On the modeling of hardening in metals during non-proportional loading J. Wang a , V. Levkovitch a , F. Reusch a , B. Svendsen a, * , J. Hue ´tink b , M. van Riel b a Chair of Mechanics, University of Dortmund, 44227 Dortmund, Germany b Netherlands Institute for Metals Research, University of Twente, 7500 AE Enschede, The Netherlands Received 12 November 2006; received in final revised form 23 July 2007 Available online 5 September 2007 Abstract The purpose of the current work is the formulation and initial application of a phenomenological model for hardening effects in metals subject to non-proportional loading histories characterized by one or more loading-path changes. This model is closely related to the incremental model of Teod- osiu and Hu [Teodosiu, C., Hu, Z., 1995. Evolution of the intragranular microstructure at moderate and large strains: modelling and computational significance. In: Shen, S.F., Dawson, P.R. (Eds.), Simulation of Materials Processing: Theory, Methods and Applications. Balkema, Rotterdam, pp. 173–182; Teodosiu, C., Hu, Z., 1998. Microstructure in the continuum modelling of plastic anisot- ropy. In: Proceedings of 19th Risø International Symposium on Material’s Science: Modelling of Structure and Mechanics of Materials from Microscale to Product. Risø National Laboratory, Ros- kilde, Denmark, pp. 149–168]. Like their model, the current model captures in particular hardening stagnation after a load reversal as well as cross-hardening after orthogonal loading-path changes. On the other hand, the two models predict qualitatively different behavior during loading-path changes which take place purely in the inelastic range. Such is the case for example during orthogonal load- ing-path changes from uniaxial tension to simple shear without release, or during monotonic simple shear, or during deep-drawing. As shown by the experimental results reported on in the current work for the mild steel DC06, significant cross-hardening can occur during continuous orthogonal load- ing-path changes. Beyond this, the current model accounts in an approximate way for the possible effects of texture development on the material behavior with the help of the plastic spin. After inves- tigating the behavior of the current model for various ideal two-stage loading histories (e.g., tension- shear), the current work ends with a comparison of standard combined hardening and current 0749-6419/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2007.08.009 * Corresponding author. Fax: +49 231 755 2688. E-mail address: bob.svendsen@udo.edu (B. Svendsen). Available online at www.sciencedirect.com International Journal of Plasticity 24 (2008) 1039–1070 www.elsevier.com/locate/ijplas