INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2007; 72:422–463 Published online 26 February 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2019 Generalized midpoint integration algorithms for J 2 plasticity with linear hardening E. Artioli 1, 2, ∗, † , F. Auricchio 1, 2 and L. Beir˜ ao da Veiga 3 1 Dipartimento di Meccanica Strutturale, Universit` a di Pavia, Via Ferrata 1, Pavia 27100, Italy 2 Istituto di Matematica Applicata e Tecnologie Informatiche del CNR, Via Ferrata 1, Pavia 27100, Italy 3 Dipartimento di Matematica ‘F. Enriques’, Universit` a di Milano, Via Saldini 50, Milano 20133, Italy SUMMARY We consider four schemes based on generalized midpoint rule and return map algorithm for the integration of the classical J 2 plasticity model with linear hardening. The comparison, aiming to establish which is the preferable scheme among the four considered, is both theoretical and numerical. On one side, extending and completing the existing results in the literature, we investigate the four schemes from the theoretical viewpoint, addressing in particular the existence of solution, long-term behaviour, accuracy and stability. On the other hand, we develop an extensive set of numerical tests, based on pointwise stress–strain loading histories, iso-error maps and initial boundary-value problems. Copyright 2007 John Wiley & Sons, Ltd. Received 15 November 2006; Revised 15 January 2007; Accepted 15 January 2007 KEY WORDS: plasticity; generalized midpoint rule; return map; second order method; linear hardening 1. INTRODUCTION In the present paper we address the analysis and the comparison of four schemes based on generalized midpoint return mapping and herein applied to the classical J 2 plasticity model with linear isotropic and kinematic hardening. Generalized midpoint algorithms are among the most cited and successful second order methods in plasticity and a total of four different schemes of this type can be found in the literature [1–3]. Nevertheless, it seems that an exhaustive theoretical and numerical comparison between the four methods is missing. Other kinds of methods exhibiting second order accuracy may be found as well. Runge–Kutta methods [4], generalized Runge–Kutta methods [5] and multi-step methods [6], for instance, share ∗ Correspondence to: E. Artioli, Istituto di Matematica Applicata e Tecnologie Informatiche del CNR, Via Ferrata 1, Pavia 27100, Italy. † E-mail: artioli@imati.cnr.it Copyright 2007 John Wiley & Sons, Ltd.