Nonlinear dynamic analysis of piezoelectric-bonded FG-CNTR composite structures using an improved FSDT theory

Abstract

In the present work, a geometrically nonlinear finite shell element is first presented to predict nonlinear dynamic behavior of piezolaminated functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shell, to enrich the existing research results on FG-CNTRC structures. The governing equations are developed via an improved first-order shear deformation theory (FSDT), in which a parabolic distribution of the transverse shear strains across the shell thickness is assumed and a zero condition of the transverse shear stresses on the top and bottom surfaces is imposed. Using a micro-mechanical model on the foundation of the developed rule of mixture, the effective material properties of the FG-CNTRC structures, which are strengthened by single-walled carbon nanotubes (SWCNTs), are scrutinized. The effectiveness of the present method is demonstrated by validating the obtained results against those of other studies from literature considering shell structures. Furthermore, some novel numerical results, including the nonlinear transient deflection of smart FG-CNTRC spherical and cylindrical shells, will be presented and can be considered for future structure design.

Appendix A: Newmark’s algorithm to solve \({\mathbf{M}}\,\varvec{\ddot{\varGamma }} + \varvec{K}_{{}}\varvec{\varGamma}\, = \varvec{F}\)

Initial acceleration:

$$\varvec{\ddot{\varGamma }}_{0} = {\mathbf{M}}^{ - 1} \left[ {{\mathbf{F}}_{0} - {\mathbf{K}}\varvec{\varGamma}_{0} } \right].$$

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New state at $t + \Delta t.$

$${\mathbf{F}}_{t + \Delta t} - \left( {{\mathbf{M}}\varvec{\ddot{\varGamma }}_{t + \Delta t} + {\mathbf{K}}\varvec{\varGamma}_{t + \Delta t} } \right) = {\mathbf{0}}.$$

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Computation of ${\bar{\mathbf{K}}}:$

$${\bar{\mathbf{K}}} = {\mathbf{K}} + \frac{1}{{\beta \Delta t^{2} }}{\mathbf{M}} + \frac{\gamma }{\beta \Delta t}{\mathbf{C}}_{{}} .$$

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Computation of ${\mathbf{R}}_{t + \Delta t}:$

$${\mathbf{R}}_{t + \Delta t} = {\mathbf{F}}_{t + \Delta t} + {\mathbf{M}}\left( {\frac{1}{{\beta \Delta t^{2} }}\varvec{\varGamma}_{t} + \frac{1}{\beta \Delta t}\varvec{\ddot{\varGamma }}_{t} + \left( {\frac{1}{2\beta } - 1} \right)\varvec{\ddot{\varGamma }}_{t} } \right).$$

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Computation of $\varvec{\varGamma}_{t + \Delta t}:$

$${\bar{\mathbf{K}}}\,\varvec{\varGamma}_{t + \Delta t} = {\mathbf{R}}_{t + \Delta t} .$$

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Computation of $\varvec{\ddot{\varGamma }}_{t + \Delta t}:$

$$\varvec{\ddot{\Gamma }}_{{t + \Delta t}} = \left[ {\left( {\varvec{\Gamma }_{{t + \Delta t}} - \varvec{\Gamma }_{t} - \Delta t\varvec{\dot{\Gamma }}_{t} } \right)\frac{1}{{\Delta t^{2} }} - \left( {\frac{1}{2} - \beta } \right)\varvec{\ddot{\Gamma }}_{t} } \right]\frac{1}{\beta },$$

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Computation of $\dot{\varvec{\varGamma }}_{t + \Delta t} :$

$$\dot{\varvec{\varGamma }}_{t + \Delta t} = \dot{\varvec{\varGamma }}_{t} + \Delta t\left( {1 - \gamma } \right)\varvec{\ddot{\varGamma }}_{t} + \gamma \Delta t\varvec{\ddot{\varGamma }}_{t + \Delta t} .$$

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Note that the Newmark parameters $\beta$ and $\gamma$ are chosen as $\beta$ = 0.25 and $\gamma$ = 0.5.