Surface free energy effects on the postbuckling behavior of cylindrical shear deformable nanoshells under combined axial and radial compressions

Abstract

In the traditional continuum mechanics, the effects of surface free energy are generally ignored. However, this cannot be the case for nanostructures because of their high surface to volume ratio; surface energy plays an important role in the mechanical responses. In the present study, the nonlinear buckling and postbuckling characteristics of cylindrical nanoshells subjected to combined axial and radial compressions are investigated in the presence of surface energy effects. To this end, Gurtin–Murdoch elasticity theory is implemented into the classical first-order shear deformation shell theory to develop an efficient size-dependent shell model incorporating surface free energy effects. Subsequently, a boundary layer theory is employed including surface effects in conjunction with the nonlinear prebuckling deformations, the large postbuckling deflections and the initial geometric imperfection. Finally, a solution methodology based on a two-stepped singular perturbation technique is utilized to obtain the size-dependent critical buckling loads and equilibrium postbuckling paths corresponding to the both axial dominated and radial dominated loading cases. It is observed that for the both axial dominated and radial dominated loading cases, surface free energy effects cause to increase the both critical buckling load and critical end-shortening of shear deformable nanoshell made of silicon.

Appendix

The sets of perturbation equations for the regular solution are:

$$O\left( { \epsilon^{0} } \right):\left\{ {\begin{array}{l} { - \frac{{\partial^{2} \overline{F}_{0} }}{{\partial X^{2} }} = \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{0} ,\overline{F}_{0} } \right) + {\mathcal{Q}}_{0} } \hfill \\ {{\mathcal{L}}_{21} \left( {\overline{F}_{0} } \right) + 2\overline{\tau } {\mathcal{L}}_{22} \left( {\overline{W}_{0} } \right) + \frac{{\partial^{2} \overline{W}_{0} }}{{\partial X^{2} }} = - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{0} ,\overline{W}_{0} } \right) - \overline{\tau } \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{0} ,\overline{W}_{0} } \right) + \overline{\tau } \vartheta_{1} \widetilde{{\mathcal{L}}}_{3} (\overline{W}_{0} ,\overline{W}_{0} )} \hfill \\ {\quad + \frac{{\overline{\tau } \vartheta_{2} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} (\overline{W}_{0} ,\overline{W}_{0} )} \hfill \\ {0 = 0} \hfill \\ {0 = 0} \hfill \\ \end{array} } \right.$$

$$O\left( { \epsilon^{1/2} } \right):\left\{ {\begin{array}{l} { - \frac{{\partial^{2} \overline{F}_{1/2} }}{{\partial X^{2} }} = \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1/2} ,\overline{F}_{0} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{0} ,\overline{F}_{1/2} } \right)} \hfill \\ {{\mathcal{L}}_{21} \left( {\overline{F}_{1/2} } \right) + 2\overline{\tau } {\mathcal{L}}_{22} \left( {\overline{W}_{1/2} } \right) + \frac{{\partial^{2} \overline{W}_{1/2} }}{{\partial X^{2} }} = - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{0} ,\overline{W}_{1/2} } \right) - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{0} ,\overline{W}_{1/2} } \right)} \hfill \\ {\quad + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} (\overline{W}_{0} ,\overline{W}_{1/2} ) + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} (\overline{W}_{0} ,\overline{W}_{1/2} )} \hfill \\ {{\mathcal{L}}_{32} \left( {\overline{\varPsi }_{{x_{1/2} }} } \right) + \vartheta_{5} \beta \frac{{\partial^{2} \overline{\varPsi }_{{y_{1/2} }} }}{\partial X\partial Y} = 0} \hfill \\ {{\mathcal{L}}_{42} \left( {\overline{\varPsi }_{{y_{1/2} }} } \right) + \vartheta_{5} \beta \frac{{\partial^{2} \overline{\varPsi }_{{x_{1/2} }} }}{\partial X\partial Y} = 0} \hfill \\ \end{array} } \right.$$

$$O\left( { \epsilon^{1} } \right):\left\{ {\begin{array}{l} { - \frac{{\partial^{2} \overline{F}_{1} }}{{\partial X^{2} }} = \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1/2} ,\overline{F}_{1/2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1} ,\overline{F}_{0} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{0} ,\overline{F}_{1} } \right) + {\mathcal{Q}}_{1} } \\ {{\mathcal{L}}_{21} \left( {\overline{F}_{2} } \right) + 2\overline{\tau } {\mathcal{L}}_{22} \left( {\overline{W}_{2} } \right) + \frac{{\partial^{2} \overline{W}_{2} }}{{\partial X^{2} }} = - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{0} ,\overline{W}_{2} } \right) - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1/2} ,\overline{W}_{3/2} } \right) - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1} ,\overline{W}_{1} } \right)} \\ {\quad - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{1/2} ,\overline{W}_{1/2} } \right) + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{0} ,\overline{W}_{1} } \right) + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{1/2} ,\overline{W}_{1/2} } \right) + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{0} ,\overline{W}_{1} } \right)} \\ {\quad + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{1/2} ,\overline{W}_{1/2} } \right)} \\ {{\mathcal{L}}_{31} \left( {\overline{W}_{0} } \right) + {\mathcal{L}}_{32} \left( {\overline{\varPsi }_{{x_{1} }} } \right) + \vartheta_{5} \beta \frac{{\partial^{2} \overline{\varPsi }_{{y_{1} }} }}{\partial X\partial Y} = 0} \\ {{\mathcal{L}}_{41} \left( {\overline{W}_{0} } \right) + {\mathcal{L}}_{42} \left( {\overline{\varPsi }_{{y_{1} }} } \right) + \vartheta_{5} \beta \frac{{\partial^{2} \overline{\varPsi }_{{x_{1} }} }}{\partial X\partial Y} = 0} \\ \end{array} } \right.$$

$$O\left( { \epsilon^{3/2} } \right):\left\{ \begin{array}{l} - \frac{{\partial^{2} \overline{F}_{3/2} }}{{\partial X^{2} }} + {\mathcal{L}}_{12} \left( {\overline{\varPsi }_{{x_{1/2} }} } \right) + {\mathcal{L}}_{13} \left( {\overline{\varPsi }_{{y_{1/2} }} } \right) = \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{0} ,\overline{F}_{3/2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1/2} ,\overline{F}_{1} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1} ,\overline{F}_{1/2} } \right) \hfill \\ + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{3/2} ,\overline{F}_{0} } \right) \hfill \\ {\mathcal{L}}_{21} \left( {\overline{F}_{3/2} } \right) + 2\overline{\tau } {\mathcal{L}}_{22} \left( {\overline{W}_{3/2} } \right) + \frac{{\partial^{2} \overline{W}_{3/2} }}{{\partial X^{2} }} = - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{0} ,\overline{W}_{3/2} } \right) - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1/2} ,\overline{W}_{1} } \right) \hfill \\ - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{0} ,\overline{W}_{3/2} } \right) - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{1/2} ,\overline{W}_{1} } \right) + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{0} ,\overline{W}_{3/2} } \right) + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{1/2} ,\overline{W}_{1} } \right) \hfill \\ + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{0} ,\overline{W}_{3/2} } \right) + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{1/2} ,\overline{W}_{1} } \right) \hfill \\ {\mathcal{L}}_{31} \left( {\overline{W}_{1/2} } \right) + {\mathcal{L}}_{32} \left( {\overline{\varPsi }_{{x_{3/2} }} } \right) + \vartheta_{5} \beta \frac{{\partial^{2} \overline{\varPsi }_{{y_{3/2} }} }}{\partial X\partial Y} = 0 \hfill \\ {\mathcal{L}}_{41} \left( {\overline{W}_{1/2} } \right) + {\mathcal{L}}_{42} \left( {\overline{\varPsi }_{{y_{3/2} }} } \right) + \vartheta_{5} \beta \frac{{\partial^{2} \overline{\varPsi }_{{x_{3/2} }} }}{\partial X\partial Y} = 0 \hfill \\ \end{array} \right.$$

$$O\left( { \epsilon^{2} } \right):\left\{ \begin{array}{l} 2\overline{\tau } {\mathcal{L}}_{11} \left( {\overline{W}_{0} } \right) + {\mathcal{L}}_{12} \left( {\overline{\varPsi }_{{x_{1} }} } \right) + {\mathcal{L}}_{13} \left( {\overline{\varPsi }_{{y_{1} }} } \right) - \frac{{\partial^{2} \overline{F}_{2} }}{{\partial X^{2} }} = \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{0} ,\overline{F}_{2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1/2} ,\overline{F}_{3/2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1} ,\overline{F}_{1} } \right) \hfill \\ + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{3/2} ,\overline{F}_{1/2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{2} + W^{*} ,\overline{F}_{0} } \right) + {\mathcal{Q}}_{2} \hfill \\ {\mathcal{L}}_{21} \left( {\overline{F}_{2} } \right) + 2\overline{\tau } {\mathcal{L}}_{22} \left( {\overline{W}_{2} } \right) + \frac{{\partial^{2} \overline{W}_{2} }}{{\partial X^{2} }} = - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{0} ,\overline{W}_{2} } \right) - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1/2} ,\overline{W}_{3/2} } \right) - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1} ,\overline{W}_{1} } \right) \hfill \\ - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{0} ,\overline{W}_{2} } \right) - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{1/2} ,\overline{W}_{3/2} } \right) - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{1} ,\overline{W}_{1} } \right) \hfill \\ + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{0} ,\overline{W}_{2} + 2W^{*} } \right) + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{1/2} ,\overline{W}_{3/2} } \right) + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{1} ,\overline{W}_{1} } \right) \hfill \\ + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{0} ,\overline{W}_{2} + 2W^{*} } \right) + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{1/2} ,\overline{W}_{3/2} } \right) + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{1} ,\overline{W}_{1} } \right) \hfill \\ {\mathcal{L}}_{31} \left( {\overline{W}_{1} } \right) + {\mathcal{L}}_{32} \left( {\overline{\varPsi }_{{x_{2} }} } \right) + \vartheta_{5} \beta \frac{{\partial^{2} \overline{\varPsi }_{{y_{2} }} }}{\partial X\partial Y} = 0 \hfill \\ {\mathcal{L}}_{41} \left( {\overline{W}_{1} } \right) + {\mathcal{L}}_{42} \left( {\overline{\varPsi }_{{y_{2} }} } \right) + \vartheta_{5} \beta \frac{{\partial^{2} \overline{\varPsi }_{{x_{2} }} }}{\partial X\partial Y} = 0 \hfill \\ \end{array} \right.$$

$$O\left( { \epsilon^{5/2} } \right):\left\{ \begin{array}{l} 2\overline{\tau } {\mathcal{L}}_{12} \left( {\overline{W}_{1/2} } \right) + {\mathcal{L}}_{12} \left( {\overline{\varPsi }_{{x_{3/2} }} } \right) + {\mathcal{L}}_{13} \left( {\overline{\varPsi }_{{y_{3/2} }} } \right) - \frac{{\partial^{2} \overline{F}_{5/2} }}{{\partial X^{2} }} = \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{0} ,\overline{F}_{5/2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1/2} ,\overline{F}_{2} } \right) \hfill \\ + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1} ,\overline{F}_{3/2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{3/2} ,\overline{F}_{1} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{2} + W^{*} ,\overline{F}_{1/2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{5/2} ,\overline{F}_{0} } \right) \hfill \\ {\mathcal{L}}_{21} \left( {\overline{F}_{5/2} } \right) + 2\overline{\tau } {\mathcal{L}}_{22} \left( {\overline{W}_{5/2} } \right) + \frac{{\partial^{2} \overline{W}_{5/2} }}{{\partial X^{2} }} = - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{0} ,\overline{W}_{5/2} } \right) - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1/2} ,\overline{W}_{2} + 2W^{*} } \right) \hfill \\ - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1} ,\overline{W}_{3/2} } \right) - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{0} ,\overline{W}_{5/2} } \right) - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{1/2} ,\overline{W}_{2} + 2W^{*} } \right) \hfill \\ - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{1} ,\overline{W}_{3/2} } \right) + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{0} ,\overline{W}_{5/2} } \right) + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{1/2} ,\overline{W}_{2} + 2W^{*} } \right) \hfill \\ + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{1} ,\overline{W}_{3/2} } \right) + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{0} ,\overline{W}_{5/2} } \right) + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{1/2} ,\overline{W}_{2} + 2W^{*} } \right) \hfill \\ + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{1} ,\overline{W}_{3/2} } \right) \hfill \\ {\mathcal{L}}_{31} \left( {\overline{W}_{3/2} } \right) + {\mathcal{L}}_{32} \left( {\overline{\varPsi }_{{x_{5/2} }} } \right) + \vartheta_{5} \beta \frac{{\partial^{2} \overline{\varPsi }_{{y_{5/2} }} }}{\partial X\partial Y} = 0 \hfill \\ {\mathcal{L}}_{41} \left( {\overline{W}_{3/2} } \right) + {\mathcal{L}}_{42} \left( {\overline{\varPsi }_{{y_{5/2} }} } \right) + \vartheta_{5} \beta \frac{{\partial^{2} \overline{\varPsi }_{{x_{5/2} }} }}{\partial X\partial Y} = 0 \hfill \\ \end{array} \right.$$

$$O\left( { \epsilon^{3} } \right):\left\{ \begin{array}{l} 2\overline{\tau } {\mathcal{L}}_{12} \left( {\overline{W}_{1} } \right) + {\mathcal{L}}_{12} \left( {\overline{\varPsi }_{{x_{2} }} } \right) + {\mathcal{L}}_{13} \left( {\overline{\varPsi }_{{y_{2} }} } \right) - \frac{{\partial^{2} \overline{F}_{3} }}{{\partial X^{2} }} = \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{0} ,\overline{F}_{3} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1/2} ,\overline{F}_{5/2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1} ,\overline{F}_{2} } \right) \hfill \\ + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{3/2} ,\overline{F}_{3/2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{2} + W^{*} ,\overline{F}_{1} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{5/2} ,\overline{F}_{1/2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{3} ,\overline{F}_{0} } \right) + {\mathcal{Q}}_{3} \hfill \\ {\mathcal{L}}_{21} \left( {\overline{F}_{3} } \right) + 2\overline{\tau } {\mathcal{L}}_{22} \left( {\overline{W}_{3} } \right) + \frac{{\partial^{2} \overline{W}_{3} }}{{\partial X^{2} }} = - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{0} ,\overline{W}_{3} } \right) - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1/2} ,\overline{W}_{5/2} } \right) - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1} ,\overline{W}_{2} + 2W^{*} } \right) \hfill \\ - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{3/2} ,\overline{W}_{3/2} } \right) - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{0} ,\overline{W}_{3} } \right) - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{1/2} ,\overline{W}_{5/2} } \right) \hfill \\ - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{1} ,\overline{W}_{2} + 2W^{*} } \right) - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{3/2} ,\overline{W}_{3/2} } \right) + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{0} ,\overline{W}_{3} } \right) \hfill \\ + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{1/2} ,\overline{W}_{5/2} } \right) + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{1} ,\overline{W}_{2} + 2W^{*} } \right) + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{3/2} ,\overline{W}_{3/2} } \right) + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{0} ,\overline{W}_{3} } \right) \hfill \\ + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{1/2} ,\overline{W}_{5/2} } \right) + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{1} ,\overline{W}_{2} + 2W^{*} } \right) + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{3/2} ,\overline{W}_{3/2} } \right) \hfill \\ {\mathcal{L}}_{31} \left( {\overline{W}_{2} } \right) + {\mathcal{L}}_{32} \left( {\overline{\varPsi }_{{x_{3} }} } \right) + \vartheta_{5} \beta \frac{{\partial^{2} \overline{\varPsi }_{{y_{3} }} }}{\partial X\partial Y} = 0 \hfill \\ {\mathcal{L}}_{41} \left( {\overline{W}_{2} } \right) + {\mathcal{L}}_{42} \left( {\overline{\varPsi }_{{y_{3} }} } \right) + \vartheta_{5} \beta \frac{{\partial^{2} \overline{\varPsi }_{{x_{3} }} }}{\partial X\partial Y} = 0 \hfill \\ \end{array} \right.$$

$$O\left( { \epsilon^{7/2} } \right):\left\{ \begin{array}{l} 2\overline{\tau } {\mathcal{L}}_{12} \left( {\overline{W}_{3/2} } \right) + {\mathcal{L}}_{12} \left( {\overline{\varPsi }_{{x_{5/2} }} } \right) + {\mathcal{L}}_{13} \left( {\overline{\varPsi }_{{y_{5/2} }} } \right) - \frac{{\partial^{2} \overline{F}_{7/2} }}{{\partial X^{2} }} = \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{0} ,\overline{F}_{7/2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1/2} ,\overline{F}_{3} } \right) \hfill \\ + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1} ,\overline{F}_{5/2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{3/2} ,\overline{F}_{2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{2} + W^{*} ,\overline{F}_{3/2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{5/2} ,\overline{F}_{1} } \right) \hfill \\ + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{3} ,\overline{F}_{1/2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{7/2} ,\overline{F}_{0} } \right) \hfill \\ {\mathcal{L}}_{21} \left( {\overline{F}_{7/2} } \right) + 2\overline{\tau } {\mathcal{L}}_{22} \left( {\overline{W}_{7/2} } \right) + \frac{{\partial^{2} \overline{W}_{7/2} }}{{\partial X^{2} }} = - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{0} ,\overline{W}_{7/2} } \right) - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1/2} ,\overline{W}_{3} } \right) - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1} ,\overline{W}_{5/2} } \right) \hfill \\ - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{3/2} ,\overline{W}_{2} + 2W^{*} } \right) - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{0} ,\overline{W}_{7/2} } \right) - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{1/2} ,\overline{W}_{3} } \right) \hfill \\ - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{1} ,\overline{W}_{5/2} } \right) - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{3/2} ,\overline{W}_{2} + 2W^{*} } \right) + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{0} ,\overline{W}_{7/2} } \right) \hfill \\ + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{1/2} ,\overline{W}_{3} } \right) + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{1} ,\overline{W}_{5/2} } \right) + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{3/2} ,\overline{W}_{2} + 2W^{*} } \right) + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{0} ,\overline{W}_{7/2} } \right) \hfill \\ + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{1/2} ,\overline{W}_{3} } \right) + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{1} ,\overline{W}_{5/2} } \right) + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{3/2} ,\overline{W}_{2} + 2W^{*} } \right) \hfill \\ {\mathcal{L}}_{31} \left( {\overline{W}_{5/2} } \right) + {\mathcal{L}}_{32} \left( {\overline{\varPsi }_{{x_{7/2} }} } \right) + \vartheta_{5} \beta \frac{{\partial^{2} \overline{\varPsi }_{{y_{7/2} }} }}{\partial X\partial Y} = 0 \hfill \\ {\mathcal{L}}_{41} \left( {\overline{W}_{5/2} } \right) + {\mathcal{L}}_{42} \left( {\overline{\varPsi }_{{y_{7/2} }} } \right) + \vartheta_{5} \beta \frac{{\partial^{2} \overline{\varPsi }_{{x_{7/2} }} }}{\partial X\partial Y} = 0 \hfill \\ \end{array} \right.$$

$$\begin{array}{l} O\left( { \epsilon^{4} } \right):\left\{ \begin{array}{l} 2\overline{\tau } {\mathcal{L}}_{12} \left( {\overline{W}_{2} } \right) + {\mathcal{L}}_{12} \left( {\overline{\varPsi }_{{x_{3} }} } \right) + {\mathcal{L}}_{13} \left( {\overline{\varPsi }_{{y_{3} }} } \right) - \frac{{\partial^{2} \overline{F}_{4} }}{{\partial X^{2} }} = \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{0} ,\overline{F}_{4} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1/2} ,\overline{F}_{7/2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1} ,\overline{F}_{3} } \right) \hfill \\ + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{3/2} ,\overline{F}_{5/2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{2} + W^{*} ,\overline{F}_{2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{5/2} ,\overline{F}_{3/2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{3} ,\overline{F}_{1} } \right) \hfill \\ + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{7/2} ,\overline{F}_{1/2} } \right) + \beta^{2} \widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{4} ,\overline{F}_{0} } \right) + {\mathcal{Q}}_{4} \hfill \\ {\mathcal{L}}_{21} \left( {\overline{F}_{4} } \right) + 2\overline{\tau } {\mathcal{L}}_{22} \left( {\overline{W}_{4} } \right) + \frac{{\partial^{2} \overline{W}_{4} }}{{\partial X^{2} }} = - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{0} ,\overline{W}_{4} } \right) - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1/2} ,\overline{W}_{7/2} } \right) - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{1} ,\overline{W}_{3} } \right) \hfill \\ - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{3/2} ,\overline{W}_{5/2} } \right) - \frac{{\beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{1} \left( {\overline{W}_{2} ,\overline{W}_{2} + 2W^{*} } \right) - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{0} ,\overline{W}_{4} } \right) \hfill \\ - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{1/2} ,\overline{W}_{7/2} } \right) - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{1} ,\overline{W}_{3} } \right) - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{3/2} ,\overline{W}_{5/2} } \right) \hfill \\ - \overline{\tau } \vartheta_{1} \beta^{2} \widetilde{{\mathcal{L}}}_{2} \left( {\overline{W}_{2} ,\overline{W}_{2} + 2W^{*} } \right) + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{0} ,\overline{W}_{4} } \right) + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{1/2} ,\overline{W}_{7/2} } \right) + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{1} ,\overline{W}_{3} } \right) \hfill \\ + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{3/2} ,\overline{W}_{5/2} } \right) + \overline{\tau } \vartheta_{2} \widetilde{{\mathcal{L}}}_{3} \left( {\overline{W}_{2} ,\overline{W}_{2} + 2W^{*} } \right) + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{0} ,\overline{W}_{4} } \right) \hfill \\ + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{1/2} ,\overline{W}_{7/2} } \right) + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{1} ,\overline{W}_{3} } \right) + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{3/2} ,\overline{W}_{5/2} } \right) \hfill \\ + \frac{{\overline{\tau } \vartheta_{3} \beta^{2} }}{2}\widetilde{{\mathcal{L}}}_{4} \left( {\overline{W}_{2} ,\overline{W}_{2} + 2W^{*} } \right) \hfill \\ {\mathcal{L}}_{31} \left( {\overline{W}_{3} } \right) + {\mathcal{L}}_{32} \left( {\overline{\varPsi }_{{x_{4} }} } \right) + \vartheta_{5} \beta \frac{{\partial^{2} \overline{\varPsi }_{{y_{4} }} }}{\partial X\partial Y} = 0 \hfill \\ {\mathcal{L}}_{41} \left( {\overline{W}_{3} } \right) + {\mathcal{L}}_{42} \left( {\overline{\varPsi }_{{y_{4} }} } \right) + \vartheta_{5} \beta \frac{{\partial^{2} \overline{\varPsi }_{{x_{4} }} }}{\partial X\partial Y} = 0 \hfill \\ \end{array} \right. \hfill \\ \hfill \\ \end{array}$$

(51)

The obtained asymptotic solution corresponding to radial dominated loading case is expressed as

$$\begin{aligned} W &= {\mathcal{A}}_{00}^{(0)} + \epsilon^{3/2} \hfill \\ & \quad \left[ {{\mathcal{A}}_{00}^{(3/2)} - {\mathcal{A}}_{00}^{(3/2)} \left( {\sin \left( {\frac{\alpha X}{\sqrt \epsilon}} \right) + { \cos }\left( {\frac{\alpha X}{\sqrt \epsilon}} \right)} \right)e^{{ - \frac{\alpha X}{\sqrt \epsilon}}} - {\mathcal{A}}_{00}^{(3/2)} \left( {\sin \left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right) + { \cos }\left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right)} \right)e^{{ - \frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}}} } \right] \hfill \\ &\quad + \epsilon^{2} \left[ {{\mathcal{A}}_{00}^{(2)} + {\mathcal{A}}_{11}^{(2)} \sin \left( {mX} \right)\sin \left( {nY} \right) - {\mathcal{A}}_{00}^{(2)} \left( {\sin \left( {\frac{\alpha X}{\sqrt \epsilon}} \right) + { \cos }\left( {\frac{\alpha X}{\sqrt \epsilon}} \right)} \right)e^{{ - \frac{\alpha X}{\sqrt \epsilon}}} - {\mathcal{A}}_{00}^{(2)} \left( {\sin \left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right) + { \cos }\left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right)} \right)e^{{ - \frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}}} } \right] \hfill \\ &\quad + \epsilon^{4} \left[ {{\mathcal{A}}_{00}^{(4)} + {\mathcal{A}}_{11}^{(4)} \sin \left( {mX} \right)\sin \left( {nY} \right) + {\mathcal{A}}_{20}^{(4)} \cos \left( {2mX} \right) + {\mathcal{A}}_{02}^{(4)} { \cos }(2nY) + {\mathcal{A}}_{22}^{(4)} \cos \left( {2mX} \right)\cos \left( {2nY} \right)} \right] + O\left( { \epsilon^{5} } \right) \hfill \\ \end{aligned}$$

(52)

$$\begin{aligned} F &= - {\mathcal{B}}_{00}^{(0)} \left( {\beta^{2} X^{2} + \left( {1 + C_{1} } \right)\frac{{Y^{2} }}{2}} \right) \hfill \\ &\quad + \epsilon^{2} \left[ { - {\mathcal{B}}_{00}^{(2)} \left( {\beta^{2} X^{2} + \left( {1 + C_{1} } \right)\frac{{Y^{2} }}{2}} \right) + {\mathcal{B}}_{11}^{(2)} \sin \left( {mX} \right)\sin \left( {nY} \right)} \right] \hfill \\ &\quad + \epsilon^{5/2} \left[ { + {\mathcal{A}}_{00}^{(3/2)} \left( {b_{10}^{(3/2)} \sin \left( {\frac{\alpha X}{\sqrt \epsilon}} \right) + b_{01}^{(3/2)} { \cos }\left( {\frac{\alpha X}{\sqrt \epsilon}} \right)} \right)e^{{ - \frac{\alpha X}{\sqrt \epsilon}}} + {\mathcal{A}}_{00}^{(3/2)} \left( {b_{10}^{(5/2)} \sin \left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right) + b_{01}^{(5/2)} { \cos }\left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right)} \right)e^{{ - \frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}}} } \right] \hfill \\ &\quad + \epsilon^{3} \left[ {{\mathcal{A}}_{00}^{(2)} \left( {b_{10}^{(3)} \sin \left( {\frac{\alpha X}{\sqrt \epsilon}} \right) + b_{01}^{(3)} { \cos }\left( {\frac{\alpha X}{\sqrt \epsilon}} \right)} \right)e^{{ - \frac{\alpha X}{\sqrt \epsilon}}} + {\mathcal{A}}_{00}^{(2)} \left( {b_{10}^{(3)} \sin \left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right) + b_{01}^{(3)} { \cos }\left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right)} \right)e^{{ - \frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}}} } \right] \hfill \\ &\quad + \epsilon^{4} \left[ { - {\mathcal{B}}_{00}^{\left( 4 \right)} \left( {\beta^{2} X^{2} + \left( {1 + C_{1} } \right)\frac{{Y^{2} }}{2}} \right) + {\mathcal{B}}_{20}^{(4)} \cos \left( {2mX} \right) + {\mathcal{B}}_{02}^{(4)} \cos \left( {2mY} \right) + {\mathcal{B}}_{22}^{(4)} \cos \left( {2mX} \right)\cos \left( {2nY} \right)} \right] + O\left( { \epsilon^{5} } \right) \hfill \\ \end{aligned}$$

(53)

$$\begin{aligned} \varPsi_{x} &= \epsilon^{2} \left[ {\left( {c_{10}^{(2)} \sin \left( {\frac{\alpha X}{\sqrt \epsilon}} \right) + c_{01}^{(2)} { \cos }\left( {\frac{\alpha X}{\sqrt \epsilon}} \right)} \right)e^{{ - \frac{\alpha X}{\sqrt \epsilon}}} + \left( {c_{10}^{(2)} \sin \left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right) + c_{01}^{(2)} { \cos }\left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right)} \right)e^{{ - \frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}}} } \right] \hfill \\ &\quad + \epsilon^{3} \left[ {{\mathcal{C}}_{11}^{\left( 3 \right)} \cos \left( {mX} \right){ \sin }\left( {nY} \right)} \right] + O\left( { \epsilon^{5} } \right) \hfill \\ \end{aligned}$$

(54)

$$\varPsi_{y} = \epsilon^{3} \left[ {{\mathcal{D}}_{11}^{(3)} \sin \left( {mX} \right){ \cos }(nY)} \right] + + O\left( { \epsilon^{5} } \right)$$

(55)

The obtained asymptotic solution corresponding to axial dominated loading case is expressed as

$$\begin{aligned} W &= {\mathcal{A}}_{00}^{(0)} + \epsilon\left[ {{\mathcal{A}}_{00}^{(1)} - {\mathcal{A}}_{00}^{(1)} \left( {\sin \left( {\frac{\alpha X}{\sqrt \epsilon}} \right) + { \cos }\left( {\frac{\alpha X}{\sqrt \epsilon}} \right)} \right)e^{{ - \frac{\alpha X}{\sqrt \epsilon}}} - {\mathcal{A}}_{00}^{(1)} \left( {\sin \left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right) + { \cos }\left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right)} \right)e^{{ - \frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}}} } \right] \hfill \\ &\quad + \epsilon^{2} \left[ {{\mathcal{A}}_{00}^{(2)} + {\mathcal{A}}_{11}^{(2)} \sin \left( {mX} \right)\sin \left( {nY} \right) + {\mathcal{A}}_{02}^{(2)} \cos \left( {2nY} \right) - \left( {{\mathcal{A}}_{00}^{(2)} + {\mathcal{A}}_{02}^{(2)} \cos \left( {2nY} \right)} \right)\left( {\sin \left( {\frac{\alpha X}{\sqrt \epsilon}} \right) + { \cos }\left( {\frac{\alpha X}{\sqrt \epsilon}} \right)} \right)e^{{ - \frac{\alpha X}{{\sqrt {\ominus \epsilon} }}}} } \right. \hfill \\ &\left. {\quad - \left( {{\mathcal{A}}_{00}^{(2)} + {\mathcal{A}}_{02}^{(2)} \cos \left( {2nY} \right)} \right)\left( {\sin \left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right) + { \cos }\left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right)} \right)e^{{ - \frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}}} } \right] \hfill \\ &\quad + \epsilon^{4} \left[ {{\mathcal{A}}_{00}^{(4)} + {\mathcal{A}}_{11}^{(4)} \sin \left( {mX} \right)\sin \left( {nY} \right) + {\mathcal{A}}_{20}^{(4)} \cos \left( {2mX} \right) + {\mathcal{A}}_{02}^{(4)} { \cos }(2nY) + {\mathcal{A}}_{13}^{(4)} \sin \left( {mX} \right)\sin \left( {3nY} \right) } \right. \hfill \\ & \left. \quad+{{\mathcal{A}}_{22}^{(4)} \cos \left( {2mX} \right)\cos \left( {2nY} \right) + {\mathcal{A}}_{04}^{(4)} { \cos }(4nY)} \right] \hfill \\ &\quad+ O\left( { \epsilon^{5} } \right) \hfill \\ \end{aligned}$$

(56)

$$\begin{aligned} F &= - {\mathcal{B}}_{00}^{(0)} \left( {\frac{{C_{2} \beta^{2} X^{2} }}{{1 + C_{2} }} + \frac{{Y^{2} }}{2}} \right) + \epsilon^{2} \left[ - {\mathcal{B}}_{00}^{(2)} \left( {\frac{{C_{2} \beta^{2} X^{2} }}{{1 + C_{2} }} + \frac{{Y^{2} }}{2}} \right) + {\mathcal{B}}_{11}^{(2)} \sin \left( {mX} \right)\sin \left( {nY} \right) + {\mathcal{B}}_{02}^{(2)} \cos \left( {2nY} \right) + {\mathcal{A}}_{00}^{(1)} \left( {b_{10}^{(2)} \sin \left( {\frac{\alpha X}{\sqrt \epsilon}} \right) + b_{01}^{(2)} { \cos }\left( {\frac{\alpha X}{\sqrt \epsilon}} \right)} \right)e^{{ - \frac{\alpha X}{\sqrt \epsilon}}} + {\mathcal{A}}_{00}^{(1)} \left( {b_{10}^{(2)} \sin \left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right) + b_{01}^{(2)} { \cos }\left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right)} \right)e^{{ - \frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}}} \right] \hfill \\ &\quad+ \epsilon^{3} \left[ {\left( {{\mathcal{A}}_{00}^{(2)} + {\mathcal{A}}_{02}^{(2)} \cos \left( {2nY} \right)} \right)\left( {b_{10}^{(3)} \sin \left( {\frac{\alpha X}{\sqrt \epsilon}} \right) + b_{01}^{(3)} { \cos }\left( {\frac{\alpha X}{\sqrt \epsilon}} \right)} \right)e^{{ - \frac{\alpha X}{\sqrt \epsilon}}} + \left( {{\mathcal{A}}_{00}^{(2)} + {\mathcal{A}}_{02}^{(2)} \cos \left( {2nY} \right)} \right)\left( {b_{10}^{(3)} \sin \left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right) + b_{01}^{(3)} { \cos }\left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right)} \right)e^{{ - \frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}}} } \right] \hfill \\ &\quad+ \epsilon^{4} \left[ { - {\mathcal{B}}_{00}^{(4)} \left( {\frac{{C_{2} \beta^{2} X^{2} }}{{1 + C_{2} }} + \frac{{Y^{2} }}{2}} \right) + {\mathcal{B}}_{20}^{(4)} \cos \left( {2mX} \right) + {\mathcal{B}}_{13}^{(4)} \sin \left( {mX} \right)\sin \left( {3nY} \right) + {\mathcal{B}}_{22}^{(4)} \cos \left( {2mX} \right)\cos \left( {2nY} \right)} \right] + O\left( { \epsilon^{5} } \right) \hfill \\ \end{aligned}$$

(57)

$$\begin{aligned} \varPsi_{x} = \epsilon^{3/2} \left[ {{\mathcal{A}}_{00}^{(1)} c_{10}^{(3/2)} \sin \left( {\frac{\alpha X}{\sqrt \epsilon}} \right)e^{{ - \frac{\alpha X}{\sqrt \epsilon}}} + {\mathcal{A}}_{00}^{(1)} c_{10}^{(3/2)} \sin \left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right)e^{{ - \frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}}} } \right] \hfill \\ &\quad+ \epsilon^{5/2} \left[ {\left( {{\mathcal{A}}_{00}^{(2)} + {\mathcal{A}}_{02}^{(2)} \cos \left( {2nY} \right)} \right)c_{10}^{(5/2)} \sin \left( {\frac{\alpha X}{\sqrt \epsilon}} \right)e^{{ - \frac{\alpha X}{\sqrt \epsilon}}} + \left( {{\mathcal{A}}_{00}^{(2)} + {\mathcal{A}}_{02}^{(2)} \cos \left( {2nY} \right)} \right)c_{10}^{(5/2)} \sin \left( {\frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}} \right)e^{{ - \frac{{\alpha \left( {\pi - X} \right)}}{\sqrt \epsilon}}} } \right] \hfill \\ &\quad+ \epsilon^{3} \left[ {{\mathcal{C}}_{11}^{(3)} \cos \left( {mX} \right){ \sin }\left( {nY} \right)} \right] + O\left( { \epsilon^{5} } \right) \hfill \\ \end{aligned}$$

(58)

$$\varPsi_{y} = \epsilon^{3} \left[ {{\mathcal{D}}_{11}^{(3)} \sin \left( {mX} \right){ \cos }(nY) + {\mathcal{D}}_{02}^{\left( 3 \right)} { \sin }(2nY)} \right] + O\left( { \epsilon^{5} } \right)$$

(59)

in which

$$\alpha = \sqrt {\frac{a}{2}} ,\quad a = \sqrt {\frac{{1 - 2\overline{\tau } \vartheta_{1} }}{{\vartheta_{1} \left( {d_{11}^{*} - e_{11}^{*} } \right)}}}$$

In the radial dominated loading case, the periodicity condition yields

$${\mathcal{A}}_{00}^{(0)} = 0$$

(60)

$${\mathcal{A}}_{00}^{(3/2)} = \frac{2}{3}3^{1/4} {\mathcal{P}}_{q} \left( {1 + C_{1} } \right)\left( {\frac{{2\vartheta_{1} - \vartheta_{2} }}{{1 - 2\overline{\tau } \vartheta_{1} }}} \right) + \frac{{2\overline{\tau } \vartheta_{6} \left( {\vartheta_{1} - \vartheta_{2} } \right)}}{{1 - 2\overline{\tau } \vartheta_{1} }} \epsilon^{ - 1/2}$$

(61)

$${\mathcal{A}}_{00}^{(2)} = 0$$

(62)

$${\mathcal{A}}_{00}^{(4)} = \frac{{2\overline{\tau } \vartheta_{2} m^{2} + \beta^{2} n^{2} \left( {1 - 2\overline{\tau } \vartheta_{1} } \right)\left( {1 + 2\ell } \right)}}{{8\left( {1 - 2\overline{\tau } \vartheta_{1} } \right)}}\left( {{\mathcal{A}}_{11}^{\left( 2 \right)} } \right)^{2}$$

(63)

In the axial dominated loading case, the periodicity condition yields

$${\mathcal{A}}_{00}^{(0)} = 0$$

(64)

$${\mathcal{A}}_{00}^{(1)} = \frac{{2{\mathcal{P}}_{x} \left( {2C_{2} \vartheta_{1} - \left( {1 + C_{2} } \right)\vartheta_{2} } \right)}}{{\left( {1 - 2\overline{\tau } \vartheta_{1} } \right)}} + \frac{{2\overline{\tau } \vartheta_{6} \left( {\vartheta_{1} - \vartheta_{2} } \right)}}{{1 - 2\overline{\tau } \vartheta_{1} }} \epsilon^{ - 1/2}$$

(65)

$${\mathcal{A}}_{00}^{(2)} = 0$$

(66)

$${\mathcal{A}}_{00}^{(4)} = \frac{{2\overline{\tau } \vartheta_{2} m^{2} + \beta^{2} n^{2} \left( {1 - 2\overline{\tau } \vartheta_{1} } \right)\left( {1 + 2\ell } \right)}}{{8\left( {1 - 2\overline{\tau } \vartheta_{1} } \right)}}\left( {{\mathcal{A}}_{11}^{\left( 2 \right)} } \right)^{2} + \beta^{2} n^{2} \left( {{\mathcal{A}}_{02}^{\left( 2 \right)} } \right)^{2}$$

(67)

The parameters in Eqs. (40–50) can be obtained as follow

$${\mathcal{P}}_{q}^{(0)} = K_{0} K_{1} + \epsilon^{2} \beta^{2} K_{4}$$

(68)

$$\begin{aligned} {\mathcal{P}}_{q}^{(2)} &= 8K_{1} K_{9} K_{14} + \frac{{8K_{1} K_{9} \left( {K_{0} K_{1} K_{13} \beta^{2} \left( {1 + C_{1} } \right) + K_{0} K_{9} K_{12} } \right)}}{{K_{0} K_{1} \beta^{2} \left( {1 + C_{1} } \right) - K_{12} }} + \frac{{4K_{1} K_{9} \left( {K_{0} K_{13} + K_{0}^{2} K_{9} } \right)}}{{K_{0} K_{1} \beta^{2} \left( {1 + C_{1} } \right) - K_{12} }} \hfill \\ &\quad + 16K_{0} K_{9} K_{10} + \frac{{4K_{0} K_{1} K_{9} K_{16} }}{{K_{0} K_{11} + K_{15} }} \hfill \\ \end{aligned}$$

(69)

$$\begin{aligned} \delta_{q}^{(0)} & = \left[ {\frac{{3^{3/4} }}{2}\overline{\tau } \vartheta_{6} \left( {\vartheta_{1} - \vartheta_{2} } \right)} \right] \epsilon^{ - 1/2} + \left[ {\frac{{\vartheta_{1} }}{2} - \vartheta_{2} - \overline{\tau } \vartheta_{2} \left( {\frac{{2\vartheta_{1} - \vartheta_{2} }}{{1 - 2\overline{\tau } \vartheta_{1} }}} \right) + \left( {\frac{{2\vartheta_{1} \vartheta_{2} - \vartheta_{2}^{2} }}{{\pi \alpha \vartheta_{1} \left( {1 - 2\overline{\tau } \vartheta_{1} } \right)}}} \right) \epsilon^{1/2} } \right]{\mathcal{P}}_{q} \left( {1 + C_{1} } \right) \\ & \quad + \left[ {\left( {\frac{{3^{1/4} \alpha \left( {2\vartheta_{1} - \vartheta_{2} } \right)^{2} }}{{6\pi \left( {1 - 2\overline{\tau } \vartheta_{1} } \right)}}} \right) \epsilon} \right]{\mathcal{P}}_{q}^{2} \left( {1 + C_{1} } \right)^{2} \\ \end{aligned}$$

(70)

$$\delta_{q}^{(2)} = \left[ {\frac{{3^{3/4} }}{32}\left( {\left( {m^{2} \left( {1 - 2\overline{\tau } \vartheta_{1} } \right) + 2\overline{\tau } \vartheta_{2} \beta^{2} n^{2} } \right) - \frac{{4\overline{\tau }^{2} \vartheta_{2}^{2} m^{2} + 2\overline{\tau } \vartheta_{2} \beta^{2} n^{2} \left( {1 - 2\overline{\tau } \vartheta_{1} } \right)}}{{1 - 2\overline{\tau } \vartheta_{1} }}} \right)} \right] \epsilon^{ - 3/2}$$

(71)

where $K_{i} (i = 0, \ldots ,17)$ are the parameters in terms of $\vartheta_{1} ,\vartheta_{2} ,\vartheta_{3} , \vartheta_{4} , \vartheta_{5} ,m, n, \beta , \ell ,C_{1}$ obtained via the sets of perturbation equations.

$${\mathcal{S}}_{1} = - \left[ {\left( {\frac{{2\vartheta_{1} - \vartheta_{2} \left( {1 + C_{1} } \right)}}{{1 - 2\overline{\tau } \vartheta_{1} }}} \right)\left( {{\mathcal{P}}_{q}^{\left( 2 \right)} } \right)} \right]$$

(72)

$${\mathcal{S}}_{2} = - \left( {\frac{{2\vartheta_{1} - \vartheta_{2} \left( {1 + C_{1} } \right)}}{{1 - 2\overline{\tau } \vartheta_{1} }}} \right)\left( {{\mathcal{P}}_{q}^{\left( 0 \right)} } \right) - \frac{{2\overline{\tau } \vartheta_{6} \left( {\vartheta_{1} - \vartheta_{2} } \right)}}{{1 - 2\overline{\tau } \vartheta_{1} }}$$

(73)

$${\mathcal{P}}_{x}^{\left( 0 \right)} = \frac{1}{2}\left\{ {K_{0} \widehat{K}_{1} + \epsilon^{2} \left( {\frac{{K_{9} - K_{3} K_{5} K_{9} + K_{2} K_{5} K_{10} - K_{4} K_{10} + K_{3} K_{4} K_{11} - K_{2} K_{11} }}{{1 - K_{3} K_{5} }}} \right)} \right\} \epsilon^{ - 1}$$

(74)

$${\mathcal{P}}_{x}^{\left( 2 \right)} = \frac{1}{2}\left\{ {\left( {\frac{{2K_{0} K_{22} \left( {K_{0} + K_{1} } \right)\left( {K_{15} + K_{16} } \right)\widehat{K}_{1} + 4K_{0} K_{17} \widehat{K}_{1} }}{{\left( {K_{7} - K_{6} K_{8} } \right)\left( {K_{15} + K_{16} } \right)}}} \right) \epsilon^{ - 1} + \left( {2K_{14} K_{22} \widehat{K}_{1} + \frac{{K_{0} K_{12} K_{22}^{2} \widehat{K}_{1} + 4K_{0} K_{13} K_{22} \widehat{K}_{1}^{2} }}{{2\left( {K_{0} \widehat{K}_{1} - K_{12} } \right)}} + \frac{{K_{0}^{2} K_{22}^{2} \widehat{K}_{1} + 4K_{0} K_{13} K_{22} \widehat{K}_{1} }}{{2\left( {K_{0} \widehat{K}_{1} - K_{12} } \right)}} + \frac{{K_{0}^{2} K_{22} \widehat{K}_{1} + 4K_{0} K_{13} \widehat{K}_{1} }}{{2\left( {K_{0} \widehat{K}_{1} - K_{12} } \right)\left( {K_{15} + K_{16} } \right)}} + \frac{{K_{0} K_{21} \widehat{K}_{3} }}{{K_{0} K_{20} \left( {K_{15} + K_{16} } \right)\widehat{K}_{1} - 4\left( {K_{15} + K_{16} } \right)^{2} }}} \right) \epsilon} \right\}$$

(75)

$${\mathcal{P}}_{x}^{\left( 4 \right)} = \frac{1}{2}\left\{ {\left( {\frac{{4K_{0} K_{18} K_{22} n^{2} \left( {K_{0} + K_{1} } \right)\widehat{K}_{1} \left( {2\widehat{K}_{1} + \widehat{K}_{2} } \right) + 8K_{0}^{2} K_{19} n^{2} \widehat{K}_{1} \left( {\widehat{K}_{2} - 1} \right)}}{{\left( {K_{7} - K_{6} K_{8} } \right)\left( {K_{0} \widehat{K}_{1} - K_{18} } \right)}} + \left( {\frac{{4K_{0} K_{17} \widehat{K}_{1} \widehat{K}_{2} }}{{\left( {K_{7} - K_{6} K_{8} } \right)\left( {K_{15} + K_{16} } \right)}}} \right)\left( {\frac{{2K_{0} K_{22} \left( {K_{0} + K_{1} } \right)\left( {K_{15} + K_{16} } \right) + 4K_{0} K_{17} }}{{\left( {K_{7} - K_{6} K_{8} } \right)\left( {K_{15} + K_{16} } \right)}}} \right) - \frac{{8K_{0}^{3} K_{17} \widehat{K}_{1} \widehat{K}_{2} }}{{\left( {K_{7} - K_{6} K_{8} } \right)^{2} \left( {K_{15} + K_{16} } \right)^{2} }} + \frac{{4K_{0}^{3} K_{22} \left( {K_{0} + K_{1} } \right)\widehat{K}_{1} + 8K_{0}^{3} K_{19} \widehat{K}_{1} }}{{\left( {K_{7} - K_{6} K_{8} } \right)^{2} \left( {K_{0} \widehat{K}_{1} - K_{18} } \right)\left( {K_{15} + K_{16} } \right)}}} \right) \epsilon^{ - 1} } \right\}$$

(76)

$$\begin{aligned} \delta_{x}^{(0)} = \overline{\tau } \vartheta_{6} \left( {\vartheta_{1} - \vartheta_{2} } \right) + {\mathcal{P}}_{x} \left( {\vartheta_{1} \left( {1 + C_{2} } \right) - - 2\vartheta_{2} C_{2} + \frac{{2\overline{\tau } \vartheta_{2} \left( {2C_{2} \vartheta_{1} - \left( {1 + C_{2} } \right)\vartheta_{2} } \right)}}{{1 - 2\overline{\tau } \vartheta_{1} }}} \right) \hfill \\ + \left( {\frac{{\varGamma \left( {2\vartheta_{1} C_{2} - \vartheta_{2} \left( {1 + C_{2} } \right)} \right)^{2} {\mathcal{P}}_{x}^{2} }}{{\pi \left( {1 - 2\overline{\tau } \vartheta_{1} } \right)}} + \frac{{2\vartheta_{2} \left( {2\vartheta_{1} C_{2} - \vartheta_{2} (1 + C_{2} )} \right){\mathcal{P}}_{x} }}{{\pi \alpha \vartheta_{1} \left( {1 - 2\overline{\tau } \vartheta_{1} } \right)}} + \frac{{2\overline{\tau } \vartheta_{2} \left( {2\vartheta_{1} C_{2} - \vartheta_{2} (1 + C_{2} )} \right){\mathcal{P}}_{x} }}{{\pi \alpha \left( {1 - 2\overline{\tau } \vartheta_{1} } \right)}}} \right) \epsilon^{1/2} \hfill \\ \end{aligned}$$

(77)

$$\delta_{x}^{(2)} = \left[ {\frac{{\left( {1 - 2\overline{\tau } \vartheta_{1} } \right)m^{2} }}{16} + \frac{{\overline{\tau } \vartheta_{2} \beta^{2} n^{2} }}{8} - \frac{{2\overline{\tau }^{2} \vartheta_{2}^{2} m^{2} + \overline{\tau } \vartheta_{2} \beta^{2} n^{2} \left( {1 - 2\overline{\tau } \vartheta_{1} } \right)}}{{8\left( {1 - 2\overline{\tau } \vartheta_{1} } \right)}}} \right] \epsilon$$

(78)

$$\begin{aligned} \delta_{x}^{(4)} & = \left[ {\frac{{\varGamma K_{0}^{2} \left( {1 - 2\overline{\tau } \vartheta_{1} } \right)}}{{8\pi^{2} \left( {K_{7} - K_{6} K_{8} } \right)^{2} }}} \right] \epsilon^{ - 3/2} + \left[ {\frac{{\overline{\tau } \vartheta_{2} K_{0}^{2} K_{22} }}{{\left( {K_{7} - K_{6} K_{8} } \right)^{2} }}} \right] \epsilon^{ - 1} \\ & \quad + \left[ {\left( {\frac{{\left( {1 - 2\overline{\tau } \vartheta_{1} } \right)m^{2} + 2\overline{\tau } \vartheta_{2} \beta^{2} n^{2} }}{4}} \right)\left( {\frac{{K_{21} \widehat{K}_{3} }}{{K_{0} K_{20} \widehat{K}_{1} - 4\left( {K_{15} + K_{16} } \right)}}} \right)^{2} + \left( {\frac{{\left( {1 - 2\overline{\tau } \vartheta_{1} } \right)m^{2} }}{2}} \right)\left( {\frac{{K_{0} K_{22} + 4K_{13} }}{{4\left( {K_{0} \widehat{K}_{1} - K_{12} } \right)}}} \right)^{2} } \right] \epsilon^{3} \\ \end{aligned}$$

(79)

where $K_{i} (i = 0, \ldots ,22)$ are the parameters in terms of $\vartheta_{1} ,\vartheta_{2} ,\vartheta_{3} , \vartheta_{4} , \vartheta_{5} ,m, n, \beta ,\text{ }\ell ,C_{2}$ obtained via the sets of perturbation equations.

$${\mathcal{S}}_{3} = - \frac{{K_{0} }}{{K_{7} + K_{6} K_{8} }} \epsilon^{ - 1} - \left( {\frac{{2C_{2} \vartheta_{1} - \left( {1 + C_{2} } \right)\vartheta_{2} }}{{1 - 2\overline{\tau } \vartheta_{1} }}} \right)\left( {2{\mathcal{P}}_{x}^{\left( 2 \right)} } \right)$$

(80)

$${\mathcal{S}}_{4} = - \left( {\frac{{2C_{2} \vartheta_{1} - \left( {1 + C_{2} } \right)\vartheta_{2} }}{{1 - 2\overline{\tau } \vartheta_{1} }}} \right)\left( {2{\mathcal{P}}_{x}^{\left( 0 \right)} } \right) - \left( {\frac{{2\overline{\tau } \vartheta_{6} \left( {\vartheta_{1} - \vartheta_{2} } \right)}}{{1 - 2\overline{\tau } \vartheta_{1} }}} \right) \epsilon^{ - 1}$$

(81)