Shape Memory Superelastic Material (LAW71)

Two phase transformations are considered:

• Transformation from austenite to martensite denoted AS (A→S)
• Transformation from martensite to austenite denoted SA (S→A)

A transformation strain is introduced dependent on the evolution of the fraction of martensite:

$${\dot{\epsilon }}^{tr}={\epsilon }_L{\dot{X}}_{m}n$$

The stress deviator and pressure are updated using the fraction of martensite and transformation strain:

$$s=2G(e-{\epsilon }_L{X}_{m}n)$$

$$p=K(tr(\epsilon )-3\alpha {\epsilon }_L{X}_m)$$

$G$ and $K$ are the shear and the bulk modulus, respectively. It is possible to define these parameters dependent on the martensite fraction. In this case, $A$ and $M$ subscripts refer to austenite and martensite, respectively. The two moduli are calculated as:

$$G=G_A+X_m(G_M-G_A)$$

Where,

$e$

Deviator of the strain

$tr$

Denotes the trace of the tensor

$Xm$

Martensite fraction

${\epsilon }_L$ and $a$

Constant material parameters

$n$

Unit vector is defined as $n=\frac{e}{‖e‖}$

Two functions are defined for the start and the final point of transformation. These functions are expressed as:

$$F_S^{SA}=F^{SA}-R_S^{SA}$$

$$F_F^{SA}=F^{SA}-R_F^{SA}$$

Where,

$$R_S^{SA}={\sigma }_S^{SA}\left(\sqrt{\frac{2}{3}}+\alpha \right)-C^{SA}{T}_S^{SA}$$

$$R_F^{SA}={\sigma }_F^{SA}\left(\sqrt{\frac{2}{3}}+\alpha \right)-C^{SA}{T}_F^{SA}$$

The loading function $F^{SA}$ is computed using the stress deviator $s$ , the pressure $p$ , and the temperature:

$$F^{SA}=‖s‖+3\alpha p-C^{SA}T$$

Where, $a$ , ${\sigma }_S^{SA}$ , ${\sigma }_F^{SA}$ , $T_S^{SA}$ , $T_F^{SA}$ , and $C^{SA}$ are the material parameters.

The conversion of austenite to martensite takes place when the following conditions are verified:

$$F_S^{SA}<0$$

$$F_F^{SAS}>0$$

$${\dot{F}}_{}^{SAS}<0$$