# Shape Memory Superelastic Material (LAW71)

This material can be used to define elements to act as a void, or empty space.

The constitutive model that is used in LAW71 to describe the shape memory alloy superelastic behavior is based on the work of Auricchio.1

## Phase Transformation Conditions

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:(1)
${\stackrel{˙}{\epsilon }}^{tr}={\epsilon }_{L}{\stackrel{˙}{X}}_{m}n$
The stress deviator and pressure are updated using the fraction of martensite and transformation strain:(2)
$s=2G\left(e-{\epsilon }_{L}{X}_{m}n\right)$
(3)
$p=K\left(tr\left(\epsilon \right)-3\alpha {\epsilon }_{L}{X}_{m}\right)$
$G$ 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:(4)
$G={G}_{A}+{X}_{m}\left({G}_{M}-{G}_{A}\right)$
(5) $K={K}_{A}+{X}_{m}\left({K}_{M}-{K}_{A}\right)$
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 $n=\frac{e}{‖e‖}$

## Transformation from Austenite to Martensite

Two functions are defined for the start and the final point of transformation. These functions are expressed as:(6)
${F}_{S}^{SA}={F}^{SA}-{R}_{S}^{SA}$
(7)
${F}_{F}^{SA}={F}^{SA}-{R}_{F}^{SA}$
Where,(8)
${R}_{S}^{SA}={\sigma }_{S}^{SA}\left(\sqrt{\frac{2}{3}}+\alpha \right)-{C}^{SA}{T}_{S}^{SA}$
(9)
${R}_{F}^{SA}={\sigma }_{F}^{SA}\left(\sqrt{\frac{2}{3}}+\alpha \right)-{C}^{SA}{T}_{F}^{SA}$
The loading function ${F}^{SA}$ is computed using the stress deviator $s$ , the pressure $p$ , and the temperature:(10)
${F}^{SA}=‖s‖+3\alpha p-{C}^{SA}T$
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:(11)
${F}_{S}^{SA}<0$
(12)
${F}_{F}^{SAS}>0$
(13)
${\stackrel{˙}{F}}_{}^{SAS}<0$
1 F. Auricchio, R.L. Taylor. Shape memory alloys: modeling and numerical simulations of the finite- strain superelastic behavior. Comput. Methods Appl. Mech. Engrg. 143 (1997) 175-194.