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SUMMARYC 0−1 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 C 0−1 patch test function is a complete quadratic polynomial that satisfies the equilibrium equations. To pass the C 0−1 patch test, the element displacement functions used to calculate strains must satisfy C 0 continuity (or weak C 0 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 C 1 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 C 0 continuity and quadratic completeness. Numerical examples show that the element (CQ12+RDKQ) passes the C 0−1 patch test and it is also more efficient than the existing available triangular and quadrilateral elements in stress concentration problems with size effects.

Two refined quadrilateral flat shell elements named RSQ20 and RSQ24 are constructed in this paper based on the refined non‐conforming element method, and the elements can satisfy the displacement compatibility requirement at the interelement of the non‐planar elements by introducing the common displacements suggested by Chen and Cheung. A refined quadrilateral plate element RPQ4 and a plane quadrilateral isoparametric element are combined to obtain the refined quadrilateral flat shell element RSQ20, and a refined quadrilateral flat shell element RSQ24 is constructed on the basis of a RPQ4 element and a quadrilateral isoparametric element with drilling degrees of freedom. The numerical examples show that the present method can improve the accuracy of shell analysis and that the two new refined quadrilateral flat shell elements are efficient and accurate in the linear analysis of some shell structures. Copyright © 2000 John Wiley & Sons, Ltd.

Based on the non-smooth nonlinear equations method for modeling three-dimensional elastic frictional contact problems (hereafter called NNEM), the extension to elastoplastic case in which the material nonlinearity is also involved is presented in this paper. Two approaches which combine two methods for solving elastoplastic problem with NNEM are proposed. A Numerical example is given to demonstrate the validation of the approaches.