IOP PUBLISHING SMART MATERIALS AND STRUCTURES Smart Mater. Struct. 19 (2010) 025009 (7pp) doi:10.1088/0964-1726/19/2/025009 An approach to modeling tensile–compressive asymmetry for martensitic shape memory alloys Wael Zaki Modeling and Simulation Unit, Henri Tudor Public Research Center, 66 rue de Luxembourg, L-4221 Esch-sur-Alzette, Luxembourg E-mail: wael.zaki@tudor.lu Received 2 September 2009, in final form 24 November 2009 Published 14 January 2010 Online at stacks.iop.org/SMS/19/025009 Abstract In this paper, the asymmetric tensile–compressive behavior of shape memory alloys is modeled based on the mathematical framework of Raniecki and Mr´ oz (2008 Acta Mech. 195 81–102). The framework allows the definition of smooth, non-symmetric, pressure-insensitive yield functions that are used here to incorporate tensile–compressive modeling capabilities into the Zaki–Moumni (ZM) model for shape memory materials. It is found that, despite some increased complexity, the generalized model is capable of producing satisfactory results that agree with uniaxial experimental data taken from the literature. 1. Introduction The asymmetric response of shape memory alloys (SMAs) in tension and compression is well established experimentally. It was reported as early as 1971 by Wasilewski who found ‘significant differences between the effects of tensile and compressive loading’ on nickel–titanium alloys. The work of Wasilewski came nearly two decades after similar observations were made for other materials capable of undergoing martensite transformations, like steel (Kulin et al 1952) and indium–thallium (Burkart and Read 1953). In a more recent paper, Liu et al (1998) sought to uncover the physical mechanisms responsible for this asymmetry in the case of martensitic NiTi SMAs; they found that, while martensite detwinning and reorientation were dominant in tension up to a certain strain level, the compressive response was governed by the formation of dislocations on the inside and on the boundaries of martensite twins. The authors gave a new definition of the process of martensite ‘reorientation’ consistent with their findings. Constitutive models for shape memory alloys rarely account for tension–compression asymmetry. This might have been justified, at some point, by the predominant use of SMA transducers in the form of simple wires. For some applications, however, such as stents and seismic retrofits, incorporating asymmetric features can probably improve the precision of numerical simulations and allow more reliable computer- assisted design and virtual prototyping of SMA devices. Gall et al (1998) conducted triaxial tensile–compressive tests on CuZnAl alloys, which allowed them to observe a variation of the critical transformation stresses with the state of the triaxial loading. The effective critical transformation stress was found to be maximal in uniaxial compression and minimal in uniaxial tension. The authors further proposed a micromechanical model that accounts for the influence of hydrostatic pressure on the effective transformation stress, which becomes ‘evident’ for extreme variation of the former. It is interesting to note that pure hydrostatic loading was practically incapable of inducing a martensitic transformation in their experiments. Similar investigations were carried out elsewhere on polycrystalline NiTi (Jacobus et al 1996). Raniecki et al (2001) extended the model of M¨ uller and Xu (1991) to simulate uniaxial nonlinear hardening and tensile–compressive asymmetry during phase change. Austenite and martensite were considered to have the same elastic stiffness. Lim and McDowell (2002) attributed the asymmetric response of textured SMAs to the existence of martensite variants with preferred crystallographic orientations 1 . Their model accounts for 24 martensite variant pairs. The interaction between these variants is accounted for using Patoor’s interaction matrix (Siredey et al 1999). 1 This interpretation is inconsistent with the experimental observations of Liu et al (1998). It is, nonetheless, the basis of many available models. 0964-1726/10/025009+07$30.00 © 2010 IOP Publishing Ltd Printed in the UK 1