Propagation stresses in phase transitions of an SMA wire: New analytical formulas based on an internal-variable model Zilong Song a , Hui-Hui Dai a,⇑ , Qing-Ping Sun b a Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong b Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong article info Article history: Received 30 June 2012 Received in final revised form 20 September 2012 Available online 22 October 2012 Keywords: Shape memory alloys (SMA) Martensitic phase transformation Tension test Analytical methods Material calibration abstract In this paper, we study the stress-induced isothermal phase transitions of a shape memory alloy wire, with a focus on homogeneous or piecewise homogeneous deformations. Based on the constitutive model in the literature with the specific Helmholz free energy and rate of mechanical dissipation, the three-dimensional model is formulated. Identifying the characteristic axial strain as the small parameter, we arrive at the asymptotic one-dimen- sional equation which involves the stress, the axial strain and the phase state variable. By considering the evolution law of the phase state variable, the stress–strain relations corre- sponding to austenite, martensite and phase transition (phase mixture) regions are obtained. Although we take the dissipation into consideration, we find that each of the pure loading and unloading processes is equivalent to the Ericksen’s bar problem with three branches in the stress–strain curve. As a result, we successfully deduce the analytical for- mulas for the nucleation stresses and propagation stresses. The analytical results reveal explicitly how such important quantities depend on the material constants and the tem- perature. We also demonstrate that they capture a number of features observed in exper- iments. As an important application, we show that these formulas can be used for calibration of the material constants (such as the difference of the thermal free energies of two phases) by comparing with the measured nominal stress–strain curves. Three exam- ples are also provided. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction Shape memory alloys (SMAs) such as Ni–Ti, Cu–Al–Ni, are well known for two major properties: shape memory effect and pseudo-elasticity. Shape memory effect means that the material can be bent or stretched at low temperature but will revert to its original shape upon heating. Pseudo-elasticity means SMAs can undergo large deformations upon loading at high tem- perature and then revert back to their original shape when the stress is removed. During the loading–unloading process, there is an energy dissipation, as indicated by the hysteresis loop in the stress–strain curve. To understand the behavior of SMAs, many works have been carried out, including constitutive modeling, experimental, analytical and numerical studies. Here, we give a review of some aspects, which we considered to be related to the present work. A large number of experiments (Tobushi et al., 1993; Shaw and Kyriakides, 1995; Shaw and Kyriakides, 1997; Tse and Sun, 2000) on pseudo-elasticity have been performed on SMA wires at different temperatures under tension. It was found that the stress-induced phase transitions are realized by the nucleation of the phase fronts in some particular region and 0749-6419/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijplas.2012.10.002 ⇑ Corresponding author. E-mail address: mahhdai@cityu.edu.hk (H.-H. Dai). International Journal of Plasticity 42 (2013) 101–119 Contents lists available at SciVerse ScienceDirect International Journal of Plasticity journal homepage: www.elsevier.com/locate/ijplas