Steinberg-Lund Hardening

Yield Stress

In the Steinberg-Lund hardening law, the yield stress is given by

      [math]\displaystyle{ \sigma_y = \left(\min\left\{\sigma_0\bigl(1 + \beta \alpha\bigr)^n,\sigma_y^{max}\right\}+\min\left\{Y_T(\dot\alpha,T),Y_P\right\}\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 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 strain rate and temperature dependent term is defined in inverse form:

      [math]\displaystyle{ {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 C₁, C₂, U_k, and Y_P 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:

      [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), G_P' and G_T' 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.

Plasticity Theory

The implementation of this hardening law is formally non-associative J₂ plasticity theory as explained here.

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 pressure units.
betahard	Yield stress hardening term β. It is dimensionless.
nhard	Exponent on cumulative equivalent plastic strain in hardening term. It is dimensionless.
GPpG0	The (Gp'/G0) ratio term for pressure dependence of shear modulus. Enter in pressure-1 units. Enter 0 to omit pressure dependence in shear modulus.
GTpG0	The (GT'/G0) ratio term for temperature dependence of shear modulus. Enter in units K-1. Enter 0 to omit temperature dependence in shear modulus.
C1SL	The C1 constant in this law entered in 1/time units.
C2SL	The C2 constant in this law entered in pressure-time units.
YP	The Peierls stress (YP) and also the maximum rate-dependent yield stress. Enter in pressure units.
UkOverk	An energy associated with forming kinks (entered as Uk/k). It has units degrees K.
yieldMax	Maximum yield stress. Enter in pressure units.

The reference temperature, T₀, is set using the simulation's stress free temperature and not set using 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 = \sum \sqrt{2\over3}\ ||d\varepsilon_p|| }[/math]

where dε_p is the incremental plastic strain tensor in one time step, the current rate- and temperature-dependent yield stress (Y_T) in pressure units, and the current plastic strain rate (dα/dt in 1/time units). These variables are stored as history variables #1, #2, and #3.