Post-critical nonlinear vibration of nonlocal strain gradient beam involving surface energy effects

Buckled beams are gaining interest as promising options for nano- or micro-electromechanical systems such as mechanical sensors, actuators, energy harvesting devices, specifically in buckling-induced smart applications. Mechanical behaviour of nanostructures is significantly influenced by long-range molecular interactions. Gradient-based higher-order continuum theories are often used to effectively predict such interactions. Owing to high surface-to-bulk ratio, stress on the material surface causes an unconventional elastic response in nanostructures. Considering the long-range interactions, surface effects, and geometric nonlinearity resulting from slenderness may provide an in-depth understanding about physical characteristics of the nanostructures. Postcritical nonlinear vibration of nano beam, however, is not explored thoroughly. This study investigates the postcritical dynamic behaviour of magneto-electro-elastic composite nano beam undergoing large-amplitude vibration. Such beam, supported on Pasternak-type substrate, is modelled using higher-order shear deformation theory together with von Ka<acute accent>rma<acute accent>n nonlinearity. Employing variational principles, governing equations for laminated magneto-electro-elastic beams are obtained. The resulting set of nonlinear partial differential equations are solved with the aid of two-step perturbation technique. Closed-form solution characterising the linear and nonlinear frequency of buckled nano beam is obtained. The effects of essential parameters, such as nonlocal and strain-gradient length-scale parameters, substrate stiffnesses, surface stress effects, and the electric and magnetic fields, are clarified.