# 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)

- ${\text{\sigma}}_{m}=\frac{1}{3}\left({\text{\sigma}}_{1}+{\text{\sigma}}_{2}+{\text{\sigma}}_{3}\right)$
- Mean (hydrostatic) stress
- ${\text{\sigma}}_{VM}=\sqrt{\frac{1}{2}\left[{\left({\text{\sigma}}_{1}-{\text{\sigma}}_{2}\right)}^{2}+{\left({\text{\sigma}}_{2}-{\text{\sigma}}_{3}\right)}^{2}+{\left({\text{\sigma}}_{3}-{\text{\sigma}}_{1}\right)}^{2}\right]}$
- von Mises stress

- In pure tension:
${\text{\sigma}}_{2}={\text{\sigma}}_{3}=0$ , then ${\sigma}^{*}=\frac{{\text{\sigma}}_{m}}{{\sigma}_{VM}}=\frac{1}{3}$

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

**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