# Ductile

The /FAIL/BIQUAD, /FAIL/JOHNSON, and /FAIL/TAB1 failure models define material failure by relating the plastic strain at failure to the stress state in the material.

These failure models are often used to describe the ductile failure of materials. The state of stress in the material can be defined by using stress triaxiality.

## Stress Triaxiality (Normalized Mean Stress)

For ductile materials, the state of stress (compression, shear, tension, and so on) of the material affects the plastic strain value at which the material will fail. An important and useful characteristic to describe the state of stress, stress triaxiality is defined as:
${\sigma }^{*}=\frac{{\text{σ}}_{m}}{{\sigma }_{VM}}$
Where,
${\text{σ}}_{m}=\frac{1}{3}\left({\text{σ}}_{1}+{\text{σ}}_{2}+{\text{σ}}_{3}\right)$
Mean (hydrostatic) stress
${\text{σ}}_{VM}=\sqrt{\frac{1}{2}\left[{\left({\text{σ}}_{1}-{\text{σ}}_{2}\right)}^{2}+{\left({\text{σ}}_{2}-{\text{σ}}_{3}\right)}^{2}+{\left({\text{σ}}_{3}-{\text{σ}}_{1}\right)}^{2}\right]}$
von Mises stress
Triaxiality values for some commons stress states can be derived as:
• In pure tension:

${\text{σ}}_{2}={\text{σ}}_{3}=0$ , then ${\sigma }^{*}=\frac{{\text{σ}}_{m}}{{\sigma }_{VM}}=\frac{1}{3}$

• In biaxial compression:

${\text{σ}}_{1}={\text{σ}}_{2}$ , and ${\text{σ}}_{3}=0$ , then ${\sigma }^{*}=\frac{{\text{σ}}_{m}}{{\sigma }_{VM}}=-\frac{2}{3}$

Stress triaxiality for various stress states:
Stress Triaxiality ${\sigma }^{*}$
Stress State
$-\frac{2}{3}$
Biaxial compression
$-\frac{1}{3}$
Uniaxial compression
0
Pure shear
$\frac{1}{3}$
Uniaxial tension
$\frac{1}{\sqrt{3}}$
Plain strain
$\frac{2}{3}$
Biaxial tension