial u}{{\partial x}}\frac{{\partial^{2} w}}{{\partial x^{2} }} + 2A_{66} \frac{{\partial^{2} w}}{\partial x\partial y}\frac{\partial u}{{\partial y}} + A_{12} \frac{{\partial^{2} w}}{{\partial y^{2} }}\frac{\partial u}{{\partial x}}$$

(47)

$$R_{3} = \left( {A_{12} + \frac{{B_{12} }}{R}} \right)\frac{\partial v}{{\partial y}}\frac{{\partial^{2} w}}{{\partial x^{2} }} + \left( {2A_{66} + \frac{{4A_{66} }}{R}} \right)\frac{{\partial^{2} w}}{\partial x\partial y}\frac{\partial v}{{\partial x}} + \left( {A_{22} + \frac{{B_{22} }}{R}} \right)\frac{{\partial^{2} w}}{{\partial y^{2} }}\frac{\partial v}{{\partial y}}$$

(48)

These symbols appeared in Eqs. (34)–(48) are defined as

$$\left\{ \begin{gathered} A_{ii} = \int_{{ - \frac{h}{2}}}^{\frac{h}{2}} {Q_{ii} (z)dz} ,\;\; \hfill \\ B_{ii} = \int_{{ - \frac{h}{2}}}^{\frac{h}{2}} {z \cdot Q_{ii} (z)dz} ,\;\; \hfill \\ D_{ii} = \int_{{ - \frac{h}{2}}}^{\frac{h}{2}} {z^{2} \cdot Q_{ii} (z)dz} \hfill \\ \end{gathered} \right.\;ij = (11,12,22,66)$$

(49)

Appendix B

For SSSS:

$$\begin{gathered} l_{11} = - \frac{{{\mkern 1mu} \left( {A_{66} {\mkern 1mu} L^{2} {\mkern 1mu} n^{2} + A_{11} {\mkern 1mu} \pi^{2} {\mkern 1mu} R^{2} {\mkern 1mu} m^{2} } \right)}}{{{\mkern 1mu} L^{2} {\mkern 1mu} R^{2} {\mkern 1mu} }}, \hfill \\ l_{12} = \frac{{{\mkern 1mu} \pi mn\left( {B_{12} + 2{\mkern 1mu} B_{66} + A_{12} {\mkern 1mu} R + A_{66} {\mkern 1mu} R} \right)}}{{{\mkern 1mu} L{\mkern 1mu} R^{2} {\mkern 1mu} }}, \hfill \\ l_{13} = \frac{{\pi {\mkern 1mu} m\left( {B_{12} {\mkern 1mu} L^{2} {\mkern 1mu} n^{2} + 2{\mkern 1mu} B_{66} {\mkern 1mu} L^{2} {\mkern 1mu} n^{2} + A_{12} {\mkern 1mu} L^{2} {\mkern 1mu} R + B_{11} {\mkern 1mu} R^{2} {\mkern 1mu} m^{2} {\mkern 1mu} \pi^{2} } \right)}}{{L^{3} {\mkern 1mu} R^{2} {\mkern 1mu} }}; \hfill \\ \end{gathered}$$

(50)

$$\begin{gathered} l_{21} = \frac{{\pi mn\left( {B_{12} + 2{\mkern 1mu} B_{66} + A_{12} {\mkern 1mu} R + A_{66} {\mkern 1mu} R} \right)}}{{L{\mkern 1mu} R^{2} {\mkern 1mu} }}, \hfill \\ l_{22} = - \frac{{\left( {A_{22} {\mkern 1mu} L^{2} {\mkern 1mu} R^{2} {\mkern 1mu} n^{2} + 2{\mkern 1mu} B_{22} {\mkern 1mu} L^{2} {\mkern 1mu} R{\mkern 1mu} n^{2} + D_{22} {\mkern 1mu} L^{2} {\mkern 1mu} n^{2} + A_{66} {\mkern 1mu} \pi^{2} {\mkern 1mu} R^{4} {\mkern 1mu} m^{2} + 4{\mkern 1mu} B_{66} {\mkern 1mu} \pi^{2} {\mkern 1mu} R^{3} {\mkern 1mu} m^{2} + 4{\mkern 1mu} D_{66} {\mkern 1mu} \pi^{2} {\mkern 1mu} R^{2} {\mkern 1mu} m^{2} } \right)}}{{L^{2} {\mkern 1mu} R^{4} }}, \hfill \\ l_{23} = - \frac{{n\left( \begin{gathered} D_{22} L^{2} n^{2} + B_{22} L^{2} R + A_{22} L^{2} R^{2} + B_{22} L^{2} Rn^{2} + B_{12} R^{3} m^{2} \pi^{2} \hfill \\ + 2B_{66} R^{3} m^{2} \pi^{2} + D_{12} R^{3} m^{2} \pi^{2} + 4D_{66} R^{2} m^{2} \pi^{2} \hfill \\ \end{gathered} \right)}}{{L^{2} R^{4} }}; \hfill \\ \end{gathered}$$

(51)

$$\begin{gathered} l_{31} = \frac{{\pi {\mkern 1mu} m\left( {B_{12} {\mkern 1mu} L^{2} {\mkern 1mu} n^{2} + 2{\mkern 1mu} B_{66} {\mkern 1mu} L^{2} {\mkern 1mu} n^{2} + A_{12} {\mkern 1mu} L^{2} {\mkern 1mu} R + B_{11} {\mkern 1mu} R^{2} {\mkern 1mu} m^{2} {\mkern 1mu} \pi^{2} } \right)}}{{L^{3} {\mkern 1mu} R^{2} {\mkern 1mu} }}, \hfill \\ l_{32} = - \frac{{n\left( \begin{gathered} D_{22} L^{2} n^{2} + B_{22} L^{2} R + A_{22} L^{2} R^{2} + B_{22} L^{2} Rn^{2} + B_{12} R^{3} m^{2} \pi^{2} \hfill \\ + 2B_{66} R^{3} m^{2} \pi^{2} + D_{12} R^{3} m^{2} \pi^{2} + 4D_{66} R^{2} m^{2} \pi^{2} \hfill \\ \end{gathered} \right)}}{{L^{2} R^{4} }}, \hfill \\ l_{33} = - \frac{{\left( \begin{gathered} D_{22} L^{4} n^{4} + A_{22} L^{4} R^{2} + 2B_{22} L^{4} Rn^{2} + D_{11} R^{4} m^{4} \pi^{4} + 2B_{12} L^{2} R^{3} m^{2} \pi^{2} \hfill \\ + L^{2} R^{4} m^{2} \pi^{2} ph - I_{0} L^{2} R^{4} V^{2} m^{2} \pi^{2} + 4D_{66} L^{2} R^{2} m^{2} n^{2} \pi^{2} \hfill \\ \end{gathered} \right)}}{{L^{4} R^{4} }}, \hfill \\ p_{3} = - \frac{{3A_{22} W_{1} n^{4} }}{{16R^{4} }} - \frac{{3A_{11} W_{1} m^{4} \pi^{4} }}{{16L^{4} }} - \frac{{\pi A_{12} W_{1} m^{2} }}{{16L^{2} R}} - \frac{{3A_{12} W_{1} m^{2} n^{2} \pi^{2} }}{{16L^{2} R^{2} }} + \frac{{A_{66} W_{1} m^{2} n^{2} \pi^{2} }}{{4L^{2} R^{2} }}, \hfill \\ p_{4} = \frac{{ - 96A_{22} L^{4} mn^{5} \pi^{2} - 96A_{11} R^{4} m^{5} n\pi^{6} - 192A_{12} L^{2} R^{2} m^{3} n^{3} \pi^{4} + 128A_{66} L^{2} R^{2} m^{3} n^{3} \pi^{4} }}{{1024L^{4} R^{4} mn\pi^{2} }}, \hfill \\ p_{1} = p_{2} = n_{1} = n_{2} = 0. \hfill \\ \end{gathered}$$

(52)

For CCSS:

$$\begin{gathered} l_{11} = - \frac{{{\mkern 1mu} \left( {A_{66} {\mkern 1mu} L^{2} {\mkern 1mu} n^{2} + 4{\mkern 1mu} A_{11} {\mkern 1mu} \pi^{2} {\mkern 1mu} R^{2} {\mkern 1mu} m^{2} } \right)}}{{{\mkern 1mu} L^{2} {\mkern 1mu} R^{2} {\mkern 1mu} }}, \hfill \\ l_{12} = - \frac{{2\pi {\mkern 1mu} mn\left( {B_{12} + 2{\mkern 1mu} B_{66} + A_{12} {\mkern 1mu} R + A_{66} {\mkern 1mu} R} \right)}}{{{\mkern 1mu} L{\mkern 1mu} R^{2} {\mkern 1mu} }}, \hfill \\ l_{13} = \frac{{2mn\pi^{2} \left( {B_{12} L^{2} n^{2} + 2B_{66} L^{2} n^{2} + A_{12} L^{2} R + 4B_{11} R^{2} m^{2} \pi^{2} } \right)}}{{8L^{3} R^{2} n\pi }}; \hfill \\ \end{gathered}$$

(53)

$$\begin{gathered} l_{21} = - \frac{{2\pi {\mkern 1mu} mn\left( {B_{12} + 2{\mkern 1mu} B_{66} + A_{12} {\mkern 1mu} R + A_{66} {\mkern 1mu} R} \right)}}{{{\mkern 1mu} L{\mkern 1mu} R^{2} {\mkern 1mu} }}, \hfill \\ l_{22} = - \frac{{6B_{22} n^{2} }}{{R^{3} }} - \frac{{3D_{22} n^{2} }}{{R^{4} }} - \frac{{4A_{66} m^{2} \pi^{2} }}{{L^{2} }} - \frac{{3A_{22} n^{2} }}{{R^{2} }} - \frac{{16B_{66} m^{2} \pi^{2} }}{{L^{2} R}} - \frac{{16D_{66} m^{2} \pi^{2} }}{{L^{2} R^{2} }}, \hfill \\ l_{23} = \frac{{A_{22} n}}{{R^{2} }} + \frac{{B_{22} n^{3} }}{{R^{3} }} + \frac{{3B_{22} n^{3} }}{{R^{3} }} + \frac{{3D_{22} n^{3} }}{{R^{4} }} + \frac{{16D_{66} m^{2} n\pi^{2} }}{{L^{2} R^{2} }} + \frac{{4D_{12} m^{2} n\pi^{2} }}{{L^{2} R}} \hfill \\ \;\;\;\;\;\; + \frac{{8B_{66} m^{2} n\pi^{2} }}{{L^{2} R}} + \frac{{8B_{12} m^{2} n\pi^{2} }}{{L^{2} R}}; \hfill \\ \end{gathered}$$

(54)

$$\begin{gathered} l_{31} = \frac{{2mn\pi^{2} \left( {B_{12} L^{2} n^{2} + 2B_{66} L^{2} n^{2} + A_{12} L^{2} R + 4B_{11} R^{2} m^{2} \pi^{2} } \right)}}{{8L^{3} R^{2} n\pi }}, \hfill \\ l_{32} = \frac{{A_{22} n}}{{R^{2} }} + \frac{{B_{22} n^{3} }}{{R^{3} }} + \frac{{3B_{22} n^{3} }}{{R^{3} }} + \frac{{3D_{22} n^{3} }}{{R^{4} }} + \frac{{16D_{66} m^{2} n\pi^{2} }}{{L^{2} R^{2} }} + \frac{{4D_{12} m^{2} n\pi^{2} }}{{L^{2} R}} + \frac{{8B_{66} m^{2} n\pi^{2} }}{{L^{2} R}} + \frac{{8B_{12} m^{2} n\pi^{2} }}{{L^{2} R}}, \hfill \\ l_{33} = \frac{{2n^{2} B_{22} }}{{R^{3} }} - \frac{{16D_{11} m^{4} \pi^{4} }}{{L^{4} }} - \frac{{4m^{2} \pi^{2} ph}}{{L^{2} }} + \frac{{A_{22} }}{{R^{2} }} + \frac{{D_{22} n^{4} }}{{R^{4} }} - \frac{{8B_{12} m^{2} \pi^{2} }}{{L^{2} R}} + \frac{{4I_{0} V^{2} m^{2} \pi^{2} }}{{L^{2} }} \hfill \\ \;\;\;\;\;\; - \frac{{16D_{22} m^{2} n^{2} \pi^{2} }}{{L^{2} R^{2} }} - \frac{{8D_{12} m^{2} n^{2} \pi^{2} }}{{L^{2} R^{2} }}, \hfill \\ p_{3} = \frac{{ - 60\pi B_{22} L^{4} mn^{4} + 30\pi A_{22} L^{4} Rmn^{2} + 96B_{12} L^{2} R^{2} m^{3} n^{2} \pi^{3} - 96B_{66} L^{2} R^{2} m^{3} n^{2} \pi^{3} }}{{36L^{4} R^{4} mn\pi^{2} }} \hfill \\ \;\;\;\;\;\; - \frac{{16\pi B_{66} m^{2} n}}{{3L^{2} R^{2} }} - \frac{{10B_{22} n^{3} }}{{3R^{4} \pi }} - \frac{{32B_{11} m^{4} \pi^{3} }}{{3L^{4} n}} - \frac{{10A_{22} n}}{{3R^{3} \pi }} - \frac{{32\pi B_{12} m^{2} n}}{{3L^{2} R^{2} }} - \frac{{16\pi A_{12} m^{2} }}{{3L^{2} Rn}} \hfill \\ \;\;\;\;\; + \frac{{5\pi A_{12} W_{1} m^{2} }}{{16L^{2} R}} - \frac{{3A_{11} W_{1} m^{4} \pi^{4} }}{{L^{4} }} - \frac{{35A_{22} W_{1} n^{4} }}{{16R^{4} }} - \frac{{15A_{12} W_{1} m^{2} n^{2} \pi^{2} }}{{4L^{2} R^{2} }} + \frac{{5A_{66} W_{1} m^{2} n^{2} \pi^{2} }}{{L^{2} R^{2} }}, \hfill \\ p_{4} = - \frac{{35A_{22} n^{4} }}{{32R^{4} }} + \frac{{3A_{11} m^{4} \pi^{4} }}{{2L^{4} }} - \frac{{13A_{12} m^{2} n^{2} \pi^{2} }}{{4L^{2} R^{2} }} + \frac{{5A_{66} m^{2} n^{2} \pi^{2} }}{{2L^{2} R^{2} }}, \hfill \\ p_{1} = p_{2} = n_{1} = n_{2} = 0. \hfill \\ \end{gathered}$$

(55)