Static stability analysis of nanoscale piezoelectric shells with flexoelectric effect based on couple stress theory

Abstract

In this article, a non-classical shell model incorporating flexoelectric effect is presented to investigate the buckling behavior of piezoelectric cylindrical nanoshell based on the couple stress theory. This non-classical shell model contains both material length scale parameter and flexoelectric parameter and can capture the size effect and piezoelectricity of nanoscale shells. The closed-form solution of the critical buckling load is achieved by energy variational approach. Results show a significant dependence of the buckling modes of the nanoshell on the flexoelectric constant as well as the material length scale parameter. It is also found that the buckling load of the nanoshell is enhanced with the consideration of the flexoelectric effect and the couple stress theory due to their enhancement on the stiffness of the nanoshell. Although the effect of flexoelectricity is more pronounced in short shells, the influence of the material length scale parameter on the buckling load is remarkable even in long shells. In addition, the effects of flexoelectricity and the material length scale parameter on the critical buckling load are more evident for thicker shells.

Appendices

Appendix 1

$$a_{1} = - \left( {\lambda + 2\mu } \right)h,\quad a_{2} = - \frac{\mu h}{{R^{2} }},\quad a_{3} = \frac{\eta h}{{R^{2} }},\quad a_{4} = \frac{\eta h}{{R^{4} }},\quad a_{5} = - \left( {\lambda + \mu } \right)\frac{h}{R},$$

$$a_{6} = - \frac{\eta h}{R},\quad a_{7} = - \frac{\eta h}{{R^{3} }},\quad a_{8} = - \frac{\lambda h}{R},\quad a_{9} = - \frac{2\eta h}{{R^{3} }},$$

$$b_{1} = - \left( {\mu + \lambda } \right)\frac{h}{R},\quad b_{2} = - \frac{\eta h}{{R^{3} }},\quad b_{3} = - \frac{\eta h}{R},\quad b_{4} = - \mu h,$$

$$b_{5} = - \left( {\lambda + 2\mu + \frac{4\eta }{{R^{2} }}} \right)\frac{h}{{R^{2} }},\quad b_{6} = \frac{\eta h}{{R^{2} }},\quad b_{7} = \eta h,\quad b_{8} = - \left( {\lambda + 2\mu } \right)\frac{h}{{R^{2} }},\quad b_{9} = \frac{6\eta h}{{R^{2} }},\quad b_{10} = \frac{4\eta h}{{R^{4} }},$$

$$c_{1} = \frac{\lambda h}{R},\quad c_{2} = \frac{2\eta h}{{R^{3} }},\quad c_{3} = \left( {\lambda + 2\mu } \right)\frac{h}{{R^{2} }},\quad c_{4} = - \frac{6\eta h}{{R^{2} }},$$

$$c_{5} = - \frac{4\eta h}{{R^{4} }},\quad c_{6} = \left( {\lambda + 2\mu } \right)\frac{h}{{R^{2} }},\quad c_{7} = - \frac{4\eta h}{{R^{2} }},\quad c_{8} = \left( {\lambda + 2\mu } \right)\frac{{h^{3} }}{12} + 4\eta h,$$

$$c_{9} = \frac{{\mu h^{3} }}{{3R^{2} }} + \frac{8\eta h}{{R^{2} }} + \frac{{\lambda h^{3} }}{{6R^{2} }},\quad c_{10} = \frac{4\eta h}{{R^{4} }} + \frac{{\left( {\lambda + 2\mu } \right)h^{3} }}{{12R^{4} }},\quad c_{11} = \frac{4fh}{\pi R},$$

$$d_{1} = \frac{ \in h}{2},\quad d_{2} = \frac{ \in h}{{2R^{2} }},\quad d_{3} = - \frac{{ \in \pi^{2} }}{2h},\quad d_{4} = \frac{4fh}{\pi R}.$$

Appendix 2

$$[S] = \left[ {\begin{array}{*{20}l} {s_{11} } &\quad {s_{12} } & \quad{s_{13} } &\quad {s_{14} } \\ {s_{21} } &\quad {s_{22} } &\quad {s_{23} } &\quad {s_{24} } \\ {s_{31} } &\quad {s_{32} } &\quad {s_{33} } &\quad {s_{34} } \\ {s_{41} } &\quad {s_{42} } &\quad {s_{43} } &\quad {s_{44} } \\ \end{array} } \right],$$

$$s_{11} = - a_{1} \frac{{m^{2} \pi^{2} }}{{L^{2} }} - a_{2} n^{2} + a_{3} \frac{{m^{2} n^{2} \pi^{2} }}{{L^{2} }} + a_{4} n^{4} ,$$

$$s_{12} = a_{5} \frac{mn\pi }{L} - a_{6} \frac{{m^{3} n\pi^{3} }}{{L^{3} }} - a_{7} \frac{{mn^{3} \pi }}{L},$$

$$s_{13} = a_{8} \frac{m\pi }{L} - a_{9} \frac{{mn^{2} \pi }}{L},$$

$$s_{21} = b_{1} \frac{mn\pi }{L} - b_{2} \frac{{mn^{3} \pi }}{L} - b_{3} \frac{{m^{3} n\pi^{3} }}{{L^{3} }},$$

$$s_{22} = - b_{4} \frac{{m^{2} \pi^{2} }}{{L^{2} }} - b_{5} n^{2} + b_{6} \frac{{m^{2} n^{2} \pi^{2} }}{{L^{2} }} + b_{7} \frac{{m^{4} \pi^{4} }}{{L^{4} }},$$

$$s_{23} = - b_{8} n + b_{9} \frac{{m^{2} n\pi^{2} }}{{L^{2} }} + b_{10} n^{3} ,$$

$$s_{31} = - c_{1} \frac{m\pi }{L} + c_{2} \frac{{mn^{2} \pi }}{L},$$

$$s_{32} = c_{3} n - c_{4} \frac{{m^{2} n\pi^{2} }}{{L^{2} }} - c_{5} n^{3} ,$$

$$s_{33} = \bar{s}_{33} - \frac{P}{2\pi R}\frac{{m^{2} \pi^{2} }}{{L^{2} }}{ + }\frac{{2\varPhi_{0} d_{31} }}{{a_{33} }}\frac{{m^{2} \pi^{2} }}{{L^{2} }} + \frac{{2\varPhi_{0} d_{31} }}{{a_{33} }}\frac{{n^{2} }}{{R^{2} }},$$

$$\bar{s}_{33} = c_{6} - c_{7} \frac{{m^{2} \pi^{2} }}{{L^{2} }} + c_{8} \frac{{m^{4} \pi^{4} }}{{L^{4} }} + c_{9} \frac{{m^{2} n^{2} \pi^{2} }}{{L^{2} }} + c_{10} n^{4} ,$$

$$s_{34} = - c_{11} \frac{{m^{2} \pi^{2} }}{{L^{2} }},$$

$$s_{43} = - d_{4} \frac{{m^{2} \pi^{2} }}{{L^{2} }},$$

$$s_{44} = - d_{1} \frac{{m^{2} \pi^{2} }}{{L^{2} }} - d_{2} n^{2} + d_{3} ,$$

$$s_{42} = 0,s_{41} = 0,s_{24} = 0,s_{14} = 0.$$