A 3-D phenomenological constitutive model for shape memory alloys under multiaxial loadings J. Arghavani a,1 , F. Auricchio b,c,d , R. Naghdabadi a,e, * , A. Reali b,d , S. Sohrabpour a a Department of Mechanical Engineering, Sharif University of Technology, 11155-9567 Tehran, Iran b Dipartimento di Meccanica Strutturale, Università degli Studi di Pavia, Italy c Center for Advanced Numerical Simulation (CeSNA), IUSS, Pavia, Italy d European Centre for Training and Research in Earthquake Engineering (EUCENTRE), Pavia, Italy e Institute for Nano-Science and Technology, Sharif University of Technology, 11155-9567 Tehran, Iran article info Article history: Received 21 June 2009 Received in final revised form 1 December 2009 Available online 4 January 2010 Keywords: Shape memory alloys Non-proportional loading Phase transformation Reorientation Internal variables abstract This paper presents a new phenomenological constitutive model for shape memory alloys, developed within the framework of irreversible thermodynamics and based on a scalar and a tensorial internal variable. In particular, the model uses a measure of the amount of stress-induced martensite as scalar internal variable and the preferred direction of variants as independent tensorial internal variable. Using this approach, it is possible to account for variant reorientation and for the effects of multiaxial non-proportional loadings in a more accurate form than previously done. In particular, we propose a model that has the prop- erty of completely decoupling the pure reorientation mechanism from the pure transfor- mation mechanism. Numerical tests show the ability to reproduce main features of shape memory alloys in proportional loadings and also to improve prediction capabilities under non-proportional loadings, as proven by the comparison with several experimental results available in the literature. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Intelligent, smart or functional materials exhibit special properties that make them a suitable choice for industrial appli- cations in many branches of engineering. Among different types of smart materials, shape memory alloys (SMAs) have un- ique features known as pseudo-elasticity, one-way and two-way shape memory effects (Duerig et al., 1990; Otsuka and Wayman, 1998). The interest in the mechanical behavior of SMAs is rapidly growing with the increasing number of potential industrial applications. Early commercialization activities, fueled by applications such as rivets, heat engines, couplings, cir- cuit breakers and automobile actuators, started in the 1970s, were intense and often highly secretive (Van Humbeeck, 1999). However, from thereon, the knowledge of SMAs has progressively spread out more and more, up to the fact that nowadays pseudo-elastic Nitinol is a common and well-known engineering material in the medical industry (Duerig et al., 1999; Kuri- bayashi et al., 2006). The origin of SMA material features is a reversible thermo-elastic martensitic phase transformation between a high sym- metry, austenitic phase and a low symmetry, martensitic phase. Austenite is a solid phase, usually characterized by a body- centered cubic crystallographic structure, which transforms into martensite by means of a lattice shearing mechanism. 0749-6419/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2009.12.003 * Corresponding author. Address: Department of Mechanical Engineering, Sharif University of Technology, 11155-9567 Tehran, Iran. Tel.: +98 21 6616 5546; fax: +98 21 6600 0021. E-mail address: naghdabd@sharif.edu (R. Naghdabadi). 1 Currently visiting Ph.D. student, Dipartimento di Meccanica Strutturale, Università degli Studi di Pavia, Italy. International Journal of Plasticity 26 (2010) 976–991 Contents lists available at ScienceDirect International Journal of Plasticity journal homepage: www.elsevier.com/locate/ijplas