Steinberg-Lund Hardening
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]
The shear modulus temperature and pressure dependence are given by the same law as in the [[]]:
[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).