View source for Johnson-Cook Hardening

In the Johnson-Cook [[Hardening Laws|hardening law]], the yield stress is given by

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\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 &alpha; is cumulative equivalent plastic strain, <math>\dot\alpha</math> is plastic strain rate, and the reduced temperature (T<sub>r</sub>) is given by:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>T_r = {T-T_0\over T_m-T_0}</math>

where T<sub>0</sub> is the reference temperature, which is equal to the current [[stress free temperature]]. For more details see paper by Johnson and Cook.<ref>G. R. Johnson and W. H. Cook, &quot;A constitutive model and data for metals subjected to large strains, high strain rates ad high temperatures,&quot; Proceedings of the 7th International Symposium on Ballistics, 541-547 (1983).</ref>

== Hardening Law Properties ==

The material parameters in this [[Hardening Laws|hardening law]] are defined by A, B, C, n, m, <math>\dot\varepsilon_p^0</math>, and T<sub>m</sub>.
These parameters are set with the following properties:

{| class="wikitable"
|-
! 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 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>\dot\varepsilon_p^0</math>) for reference yield stress in A. It has units sec<sup>-1</sup>.
|-
| Tmjc || The material's melting point (T<sub>m</sub>). 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, T<sub>0</sub>, is set using the simulation's [[stress free temperature]] and not set by using [[Hardening Laws|hardening law]] properties.

== History Data ==

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

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\alpha = \sum \sqrt{2\over3}\ ||d\varepsilon_p||</math>

where d&epsilon;<sub>p</sub> is the incremental plastic strain tensor in one time step.

== References ==

<references/>