The Johnson-Cook plasticity model is a particular type of Mises plasticity model with
analytical forms of the hardening law and rate dependence.
The Johnson-Cook plasticity model:
is suitable for high-strain-rate deformation of many materials, including most metals;
is typically used in adiabatic transient dynamic simulations;
can be used in conjunction with the Johnson-Cook dynamic failure model in Abaqus/Explicit;
can be used to specify the plastic response for models defined using the parallel
rheological framework (Parallel Rheological Framework) in Abaqus/Standard;
can be used in conjunction with the tensile failure model to model tensile spall or a
pressure cutoff in Abaqus/Explicit;
can be used in conjunction with the progressive damage and failure models (Progressive Damage and Failure)
to specify different damage initiation criteria and damage evolution laws that allow for
the progressive degradation of the material stiffness and the removal of elements from
the mesh; and
A Mises yield surface with associated flow is used in the Johnson-Cook
plasticity model.
Johnson-Cook Hardening
Johnson-Cook hardening is a particular type of isotropic hardening where the
static yield stress, σ0,
is assumed to be of the form
σ0=[A+B(ˉεpl)n](1-ˆθm),
where ˉεpl
is the equivalent plastic strain and A,
B, n, and m are
material parameters. ˆθ
is the nondimensional temperature defined as
ˆθ≡{0for
where is the current temperature, is the melting temperature, and is the transition temperature defined as the one at or below which there
is no temperature dependence of the yield stress. The material parameters
A, B, and n must be
measured at or below the transition temperature. The material parameter
m should be determined based on measurements above the transition
temperature. Temperature dependency of is ignored if you specify a zero value or if you do not specify a value
for m.
When ,
the material will be melted and will behave like a fluid; there will be no
shear resistance since .
The hardening memory will be removed by setting the equivalent plastic strain
to zero. If backstresses are specified for the model, these will also be set to
zero.
If you include annealing behavior in the material definition and the
annealing temperature is defined to be less than the melting temperature
specified for the metal plasticity model, the hardening memory will be removed
at the annealing temperature and the melting temperature will be used strictly
to define the hardening function. Otherwise, the hardening memory will be
removed automatically at the melting temperature. If the temperature of the
material point falls below the annealing temperature at a subsequent point in
time, the material point can work harden again. For more details, see
Annealing or Melting.
You provide the values of A, B,
n, m, ,
and
as part of the metal plasticity material definition.
Abaqus/Explicit
provides a dynamic failure model specifically for the Johnson-Cook plasticity
model, which is suitable only for high-strain-rate deformation of metals. This
model is referred to as the “Johnson-Cook dynamic failure model.”
Abaqus/Explicit
also offers a more general implementation of the Johnson-Cook failure model as
part of the family of damage initiation criteria, which is the recommended
technique for modeling progressive damage and failure of materials (see
About Damage and Failure for Ductile Metals).
The Johnson-Cook dynamic failure model is based on the value of the equivalent
plastic strain at element integration points; failure is assumed to occur when
the damage parameter exceeds 1. The damage parameter, ,
is defined as
where
is an increment of the equivalent plastic strain,
is the strain at failure, and the summation is performed over all increments in
the analysis. The strain at failure, ,
is assumed to be dependent on a nondimensional plastic strain rate,
;
a dimensionless pressure-deviatoric stress ratio,
(where p is the pressure stress and q
is the Mises stress); and the nondimensional temperature,
,
defined earlier in the Johnson-Cook hardening model. The dependencies are
assumed to be separable and are of the form
where –
are failure parameters measured at or below the transition temperature,
,
and
is the reference strain rate. You provide the values of
–
when you define the Johnson-Cook dynamic failure model. This expression for
differs from the original formula published by
Johnson
and Cook (1985) in the sign of the parameter .
This difference is motivated by the fact that most materials experience an
increase in
with increasing pressure-deviatoric stress ratio; therefore,
in the above expression will usually take positive values.
When this failure criterion is met, the deviatoric stress components are set
to zero and remain zero for the rest of the analysis. Depending on your choice,
the pressure stress may also be set to zero for the rest of calculation (if
this is the case, you must specify element deletion and the element will be
deleted) or it may be required to remain compressive for the rest of the
calculation (if this is the case, you must choose not to use element deletion).
By default, the elements that meet the failure criterion are deleted.
The Johnson-Cook dynamic failure model is suitable for high-strain-rate deformation of metals;
therefore, it is most applicable to truly dynamic situations. For quasi-static problems that
require element removal, the progressive damage and failure models or the Gurson metal
plasticity model (Porous Metal Plasticity) are recommended.
The use of the Johnson-Cook dynamic failure model requires the use of
Johnson-Cook hardening but does not necessarily require the use of Johnson-Cook
strain rate dependence. However, the rate-dependent term in the Johnson-Cook
dynamic failure criterion will be included only if Johnson-Cook strain rate
dependence is defined. The Johnson-Cook damage initiation criterion described
in
Damage Initiation for Ductile Metals
does not have these limitations.
Input File Usage
Use both of the
following options:
PLASTIC, HARDENING=JOHNSON COOKSHEAR FAILURE, TYPE=JOHNSON COOK,
ELEMENT DELETION=YES or NO
Abaqus/CAE Usage
Johnson-Cook dynamic failure
is not supported in
Abaqus/CAE.
Progressive Damage and Failure
The Johnson-Cook plasticity model can be used in conjunction with the
progressive damage and failure models discussed in
About Damage and Failure for Ductile Metals.
The capability allows for the specification of one or more damage initiation
criteria, including ductile, shear, forming limit diagram
(FLD), forming limit stress diagram
(FLSD), Müschenborn-Sonne forming limit
diagram (MSFLD), and, in
Abaqus/Explicit,
Marciniak-Kuczynski (M-K) criteria. After
damage initiation, the material stiffness is degraded progressively according
to the specified damage evolution response. The models offer two failure
choices, including the removal of elements from the mesh as a result of tearing
or ripping of the structure. The progressive damage models allow for a smooth
degradation of the material stiffness, making them suitable for both
quasi-static and dynamic situations. This is a great advantage over the dynamic
failure models discussed above.
Property module: material editor: MechanicalDamage for Ductile Metalsdamage initiation type: specify the damage initiation criterion: SuboptionsDamage Evolution: specify the damage evolution parameters
Tensile Failure
In
Abaqus/Explicit
the tensile failure model can be used in conjunction with the Johnson-Cook
plasticity model to define tensile failure of the material. The tensile failure
model uses the hydrostatic pressure stress as a failure measure to model
dynamic spall or a pressure cutoff and offers a number of failure choices
including element removal. Similar to the Johnson-Cook dynamic failure model,
the
Abaqus/Explicit
tensile failure model is suitable for high-strain-rate deformation of metals
and is most applicable to truly dynamic problems. For more details, see
Dynamic Failure Models.
Property module: material editor: MechanicalPlasticityPlastic: Hardening: Johnson-Cook: SuboptionsTensile Failure
Heat Generation by Plastic Work
Abaqus
allows for an adiabatic thermal-stress analysis (Adiabatic Analysis),
a fully coupled temperature-displacement analysis (Fully Coupled Thermal-Stress Analysis),
or a fully coupled thermal-electrical-structural analysis (Fully Coupled Thermal-Electrical-Structural Analysis)
to be performed in which heat generated by plastic straining of a material is
calculated. This method is typically used in the simulation of bulk metal
forming or high-speed manufacturing processes involving large amounts of
inelastic strain, where the heating of the material caused by its deformation
is an important effect because of temperature dependence of the material
properties. Since the Johnson-Cook plasticity model is motivated by
high-strain-rate transient dynamic applications, temperature change in this
model is generally computed by assuming adiabatic conditions (no heat transfer
between elements). Heat is generated in an element by plastic work, and the
resulting temperature rise is computed using the specific heat of the material.
This effect is introduced by defining the fraction of the rate of inelastic
dissipation that appears as a heat flux per volume.
Input File Usage
Use all of the
following options in the same material data block:
Use all of the following
options in the same material definition:
Property module: material editor:
MechanicalPlasticityPlastic: Hardening: Johnson-CookThermalSpecific HeatGeneralDensityThermalInelastic Heat Fraction
Initial Conditions
When we need to study the behavior of a material that has already been
subjected to some work hardening, initial equivalent plastic strain values can
be provided to specify the yield stress corresponding to the work hardened
state (see
Initial Conditions).
An initial backstress, ,
can also be specified. The backstress
represents a constant kinematic shift of the yield surface, which can be useful
for modeling the effects of residual stresses without considering them in the
equilibrium solution.
Load module: Create Predefined Field: Step: Initial, choose Mechanical for the Category and Hardening for the Types for Selected Step
Elements
The Johnson-Cook plasticity model can be used with any elements in
Abaqus that
include mechanical behavior (elements that have displacement degrees of
freedom).
Equivalent plastic strain,
where
is the initial equivalent plastic strain (zero or user-specified; see
Initial Conditions).
STATUS
Status of element. The status of an element is 1.0 if the element is active
and 0.0 if the element is not.
STATUSMP
Status of each material point in the element (1.0 if a material point is
active, 0.0 if it is not).
Abaqus/Explicit
only.
YIELDS
Yield stress, .
References
Johnson, G.R., and W. H. Cook, “Fracture
Characteristics of Three Metals Subjected to Various Strains, Strain rates,
Temperatures and Pressures,” Engineering
Fracture
Mechanics, vol. 21, no. 1, pp. 31–48, 1985.