Difference between revisions of "Steinberg-Lund Hardening"
(Created page with "In the Steinberg-Lund hardening law, the yield stress is given by <math>\sigma_y = \left(Y_T(\dot\alpha,T)+\sigma_0\bigl(1 + \beta \alpha^n\bigr)\right){G(...") |
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<math>{d\alpha(Y_T,T)\over dt} = \left[{C_2\over Y_T} + {1\over C_1}\exp\left({2U_k\over kT}\left(1 - {Y_T\over Y_P}^2\right)\right)\right]^{-1}</math> | <math>{d\alpha(Y_T,T)\over dt} = \left[{C_2\over Y_T} + {1\over C_1}\exp\left({2U_k\over kT}\left(1 - {Y_T\over Y_P}^2\right)\right)\right]^{-1}</math> | ||
The shear modulus temperature and pressure dependence are given by the same | where C<sub>1</sub>, C<sub>2</sub>, U<sub>k</sub>, and Y<sub>P</sub> are hardening law parameters. | ||
The shear modulus temperature and pressure dependence are given by the same function as in the [[Stenberg-Cochran-Guinan Hardening]] law: | |||
<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> | ||
where J is the relative volume change (V/V<sub>0</sub>), G<sub>P</sub>' and G<sub>T</sub>' are coefficients for pressure and temperature affects, T is current temperature, and T<sub>0</sub> is a reference temperature. For more details, see paper by Steinberg | where J is the relative volume change (V/V<sub>0</sub>), G<sub>P</sub>' and G<sub>T</sub>' are coefficients for pressure and temperature affects, T is current temperature, and T<sub>0</sub> is a reference temperature. For more details, see paper by Steinberg and Lund<ref>D. J. Steinberg and C. M. Lund, "A constitutive model for strain rates from 10-4 to 106," J. Appl. Phys., 65, 1528-1533 (1989).</ref>. | ||
== Hardening Law Properties == | == Hardening Law Properties == | ||
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| GTpG0 || The (G<sub>T</sub>'/G<sub>0</sub>) ratio term for temperature dependence of shear modulus. Enter in units MPa<sup>-1</sup>. Enter 0 to omit temperature dependence in shear modulus. | | GTpG0 || The (G<sub>T</sub>'/G<sub>0</sub>) ratio term for temperature dependence of shear modulus. Enter in units MPa<sup>-1</sup>. Enter 0 to omit temperature dependence in shear modulus. | ||
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| C1SL || The C<sub>1</sub> constant in this law entered in units of 1/sec. | |||
|- | |||
| C2SL || The C<sub>2</sub> constant in this law entered in units of MPa-sec. | |||
|- | |||
| YP || The Peierls stress (Y<sub>P</sub>) and also the maximum rate-dependent yield stress. Enter in units of MPa. | |||
|- | |||
| Uk || An energy associated with forming kinks (U<sub>k</sub>). It has units eV. | |||
|- | |- | ||
| yieldMax || Maximum yield stress. Enter in units of MPa. | | yieldMax || Maximum yield stress. Enter in units of MPa. | ||
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== History Data == | == History Data == | ||
This [[Hardening Laws|hardening law]] defines | This [[Hardening Laws|hardening law]] defines three history variables, wihich are the cumulative equivalent plastic strain (absolute) defined as: | ||
<math>\alpha = \sqrt{2\over3}\ ||d\varepsilon_p||</math>, | |||
the current rate- and temperature-dependent yield stress (Y<sub>T</sub>) in MPa, and the current equivalent plastic strain rate (dα/dt in 1/sec). These variables are stored as as history variables #1, #2, and #3. | |||
<references/> | <references/> | ||
Revision as of 16:41, 21 May 2013
In the Steinberg-Lund hardening law, the yield stress is given by
[math]\displaystyle{ \sigma_y = \left(Y_T(\dot\alpha,T)+\sigma_0\bigl(1 + \beta \alpha^n\bigr)\right){G(T,P)\over G_0} }[/math]
where [math]\displaystyle{ Y_T(\dot\alpha,T) }[/math] is a strain rate and temperature dependent term, 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 strain rate and temperature dependent term is defined in inverse form:
[math]\displaystyle{ {d\alpha(Y_T,T)\over dt} = \left[{C_2\over Y_T} + {1\over C_1}\exp\left({2U_k\over kT}\left(1 - {Y_T\over Y_P}^2\right)\right)\right]^{-1} }[/math]
where C1, C2, Uk, and YP are hardening law parameters. The shear modulus temperature and pressure dependence are given by the same function as in the Stenberg-Cochran-Guinan Hardening law:
[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 and Lund[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. |
| C1SL | The C1 constant in this law entered in units of 1/sec. |
| C2SL | The C2 constant in this law entered in units of MPa-sec. |
| YP | The Peierls stress (YP) and also the maximum rate-dependent yield stress. Enter in units of MPa. |
| Uk | An energy associated with forming kinks (Uk). It has units eV. |
| 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 three history variables, wihich are the cumulative equivalent plastic strain (absolute) defined as:
[math]\displaystyle{ \alpha = \sqrt{2\over3}\ ||d\varepsilon_p|| }[/math],
the current rate- and temperature-dependent yield stress (YT) in MPa, and the current equivalent plastic strain rate (dα/dt in 1/sec). These variables are stored as as history variables #1, #2, and #3.
- ↑ D. J. Steinberg and C. M. Lund, "A constitutive model for strain rates from 10-4 to 106," J. Appl. Phys., 65, 1528-1533 (1989).