View source for Steinberg-Lund Hardening

In the Steinberg-Lund [[Hardening Laws|hardening law]], the yield stress is given by

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\sigma_y = \left(\min\left\{Y_T(\dot\alpha,T),Y_P\right\}+\min\left\{\sigma_0\bigl(1 + \beta \alpha\bigr)^n,\sigma_y^{max}\right\}\right){G(T,P)\over G_0}</math>

where <math>Y_T(\dot\alpha,T)</math> is a strain rate and temperature dependent term, &sigma;<sub>0</sub> is the initial yield stress, &beta; and n are hardening law properties, &alpha; 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 strain rate and temperature dependent term is defined in inverse form:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>{d\alpha(Y_T,T)\over dt} = \left[{1\over C_1}\exp\left({2U_k\over kT}\left(1 - {Y_T\over Y_P}\right)^2\right)+{C_2\over Y_T} \right]^{-1}</math>

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 is given by the same function as in the [[Steinberg-Cochran-Guinan Hardening]] law:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<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 and Lund<ref>D. J. Steinberg and C. M. Lund, &quot;A constitutive model for strain rates from 10-4 to 106,&quot; J. Appl. Phys., 65, 1528-1533 (1989).</ref>.

== Hardening Law Properties ==

This [[Hardening Laws|hardening law]] can set the following properties:

{| class="wikitable"
|-
! Property !! Description 
|-
| yield ||  Initial yield stress (&sigma;<sub>0</sub> at zero pressure and the reference temperature). Enter in units of MPa.
|-
| betahard || Yield stress hardening term &beta;. It is dimensionless.
|-
| nhard || Exponent on cumulative equivalent plastic strain in hardening term. It is dimensionless.
|-
| GPpG0 || The (G<sub>p</sub>'/G<sub>0</sub>) ratio term for pressure dependence of shear modulus. Enter in units MPa<sup>-1</sup>. Enter 0 to omit pressure 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.
|-
| 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.
|}

The reference temperature, T<sub>0</sub>, is set using the simulation's [[stress free temperature]] and not set using [[Hardening Laws|hardening law]] properties.

== History Data ==

This [[Hardening Laws|hardening law]]  defines three history variables, wihich are the cumulative equivalent plastic strain (absolute) defined as:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\alpha = \sum \sqrt{2\over3}\ ||d\varepsilon_p||</math>

where d&epsilon;<sub>p</sub> is the incremental plastic strain tensor in one time step,
the current rate- and temperature-dependent yield stress (Y<sub>T</sub>) in MPa, and the current plastic strain rate (d&alpha;/dt in 1/sec). These variables are stored as history variables #1, #2, and #3.

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