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.

本文考虑了形状记忆合金细长圆柱体的应力诱导相变，并分析了相变过程和相关的不稳定性。采用带有内部变量的三维（3D）现象学模型，该模型简化为在三个区域（奥氏体，马氏体和相变区域）中两个应变的一维系统。提出了合适的边界条件和界面条件。从理论上讲，这是一个自由边界问题，因为相界面的位置和形状是未知的。然后，我们关注平面界面（在能量上受青睐），以及中间发生相变时的对称情况。对于给定的施加应力，可以通过解析或半解析的方式获得两区域解，三区域解以及它们之间的连接解，包括许多period-k解决方案。两区域解决方案表明，相变不是在一个点发生，而是在一个小区域同时发生。通过分析发现相变区的宽度，揭示了材料和几何参数的作用。三区域解表示实验中的局部非均匀变形，并捕获到应力在变换前沿传播期间几乎保持在麦克斯韦应力处。对于位移控制过程，过渡过程由应力-应变曲线演示，该曲线捕获了应力下降/上升以及从均匀变形到周期1局部非均匀变形的过渡。当半径小于临界值（由材料常数决定）时，由于回弹分叉中溶液的转变，应力下降非常明显。这些特征与实验观察结果很好地吻合，并阐明了数值模拟的困难。