Size-dependent nonlinear pull-in instability of the clamped cylindrical thin micro-/nanoshell based on the non-classical theories

In recent decades, non-classical continuum theories have been commonly utilized to model material discontinuities in micro-/nanoscales. In the present study, modified strain gradient theory (MSGT) together with the fully clamped cylindrical thin-shell model has been adopted to investigate the size-dependent nonlinear pull-in instability of thin cylindrical nano-/microshell. The modified strain gradient theory in cylindrical coordinates and virtual work principle has been applied to derive the constitutive size-dependent equations of motion. In this study, the linear strains have been utilized to establish a system of three-coupled partial differential equations. Further, effect of van der Waals force has been included in the nonlinear governing equations of the systems, and extended Kantorovich method has been used to solve the nonlinear differential equations that in turn lead to achieving the pull-in parameters of the microshell. From theoretical point of view, effect of van der Waals attraction, the size dependency, and the importance of coupling between them on the instability performance have been discussed and the pull-in parameters, i.e., critical mid-deflection and instability voltage have been determined. It was found that comparable pull-in parameter of MSGT to those obtained by the classic theory, and modified couple stress theory depends only on the length scale of the microshell.