A refined nonconforming quadrilateral element for couple stress/strain gradient elasticity

Abstract

AbstractC0−1 patch test (Int. J. Numer. Meth. Engng 2004; 61:433–454) proposed by Soh and Chen is a reliable method to ensure convergence of nonconforming finite element for the couple stress/strain gradient elasticity. The C0−1 patch test function is a complete quadratic polynomial that satisfies the equilibrium equations. To pass the C0−1 patch test, the element displacement functions used to calculate strains must satisfy C0 continuity (or weak C0 continuity) and quadratic completeness. In this paper, a 24‐DOF (degrees of freedom) quadrilateral element (CQ12+RDKQ) for the couple stress/strain gradient elasticity is developed by combining the refined thin plate element RDKQ and the nonconforming element CQ12. The element RDKQ, which satisfies weak C1 continuity, is used to calculate strain gradients, whereas strains are computed by the element CQ12, which is established based on an extended variational functional and satisfies weak C0 continuity and quadratic completeness. Numerical examples show that the element (CQ12+RDKQ) passes the C0−1 patch test and it is also more efficient than the existing available triangular and quadrilateral elements in stress concentration problems with size effects. Copyright © 2010 John Wiley & Sons, Ltd.