Analysis of strain gradient Reissner–Mindlin plates using a C0 four-node quadrilateral element

Abstract

The weak form of Reissner–Mindlin plate model in the context of the classical continuum theory requires C0-continuity of the lateral deflection as well as rotation fields. However, due to the presence of higher-order derivatives, the same plate model based on strain gradient elasticity demands C1-continuity of interpolation functions for a standard finite element (FE) formulation. On the other hand, it is known that C0 interpolation functions result in more stable and cost-effective elements. Accordingly, a four-node quadrilateral gradient-enhanced plate element using C0 interpolation of the field variables is introduced. The main idea is to start with an eight-node quadrilateral element on which any field variable and its spatial derivatives are independently interpolated via standard C0 shape functions. By introducing appropriate algebraic constraints, the degrees of freedom corresponding to the midside nodes of the eight-node element are then eliminated and a quadrilateral element with four nodes on its corners is constructed. Accuracy and performance of the proposed element in several examples and for a wide range of parameters is investigated. It is shown that the new element can successfully capture the size-dependent behaviour of plates at small scales, has a proper rank (contains no spurious zero-energy modes), passes the patch test for thin as well as thick plates in an arbitrary mesh, and is free of shear locking. Furthermore, the introduced element can reproduce the results of the Reissner–Mindlin plate model based on the classical continuum theory when the plate thickness is far greater than the material length scale parameter.