Difference between revisions of "Steinberg-Cochran-Guinan Hardening"
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In the Steinberg-Cochran-Guinan [[Hardening Laws|hardening law]], the yield stress is given by | In the Steinberg-Cochran-Guinan [[Hardening Laws|hardening law]], the yield stress is given by | ||
<math>\sigma_y = \sigma_0\bigl(1 + \beta \ | <math>\sigma_y = \sigma_0\bigl(1 + \beta \alpha^n){G(T,P)\over G_0}</math> | ||
where σ<sub>0</sub> is the initial yield stress, β and n are hardening law properties, | where σ<sub>0</sub> is the initial yield stress, β and n are hardening law properties, α is the cumulative 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: | ||
<math>{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> | <math>{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> | ||
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! Property !! Description | ! Property !! Description | ||
|- | |- | ||
| yield || Initial yield stress (& | | yield || Initial yield stress (σ<sub>0</sub> at zero pressure and the reference temperature. Enter in units of MPa. | ||
|- | |- | ||
| betahard || Yield stress hardening term &beta. It is dimensionless. | | betahard || Yield stress hardening term β. It is dimensionless. | ||
|- | |- | ||
| nhard || Exponent on cumulative plastic strain in hardening term. It is dimensionless. | | nhard || Exponent on cumulative plastic strain in hardening term. It is dimensionless. | ||
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== History Data == | == History Data == | ||
This [[Hardening Laws|hardening law]] | 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 | ||
<math>\sqrt{2\over3}\ ||d\varepsilon_p||</math> | <math>\alpha = \sqrt{2\over3}\ ||d\varepsilon_p||</math> | ||
<references/> | <references/> | ||
Revision as of 12:18, 21 May 2013
In the Steinberg-Cochran-Guinan hardening law, the yield stress is given by
[math]\displaystyle{ \sigma_y = \sigma_0\bigl(1 + \beta \alpha^n){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 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].
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 in the 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 = \sqrt{2\over3}\ ||d\varepsilon_p|| }[/math]
- ↑ 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).