Johnson-Cook Plasticity

A Mises yield surface with associated flow is used in the Johnson-Cook plasticity model.

Johnson-Cook hardening is a particular type of isotropic hardening where the static yield stress, ${\sigma }^0$, is assumed to be of the form

where ${\overline{\epsilon }}^{pl}$ is the equivalent plastic strain and A, B, n, and m are material parameters. $\widehat{\theta }$ is the nondimensional temperature defined as

where $\theta$ is the current temperature, ${\theta }_{\mathrm{melt}}$ is the melting temperature, and ${\theta }_{\mathrm{transition}}$ 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 ${\sigma }^0$ is ignored if you specify a zero value or if you do not specify a value for m.

When $\theta \ge {\theta }_{\mathrm{melt}}$, the material will be melted and will behave like a fluid; there will be no shear resistance since ${\sigma }^0=0$. 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, ${\theta }_{\mathrm{melt}}$, and ${\theta }_{\mathrm{transition}}$ as part of the metal plasticity material definition.

Johnson-Cook strain rate dependence assumes that

and

where

$\overline{\sigma }$

is the yield stress at nonzero strain rate;

${\dot{\overline{\epsilon }}}^{pl}$

is the equivalent plastic strain rate;

${\dot{\epsilon }}_0$ and C

are material parameters measured at or below the transition temperature, ${\theta }_{\mathrm{transition}}$;

${\sigma }^0\left({\overline{\epsilon }}^{pl},\theta \right)$

is the static yield stress; and

$R\left({\dot{\overline{\epsilon }}}^{pl}\right)$

is the ratio of the yield stress at nonzero strain rate to the static yield stress (so that $R\left({\dot{\epsilon }}_0\right)=1.0$).

The yield stress is, therefore, expressed as

You provide the values of C and ${\dot{\epsilon }}_0$ when you define Johnson-Cook rate dependence.

The use of Johnson-Cook hardening does not necessarily require the use of Johnson-Cook strain rate dependence.

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, $\omega$, is defined as

where $\mathrm{\Delta }{\overline{\epsilon }}^{pl}$ is an increment of the equivalent plastic strain, ${\overline{\epsilon }}_f^{pl}$ is the strain at failure, and the summation is performed over all increments in the analysis. The strain at failure, ${\overline{\epsilon }}_f^{pl}$, is assumed to be dependent on a nondimensional plastic strain rate, ${\dot{\overline{\epsilon }}}^{pl}/{\dot{\epsilon }}_0$; a dimensionless pressure-deviatoric stress ratio, $p/q$ (where p is the pressure stress and q is the Mises stress); and the nondimensional temperature, $\widehat{\theta }$, defined earlier in the Johnson-Cook hardening model. The dependencies are assumed to be separable and are of the form

where $d_1$–$d_5$ are failure parameters measured at or below the transition temperature, ${\theta }_{\mathrm{transition}}$, and ${\dot{\epsilon }}_0$ is the reference strain rate. You provide the values of $d_1$–$d_5$ when you define the Johnson-Cook dynamic failure model. This expression for ${\overline{\epsilon }}_f^{pl}$ differs from the original formula published by Johnson and Cook (1985) in the sign of the parameter $d_3$. This difference is motivated by the fact that most materials experience an increase in ${\overline{\epsilon }}_f^{pl}$ with increasing pressure-deviatoric stress ratio; therefore, $d_3$ 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.

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.

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.

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.