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.
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
J
2 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
J
2 plasticity problem, attention is focused on a straightforward extension of the linear problem without reference to additional postulates. In case of
J
2 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.
The objective of the paper is to set forth, in a consistent manner, reasons for the appearance of excessive energy in the four-noded membrane quadrilateral element and to propose a formulation ...leading to simple and reliable elements that are less sensitive to distortions of the geometrical shape. By presenting the differential geometry, emphasis is placed upon those geometrical attributes which are inherently related to the quadrilateral. A modified version of the functional of Hu-Washizu is employed for the discretization. Appropriate approximations for the displacements/rotations are chosen and the physical meaning of the various parameters is identified. A systematic procedure is followed for the approximation of stress and strain. The convergence of the formulation is investigated by examining the inf–sup condition and applying the patch test. Also, results of numerical examples and comparisons with other elemental formulations are presented.
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