Johnson-Cook Hardening
In the Johnson-Cook hardening law, the yield stress is given by
[math]\displaystyle{ \sigma_y = \left(A + B\varepsilon_p^n\right)\left(1 + C \ln {\dot\varepsilon_p\over \dot\varepsilon_p^0}\right)\left(1-T_r^m\right) }[/math]
where [math]\displaystyle{ \varepsilon_p }[/math] is equivalent plastic strain, [math]\displaystyle{ \dot\varepsilon_p }[/math] is plastic strain rate, and the reduced temperature (Tr) is given by:
[math]\displaystyle{ T_r = {T-T_0\over T_m-T_0} }[/math]
where T0 is the reference temperature, which is equal to the current stress free temperature. For more details see paper by Johnson and Cook[1]
Hardening Law Properties
The material parameters in this hardening law are defined by A, B, C, n, m, [math]\displaystyle{ \dot\varepsilon_p }[/math], and Tm. These parameters are set with the following properties:
| Property | Description |
|---|---|
| Ajc | Parameter A and equal to the initial yield stress at the reference strain rate and the reference temperature. Enter in units of MPa. |
| Bjc | The hardening term B. Enter in units of MPa. |
| njc | Exponent on cumulative plastic strain in hardening term (n). It is dimensionless. |
| Cjc | Coefficient for rate-dependent term (C). It is dimensionless |
| ep0jc | Reference strain rate ([math]\displaystyle{ \dot\varepsilon_p }[/math]) for reference yield stress in A. It has units sec-1. |
| Tmjc | The material's melting point (Tm). Above this temperature the yield strength will be zero. It has units degree K. |
| mjc | Exponent on reduced temperature that defines the temperature dependence of the yield stress. |
The reference temperature, T0, is set using the simulation's stress free temperature and not in the hardening law properties.
History Data
This hardening law define one history variable, which is stored as history variable #1. It stores the the cumulative equivalent plastic strain (absolute) defined as the sum of
[math]\displaystyle{ \sqrt{2\over3}\ ||d\varepsilon_p|| }[/math]
over each time step.
- ↑ G. R. Johnson and W. H. Cook, "A constitutive model and data for metals subjected to large strains, high strain rates ad high temperatures," Proceedings of the 7th International Symposium on Ballistics, 541-547 (1983).