/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;

ε_{f} = \underset{\begin{array}{l} I n f l u e n c e \begin{matrix}  \end{matrix} o f \\ \begin{matrix}  \end{matrix} t r i a x i a l i t y \end{array}}{\underset{︸}{[D_{1} + D_{2} \exp (D_{3} σ^{*})]}} \underset{\begin{array}{l} I n f l u e n c e \begin{matrix}  \end{matrix} o f \\ \begin{matrix}  \end{matrix} s t r a i n \begin{matrix}  \end{matrix} r a t e \end{array}}{\underset{︸}{[1 + D_{4} \ln ({\dot{ε}}^{*})]}} \underset{\begin{matrix} \begin{array}{l} I n f l u e n c e \begin{matrix}  \end{matrix} o f \\ t e m p e r a t u r e \end{array} \end{matrix}}{\underset{︸}{[1 + D_{5} T^{*}]}}

Where,

$ε_{f}$: Plastic failure strain
${\dot{ε}}^{*} = \frac{\dot{ε}}{{\dot{ε}}_{0}}$: 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:

Figure 1. Example Plot of a Johnson-Cook Failure Model

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:

ε_{f} = D_{1} + D_{2} \cdot exp (D_{3} \cdot σ^{*})

Using 3 failure data points from test:

$ε_{f} = 0.1585$ by uniaxial tension ( $σ^{*} = 1 / 3$ )
$ε_{f} = 0.19$ by pure shear ( $σ^{*} = 0$ )
$ε_{f} = 0.2419$ by uniaxial compression ( $σ^{*} = - 1 / 3$ )

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

{\begin{matrix} 0.1585 = D_{1} + D_{2} \cdot exp (D_{3} \cdot \frac{1}{3}) \\ 0.19 = D_{1} + D_{2} \cdot exp (D_{3} \cdot 0) \\ 0.2419 = D_{1} + D_{2} \cdot exp (D_{3} \cdot - \frac{1}{3}) \end{matrix}

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{Δ ε_{p}}{ε_{f}} \geq 1

What happens when

D \geq 1

depends on the values of element failure flags (I_{fail_sh} andI_{fail_so}) and XFEM formulation flag (I_xfem). When the XFEM formulation is not used (I_xfem=0), the following table summarizes the different element failure flag options:

Table 1. Element Failure Options
Element	Element Failure Flag	If $D \geq 1$	Failure Behavior
Shell	`I`_{fail_sh}=1 (Default)	In 1 IP or layer	Element deleted
Shell	`I`_{fail_sh}=2	In 1 IP or layer	Stress tensor set to zero in IP or layer
Shell	`I`_{fail_sh}=2	All IP or layer	Element deleted
Solid	`I`_{fail_sh}=1 (Default)	In 1 IP	Element deleted
Solid	`I`_{fail_sh}=2	In 1 IP	Stress tensor set to zero in IP
Solid	`I`_{fail_sh}=2	All IP	Stress tensor set to zero in element

Details on the XFEM formulation (I_xfem=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.