Falk's One-Dimensional Shape Memory Alloy (SMA) Model

This repository contains the implementation of the one-dimensional shape memory alloy (SMA) phenomenological model proposed by Falk [1, 2].

The model is based on a Landau-type free energy function that describes the phase transformation behavior of SMA materials.

The free energy is defined as a function of strain and temperature such that

$$ \rho\psi(\varepsilon, T) = \frac{a}{2}\left(T - T_m\right)\varepsilon^2 -\frac{b}{4}\varepsilon^4 + \frac{b^2}{24a\left(T_a - T_m\right)}\varepsilon^6 $$

1. High temperatures ($T > T_a$):
   The energy has a single minimum at zero strain associated with the stable austenite (A) phase.

2. Intermediate temperatures ($T_m < T < T_a$):
   The energy exhibits three distinct minima:

   • One at zero strain corresponding to stable austenite (A).
   • Two symmetric minima at non-zero strains representing detwinned martensite induced by compression ($M^−$) and tension ($M^+$).

3. Low temperatures ($T < T_m$):
   The energy has two symmetric minima at non-zero strains associated with stable martensite phases ($M^-$ and $M^+$), while the austenite phase becomes unstable.

The parameter $\alpha$ from reference [1] is modified as done in Savi & Braga [2]:

$$\alpha \propto \frac{b^2}{a (T_a - T_m)} $$

This modification defines the temperature $T_a$, above which the austenite phase is stable and the free energy has only one minimum at zero strain.

Output Example

References

[1] Falk, F. (1980). Model free energy, mechanics and thermodynamics of shape memory alloys. Acta Metallurgica, 28, 1773--1780.

[2] Falk, F. (1983). One-dimensional model of shape memory alloys.Archives of Mechanics, 35(1), 63--84.

[3] Savi, M.A. & Braga, A.M.B. (1993). Chaotic vibrations of an oscillator with shape memory. Journal of the Brazilian Society of Mechanical Sciences, 15(1), 1--20.

📦 Requirements

• Python ≥ 3.12
• NumPy
• Matplotlib

⚙️ Installation

Using uv (recommended)

uv sync
uv run python falk_SMA.py

Or using pip

pip install numpy matplotlib
python falk_SMA.py