Second-order accurate integration algorithms for von-Mises plasticity with a nonlinear kinematic hardening mechanism E. Artioli b,c, * , F. Auricchio b,c , L. Beira ˜o da Veiga a a Dipartimento di Matematica, Universita ` di Milano, Italy b Dipartimento di Meccanica Strutturale, Universita ` di Pavia, Italy c IMATI-CNR, Pavia, Via Ferrata 1, I-27100, Italy Received 30 May 2006; received in revised form 10 October 2006; accepted 16 October 2006 Abstract Two second-order numerical schemes for von-Mises plasticity with a combination of linear isotropic and nonlinear kinematic hard- ening are presented. The first scheme is the generalized midpoint integration procedure, originally introduced by Ortiz and Popov in 1985, detailed and applied here to the case of Armstrong–Frederick nonlinear kinematic hardening. The second algorithm is based on the constitutive model exponential-based reformulation and on the integration procedure previously introduced by the authors in the context of linearly hardening materials. There are two main targets to the work. Firstly, we wish to extensively test the generalized midpoint procedure since in the scientific literature no thorough numerical testing campaign has been carried out on this second-order algorithm. Secondly, we wish to extend the exponential-based integration technique also to nonlinear hardening materials. A wide numerical investigation is carried out in order to compare the performance of the two methods. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Plasticity; Exponential-based integration algorithm; Return map; Second-order method; Armstrong–Frederick constitutive model; Nonlinear kinematic hardening 1. Introduction In the present paper we address the numerical solution algorithm for a von-Mises elastoplastic model with a com- bination of linear isotropic hardening and nonlinear Arm- strong–Frederick kinematic hardening in the realm of small deformations [3]. The main aim of the present contribution is to develop and to test two second-order schemes for the aforemen- tioned model. To the authors’ knowledge, in the scientific literature, the issue on second-order accurate integration schemes which allow for nonlinear kinematic hardening is not studied, while a number of first-order accurate proce- dures are known [6,4,24]. The first numerical scheme considered here follows from the methods proposed in the work of Ortiz and Popov [15]. These methods adopt a single-step generalized midpoint integration rule for approximating the time derivatives and make use of a return map algorithm for the solution of the ensuing nonlinear algebraic system. Accordingly, the first method object of study is labeled MPTnl and rep- resents a special case of generalized midpoint integration algorithm. This scheme shows yield consistency at the end of each step and is second-order accurate. The second scheme introduced is based on a quasi-linear reformulation of the constitutive model combined with an exponential-based time integration method. The proposed integration scheme, labeled ESC 2 nl, is achieved in the spirit of [2]. It is noted that exponential-based integration 0045-7825/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2006.10.002 * Corresponding author. Address: IMATI-CNR, Pavia, Via Ferrata 1, I-27100, Italy. Tel.: +39 0382 548229; fax: +39 0382 548300. E-mail address: artioli@imati.cnr.it (E. Artioli). www.elsevier.com/locate/cma Comput. Methods Appl. Mech. Engrg. 196 (2007) 1827–1846