A variationally consistent generalized variable formulation for enhanced strain finite elements

Abstract

The so‐called enhanced strain finite elements are based on the enrichment of the standard compatible strain field by the introduction of additional, non‐compatible strains. This class of elements can be derived starting from a partial Hu–Washizu variational principle. However, since in the original enhanced strain formulation the stress field is eliminated from the formulation, a separate least‐squares procedure had to be implemented for a variational derivation of the stress field. A three‐field generalized variable approach incorporating strain enhancement is proposed in the present paper within the context of linear elastic structural problems. It is shown how the original two‐field enhanced strain method can be naturally recovered by suitably choosing the strain model. For this case a straightforward, but still variationally consistent stress recovery is proposed. Copyright © 2000 John Wiley & Sons, Ltd.

A Hu–Washizu variational approach to self-stabilized virtual elements: 2D linear elastostatics

“…An original construction of the hourglass matrix H, based on the natural approach of Argyris [12,15,16], is detailed in "Appendix A". The resulting stabilization stiffness matrix is identical to the one originally proposed in [1,17].…”

“…The key operator in the present VEM formulation is the compatibility matrix C defined in (18), projecting the symmetric part of the displacement gradient field onto the space P k−1 of polynomials of degree up to k − 1 of the approximate strain field. The computation of C requires the computation of the symmetric and invertible matrix G, defined in (16), and of A, defined in (19). The matrix G is directly computable once the degree of accuracy k of the method is defined, by computing the integrals by means of a subtriangulation technique.…”

“…To define the hourglass modes ûH and the hourglass matrix H in (28) it is possible to proceed as follows (see, e.g [12,16]). According to the natural approach proposed by Argyris [15], the element nodal displacements û can be expressed as a linear combination of n R rigid body modes and n D natural or straining modes, through a non-singular matrix T. In the case that n H hourglass modes are also allowed by the element kinematics, the vector of displacement parameters û can be expressed as:…”

“…For the connection between the VEM and the finite element hourglass control techniques, see [14]. In this work we present an orig-inal derivation of the stabilization stiffness matrix, departing from Argyris' notion of natural strains [15,16]. The key ingredient is the hourglass matrix, which is constructed based on a decomposition of the discretized displacement modes into a rigid body and pure deformation part and into an hourglass part.…”

Performance of mixed and enhanced finite elements for strain localization in hypoplasticity

“…Therefore, Simo and Rifai 35 suggested a procedure for stress recovery based on a least‐square optimization. The evaluation of the discrete stress field can be referred to 37–39 and has been debated. Bischoff et al 40 showed that the stress field calculated from an elastic constitutive law converges to the correct solution, which satisfies the orthogonality assumption (61).…”