Composite finite elements for rigid-plastic analysis

In the conventional rigid-plastic finite element analysis, quadrilateral elements or brick-type elements are widely used for two- or three-dimensional simulations in rolling or forging. Through some mathematical studies and numerical tests of errors, the quadrilateral element family, especially the 4-node element with bilinear interpolation for incremental displacement or velocity and constant for pressure, is found to be unsuitable for elasto-plastic or rigid-plastic analyses with large deformations; this type of element often causes large local errors for shear components of strains and stresses, especially in stress concentration problems or in plastic forming problems with local distortion of elements. An alternative family of elements will be proposed and discussed for two- and three-dimensional rigid-plastic simulation in rolling or forging. Since the LBB condition for incompressibility is imposed on the employed velocity interpolations in construction of the adequate finite elements models, the use of conventional triangular elements is prohibited in the rigid-plastic plane strain, axisymmetric or three-dimensional analyses, where the incompressibility condition of plastic strain or strain rate or the volume constancy should be explicitly evaluated by the variational equation or the weak form. Our approach is based on the mixed finite element method where both velocity and pressure are interpolated to satisfy the LBB condition and to become robust for local element distortion. In the present paper, a list of candidate mixed type finite element pairs other than the four-node element will be shown with some comments on their characteristics and features. Special attention will be paid to the composite-type finite element schemes for two- and three-dimensional problems; in particular, the inner-mode composite element family is described with respect to their fundamental behavior in rigid-plasticity. In principle, our developing elements are free from shear/membrane locking and from contamination by zero-energy modes, and stable and robust against the folding behavior of the corner points. Furthermore, the introduced pressure can be excluded from the final element equations by the static condensation or the penalty function method. Through the basic numerical tests for upsetting problems, accuracy and convergency will be evaluated for these elements. Finally, some further examples on the complete automatic mesh control will be employed to show the potential of these elements for adaptive mesh control.