A size-dependent nonlinear third-order shear-deformable dynamic model for a microplate on an elastic medium

This paper develops a size-dependent nonlinear third-order shear-deformable model for the dynamic analysis of microplates. Taking into account in-plane and out-of-plane displacements and inertia as well as rotations (via using the third-order shear deformation theory) and the modified couple stress theory, the Lagrange equations are employed to derive the equations of motion. An assumed-mode technique is applied to the expressions for the elastic strain energy of the microplate, the elastic potential energy due to the translational springs, the kinetic energy of the microplate, the energy dissipation function due to damping, and the work of a harmonically varying external loading on the microplate; these expressions are then inserted in the Lagrange equations in order to obtain the discretised equations of motion as nonlinear coupled functions of generalized coordinates. The pseudo-arclength continuation technique and a direct time-integration are employed to solve these equations and to obtain the generalized coordinates, hence system responses, numerically. Apart from the nonlinear analysis, a linear analysis is conducted by means of an eigenvalue analysis. The motion behaviour of the system is analysed and the importance of employing the modified couple stress theory, rather than the classical continuum theory, is discussed.