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This paper considers the stress-induced phase transitions of shape memoryalloy slender cylinder, and analytically studies the phase transition processand the associated instability. A three-dimensional (3D) phenomenological modelwith an internal variable is adopted, which is simplified to a 1D system of twostrains in three regions (austenite, martensite and phase transition regions).Suitable boundary conditions and interface conditions are proposed.Theoretically, it is a free boundary problem, as the position and shape ofphase interfaces are unknown. We then focus on planar interfaces (which areenergetically favored), and a symmetric case when phase transition occurs inthe middle. For given applied stress, two-region solutions, three-regionsolutions and the connecting solutions between them are obtained analyticallyor semi-analytically, including many period-k solutions.  Two-region solutions show that phase transition does not take place at onepoint, but simultaneously in a small region. The width of phase transitionregion is found analytically, revealing the roles of the material andgeometrical parameters. Three-region solutions represent localizedinhomogeneous deformations in experiments, and capture that the stress staysalmost at the Maxwell stress during propagation of transformation front. Fordisplacement-controlled process, the transition process is demonstrated by thestress-strain curve, which captures the stress drop/rise and the transitionfrom homogeneous deformations to period-1 localized inhomogeneous deformations.When the radius is smaller than a critical value (given by material constants),the stress drop is very sharp due to transition of solutions in a snap-backbifurcation. These features show good agreement with experimental observationsand shed light on the difficulties of numerical simulations.

Analytical study on phase transition of shape memory alloy wire under uniaxial tension