Quadrilateral elements for the solution of elasto‐plastic finite strain problems

In this paper two plane strain quadrilateral elements with two and four variables, are proposed. These elements are applied to the analysis of finite strain elasto‐plastic problems. The elements are based on the enhanced strain and B‐bar methodologies and possess a stabilizing term. The pressure and dilatation fields are assumed to be constant in each element's domain and the deformation gradient is enriched with additional variables, as in the enhanced strain methodology. The formulation is deduced from a four‐field functional, based on the imposition of two constraints: annulment of the enhanced part of the deformation gradient and the relation between the assumed dilatation and the deformation gradient determinant. The discretized form of equilibrium is presented, and the analytical linearization is deduced, to ensure the asymptotically quadratic rate of convergence in the Newton–Raphson method. The proposed formulation for the enhanced terms is carried out in the isoparametric domain and does not need the usually adopted procedure of evaluating the Jacobian matrix in the centre of the element. The elements are very effective for the particular class of problems analysed and do not present any locking or instability tendencies, as illustrated by various representative examples. Copyright © 2001 John Wiley & Sons, Ltd.