Nonlinear finite element free and forced vibrations of cellular plates having lattice-type metamaterial cores : a strain gradient plate model approach

The main objective of this paper is to develop a theoretically and numerically reliable and efficient methodology based on combining a finite element method and a strain gradient shear deformation plate model accounting for the nonlinear free and forced vibrations of cellular plates having lattice-type metamaterial cores. The proposed model provides a modelling framework computationally more efficient than framework of the conventional three-dimensional elasticity for the simulation of the underlying nonlinear dynamics of cellular plates with microarchitectures. The corresponding governing equations follow Mindlin's strain gradient elasticity theory including the micro-inertia effect applied to the first-order shear deformation plate theory along with the nonlinear von Kármán kinematics. Standard and higher-order computational homogenization methods determine the classical and strain gradient material constants, respectively. A higher-order C¹-continuous 6-node finite element is adopted for the discretization in the spatial domain, and an arc-length continuation technique along with time-periodic discretization is implemented to solve the resulting nonlinear time-dependent problem. Through a set of comparative studies with 3D full-field models as reliable validation references for triangularly prismatic example cores, the accuracy and efficacy of the proposed dimension reduction methodology are demonstrated for a diverse range of problem parameters for analyzing the large-amplitude dynamic structural response. The results show that the proposed strain gradient plate model can accurately capture the microarchitecture effects of cellular plates with low computational costs, which signifies a relevant contribution particularly to the computational analysis of nonlinear dynamics.