Finite Element Application of the Hellinger-Reissner Variational Theorem.

摘要

The Hellinger-Reissner Variational Theorem of linear elastostatics is used to construct a Finite Element Method of solution for boundary value problems in generalized plane stress. For the examples considered, the present application of the Hellinger-Reissner Theorem gives a more accurate field description of both displacement and stress than do existing applications of displacement models. The mixed model yields solutions in which all components of the stress tensor are continuous from element to element, eliminating the histograph distribution often found in displacement models. The displacement vector obtained from the mixed model is considerably more accurate than that given by comparable displacement models. Results for the stress field are superior, but tend to be dependent on material properties and mesh configuration. In part, this can be attributed to identically satisfying the displacement boundary conditions. Since the stress distribution is the desired result, it would be advantageous to bias the model in this direction. An alternate form of the functional which accomplishes this is currently being studied. A stiffness matrix for plane and axisymmetric solids is discussed and several examples are presented with comparisons made to a frequently used displacement model. What has been clearly demonstrated here is that Finite Element Methods can be successfully used with mixed variational theorems for direct solutions to boundary value problems in mechanics.