Johnson-Cook Hardening

In the Johnson-Cook hardening law, the yield stress is given by

      [math]\displaystyle{ \sigma_y = \left(A + B\alpha^n\right)\left(1 + C \ln {\dot\alpha\over \dot\varepsilon_p^0}\right)\left(1-T_r^m\right) }[/math]

where α is cumulative equivalent plastic strain, [math]\displaystyle{ \dot\alpha }[/math] is plastic strain rate, and the reduced temperature (T_r) 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.

Hardening Law Properties

The material parameters in this hardening law are defined by A, B, C, n, m, [math]\displaystyle{ \dot\varepsilon_p^0 }[/math], and T_m. 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 pressure units.
Bjc	The hardening term B. Enter in pressure units.
njc	Exponent on cumulative equivalent 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^0 }[/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 set by using hardening law properties.

History Data

This hardening law defines one history variable, which is stored as history variable #1. It stores the the cumulative equivalent plastic strain (absolute) defined as

      [math]\displaystyle{ \alpha = \sum \sqrt{2\over3}\ ||d\varepsilon_p|| }[/math]

where dε_p is the incremental plastic strain tensor in one time step.