Nitinol (Superelasticity)

The superelastic model is based on the uniaxial stress-strain response of phase transforming materials. Such materials (e.g., Nitinol) are in the austenite phase under no loading conditions. Austenite is assumed to follow isotropic linear elasticity. On loading the material, the austenite phase starts transforming into martensite beyond a certain stress. Martensite is also assumed to follow isotropic linear elasticity. During the phase transformation, elastic properties are calculated from the elastic constants of austenite and martensite, following the rule of mixtures:

E = E_{A} + ζ (E_{M} - E_{A})

ν = ν_{A} + ζ (ν_{M} - ν_{A})

where $ζ$ is the fraction of martensite, $E_{A}$ is the Young's modulus of austenite, $E_{M}$ is the Young's modulus of martensite, $ν_{A}$ is the Poisson's ratio of austenite, and $ν_{M}$ is the Poisson's ratio of martensite. After a certain stress, austenite is completely transformed into martensite, which deforms elastically thereafter. Therefore, the deformation follows the elastic constants of austenite when the fraction of martensite is zero and follows the elastic constants of martensite if the fraction of martensite is one (full transformation). On unloading, martensite transforms back into austenite and the transformation strain is fully recovered. However, the stress at which the reverse transformation occurs is different from the stress at which the austenite to martensite transformation occurred.

Superelasticity

You can define the elastic properties of martensite, the critical stress levels for forward and reverse transformation, and the variation of transformation plateau with temperature. You can define elastic properties of austenite in the elastic material option. Superelasticity supports both associated and nonassociated flow rules:

Associated: When using associated flow the volumetric transformation strain, $ε_{V}^{L}$ , is assumed to be equal to the uniaxial transformation strain, $ε^{L}$ .
Nonassociated: When using nonassociated flow the volumetric transformation strain, $ε_{V}^{L}$ , must be specified independently from the uniaxial transformation strain, $ε^{L}$ .

Table 1. Flow rule=Associated
Input Data	Description
E_m	Young's modulus of martensite, $E_{M}$ .
Nu_m	Poisson's ratio of martensite, $ν_{M}$ .
Episilon_L	Uniaxial transformation strain, $ε^{L}$ .
Stress_S_tL	Stress at which the transformation begins during loading in tension, $σ_{t L}^{S}$
Stress_E_tL	Stress at which the transformation ends during loading in tension, $σ_{t L}^{E}$
Stress_S_tU	Stress at which the reverse transformation begins during unloading in tension, $σ_{t U}^{S}$
Stress_E_tU	Stress at which the reverse transformation ends during unloading in tension, $σ_{t U}^{E}$
Stress_S_cL	Stress at which the transformation begins during loading in compression, as a positive value, $σ_{c L}^{S}$
T_0	Reference temperature, $T_{0}$
DeltaS/DeltaT_L	Slope of the stress versus temperature curve for loading, ${(\frac{δ σ}{δ T})}_{L}$
DeltaS/DeltaT_U	Slope of the stress versus temperature curve for loading, ${(\frac{δ σ}{δ T})}_{U}$

Table 2. Flow rule=Nonassociated
Input Data	Description
E_m	Young's modulus of martensite, $E_{M}$ .
Nu_m	Poisson's ratio of martensite, $ν_{M}$ .
Episilon_L	Uniaxial transformation strain, $ε^{L}$ .
Episilon_VL	Volumetric transformation strain, $ε_{V}^{L}$ .
Stress_S_tL	Stress at which the transformation begins during loading in tension, $σ_{t L}^{S}$
Stress_E_tL	Stress at which the transformation ends during loading in tension, $σ_{t L}^{E}$
Stress_S_tU	Stress at which the reverse transformation begins during unloading in tension, $σ_{t U}^{S}$
Stress_E_tU	Stress at which the reverse transformation ends during unloading in tension, $σ_{t U}^{E}$
Stress_S_cL	Stress at which the transformation begins during loading in compression, as a positive value, $σ_{c L}^{S}$
T_0	Reference temperature, $T_{0}$
DeltaS/DeltaT_L	Slope of the stress versus temperature curve for loading, ${(\frac{δ σ}{δ T})}_{L}$
DeltaS/DeltaT_U	Slope of the stress versus temperature curve for loading, ${(\frac{δ σ}{δ T})}_{U}$

Superelastic Hardening

The plasticity model for superelastic materials is based on the uniaxial stress-strain response. Such materials (e.g., Nitinol) are in the austenite phase under no loading conditions. Austenite is assumed to follow isotropic linear elasticity. On loading the material, the austenite phase starts transforming into martensite beyond a certain stress. Martensite is assumed to follow an elastoplastic response, with elasticity characterized by the linear elastic model and the plastic behavior represented by the Drucker Prager model. Martensite exhibits plastic behavior after full transformation.

Note: The hardening data for superelastic materials is specified by providing the yield stress as a function of total strain. This is in contrast to hardening data for many other types of materials that specify yield stress as a function of plastic strain.


Input Data	Description
Yield Stress	The true - or logarithmic - stress value at the corresponding total strain.
Total Strain	Total stran .

Superelastic Hardening Modifications

It is observed that the transformation stress levels decrease with an increase in the plastic strain. There are two ways to specify this variation in the transformation plateau with plastic strain. You can either specify the data describing the change in transformation stress levels as a function of the plastic strain, using the built-in functionality, or you can use a user subroutine to specify this dependency.

Table 3. Builtin
Input Data	Description
Stress_S_tl	Stress at which the transformation begins during loading in tension, $σ_{t L}^{S}$
Stress_E_tl	Stress at which the transformation ends during loading in tension, $σ_{t L}^{E}$
Stress_S_tU	Stress at which the reverse transformation begins during unloading in tension, $σ_{t U}^{S}$
Stress_E_tU	Stress at which the reverse transformation ends during unloading in tension, $σ_{t U}^{E}$
Plastic Strain	Stress at which the transformation begins during loading in compression, as a positive value, $σ_{c L}^{S}$

The user subroutine USUPERELASHARDMOD is used for implicit time integration simulations and VUSUPERELASHARDMOD is used for explicit time integration simulations.

Table 4. User
Input Data	Description
User parameters	User defined material parameters.