# /FAIL/JOHNSON

The Johnson-Cook failure model is often used to describe the ductile failure of metals. It uses a Johnson-Cook equation to define failure strain as a function of stress triaxiality.

In the Johnson-Cook failure model, there are three parts to the failure model;

${\epsilon }_{f}=\underset{\begin{array}{l}Influence\begin{array}{c}\end{array}of\\ \begin{array}{c}\end{array}triaxiality\end{array}}{\underbrace{\left[{D}_{1}+{D}_{2}\mathrm{exp}\left({D}_{3}{\sigma }^{*}\right)\right]}}\underset{\begin{array}{l}Influence\begin{array}{c}\end{array}of\\ \begin{array}{c}\end{array}strain\begin{array}{c}\end{array}rate\end{array}}{\underbrace{\left[1+{D}_{4}\mathrm{ln}\left({\stackrel{˙}{\epsilon }}^{*}\right)\right]}}\underset{\begin{array}{c}\begin{array}{l}Influence\begin{array}{c}\end{array}of\\ temperature\end{array}\end{array}}{\underbrace{\left[1+{D}_{5}{T}^{*}\right]}}$

Where,
${\epsilon }_{f}$
Plastic failure strain
${\stackrel{˙}{\epsilon }}^{*}=\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}$
Current strain rate divided by the input reference strain rate
${T}^{*}$
Computed in the material law or /HEAT/MAT
Ignoring the influence of strain rate and temperature a plot of the Johnson-Cook failure is:

Plastic strains above the curve represent material fracture and below the curve no material fracture.

In a simple case where only the triaxiality influence is considered, the failure strain is:

${\epsilon }_{f}={\text{D}}_{1}+{\text{D}}_{2}\cdot \text{exp}\left({\text{D}}_{3}\cdot {\sigma }^{*}\right)$

Using 3 failure data points from test:
• ${\epsilon }_{f}=0.1585$ by uniaxial tension ( ${\sigma }^{*}=1/3$ )
• ${\epsilon }_{f}=0.19$ by pure shear ( ${\sigma }^{*}=0$ )
• ${\epsilon }_{f}=0.2419$ by uniaxial compression ( ${\sigma }^{*}=-1/3$ )

The parameters ${\text{D}}_{1}$ , ${\text{D}}_{2}$ and ${\text{D}}_{3}$ could be calculated analytically by solving the following equations:

$\left\{\begin{array}{c}0.1585={\text{D}}_{1}+{\text{D}}_{2}\cdot \text{exp}\left({\text{D}}_{3}\cdot \frac{1}{3}\right)\\ 0.19={\text{D}}_{1}+{\text{D}}_{2}\cdot \text{exp}\left({\text{D}}_{3}\cdot 0\right)\\ 0.2419={\text{D}}_{1}+{\text{D}}_{2}\cdot \text{exp}\left({\text{D}}_{3}\cdot -\frac{1}{3}\right)\end{array}$

## Element Failure treatment

A cumulative damage method is used to sum the amount of plastic strain that has occurred in the element using:

$D=\sum \frac{\text{Δ}{\epsilon }_{p}}{{\epsilon }_{f}}\ge 1$

What happens when $D\ge 1$ depends on the values of element failure flags (Ifail_sh andIfail_so) and XFEM formulation flag (Ixfem). When the XFEM formulation is not used (Ixfem=0), the following table summarizes the different element failure flag options:
Table 1. Element Failure Options
Element Element Failure Flag If $D\ge 1$ Failure Behavior
Shell Ifail_sh=1

(Default)

In 1 IP or layer Element deleted
Shell Ifail_sh=2 In 1 IP or layer Stress tensor set to zero in IP or layer
Shell Ifail_sh=2 All IP or layer Element deleted
Solid Ifail_sh=1

(Default)

In 1 IP Element deleted
Solid Ifail_sh=2 In 1 IP Stress tensor set to zero in IP
Solid Ifail_sh=2 All IP Stress tensor set to zero in element

Details on the XFEM formulation (Ixfem=1), can be found in /FAIL/JOHNSON.

The damage, $D$ , can be plotted in animation files using /ANIM/SHELL/DAMA or /ANIM/BRICK/DAMA. This will show the risk of material damage.