A novel hybrid 8-node plate element for capturing the nonlocal effect based on the Hellinger-Reissner variational principle

Abstract

The Hellinger-Reissner variational principle based hybrid finite element method (FEM) is developed and applied to study the nonlocal mechanics of plates and beams at a micro/nano-scale. For this purpose, a plane 8-node plate element termed as MHAS-24β with 24 independent internal force parameters is proposed to modelling the mechanical behaviors including static bending, free vibration and buckling. The Mindlin plate theory allows the use of generalized displacement to satisfy \({\text{C}}^{\text{0}}\) continuity requirements, making it applicable to plates with different thicknesses. To overcome shear locking, the assumed strain method (ASM) is adopted to modify original strains. The polynomials for internal forces are assumed to be related to the highest-order derivatives of variables, making them complete and capable of avoiding the zero-energy mode. The posterior error estimation indicates that the convergence order is not affected by the nonlocal parameter, plate thickness or element shape. The MHAS-24β element effectively captures the nonlocal effect and outperforms the displacement-type FEM and the low-order hybrid FEM described in previous literature.