Nonlocal Strain Gradient Pull-in Study of Nanobeams Considering Various Boundary Conditions

Abstract

The main objective of this investigation is to study the size-dependent dynamic pull-in instability of nanobeams based on the nonlocal strain gradient theory (NLSGT) and Euler–Bernoulli beam model. To this end, the partial differential equation is obtained based on the NLSGT considering the electrostatic, fringing field, and intermolecular nonlinear forces. Then, the Galerkin method and the homotopy analysis method (HAM) were employed to solve the nonlinear governing equation. To validate the proposed results, the non-dimensional natural frequency and pull-in voltage are compared with the previously published results. Likewise, the analytical results of the HAM are compared with those obtained based on the Runge–Kutta numerical method. Besides, the impacts of the NLSGT, strain gradient theory, nonlocal theory, and classical theory on the dynamic behavior of nanobeams are investigated in the same situation. The pull-in voltage is also presented and the effects of electrostatic forces, fringing field, and initial gap are discussed in detail for different boundary conditions.

Appendices

Appendix 1

The first five consecutive Taylor’s series of parameters of the fringing field, electrostatic force, and van der Waals:

$$b_{0} = \alpha \beta + \beta + \theta ,$$

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$$b_{1} = \alpha \beta + 2\beta + 3$$

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$$b_{2} = \alpha \beta + 3\beta + 6\theta ,$$

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$$b_{3} = \alpha \beta + 4\beta + 10\theta ,$$

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$$b_{4} = \alpha \beta + 5\beta + 15\theta ,$$

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Appendix 2

Parameters of Eq. (14) for C–C boundary supports:

$$d_{0} \cong - 0.8309\theta - 0.8309\beta - 0.8309\alpha \beta ,$$

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$$\begin{aligned} d_{1} &\cong - 12.3025\mu^{2} \alpha \beta - \alpha \beta - 24.605\beta \mu^{2} - 36.9075\theta \mu^{2} \\ &\quad+ 500.551 - 2\beta - 3\theta + 6157.9816\eta^{2} , \end{aligned}$$

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$$\begin{aligned} d_{2} &\cong - 19.1404\mu^{2} \alpha \beta - 3.9882\beta - 57.4211\beta \mu^{2} \\ &\quad- 114.8423\theta \mu^{2} + - 3.9882\beta - 7.9763\theta , \end{aligned}$$

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$$\begin{aligned} d_{3} &\cong - 112.8519\mu^{2} \beta - 282.1296\theta \mu^{2} - 1.8519\alpha \beta \\ &\quad- 28.213\mu^{2} \alpha \beta - 7.4077\beta - 18.5191\theta , \end{aligned}$$

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$$\begin{aligned} d_{4} &\cong - 41.5719\mu^{2} \alpha \beta - 207.8593\mu^{2} \beta - 623.5779\mu^{2} \theta \\ &\quad- 2.6511\alpha \beta - 13.2556\beta - 39.7668\theta , \end{aligned}$$

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$$M \cong 1 + 12.3025\mu^{2} + 12.3025 \chi + 500.5510 \mu^{2} \chi .$$

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Parameters of Eq. (14) for C–S boundary supports:

$$d_{0} \cong - 0.86\theta - 0.86\beta - 0.86\alpha \beta ,$$

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$$\begin{aligned} d_{1} &\cong - 11.5132\mu^{2} \alpha \beta - \alpha \beta - 23.0264\mu^{2} - 34.5396\theta \mu^{2} \\ &\quad+ 237.8415 - 2.0002\beta - 3\theta + 2738.0356\eta^{2} , \end{aligned}$$

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$$\begin{aligned} d_{2} &\cong - 15.6908\mu^{2} \alpha \beta - 1.2718\alpha \beta - 47.0724\beta \mu^{2} \\ &\quad- 94.1448\theta \mu^{2} - 3.8153\beta - 7.6307\theta , \end{aligned}$$

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$$\begin{aligned} d_{3} &\cong - 85.5997\mu^{2} \beta - 213.9993\theta \mu^{2} - 1.6895\alpha \beta \\ &\quad- 21.3999\mu^{2} \alpha \beta - 6.7579\beta - 16.8948\theta , \end{aligned}$$

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$$\begin{aligned} d_{4} &\cong - 29.5791\mu^{2} \alpha \beta - 147.8955\mu^{2} \beta - 443.6866\mu^{2} \theta \\ &\quad- 2.3033\alpha \beta - 11.5167\beta - 34.5501\theta , \end{aligned}$$

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$$M \cong 1 + 11.5132\mu^{2} + 11.5132 \chi + 237.8415 \mu^{2} \chi .$$

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Parameters of Eq. (14) for S–S boundary supports:

$$d_{0} \cong - 0.6366\theta - 0.6366\beta - 0.6366\alpha \beta ,$$

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$$\begin{aligned} d_{1} &\cong - 4.9348\mu^{2} \alpha \beta - 0.5\alpha \beta - 9.8696\beta \mu^{2} - 14.8044\theta \mu^{2} \\ &\quad+ 48.7046 - \beta - 1.5\theta + 480.6946\eta^{2} , \end{aligned}$$

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$$\begin{aligned} d_{2} &\cong - 4.1888\mu^{2} \alpha \beta - 0.4244\alpha \beta - 12.5664\beta \mu^{2} \\ &\quad- 25.1327\theta \mu^{2} - 1.2732\beta - 2.5465\theta , \end{aligned}$$

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$$\begin{aligned} d_{3} &\cong - 14.8044\mu^{2} \beta - 37.011\theta \mu^{2} - 0.375\alpha \beta \\ &\quad- 3.7011\mu^{2} \alpha \beta - 1.5\beta - 3.75\theta , \end{aligned}$$

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$$\begin{aligned} d_{4} &\cong - 3.351\mu^{2} \alpha \beta - 16.7552\mu^{2} \beta - 50.2655\mu^{2} \theta \\ &\quad- 0.3395\alpha \beta - 1.6977\beta - 5.093\theta , \end{aligned}$$

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$$M \cong 0.5 + 4.9348\mu^{2} + 4.9348 \chi + 48.7046 \mu^{2} \chi .$$

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