An enhanced strain 3D element for large deformation elastoplastic thin-shell applications

Abstract

In this work a previously proposed solid-shell finite element, entirely based on the Enhanced Assumed Strain (EAS) formulation, is extended in order to account for large deformation elastoplastic thin-shell problems. An optimal number of 12 enhanced (internal) variables is employed, leading to a computationally efficient performance when compared to other 3D or solid-shell enhanced elements. This low number of enhanced variables is sufficient to (directly) eliminate either volumetric and transverse shear lockings, the first one arising, for instance, in the fully plastic range, whilst the last appears for small thickness' values. The enhanced formulation comprises an additive split of the Green-Lagrange material strain tensor, turning the inclusion of nonlinear kinematics a straightforward task. Finally, some shell-type numerical benchmarks are carried out with the present formulation, and good results are obtained, compared to well-established formulations in the literature. Keywords Solid-shell elements, Enhanced strains, Volumetric and transverse shear lockings, Geometric and material nonlinearities, Thin shells 1 Introduction Finite element analysis of shell structures goes back in time until the onset of the so-called degenerated approach in works of Ahmad et al. [1] and Zienkiewicz et al. [101], as well as in early papers of Ramm [78], and afterwards with Hughes and Liu [55] and Hughes and Carnoy [57], among others. Soon it was verified that brick elements were prone to the appearance of volumetric and transverseshear locking effects. The first one is characteristic of common metal plasticity models, where plastic deformation is taken to be isochoric or, in other words, incompressible [19]. The second one comes from the analysis of thin shells, were the limit between ''thick'' and ''thin'' geometries is somewhat difficult to establish, with the occurrence of locking not only strictly relying on thickness/length ratios, as demonstrated by Chapelle, Bathe and co-workers [11,12,29,30,31].In order to circumvent these parasitic phenomena, selective reduced integration (or, equivalently, u/p formulation, mean-dilatation technique and B-bar methods) -for volumetric locking -and the ''mixed interpolation of tensorial components''/assumed strain method -for transverse shear locking -had arisen as possible and successful techniques. In the literature see, for instance, references [4,45,53,54,61,68,69,74,85,86,96] for the grounds of computational treatment of incompressibility, in elastic and elastoplastic finite element cases, and also [9,38,56,67] for earlier works dealing with transverse shear locking.In the specific case of shell elements, original planestress assumptions were enough to avoid or postpone incompressibility issues in the nonlinear material range [5,47,55,78,88], although at the expense of a rotation tensor inclusion. As more generality was needed, higher order theories including thickness change via extensible director fields and/or ''layerwise'' approaches were developed, including (or not) rotatio...

A new one‐point quadrature enhanced assumed strain (EAS) solid‐shell element with multiple integration points along thickness—part II: nonlinear applications

“…Considering the interpolation assumptions of Equations (25)- (28) the second term on the right-hand side of the linearized weak form, Equation (23), can be rewritten (dropping the iteration indices) following references [8,16,24,37] as:…”

“…The adopted procedure introduces an additive split of the Green-Lagrange strain tensor for the enhanced strain part, which has proved to be as accurate as the multiplicative decomposition of the deformation gradient, although more simple and efficient, as stated for instance in References [8,9,16] and, more recently, in References [18,46], in shell formulations within composites and anisotropic materials. The physical stabilization procedure adopted in linear applications to correct the rank-deficiency of the stiffness matrix [3] is now extended to the internal force vector.…”

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].…”

A 4-node 3D-shell element to model shell surface tractions and incompressible behavior

“…To improve the element behavior and computational effectiveness, instead of displacement degrees of freedom, enhanced strains have been used, see for example Refs. [12,19,20], and the references therein. However, enhanced strain formulations can be unstable, see Refs.…”