  This paper considers the stress-induced phase transitions of shape memory
alloy slender cylinder, and analytically studies the phase transition process
and the associated instability. A three-dimensional (3D) phenomenological model
with an internal variable is adopted, which is simplified to a 1D system of two
strains 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 of
phase interfaces are unknown. We then focus on planar interfaces (which are
energetically favored), and a symmetric case when phase transition occurs in
the middle. For given applied stress, two-region solutions, three-region
solutions and the connecting solutions between them are obtained analytically
or semi-analytically, including many period-k solutions.
  Two-region solutions show that phase transition does not take place at one
point, but simultaneously in a small region. The width of phase transition
region is found analytically, revealing the roles of the material and
geometrical parameters. Three-region solutions represent localized
inhomogeneous deformations in experiments, and capture that the stress stays
almost at the Maxwell stress during propagation of transformation front. For
displacement-controlled process, the transition process is demonstrated by the
stress-strain curve, which captures the stress drop/rise and the transition
from 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-back
bifurcation. These features show good agreement with experimental observations
and shed light on the difficulties of numerical simulations.
