Modeling the instability of electrostatic nano-bridges and nano-cantilevers using modified strain gradient theory

Experimental observations reveal that the physical response of nano-structures is size-dependent. Herein, modified strain gradient theory in conjunction with Euler–Bernoulli beam theory have been used for mathematical modeling of the size dependent instability of nanostructures. Effect of van der Waals intermolecular force has been included in the mathematical model. Two common beam-type systems including double-clamped nano-bridge and clamped-free nano-cantilever have been investigated. Three approaches including using an approximated differential transformation method (DTM), applying iterative numerical method and developing a simple lumped parameter model have been employed to solve the governing equations of the systems. The obtained results have been compared with those obtained via numerical method. Furthermore a lumped parameter model has been developed to simply explain the physical performance of the systems without mathematical complexity. The pull-in parameters of the nanostructures as basic design parameters have been calculated. Effect of the size dependency on the pull-in performance has been discussed for both nano-structures.