Nonlinear Bending Analysis of First-Order Shear Deformable Microscale Plates Using a Strain Gradient Quadrilateral Element

Based on Mindlin's strain gradient elasticity and first-order shear deformation plate theory, a size-dependent quadrilateral plate element is developed in this paper to study the nonlinear static bending of microplates. In comparison with the classical first-order shear deformable quadrilateral plate element, the proposed element needs 15 additional nodal degrees-of-freedom (DOF) including derivatives of lateral deflection and rotations with respect to coordinates, which means a total of 20DOFs per node. Also, the developed strain gradient-based finite-element formulation is general so that it can be reduced to that on the basis of modified couple stress theory (MCST) and modified strain gradient theory (MSGT). In the numerical results, the nonlinear bending response of microplates for different boundary conditions, length-scale factors, and geometrical parameters is studied. It is revealed that by the developed nonclassical finite-element approach, the nonlinear behavior of microplates with the consideration of strain gradient effects can be accurately studied.