Steinberg-Cochran-Guinan Hardening

Yield Stress

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

      [math]\displaystyle{ \sigma_y = \min\left\{\sigma_0\bigl(1 + \beta \alpha\bigr)^n, \sigma_y^{max}\right\}{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 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 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), 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, Cochran, and Guinan.

The Steinberg-Lund Hardening law is similar to this law but adds a a strain rate- and temperature-dependent term.

Plasticity Theory

This law (and the Steinberg-Lund Hardening law) are implemented use J₂ plasticity theory. In associative, J₂ plasticity theory, the yield function has to depend only on deviatoric stress and cannot depend on pressure, but these laws have yield strengths that depend on pressure. They can, however be used in J₂ plasticity as non-associative laws. The plastic flow is determined by the failure surface defined by:

      [math]\displaystyle{ \Phi = \|{\bf s}\| - \sqrt{2\over3}\min\left\{\sigma_0\bigl(1 + \beta \alpha\bigr)^n, \sigma_y^{max}\right\}{G(T,P)\over G_0} }[/math]

where s is the deviatoric stress. The plastic strain and forces, however, are determined by a different plastic potential of:

      [math]\displaystyle{ \Psi = \|{\bf s}\| - \sqrt{2\over3}\min\left\{\sigma_0\bigl(1 + \beta \alpha\bigr)^n, \sigma_y^{max}\right\} }[/math]

Because these two functions different, the implementation is formally non-associative plasticity.

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 plastic strain in hardening term. It is dimensionless.
GPpG0	The (Gp'/G0) ratio term for pressure dependence of shear modulus. Enter in units 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.
yieldMax	Maximum yield stress. Enter in pressure units.

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

where dε_p is the incremental plastic strain tensor in one time step.