On numerically accurate finite element solutions in the fully plastic range

It is often found that tangent-stiffness finite element solutions for elastic-plastic materials exhibit much too stiff a response in the fully plastic range. This is most striking for the perfectly plastic material idealization, in which case a limit load exists within conventional small displacement gradient assumptions. However, finite element solutions often exceed the limit load by substantial amounts, and in some cases have no limit load at all. It is shown that a cause of this inaccuracy is that incremental deformation fields of typical two and three-dimensional finite elements are highly constrained at or near the limit load. This is shown to enforce unreasonable kinematic constraints on the modes of deformation which assemblages of elements are capable of exhibiting. A general criterion for testing a mesh with topologically similar repeat units is given, and the analysis shows that only a few conventional element types and arrangements are, or can be made, suitable for computations in the fully plastic range. Further, a new variational principle, which can easily and simply be incorporated into an existing finite element program, is presented. This allows accurate computations to be made even for element designs that would not normally be suitable. Numerical results are given for three plane strain problems, namely pure bending of a beam, a thick-walled tube under pressure, and a deep double edge cracked tensile specimen. These illustrate the effects of various element designs and of the new variational procedure. An appendix extends the discussion to elastic-plastic computation at finite strain.