Using Hardening Laws

Some materials allow the use of hardening laws. The material class defines the elastic equation of state and the hardening law implements the yield criterion. Hardening laws are assigned to materials within their material definition:

Material (matid),(name),(type)
      . . .
   Hardening (Law_Name or number)
   (Law properties)
      . . .
Done

where the plasticLaw command selects the law (using the law name or number from the list below) and each material can have only one law. The law selection is followed by commands to set any properties specific to that law. The following list has the available hardening laws with the number and name. Click any one to see the law and the properties needed for that law.

  1. Linear - Linear hardening
  2. Nonlinear - Nonlinear hardening
  3. JohnsonCook - Johnson Cook hardening law
  4. SCGL - Stenberg-Cochran-Guinan hardening
  5. SL - Steinberg-Lund hardening
  6. Nonlinear2 - Alternate nonlinear hardening
  7. DDB-PPM - A dislocation density based polycrystal plasticity model

More documentation on these hardening laws is available on the OSUPDocs wiki.

With these laws, the material yields when f=0 according to the plastic potential

f = ||s|| - sqrt(2/3) * σY

where s is deviatoric stress and σY is the current yield stress, which is determined by the hardening law.

Linear Hardening

In linear hardening the yield stress is given by

σY = σY0 + Epα = σY0(1 + Kα)

where σY0 is the initial yield stress, Ep is the plastic modulus, K is hardening coefficient, and α is an internal variable that tracks cumulative, equivalent plastic strain (see history variable below).

The input properties are:

History Data for Linear Hardening

The one history variable is the cumulative equivalent plastic strain, α, defined as the sum of sqrt(2/3)||dεp||. This variable can be archived as history variable 1.

Nonlinear Hardening

In nonlinear hardening the yield stress is given by

σY = σY0(1 + Kα)n

where K and n are dimensionless hardening parameters.

The input properties are:

History Data for Nonlinear Hardening

The one history variable is the cumulative equivalent plastic strain, α, defined as the sum of sqrt(2/3)||dεp||. This variable can be archived as history variable 1.

Alternate Nonlinear Hardening

In this alternate nonlinear hardening the yield stress is given by

σY = σY0(1 + Kαn)

where K and n are dimensionless hardening parameters. The input properties and history data are the same as for the other nonlinear hardening law.

Johnson-Cook Hardening

The empirical Johnson-Cook hardening law is:

σy = (Ajc + Bjc εpnjc) [1 + Cjc ln(dεp/ep0jc) + Djc ln(max(dεp/ep0jc,1))n2jc] (1 - Trmjc)

where εp is equivalent plastic strain, dεp is plastic strain rate, and the reduced temperature (Tr) is given by

Tr = (T - T0) / (Tmjc - T0)

and T0 is the current stress free temperature. See Johnson and Cook (1983) for details. The law properties are:

History Data for Johnson-Cook Material

The one history variable is the cumulative equivalent plastic strain, α, defined as the sum of sqrt(2/3)||dεp||. This variable can be archived as history variable 1.

Steinberg-Cochran-Lund Hardening

For plasticity, the yielding (or strength) of this material is determined by analysis from Steinberg, Cochran, and Guinan. Both the shear modulus and the yield stress change. The shear modulus is given by

G = G0 [ 1 + (Gp'/G0)(P/η1/3) + (GT'/G0)(T-T0) ]

where η = 1/(1-x) and x = -ΔV/V0. Here T0 is the reference temperature for the material that is set using the stress free temperature command.

This yield stress is given by the hardening law:

σy = σ0 (1 + β εpn) G

where εp is the equivalent plastic strain and G is current shear modulus from above. The yield stress is limited to a maximum yield stress. See Steinberg, Cochran, and Guinan (1989) for details. The properties for this law are:

Note that plasticity is implemented using J2 flow theory, which is a theory that assumes f depends on ||s|| and is independent of pressure. The yield stress, however, has a pressure-dependent term (see (Gp'/G0) term). The effect may be small and it appears that other implementations of this material take this same approach.

History Data for SCGL Hardening Law

The history variable is the cumulative equivalent plastic strain (absolute) defined as the sum of sqrt(2/3)||dεp||. This variable can be archived as history variable 1.

Steinberg-Lund Material

This hardening law has a rate- and temperature-dependent yield stress given by:

σy = {YT(dεp/dt,T) + σ0(1 + β εpn)} G

where G is given by same expression as for the SCGL law. This law is identical to the SCGL law except for the new rate- and temperature-dependent term YT(dεp/dt,T), which is only defined in inverse form:

p(YT,T)/dt = { (C2/YT) + (1/C1)exp[(2Uk/(kT))(1-(YT/YP))2] }-1

where εp is the equivalent plastic strain. This equation is numerically inverted to find YT as a function of strain rate and YT is limited to ≤ YP. See Steinberg and Lund (1989) for details.

Note that plasticity is implemented using J2 flow theory, and therefore has the same concerns as the SCGL hardening law when the pressure-dependent term is non-zero.

History Data for Steinberg-Lund Hardening Law

The three history variables are the cumulative equivalent plastic strain (absolute) defined as the sum of sqrt(2/3)||dεp||, the current rate- and temperature-dependent yield stress (YT) in pressure units, and the current equivalent plastic strain rate (dεp/dt in 1/sec). These variables can be archived as history variables 1, 2, and/or 3.

Dislocation Density Law

This hardening law is a dislocation density based polycrystal plasticity model. The documentation for ths law is currently all on the OSUPDocs wiki.