Studying the effects of small scale and Casimir force on the non-linear pull-in instability and vibrations of FGM microswitches under electrostatic actuation

• Developing a non-classical non-linear beam model for FGM microswitches.
• Considering size effects based on the most general form of SGT.
• Considering the shear deformation effects based on the Timoshenko beam theory.
• Deriving the governing equations including the Casimir force effect using a variational approach.
• Solving the pull-in and free vibration problems of electrostatically actuated microswitches.
• Studying the effects of length scale parameter, Casimir force, types of boundary conditions and other parameters in the results.

The size-dependent non-linear pull-in instability and free vibration of electrostatically actuated microswitches with the consideration of Casimir force effect are studied using a numerical solution approach. To this end, a non-classical non-linear beam model is developed based on Mindlin׳s strain gradient elasticity and the Timoshenko beam theory. The geometric non-linearity is taken into account according to the von Kármán hypothesis. Also, the microswitches are assumed to be made of functionally graded materials (FGMs). To obtain the size-dependent governing equations and boundary conditions, the virtual work principle is applied. The presented equations can be simply reduced to those on the basis of modified versions of strain gradient and couple stress theories (MSGT and MCST) as well as the classical elasticity theory. For solving the problem, the generalized differential quadrature (GDQ) method and the pseudo arc-length continuation technique are employed. In the numerical results, the influences of different parameters such as length scale parameter, Casimir force, material gradient index and geometrical properties on the pull-in instability and free vibration of actuated microswitches are examined.