A new computationally efficient finite element formulation for nanoplates using second-order strain gradient Kirchhoff's plate theory

Abstract

The strain gradient nonlocal theory is important to include the size effects of nanostructures in classical continuum theory with the corresponding development of computationally efficient numerical tool such as finite elements for the analysis of such structures with different boundary conditions. However, there is no literature on the finite element formulation of second-order strain gradient elastic plates. The weak form of the governing equation of motion of the Kirchhoff nanoplate using second-order positive/negative strain gradient nonlocal theories requires C2 continuity of transverse displacement. In this paper, a new computationally efficient nonconforming finite element formulation for the modelling of nanoplates using second-order positive/negative strain gradient nonlocal theories is presented. The performance of the developed finite element is compared with conforming finite element for rectangular isotropic Kirchhoff nanoplates with different boundary conditions. Analytical solution for static bending, free vibration, and buckling under biaxial in-plane compressive loading are also obtained for rectangular all edges simply supported isotropic Kirchhoff nanoplate for the comparison purpose. The nonconforming element is found to be computationally more efficient than the conforming element with better accuracy and convergence rate. The negative strain gradient model predicts results matching with the experimental results available in the literature.