Within the framework of plane strain nonlinear elasticity, we present a discussion on the stability prop- erties 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. 1 Introduction The incentive for the present work emanates from the unsolved problem of artificial instabilities encountered in large strain nonlinear finite element analyses in the field of solid and structural mechanics. The problem that is addressed is briefly summarized in the following. Standard Galerkin finite elements may suffer from locking and perform too stiff in certain situations. One of the first ideas to remove locking was to use reduced integration but it was soon detected that this may lead to artificial instabilities [1, 2]. Such artificial instabilities are often denoted as hourglassing , because in the pattern of the artificial buckling modes, pairs of elements resemble the shape of an hourglass. Later, hourglass stabilization has been developed to alleviate this problem, see e.g. [3, 4]. In the meantime, numerous alternative “unlocking” schemes have been developed, which are mostly stable in the linear case. For the important class of mixed finite elements in the linear regime, numerical stability can be guaranteed by the celebrated inf-sup- condition [5]. In the context of nonlinear problems, however, the flexibility introduced by unlocking schemes may give rise to unphysical unstable configurations, even if full integration is used and the elements are stable in the linear case. This was first described by Wriggers & Reese [6] and De Souza Neto et al. [7]. The problem appears to be intrinsic to many typical approaches that are taken to remove locking and is not an exclusive quality of only a few particular strategies in finite element technology. To the authors’ best knowledge, in contrast to the linear case, there is no mathematical theory for numerical stability for general nonlinear problems. One of the reasons is that the exact solution may contain physical instabilities, such that naively requiring the finite element model to be stable, over all ranges of deformation, is not appropriate. Accordingly, finite element models should be able to capture physical instabilities, without presenting extra unphysical instabilities associated to numerical 1 pathologies. The delicate distinction of pathological numerical instabilities and physical instabilities is one of the key challenges tackled in this paper. Similar to physical instabilities, artificial, unphysical instabilities can be induced by geometric or material nonlinearity, see Reese et al. [8, 9]. When trying to analyze the underlying effects, it is crucial to separate these two classes of nonlinear problems. In the following, we will therefore refer to geometric hourglassing for the former and material hourglassing for the latter case. Despite the considerable amount of literature, the development of nonlinear finite elements has not reached the same level of maturity as it is the case in the linear regime. The following literature review is restricted to some selected works and we refer to the recent overview by Pfefferkorn & Betsch [10] as well as Chapter 10 of Wriggers [11] for further