Artificial instabilities of finite elements for nonlinear elasticity: Analysis and remedies

Abstract

Within the framework of plane strain nonlinear elasticity, we present a discussion on the stability properties of various Enhanced Assumed Strain (EAS) finite element formulations with respect to physical and artificial (hourglassing) instabilities. By means of a linearized buckling analysis we analyze the influence of element formulations on the geometric stiffness and provide new mechanical insights into the hourglassing phenomenon. Based on these findings, a simple strategy to avoid hourglassing for compression problems is proposed. It is based on a modification of the discrete Green-Lagrange strain, simple to implement and generally applicable. The stabilization concept is tested for various popular element formulations (namely EAS elements and the assumed stress element by Pian and Sumihara). A further aspect of the present contribution is a discussion on proper benchmarking of finite elements in the context of hourglassing. We propose a simple bifurcation problem for which analytical solutions are readily available in the literature. It is tailored for an in-depth stability analysis of finite elements and allows a reliable assessment of its stability properties.

On the stability of mixed polygonal finite element formulations in nonlinear analysis

“…6 In a finite deformation state, however, EAS elements have shown to become unstable and induce hourglass modes as well. 7 Effective hourglass stabilization techniques for EAS elements were developed in for example, References 8 and 9. In a recent publication by Pfefferkorn and Betsch, 10 the hourglass modes are eliminated by enhancing the spatial displacement gradient.…”

“…Notable works include publications dealing with the closely related concepts of incompatible modes 5 and enhanced assumed strain (EAS) 6 . In a finite deformation state, however, EAS elements have shown to become unstable and induce hourglass modes as well 7 . Effective hourglass stabilization techniques for EAS elements were developed in for example, References 8 and 9.…”