Response of a nonlinear elastic general Cosserat brick element in simulations typically exhibiting locking and hourglassing

Abstract

A comparison between the recently developed Cosserat brick element (see [9]) and other standard elements known from the literature is presented in this paper. The Cosserat brick element uses a director vector formulation based on the theory of a Cosserat point. The strain energy for a hyperelastic element is split additively into parts for homogeneous and inhomogeneous deformations respectively. The kinetic response due to inhomogeneous deformations uses constitutive constants that are determined by analytical solutions of a rectangular parallelepiped to the deformation modes of bending, torsion and hourglassing. Standard tests are performed which typically exhibit hourglassing or locking for many other finite elements. These tests include problems for beam and plate bending, shell structures and nearly incompressible materials, as well as for buckling under high pressure loads. For all these critical tests the Cosserat brick element exhibits robustness and reliability. Moreover, it does not depend on user-tuned stabilization parameters. Thus, it shows promise of being a truly user-friendly element for problems in nonlinear elasticity.

An improved Cosserat point element for axisymmetric problems in nonlinear elasticity: Comparison with other element formulations

“…It was shown in [11] that the accuracy of the brick CPE developed in [7] degraded with increased irregularity of the reference shapes of the elements. Motivated by this problem, Jabareen and Rubin [17] proposed an improved functional form for which included dependence on the reference geometry of the element.…”

Influence of Membrane Stresses on Postbuckling of Rectangular Plates Using a Nonlinear Elastic 3-D Cosserat brick Element

“…[11][12][13]). This 3-D Cosserat element has recently been shown to perform well for beams, plates and shells when only one element is used to discretize the thickness of these structures [7]. In particular, the element does not exhibit locking when it becomes thin or when the material is nearly incompressible.…”

A generalized Cosserat point element (CPE) for isotropic nonlinear elastic materials including irregular 3-D brick and thin structures

“…Also, it can be shown [Loehnert et al 2005] that d 1/2 T is equal to the volume integral of the three-dimensional Cauchy stress 22) where dv * is the current element of volume.…”

“…Special methods based on enhanced strains, incompatible modes and reduced integration with hourglass control have been developed to overcome these problems; see, e.g., [Belytschko et al 1984;Simo and Rifai 1990;Simo and Armero 1992;Belytschko and Bindeman 1993;Simo et al 1993;Bonet and Bhargava 1995;Reese and Wriggers 1996;2000;Hutter et al 2000]. However, these improved formulations can exhibit unphysical hourglassing in regions experiencing combined high compression with bending; see, for example, [Reese and Wriggers 1996;2000;Loehnert et al 2005;Jabareen and Rubin 2007a;2008c]. Furthermore, it was shown in [Jabareen and Rubin 2007a] that some of these improved element formulations in commercial codes can exhibit inelastic response even though they attempt to model a hyperelastic material with a strain energy function.…”

“…Loehnert et al [2005] implemented the CPE formulation into the finite element code FEAP [Taylor 2005] and considered specific example problems which showed that the response of the original CPE was robust and locking free for thin structures. However, it was also shown that the accuracy of the original CPE degraded with increased irregularity of the reference element shape.…”