Surface stress effect on the nonlinear free vibrations of functionally graded composite nanoshells in the presence of modal interaction

Abstract

As one of the innovative materials, functionally graded (FG) composite materials have the capability to vary microstructure and design attributes from one side to other representing the desired material properties. The prime aim of this work is to analyze the surface stress effect on the nonlinear free vibration response of FG cylindrical nanoshells incorporating various modal interactions. To this end, the Gurtin–Murdoch theory of elasticity together with the von Karman geometrical nonlinearity is implemented to the classical shell theory to construct an efficient size-dependent shell model. In order to take the modal interactions between the main oscillation mode and various symmetric vibration modes, the lateral deflection of the FG nanoshell is expressed as combination of the simple main vibration mode and convergent symmetric modes. Thereafter, the solution of problem is considered as the summation of the homogenous and particular parts to put the Galerkin technique to use. Finally, the multiple time-scales method is employed to achieve analytical expression for the surface elastic-based frequency response of FG nanoshells. It is displayed that in the presence of modal interaction, by increasing the shell deflection, the value of the frequency ratio decreases while in the absence of modal interaction, it enhances.

Appendix

$$\begin{aligned} c_{1} \left(t \right) & = \eta \pi^{2} {\mathcal{W}}_{1,n}/\left[{\left({\xi^{2} \eta^{2} \pi^{4} + \eta^{2} n^{4}/\xi^{2}} \right)\bar{\varUpsilon}_{1} + \eta^{2} \pi^{2} n^{2} \left({\bar{\varUpsilon}_{2} - 2\bar{\varUpsilon}_{5}} \right)} \right] \\ c_{2} \left(t \right) & = {\mathcal{W}}_{1,0}/\left({\xi^{2} \eta \pi^{2} \bar{\varUpsilon}_{1}} \right) \\ c_{3} \left(t \right) & = {\mathcal{W}}_{3,0}/\left({9\xi^{2} \eta \pi^{2} \bar{\varUpsilon}_{1}} \right) \\ c_{4} \left(t \right) & = {\mathcal{W}}_{5,0}/\left({25\xi^{2} \eta \pi^{2} \bar{\varUpsilon}_{1}} \right) \\ c_{5} \left(t \right) & = n^{2} {\mathcal{W}}_{1,n}^{2}/\left({32\xi^{2} \pi^{2} \bar{\varUpsilon}_{1}} \right) \\ c_{6} \left(t \right) & = \xi^{2} \pi^{2} {\mathcal{W}}_{1,n} {\mathcal{W}}_{1,0}/\left({2n^{2} \bar{\varUpsilon}_{1}} \right) \\ c_{7} \left(t \right) & = n^{2} \eta^{2} \pi^{2} {\mathcal{W}}_{1,n} \left({{\mathcal{W}}_{1,0} - 9{\mathcal{W}}_{3,0}} \right)/\left[{2\left({16\xi^{2} \eta^{2} \pi^{4} + \eta^{2} n^{4}/\xi^{2}} \right)\bar{\varUpsilon}_{1} + 8\eta^{2} \pi^{2} n^{2} \left({\bar{\varUpsilon}_{2} - 2\bar{\varUpsilon}_{5}} \right)} \right] \\ c_{8} \left(t \right) & = n^{2} \eta^{2} \pi^{2} {\mathcal{W}}_{1,n} \left({9{\mathcal{W}}_{3,0} - 25{\mathcal{W}}_{5,0}} \right)/\left[{2\left({256\xi^{2} \eta^{2} \pi^{4} + \eta^{2} n^{4}/\xi^{2}} \right)\bar{\varUpsilon}_{1} + 32\eta^{2} \pi^{2} n^{2} \left({\bar{\varUpsilon}_{2} - 2\bar{\varUpsilon}_{5}} \right)} \right] \\ c_{9} \left(t \right) & = 25n^{2} \eta^{2} \pi^{2} {\mathcal{W}}_{1,n} {\mathcal{W}}_{5,0}/\left[{2\left({1296\xi^{2} \eta^{2} \pi^{4} + \eta^{2} n^{4}/\xi^{2}} \right)\bar{\varUpsilon}_{1} + 36\eta^{2} \pi^{2} n^{2} \left({\bar{\varUpsilon}_{2} - 2\bar{\varUpsilon}_{5}} \right)} \right] \\ c_{10} \left(t \right) & = - \,\xi^{2} \pi^{2} {\mathcal{W}}_{1,n}^{2}/\left({32n^{2} \bar{\varUpsilon}_{1}} \right). \\ \end{aligned}$$