Size-dependent dynamic instability of double-clamped nanobeams under dispersion forces in the presence of thermal stress effects

Abstract

The objective of present study is to investigate the small-scale effects on the dynamic instability of nanoswitches subjected to electrostatic and intermolecular forces in the presence of thermal stress effects. To this end, Eringen’s nonlocal elasticity theory is applied along with the Euler–Bernoulli beam model and the equilibrium equation is derived by considering thermal stress effects. The dynamic governing equation, which is extremely nonlinear due to the intermolecular and electrostatic attraction forces and also thermal effects, is solved numerically by reduced order method. The accuracy of the solution is examined by comparing the obtained results with the existing numerical and analytical models. Finally, a comprehensive study is carried out to determine the influence of nonlocal parameters on the dynamic pull-in instability characteristics of double clamped nanobeam in the presence of thermal and dispersion forces. It is found that the pull-in parameters are conspicuously changed in accordance with the variations of nonlocal parameter as well as temperature changes by considering molecular attraction.

Appendix

1.1 The governing equation of van der Waals attraction

$$\begin{aligned} & {\text{u}}^{\prime\prime}_{\text{n}} + \upomega_{\text{n}}^{2} {\text{u}}_{\text{n}} - 5(1 + \upmu ({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} ))\,\upomega_{1}^{2} {\text{u}}_{1}^{2} \int_{0}^{1} {{\text{q}}_{1}^{3} } {\text{dx}} + 10(1 + \upmu ({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} ))\upomega_{1}^{2} {\text{u}}_{1}^{3} \int_{0}^{1} {{\text{q}}_{1}^{4} } {\text{dx}} \hfill \\ & \quad - 10(1 + \upmu ({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} ))\upomega_{1}^{2} {\text{u}}_{1}^{4} \int_{0}^{1} {{\text{q}}_{1}^{5} } {\text{dx}} + 5(1 + \upmu ({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} ))\upomega_{1}^{2} {\text{u}}_{1}^{5} \int_{0}^{1} {{\text{q}}_{1}^{6} } {\text{dx}} - (1 + \upmu ({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} ))\upomega_{1}^{2} {\text{u}}_{1}^{6} \int_{0}^{1} {{\text{q}}_{1}^{7} } {\text{dx}} \hfill \\ & \quad - 5{\text{u}}^{\prime\prime}_{1} {\text{u}}_{1} \int_{0}^{1} {{\text{q}}_{1}^{3} } {\text{dx}} + 10{\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{2} \int_{0}^{1} {{\text{q}}_{1}^{4} } {\text{dx}} - 10{\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{3} \int_{0}^{1} {{\text{q}}_{1}^{5} } {\text{dx}} + 5{\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{4} \int_{0}^{1} {{\text{q}}_{1}^{6} } {\text{dx}} - {\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{5} \int_{0}^{1} {{\text{q}}_{1}^{7} } {\text{dx}} - \upmu {\text{u}}^{\prime\prime}_{1} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1} } {\text{dx}} \hfill \\ & \quad + 5\upmu {\text{u}}^{\prime\prime}_{1} {\text{u}}_{1} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{2} } {\text{dx}} - 10\upmu {\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{2} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{3} } {\text{dx}} + 10\upmu {\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{3} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{4} } {\text{dx}} - 5\upmu {\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{4} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{5} } {\text{dx}} + \upmu {\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{5} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{6} } {\text{dx}} \hfill \\ & \quad + \upmu {\text{V}}^{2} (6 + 2\frac{{0.65{\text{d}}}}{\text{b}}){\text{u}}_{1}^{2} \int_{0}^{1} {\dot{\text{q}}_{1}^{2} {\text{q}}_{1} } {\text{dx}} - \upmu {\text{V}}^{2} (6 + 4\frac{{0.65{\text{d}}}}{\text{b}}){\text{u}}_{1}^{3} \int_{0}^{1} {\dot{\text{q}}_{1}^{2} {\text{q}}_{1}^{2} } {\text{dx}} + \left[ {\upmu {\text{V}}^{2} (2 + \frac{{0.65{\text{d}}}}{\text{b}}) - {\text{N}}_{\text{t}}^{*} } \right]{\text{u}}_{1} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{2} } {\text{dx}} \hfill \\ & \quad + \left[ {\upmu {\text{V}}^{2} (4 + 3\frac{{0.65{\text{d}}}}{\text{b}}) - 5({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} )} \right]{\text{u}}_{1}^{2} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{2} } {\text{dx}} + \left[ {\upmu {\text{V}}^{2} (2 + 3\frac{{0.65{\text{d}}}}{\text{b}}) - 10({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} )} \right]{\text{u}}_{1}^{3} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{3} } {\text{dx}} \hfill \\ & \quad + 2\upmu {\text{V}}^{2} \frac{{0.65{\text{d}}}}{\text{b}}{\text{u}}_{1}^{4} \int_{0}^{1} {\dot{\text{q}}_{1}^{2} {\text{q}}_{1}^{3} } {\text{dx}} - \left[ {\upmu {\text{V}}^{2} \frac{{0.65{\text{d}}}}{\text{b}} - 10({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} )} \right]{\text{u}}_{1}^{4} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{4} } {\text{dx}} + 5({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} ){\text{u}}_{1}^{5} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{5} } {\text{dx}} \hfill \\ & \quad + ({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} ){\text{u}}_{1}^{6} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{6} } {\text{dx}} + 12\upmu \uplambda_{3} {\text{u}}_{1}^{2} \int_{0}^{1} {\dot{\text{q}}_{1}^{2} {\text{q}}_{1} } {\text{dx}} + 3\upmu \uplambda_{3} {\text{u}}_{1} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1} } {\text{dx}} - 3\upmu \uplambda_{3} {\text{u}}_{1}^{2} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{2} } {\text{dx}} \hfill \\ & \quad - {\text{V}}^{2} (1 + \frac{{0.65{\text{d}}}}{\text{b}})\int_{0}^{1} {{\text{q}}_{1} } {\text{dx}} + {\text{V}}^{2} (3 + 4\frac{{0.65{\text{d}}}}{\text{b}}){\text{u}}_{1} \int_{0}^{1} {{\text{q}}_{1}^{2} } {\text{dx}} - {\text{V}}^{2} (3 + 6\frac{{0.65{\text{d}}}}{\text{b}}){\text{u}}_{1}^{2} \int_{0}^{1} {{\text{q}}_{1}^{3} } {\text{dx}} \hfill \\ & \quad + {\text{V}}^{2} (1 + 4\frac{{0.65{\text{d}}}}{\text{b}}){\text{u}}_{1}^{3} \int_{0}^{1} {{\text{q}}_{1}^{4} } {\text{dx}} - {\text{V}}^{2} \frac{{0.65{\text{d}}}}{\text{b}}{\text{u}}_{1}^{4} \int_{0}^{1} {{\text{q}}_{1}^{5} } {\text{dx}} - \uplambda_{3} \int_{0}^{1} {{\text{q}}_{1} } {\text{dx}} + 2\uplambda_{3} {\text{u}}_{1} \int_{0}^{1} {{\text{q}}_{1}^{2} } {\text{dx}} - \uplambda_{3} {\text{u}}_{1}^{2} \int_{0}^{1} {{\text{q}}_{1}^{3} } {\text{dx}} = 0. \hfill \\ \end{aligned}$$

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1.2 The governing equation of Casimir attraction

$$\begin{aligned} & {\text{u}}_{\text{n}}^{\prime } + \upomega_{\text{n}}^{2} {\text{u}}_{\text{n}} - 6(1 + \upmu ({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} ))\,\upomega_{1}^{2} {\text{u}}_{1}^{2} \int_{0}^{1} {{\text{q}}_{1}^{3} } {\text{dx}} + 15(1 + \upmu ({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} ))\upomega_{1}^{2} {\text{u}}_{1}^{3} \int_{0}^{1} {{\text{q}}_{1}^{4} } {\text{dx}} \hfill \\ & \quad - 20(1 + \upmu ({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} ))\upomega_{1}^{2} {\text{u}}_{1}^{4} \int_{0}^{1} {{\text{q}}_{1}^{5} } {\text{dx}} + 15(1 + \upmu ({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} ))\upomega_{1}^{2} {\text{u}}_{1}^{5} \int_{0}^{1} {{\text{q}}_{1}^{6} } {\text{dx}} - 6(1 + \upmu ({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} ))\upomega_{1}^{2} {\text{u}}_{1}^{6} \int_{0}^{1} {{\text{q}}_{1}^{7} } {\text{dx}} \hfill \\ & \quad + (1 + \upmu ({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} ))\upomega_{1}^{2} {\text{u}}_{1}^{7} \int_{0}^{1} {{\text{q}}_{1}^{8} } {\text{dx}} - 6{\text{u}}_{1}^{\prime } {\text{u}}_{1} \int_{0}^{1} {{\text{q}}_{1}^{3} } {\text{dx}} + 15{\text{u}}_{1}^{\prime } {\text{u}}_{1}^{2} \int_{0}^{1} {{\text{q}}_{1}^{4} } {\text{dx}} - 20{\text{u}}_{1}^{\prime } {\text{u}}_{1}^{3} \int_{0}^{1} {{\text{q}}_{1}^{5} } {\text{dx}} + 15{\text{u}}_{1}^{\prime } {\text{u}}_{1}^{4} \int_{0}^{1} {{\text{q}}_{1}^{6} } {\text{dx}} \hfill \\ & \quad - 6{\text{u}}_{1}^{\prime } {\text{u}}_{1}^{5} \int_{0}^{1} {{\text{q}}_{1}^{7} } {\text{dx}} + {\text{u}}_{1}^{\prime } {\text{u}}_{1}^{6} \int_{0}^{1} {{\text{q}}_{1}^{8} } {\text{dx}} - \upmu {\text{u}}_{1}^{\prime } \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1} } {\text{dx}} + 6\upmu {\text{u}}_{1}^{\prime } {\text{u}}_{1}^{{}} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{2} } {\text{dx}} - 15\upmu {\text{u}}_{1}^{\prime } {\text{u}}_{1}^{2} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{3} } {\text{dx}} + 20\upmu {\text{u}}_{1}^{\prime } {\text{u}}_{1}^{3} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{4} } {\text{dx}} \hfill \\ & \quad - 15\upmu {\text{u}}_{1}^{\prime } {\text{u}}_{1}^{4} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{5} } {\text{dx}} + 6\upmu {\text{u}}_{1}^{\prime } {\text{u}}_{1}^{5} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{6} } {\text{dx}} - \upmu {\text{u}}_{1}^{\prime } {\text{u}}_{1}^{6} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{7} } {\text{dx}} + \upmu {\text{V}}^{2} (6 + 2\frac{{0.65{\text{d}}}}{\text{b}}){\text{u}}_{1}^{2} \int_{0}^{1} {\dot{\text{q}}_{1}^{2} {\text{q}}_{1}^{{}} } {\text{dx}} \hfill \\ & \quad - \upmu {\text{V}}^{2} (12 + 6\frac{{0.65{\text{d}}}}{\text{b}}){\text{u}}_{1}^{3} \int_{0}^{1} {\dot{\text{q}}_{1}^{2} {\text{q}}_{1}^{2} } {\text{dx}} + \upmu {\text{V}}^{2} (6 + 6\frac{{0.65{\text{d}}}}{\text{b}}){\text{u}}_{1}^{4} \int_{0}^{1} {\dot{\text{q}}_{1}^{2} {\text{q}}_{1}^{3} } {\text{dx}} \hfill \\ & \quad + \left[ {\upmu {\text{V}}^{2} (12 + \frac{{0.65{\text{d}}}}{\text{b}}) - ({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} )} \right]{\text{u}}_{1}^{{}} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{2} } {\text{dx}} - \left[ {\upmu {\text{V}}^{2} (36 + 4\frac{{0.65{\text{d}}}}{\text{b}}) - 6({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} )} \right]{\text{u}}_{1}^{2} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{2} } {\text{dx}} \hfill \\ & \quad + \left[ {\upmu {\text{V}}^{2} (6 + 6\frac{{0.65{\text{d}}}}{\text{b}}) - 15({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} )} \right]{\text{u}}_{1}^{3} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{3} } {\text{dx}} - \left[ {\upmu {\text{V}}^{2} (12 + 4\frac{{0.65{\text{d}}}}{\text{b}}) - 20({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} )} \right]{\text{u}}_{1}^{4} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{4} } {\text{dx}} \hfill \\ & \quad - 2\upmu {\text{V}}^{2} \frac{{0.65{\text{d}}}}{\text{b}}{\text{u}}_{1}^{5} \int_{0}^{1} {\dot{\text{q}}_{1}^{2} {\text{q}}_{1}^{4} } {\text{dx}} + (\upmu {\text{V}}^{2} \frac{{0.65{\text{d}}}}{\text{b}} - 15({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} )){\text{u}}_{1}^{5} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{5} } {\text{dx}} + 6({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} ){\text{u}}_{1}^{6} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{6} } {\text{dx}} \hfill \\ & \quad - 6({\text{N}}_{\text{t}}^{*} + {\text{N}}_{\text{e}}^{*} ){\text{u}}_{1}^{7} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{7} } {\text{dx}} + 20\upmu \uplambda_{4} {\text{u}}_{1}^{2} \int_{0}^{1} {\dot{\text{q}}_{1}^{2} {\text{q}}_{1}^{{}} } {\text{dx}} + 4\upmu \uplambda_{4} {\text{u}}_{1}^{{}} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{{}} } {\text{dx}} - 4\upmu \uplambda_{4} {\text{u}}_{1}^{2} \int_{0}^{1} {\ddot{\text{q}}_{1} {\text{q}}_{1}^{2} } {\text{dx}} \hfill \\ & \quad - {\text{V}}^{2} (1 + \frac{{0.65{\text{d}}}}{\text{b}})\int_{0}^{1} {{\text{q}}_{1}^{{}} } {\text{dx}} + {\text{V}}^{2} (4 + 5\frac{{0.65{\text{d}}}}{\text{b}}){\text{u}}_{1}^{{}} \int_{0}^{1} {{\text{q}}_{1}^{2} } {\text{dx}} - {\text{V}}^{2} (6 + 10\frac{{0.65{\text{d}}}}{\text{b}}){\text{u}}_{1}^{2} \int_{0}^{1} {{\text{q}}_{1}^{3} } {\text{dx}} \hfill \\ & \quad + {\text{V}}^{2} (4 + 10\frac{{0.65{\text{d}}}}{\text{b}}){\text{u}}_{1}^{3} \int_{0}^{1} {{\text{q}}_{1}^{4} } {\text{dx}} - {\text{V}}^{2} (1 + 5\frac{{0.65{\text{d}}}}{\text{b}}){\text{u}}_{1}^{4} \int_{0}^{1} {{\text{q}}_{1}^{5} } {\text{dx}} + {\text{V}}^{2} \frac{{0.65{\text{d}}}}{\text{b}}{\text{u}}_{1}^{5} \int_{0}^{1} {{\text{q}}_{1}^{6} } {\text{dx}} \hfill \\ & \quad - \uplambda_{4} \int_{0}^{1} {{\text{q}}_{1}^{{}} } {\text{dx}} + 2\uplambda_{4} {\text{u}}_{1}^{{}} \int_{0}^{1} {{\text{q}}_{1}^{2} } {\text{dx}} - \uplambda_{4} {\text{u}}_{1}^{2} \int_{0}^{1} {{\text{q}}_{1}^{3} } {\text{dx}} = 0. \hfill \\ \end{aligned}$$

(27)