A Systematic Construction of B-Bar Functions for Linear and Non-linear Mixed-enhanced Finite Elements for Plane Elasticity Problems

In a previous paper a modified Hu–Washizu variational formulation has been used to derive an accurate four node plane strain/stress finite element denoted QE2. For the mixed element QE2 two enhanced strain terms are used and the assumed stresses satisfy the equilibrium equations  a priori  for the linear elastic case. In this paper an alternative approach is discussed. The new formulation leads to the same accuracy for linear elastic problems as the QE2 element; however it turns out to be more efficient in numerical simulations, especially for large deformation problems. Using orthogonal stress and strain functions we derive  B̄  functions which avoid numerical inversion of matrices. The  B̄ -strain matrix is sparse and has the same structure as the strain matrix  B  obtained from a compatible displacement field. The implementation of the derived mixed element is basically the same as the one for a compatible displacement element. The only difference is that we have to compute a  B̄ -strain matrix instead of the standard  B -matrix. Accordingly, existing subroutines for a compatible displacement element can be easily changed to obtain the mixed-enhanced finite element which yields a higher accuracy than the Q4 and QM6 elements.