An improved quadrilateral finite element for nonlinear second-order strain gradient elastic Kirchhoff plates

Abstract

In this paper, a geometrically nonlinear size-dependent Elastic Kirchhoff nanoplate model is developed based on the second-order negative strain gradient nonlocal theory useful for capturing the size effects. Taking the mid-plane stretching into consideration as the source of the nonlinear behaviour, weak form of the governing partial differential equations of equilibrium and the relevant classical/non-classical boundary conditions are derived using the variational method. For finite element analysis, the weak form requires C1 continuity of the in-plane displacements and C2 continuity of the transverse displacement. In the present work, a new computationally efficient subparametric nonconforming 4-noded finite element of arbitrary quadrilateral shape is presented for the first time for the modelling of nanoplates using the second-order negative strain gradient theory. The performance of the developed finite element is investigated for static bending of rectangular nanoplates with all edges simply supported and all edges clamped boundary conditions. The proposed element is found to be accurate and depicts good convergence characteristics for rectangular and non-rectangular meshes. The strain gradient model with negative nonlocal coefficient predicts results matching with those from the lattice model available in the literature.