Main Help → All Commands → Defining Materials → 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.
Linear
- Linear hardeningNonlinear
- Nonlinear hardeningJohnsonCook
- Johnson Cook hardening lawSCGL
- Stenberg-Cochran-Guinan hardeningSL
- Steinberg-Lund hardeningNonlinear2
- Alternate nonlinear hardeningDDB-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.
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:
yield
- The initial yield stress (in pressure units). This stress corresponds to the axial stress at yield during uniaxial, 3D loading.Ep
- The plastic modulus (in pressure units). This modulus is the slope of total stress as a function of plastic strain during uniaxial, 3D loading. The default is 0.0 which results in an elastic-perfectly plastic material or a material with no work hardening. Ep
can only be non-negative. To modeling softening with negative Ep
, omit this parameter and enter a negtative Khard
instead.Khard
-
Alternatively, you can enter this dimensionless parameter for hardening. It is only used in Ep is not entered and when entered, it is convert to Ep using Ep = σY0K.yieldMin
-
The minimum yield stress (in pressure units with default of 0). A minimum stress is only used when modeling softening by entering Khard
< 0.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.
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:
yield
-
The initial yield stress (in pressure units). This stress corresponds to the axial stress at yield during uniaxial, 3D loading.Khard
-
The dimensionless coefficient for nonlinear hardening.nhard
-
The dimensionless exponent parameter in the nonlinear hardening. If n=1, it is more efficient to use linear hardening stead.yieldMin
-
The minimum yield stress (in pressure units with default of 0). A minimum stress is only used when modeling softening by entering Khard
< 0.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.
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.
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:
Ajc
- The initial yield stress at reference rate and temperature in pressure units.Bjc
- Hardening term in the yield stress in pressure units.njc
- Exponent for power-law hardening term. It is dimensionless.Cjc
- Coefficient for rate-dependent term. It is dimensionlessDjc
- Coefficient for second rate-dependent term. It is dimensionlessn2jc
- Exponent on second rate-dependent term with D. It is dimensionlessep0jc
- Reference strain rate for reference yield stress in Ajc
in 1/time units. (default is 1)Tmjc
- The material's melting point. It has units degree K.mjc
- Exponent in the thermal term. It is dimensionless.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.
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:
yield
- Initial yield stress (σ0) at zero pressure and the reference temperature in pressure units.yieldMax
- Maximum yield stress in pressure units.betahard
- Yield stress hardening term. It is dimensionless.nhard
- Exponent on cumulative plastic strain in hardening term. It is dimensionless.GPpG0
- Gp'/G0 term for pressure dependence of shear modulus and yield stress in Steinberg, Cochran, and Guinan form in 2/time units. Enter 0 to omit pressure dependence in shear modulus.GTpG0
- GT'/G0 term for temperature dependence of shear modulus and yield stress in Steinberg, Cochran, and Guinan form. Enter in units K-1. Enter 0 to omit temperature dependence in shear modulus.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.
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.
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:
dεp(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.
C1SL
- Material constant in Steinberg-Lund formulation entered in units of 1/time units.C2SL
- Material constant in Steinberg-Lund formulation entered in units of pressure unit-time units.YP
- The Peierls stress and also the maximum rate-dependent yield stress in pressure units.UkOverk
- an energy associated with forming kinks (enter Uk/k in degrees K)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.
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.
This hardening law is a dislocation density based polycrystal plasticity model. The documentation for ths law is currently all on the OSUPDocs wiki.