Difference between revisions of "Johnson-Cook Hardening"
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<math>\sigma_y = \left(A + B\alpha^n\right)\left(1 + C \ln {\dot\alpha\over \dot\varepsilon_p^0} | <math>\sigma_y = \left(A + B\alpha^n\right)\left(1 + C \ln {\dot\alpha\over \dot\varepsilon_p^0} | ||
+D \ln \left({\rm max}\left({\dot\alpha\over \dot\varepsilon_p^0},1\right)\right)^{ | +D \ln \left({\rm max}\left({\dot\alpha\over \dot\varepsilon_p^0},1\right)\right)^{n_2}\right)\left(1-T_r^m\right)</math> | ||
where α is cumulative equivalent plastic strain, <math>\dot\alpha</math> is plastic strain rate, and the reduced temperature (T<sub>r</sub>) is given by: | where α is cumulative equivalent plastic strain, <math>\dot\alpha</math> is plastic strain rate, and the reduced temperature (T<sub>r</sub>) is given by: | ||
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| Cjc || Coefficient for rate-dependent term (C). It is dimensionless | | Cjc || Coefficient for rate-dependent term (C). It is dimensionless | ||
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| Djc || Coefficient for rate-dependent term (D). It is dimensionless | | Djc || Coefficient for second rate-dependent term (D). It is dimensionless | ||
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| n2jc || Exponent on rate dependent | | n2jc || Exponent on second rate-dependent term with D (n<sub>2</sub>). It is dimensionless | ||
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| ep0jc || Reference strain rate (<math>\dot\varepsilon_p^0</math>) for reference yield stress in A. It has units [[ConsistentUnits Command#Legacy and Consistent Units|1/time units]]. | | ep0jc || Reference strain rate (<math>\dot\varepsilon_p^0</math>) for reference yield stress in A. It has units [[ConsistentUnits Command#Legacy and Consistent Units|1/time units]]. (default is 1) | ||
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| Tmjc || The material's melting point (T<sub>m</sub>). Above this temperature the yield strength will be zero. It has units degree K. | | Tmjc || The material's melting point (T<sub>m</sub>). Above this temperature the yield strength will be zero. It has units degree K. | ||
Latest revision as of 14:26, 14 August 2019
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} +D \ln \left({\rm max}\left({\dot\alpha\over \dot\varepsilon_p^0},1\right)\right)^{n_2}\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 (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, D, n, n2, m, [math]\displaystyle{ \dot\varepsilon_p^0 }[/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 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 |
| Djc | Coefficient for second rate-dependent term (D). It is dimensionless |
| n2jc | Exponent on second rate-dependent term with D (n2). It is dimensionless |
| ep0jc | Reference strain rate ([math]\displaystyle{ \dot\varepsilon_p^0 }[/math]) for reference yield stress in A. It has units 1/time units. (default is 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 dεp is the incremental plastic strain tensor in one time step.
References
- ↑ 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).