Plasticity with Johnson-Cook hardening.

In some sense this is a special case of the previous section, however, since it has been implemented as a Abaqus Material User Subroutine in CalculiX, establishing a relationship between the corotational Cauchy stress and the corotational logarithmic strain it can also be used for large deformations.

In principal, the Johnson-Cook model [43] proposes a yield curve for a conventional plasticity model with isotropic hardening in the form:

$$\sigma_{\text{vm}} = \left [A+B \epsilon_p^n \right ] \left [1+C ... ...dot{\epsilon}_p}{\dot{\epsilon}_0} \right ) \right ] \left [1-(T^*)^m \right ],$$ (385)

where

$$T^* = 0 \;\;\; \; T<T_0$$

$$T^* = \left(\frac{T-T_0}{T_m-T_0}\right) \;\;\; \; T_0 \le T \le T_m$$

$$T^* = 1 \;\;\; \; T>T_m$$ (386)

Here, $A,B,C,n,m,\dot{\epsilon_0},T_0$ and $T_m$ are material constants. The constant $T_0$ has the physical meaning of transition temperature, i.e. the temperature above which the yield surface starts to shrink and $T_m$ has the physical meaning of melt temperature, i.e. the temperature above which the yield surface is reduced to zero. The model is meant to describe highly dynamical phenomena such as explosions, bird strike in a jet engine etc.

For $\dot{\epsilon}_p < \dot{\epsilon}_0$ the logarithm becomes negative and also for small $\epsilon_p$ convergence seems more difficult. Therefore, Bernhardi and co-workers [57] have modified the above law for the following special ranges of $\epsilon_p$ and $\dot{\epsilon}_p$ to:

$$\sigma_{\text{vm}} = \left [A+B \epsilon_p \delta ^{n-1} \right ] \left [1+C \ln \left... ...ght ]f(T^*) \;\;\; \dot{\epsilon}_p \ge \dot{\epsilon}_0, \epsilon_p \le \delta$$

$$\sigma_{\text{vm}} = \left [A+B \epsilon_p^n \right ] \left [1+C \left( \frac{\dot{\ep... ... \right ]f(T^*) \;\;\; \dot{\epsilon}_p < \dot{\epsilon}_0, \epsilon_p > \delta$$

$$\sigma_{\text{vm}} = \left [A+B \epsilon_p \delta ^{n-1} \right ] \left [1+C \left( \f... ...right ]f(T^*) \;\;\; \dot{\epsilon}_p < \dot{\epsilon}_0, \epsilon_p \le \delta$$ (387)

where $f(T^*):=1-(T^*)^m$ and $\delta$ is some small number. In CalculiX $\delta=10^{-4}$ was taken.

In the input deck the Johnson-Cook model is activated by the HARDENING=JOHNSON COOK parameter on the *PLASTIC card. Underneath this card the rate-independent terms are defined, i.e. the parameters $A,B,n,m,T_m$ and $T_0$. If $T_m=T_0$ no temperature dependence is taken into account. The rate dependence has to be defined using a *RATE DEPENDENT card with the parameter TYPE=JOHNSON COOK, listing underneath the parameters $C$ and $\dot{\epsilon}_0$.

The name of a material with Johnson-Cook hardening is not allowed to contain more than 69 characters.