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

Abstract

SUMMARYIn this paper two plane strain quadrilateral elements with two and four variables, are proposed. These elements are applied to the analysis of ÿnite 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 ÿelds 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-ÿeld 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 e ective for the particular class of problems analysed and do not present any locking or instability tendencies, as illustrated by various representative examples.

On the use of an enhanced transverse shear strain shell element for problems involving large rotations

“…A set of additive strain terms was defined based on the analysis of the null transverse shear strain subspace, inherent to each finite element formulation. Correction of locking was then related to the dimension of this subspace, with the proposed enhanced transverse strain field being responsible for its enlargement, in a way similar to previous works of the authors for 2D problems (César de Sá and Natal Jorge [16], César de Sá et al [17]). The shell element then obtained employs a full numerical integration rule without including artificial interpolation points for the strain field correction in the formulation.…”

On the comparison of two solid-shell formulations based on in-plane reduced and full integration schemes in linear and non-linear applications

“…In this purpose, an analysis of the efficiency of the EAS method using the framework of the subspace analysis is now underway and will be reported in future publications. This method was initially developed by César de Sà and Owen [54] for 2D elements and successfully applied later for 2D plane strain quadrilateral in [55,56], shell elements [43,48], 3D solid-shell elements by Alves de Sousa et al [25] and Caseiro et al [40].…”

Enhanced transverse shear strain shell formulation applied to large elasto‐plastic deformation problems

“…Departing from the conventional degenerated approach applied to bilinear (four-node) fully integrated shell elements, a complete analysis of the null transverse shear strain subspace was performed in Reference [29]. Along with the present work, this reference points to previous developments by the authors in EASbased two dimension, shell and solid-shell finite elements technology [30][31][32][33][34][35][36].…”

“…After detailing the relevant aspects of the first enhanced interpolation matrix (M ), it is necessary to establish the topology of the enhanced matrix (M ) in (31). This term is intended to affect only the in-plane strain field, improving the element performance in membranedominated situations and being decoupled from the transverse shear strain enhancement.…”