Static and Dynamic Pull-In Instability of Nano-Beams Resting on Elastic Foundation Based on the Nonlocal Elasticity Theory

Abstract

This paper provides the static and dynamic pull-in behavior of nano-beams resting on the elastic foundation based on the nonlocal theory which is able to capture the size effects for structures in micron and sub-micron scales. For this purpose, the governing equation of motion and the boundary conditions are driven using a variational approach. This formulation includes the influences of fringing field and intermolecular forces such as Casimir and van der Waals forces. The differential quadrature (DQ) method is employed as a high-order approximation to discretize the governing nonlinear differential equation, yielding more accurate results with a considerably smaller number of grid points. In addition, a powerful analytical method called parameter expansion method (PEM) is utilized to compute the dynamic solution and frequency-amplitude relationship. It is illustrated that the first two terms in series expansions are sufficient to produce an acceptable solution of the mentioned structure. Finally, the effects of basic parameters on static and dynamic pull-in instability and natural frequency are studied.

Appendix A

$$\beta_{i} = \frac{{\beta^{\prime}_{i} }}{{\left( {1 - \left( {e_{0} a} \right)^{ * 2} \int\limits_{0}^{1} {\phi \phi^{\prime\prime}d\xi } } \right)}},\,\,\,\,i = 0,1,2,3,4$$

For van der Waals intermolecular force:

$$\begin{aligned} \beta^{\prime}_{0} = - \left( {(V_{0}^{ * } )^{2} \left( {1 + \hat{f}^{ * } } \right) + \varLambda^{ * } } \right)\int\limits_{0}^{1} {\phi d\xi } \hfill \\ \beta^{\prime}_{1} = \lambda^{4} - \left( {(V_{0}^{ * } )^{2} \left( {2 + \hat{f}^{ * } } \right) + 3\varLambda^{ * } } \right)\int\limits_{0}^{1} {\left( {\phi^{2} - \left( {e_{0} a} \right)^{ * 2} \phi \phi^{\prime\prime}} \right)d\xi } - f_{i} \int\limits_{0}^{1} {\left( {\phi \phi^{\prime\prime} - \left( {e_{0} a} \right)^{ * 2} \phi \phi^{\left( 4 \right)} } \right)d\xi } \hfill \\ \beta^{\prime}_{2} = - \left( {(V_{0}^{ * } )^{2} \left( {3 + \hat{f}^{ * } } \right) + 6\varLambda^{ * } } \right)\int\limits_{0}^{1} {\left( {\phi^{3} - \left( {e_{0} a} \right)^{ * 2} \phi \left( {\phi^{2} } \right)^{\prime \prime } } \right)d\xi } \hfill \\ \beta^{\prime}_{3} = - \left( {(V_{0}^{ * } )^{2} \left( {4 + \hat{f}^{ * } } \right) + 10\varLambda^{ * } } \right)\int\limits_{0}^{1} {\left( {\phi^{4} - \left( {e_{0} a} \right)^{ * 2} \phi \left( {\phi^{3} } \right)^{\prime \prime } } \right)d\xi } \hfill \\ - \alpha \left[ {\left( {\int\limits_{0}^{1} {\left( {\phi \phi^{\prime\prime}\left[ {\int\limits_{0}^{1} {\phi^{\prime 2} d\xi } } \right] - \left( {e_{0} a} \right)^{ * 2} \phi \left( {\phi^{\prime\prime}\left[ {\int\limits_{0}^{1} {\phi^{\prime 2} d\xi } } \right]} \right)^{\prime \prime } } \right)d\xi } } \right)} \right] \hfill \\ \beta^{\prime}_{4} = - \left( {(V_{0}^{ * } )^{2} \left( {5 + \hat{f}^{ * } } \right) + 15\varLambda^{ * } } \right)\int\limits_{0}^{1} {\left( {\phi^{5} - \left( {e_{0} a} \right)^{ * 2} \phi \left( {\phi^{4} } \right)^{\prime \prime } } \right)d\xi } \hfill \\ \end{aligned}$$

(A-1)

For Casimir intermolecular force:

$$\begin{aligned} \beta^{\prime}_{0} = - \left( {(V_{0}^{ * } )^{2} \left( {1 + \hat{f}^{ * } } \right) + (\hbar c)^{ * } } \right)\int\limits_{0}^{1} {\phi d\xi } \hfill \\ \beta^{\prime}_{1} = \lambda^{4} - \left( {(V_{0}^{ * } )^{2} \left( {2 + \hat{f}^{ * } } \right) + 4(\hbar c)^{ * } } \right)\int\limits_{0}^{1} {\left( {\phi^{2} - \left( {e_{0} a} \right)^{*2} \phi \phi^{\prime\prime}} \right)d\xi } - f_{i} \int\limits_{0}^{1} {\left( {\phi \phi^{\prime\prime} - \left( {e_{0} a} \right)^{*2} \phi \phi^{\left( 4 \right)} } \right)d\xi } \hfill \\ \beta^{\prime}_{2} = - \left( {(V_{0}^{ * } )^{2} \left( {3 + \hat{f}^{ * } } \right) + 10(\hbar c)^{ * } } \right)\int\limits_{0}^{1} {\left( {\phi^{3} - \left( {e_{0} a} \right)^{ * 2} \phi \left( {\phi^{2} } \right)^{\prime \prime } } \right)d\xi } \hfill \\ \beta^{\prime}_{3} = - \left( {(V_{0}^{ * } )^{2} \left( {4 + \hat{f}^{ * } } \right) + 20(\hbar c)^{ * } } \right)\int\limits_{0}^{1} {\left( {\phi^{4} - \left( {e_{0} a} \right)^{ * 2} \phi \left( {\phi^{3} } \right)^{\prime \prime } } \right)d\xi } \hfill \\ - \alpha \left[ {\left( {\int\limits_{0}^{1} {\left( {\phi \phi^{\prime\prime}\left[ {\int\limits_{0}^{1} {\phi^{\prime 2} d\xi } } \right] - \left( {e_{0} a} \right)^{ * 2} \phi \left( {\phi^{\prime\prime}\left[ {\int\limits_{0}^{1} {\phi^{\prime 2} d\xi } } \right]} \right)^{\prime \prime } } \right)d\xi } } \right)} \right] \hfill \\ \beta^{\prime}_{4} = - \left( {(V_{0}^{ * } )^{2} \left( {5 + \hat{f}^{ * } } \right) + 35(\hbar c)^{ * } } \right)\int\limits_{0}^{1} {\left( {\phi^{5} - \left( {e_{0} a} \right)^{ * 2} \phi \left( {\phi^{4} } \right)^{\prime \prime } } \right)d\xi } \hfill \\ \end{aligned}$$

(A-2)