Variational formulation and differential quadrature finite element for freely vibrating strain gradient Kirchhoff plates

In this paper, we apply the energy variational principle to arrive at the differential equation of motion and all appropriate boundary conditions for strain gradient Kirchhoff micro-plates. The resulting sixth-order boundary value problem of free vibration is solved by a thirty-six-DOF four-node differential quadrature plate finite element. The C²-continuity condition of the deflection is guaranteed by devising a sixth-order differential quadrature- based geometric mapping scheme that can transform the displacement parameters at Gauss-Lobatto quadrature points into those at element nodes. The total potential energy of a generic micro-plate element is firstly discretized in terms of nodal parameters. It is then minimized to obtain the formulation of element stiffness and mass matrices. For comparison reasons, a Hermite interpolation-based strain gradient finite element is provided. With the help of the symbolic computation system Maple, the explicit algebraic relationship between the stiffness (or mass) matrices of two types of elements is derived. Convergence and comparison studies are conducted to show the efficacy of our element in the free vibration analysis of macro/micro- plates. Finally, we apply the developed method to study the size-dependent vibration behavior of micro-plates with uniform or stepped thickness. Numerical examples reveal that strain gradient effects can change the vibration mode shapes, not the vibration frequencies alone.