# 4.37.2. Johnson-Cook Failure

The Johnson-Cook model [[1]] is implemented with the form,

(4.259)\[d = \int_0^{\bar{\varepsilon}^p} \frac{ d\bar{\varepsilon}^{p} }{ \left(D_1 + D_2 \exp(D_3 \eta)\right) \left(1 + D_4 \langle \ln \frac{\dot{\bar{\varepsilon}}^p }{\dot{\varepsilon}_0 } \rangle \right) \left( 1 + D_5 T^* \right) },\]

where \(\{D_1, D_2, D_3, D_4, D_5\}\) are fitting constants and \(\dot{\varepsilon}_0\) is a reference strain rate. The term \(\eta\) represents stress triaxiality, the ratio of mean hydrostatic stress to von-Mises stress: \(\eta = \frac{p}{\sigma_{vm}}\). The term \(T^*\) represents the homologous temperature, given as a function of the temperature \(T\) by,

\[T^* = \frac{T - T_{\text{ref}} }{T_{\text{melt}} - T_{\text{ref}}},\]

where \(T_{\text{ref}}\) is a reference temperature and \(T_{\text{melt}}\) is the melting temperature.

The Johnson-Cook failure model form (4.259) is formulated as a multiplicative combination of triaxiality, strain-rate, and temperature effects, and the denominator may be interpreted as the critical failure strain. The failure process initiates once the total quantity reaches \(d = 1\).

## 4.37.2.1. User Guide

    #
    # JOHNSON_COOK_FAILURE Failure model definitions
    #
    JOHNSON COOK D1       = <real>
    JOHNSON COOK D2       = <real>
    JOHNSON COOK D3       = <real>
    JOHNSON COOK D4       = <real>
    JOHNSON COOK D5       = <real>
    #
    #Following Johnson-Cook parameters can only be defined once.  As such, only
    #  needed if not previously defined via Johnson-Cook multipliers
    #  w/ flow-stress hardening.  Does need to be defined
    #  w/ Decoupled Flow Stress
    #
    REFERENCE RATE        = <real>
    REFERENCE TEMPERATURE = <real>
    MELTING TEMPERATURE   = <real>

[1]

G.R. Johnson and W.H. Cook. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Engineering Fracture Mechanics, 21(1):31–48, 1985.