Reduction of excessive energy in the four-noded membrane quadrilateral element. Part II: Near incompressibility and J2 plasticity Original Research Article

The elemental formulation presented in Part I of this study [E. Paraskevopoulos, D. Talaslidis, Reduction of excessive energy in the four-noded membrane quadrilateral element. Part I: Linear theory-compressible materials, Comput. Methods Appl. Mech. Engrg. 194 (2005) 3771–3796] is extended in a straightforward manner to problems with nearly incompressible materials and to J2 plastic flow-problems. Sources of excessive energy resulting from the incompressibility constraint as well as the coupling between the deviatoric components of strain are examined and conclusions are drawn concerning the selection of appropriate approximations for the field variables. A modified version of the Hu-Washizu principle is employed that utilizes the orthogonality between deviatoric and volumetric terms and incorporates part of the plane strain conditions. Furthermore, satisfaction of the patch test is analytically verified. In deriving the weak form of the equations governing the J2 plasticity problem, attention is focused on a straightforward extension of the linear problem without reference to additional postulates. In case of J2 plasticity, the current formulation incorporates an important modification that leads to further simplifications: the continuous, linear functions employed for the approximations are replaced by the Heaviside function. Finally, results of numerical examples and comparisons with other formulations are presented.