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

## TL;DR

This paper analytically investigates the stress-induced phase transition and instability in shape memory alloy wires under uniaxial tension, revealing detailed transition mechanisms and their dependence on material and geometric parameters.

## Contribution

It introduces a simplified 1D analytical model for phase transitions in SMA wires, providing explicit solutions and insights into the transition process and associated instabilities.

## Key findings

- Phase transition occurs over a small region, not at a single point.
- The width of the transition region depends on material and geometric parameters.
- Stress-strain curves show sharp drops and bifurcations consistent with experiments.

## Abstract

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.