INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2010; 84:1490–1518 Published online 22 June 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.2947 Fast methods for determining instabilities of elastic–plastic damage models through closed-form expressions Liang Xue ∗, † and Ted Belytschko ‡ Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208-3111, U.S.A. SUMMARY Fast methods for determining the onset of instability for elastic–plastic damage models under multiaxial loading conditions are developed. On the basis of the general Hadamard instability criterion, we derive a closed-form expression to determine the onset of instability, the bifurcation directions and the polarization vector. The results of the present analytical method are compared with solutions using a numerical minimization method. Excellent agreement and significantly faster computations are achieved for the onset of instability and the directions of the instability mode and the polarization vector when using these new methods. Copyright  2010 John Wiley & Sons, Ltd. Received 10 February 2010; Revised 16 April 2010; Accepted 19 April 2010 KEY WORDS: material instability condition; damage plasticity model; localized necking; material bifur- cation; Hadamard stability . 1. INTRODUCTION An increasingly common approach to modeling failure is to inject discontinuities into the continuum model when the constitutive model loses stability, see for example Belytschko et al. [1]. The loss of well-posedness is then avoided and a physically reasonable consequence of material instability, a crack or shear band, can then be modeled by a cohesive law across this discontinuity. This approach of course assumes that the constitutive equation is valid until the onset of material instability. For example, ductile materials subjected to large plastic deformation may become unstable at some point along the loading path when the destabilizing effect from damage-induced weakening becomes comparatively important; fracture or shear banding may then occur. In this present paper, we focus on the planar discontinuous bifurcation as proposed by Hadamard [2]. The Hadamard instability condition is understood as the onset of bifurcation to a localized ∗ Correspondence to: Liang Xue, Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208-3111, U.S.A. † E-mail: l-xue@northwestern.edu ‡ Walter P. Murphy Professor of Computational Mechanics. Copyright  2010 John Wiley & Sons, Ltd.