Difference between revisions of "Steinberg-Cochran-Guinan Hardening"
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<math>\sigma_y = \min\left\{\sigma_0\bigl(1 + \beta \alpha\bigr)^n{G(T,P)\over G_0 | <math>\sigma_y = \min\left\{\sigma_0\bigl(1 + \beta \alpha\bigr)^n, \sigma_y^{max}\right\}{G(T,P)\over G_0}</math> | ||
where σ<sub>0</sub> is the initial yield stress, β and n are hardening law properties, α is the cumulative equivalent plastic strain, G(T,P) is the shear modulus (which may depend on temperature and pressure), and G<sub>0</sub> is the initial shear modulus. The shear modulus temperature and pressure dependence are given by: | where σ<sub>0</sub> is the initial yield stress, β and n are hardening law properties, α is the cumulative equivalent plastic strain, G(T,P) is the shear modulus (which may depend on temperature and pressure), and G<sub>0</sub> is the initial shear modulus. The shear modulus temperature and pressure dependence are given by: | ||
Revision as of 16:31, 29 May 2013
In the Steinberg-Cochran-Guinan hardening law, the yield stress is given by
[math]\displaystyle{ \sigma_y = \min\left\{\sigma_0\bigl(1 + \beta \alpha\bigr)^n, \sigma_y^{max}\right\}{G(T,P)\over G_0} }[/math]
where σ0 is the initial yield stress, β and n are hardening law properties, α is the cumulative equivalent plastic strain, G(T,P) is the shear modulus (which may depend on temperature and pressure), and G0 is the initial shear modulus. The shear modulus temperature and pressure dependence are given by:
[math]\displaystyle{ {G(T,P)\over G_0} = 1 + {G_P'\over G_0} P J^{1/3} + {G_T'\over G_0}(T-T_0) }[/math]
where J is the relative volume change (V/V0), GP' and GT' are coefficients for pressure and temperature affects, T is current temperature, and T0 is a reference temperature. For more details, see paper by Steinberg, Cochran, and Guinan.[1]
The Steinberg-Lund Hardening law is similar to this law but adds a a strain rate- and temperature-dependent term.
Hardening Law Properties
This hardening law can set the following properties:
| Property | Description |
|---|---|
| yield | Initial yield stress (σ0 at zero pressure and the reference temperature). Enter in units of MPa. |
| betahard | Yield stress hardening term β. It is dimensionless. |
| nhard | Exponent on cumulative plastic strain in hardening term. It is dimensionless. |
| GPpG0 | The (Gp'/G0) ratio term for pressure dependence of shear modulus. Enter in units MPa-1. Enter 0 to omit pressure dependence in shear modulus. |
| GTpG0 | The (GT'/G0) ratio term for temperature dependence of shear modulus. Enter in units MPa-1. Enter 0 to omit temperature dependence in shear modulus. |
| yieldMax | Maximum yield stress. Enter in units of MPa. |
The reference temperature, T0, is set using the simulations stress free temperature and not set 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
- ↑ D. J. Steinberg S. G. Cochran, and M. W. Guinan, "A constitutive model for metals applicable at high strain rates," J. Appl. Phys., 51, 1498-1504 (1989).