**Nitinol Model Formulation**

Since Nitinol is usually used for its ability to undergo finite strains, the large strain theory utilizing logarithmic strains along with the updated Lagrangian formulation is employed for this model.

The constitutive model is, thus, constructed to relate the logarithmic strains and the Kirchhoff stress components. However, ultimately the constitutive matrix and the stress vector are both transformed to present the Cauchy (true) stresses.

σ |
Initial and Final yield stress for tensile loading [SIGT_S1, SIGT_F1] |

σ |
Initial and Final yield stress for tensile unloading [SIGT_S2, SIGT_F2] |

σ |
Initial and Final yield stress for compressive loading [SIGC_S1, SIGC_F1] |

σ |
Initial and Final yield stress for compressive unloading [SIGC_S2, SIGC_F2] |

e |
(Maximum Tensile Plastic Strain) *(3/2) |

The exponential flow rule, utilizes additional input constants, β^{t1}, β^{t2}, β^{c1}, β^{c2}:

β |
material parameter, measuring the speed of transformation for tensile loading, [BETAT_1] |

β |
material parameter, measuring the speed of transformation for tensile unloading, [BETAT_2] |

β |
material parameter, measuring the speed of transformation for compressive loading, [BETAC_1] |

β |
material parameter, measuring the speed of transformation for compressive unloading, [BETAC_2] |

To model the possibility of pressure-dependency of the phase-transformation, a Drucker-Prager-type loading function is used for the yield criterion:

F(τ) = sqrt(2)*σ(bar) + 3*α*pF - R_{I}^{f} = 0

where

σ(bar) = effective stress

p = mean stress (or hydrostatic pressure)

α = sqrt(2/3) (σ_{s}^{c1} - σ_{s}^{t1} ) / (σ_{s}^{c1} - σ_{s}^{t1})

R_{f}^{I }= [ σ_{f}^{I}(sqrt(2/3) + α)], I = 1 for loading and 2 for unloading

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