Johnson-Cook Hardening

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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.

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.