Nonlinear dynamical behaviour of geometrically imperfect microplates based on modified couple stress theory

The nonlinear dynamical behaviour of a geometrically imperfect microplate is examined based on the modified couple stress theory. The microplate is modelled by means of the von Kármán plate theory and Kirchhoff's hypotheses retaining all in-plane and out-of-plane displacements and inertia. An initial imperfection in the out-of-plane direction is taken into account and the equations of motion for the in-plane and out-of-plane motions are obtained by means of an energy method based on the Lagrange equations. This operation gives three sets of second-order nonlinear ordinary differential equations with coupled terms for two in-plane motions and one out-of-plane motion. These sets are transformed into double-dimensional sets of first-order nonlinear ordinary differential equations which are solved numerically through use of a continuation technique. Apart from the nonlinear analysis, an eigenvalue analysis is also conducted to obtain the linear natural frequencies of the system with different amplitudes of the geometric imperfection. The effect of the amplitude of the geometric imperfection and thickness of the microplate as well as the forcing frequency on the response of the system is highlighted. Finally, a comparison is made between the responses of the system based on the modified couple stress and classical continuum mechanics theories so as to highlight the importance of taking into account small-size effects.