Nonlinear Finite Element Analysis of Free and Forced Vibrations of Cellular Plates Having Triangularly Prismatic Metamaterial Cores: A Strain Gradient Plate Model Approach

Abstract

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 equitriangularly prismatic metamaterial cores. The proposed model based on the nonlinear finite element strain gradient elasticity is developed for the first time to provide a computationally efficient framework for the simulation of the underlying nonlinear dynamics of cellular plates with advanced microarchitectures. The corresponding governing equations follow Mindlin’s SG 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 \({C}^{1}\)-continuous 6-node finite element is adopted for the discretization of the governing variational formulation with respect to 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 references, 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.