# Zhao Plasticity Model (LAW48)

The elasto-plastic behavior of material with strain rate dependence is given by Zhao formula:
^{1}
^{2}

- ${\epsilon}_{p}$
- Plastic strain
- $\dot{\epsilon}$
- Strain rate
- $A$
- Yield stress
- $B$
- Hardening parameter
- $n$
- Hardening exponent
- $C$
- Relative strain rate coefficient
- $D$
- Strain rate plasticity factor
- $m$
- Relative strain rate exponent
- $E$
- Strain rate coefficient
- $k$
- Strain rate exponent

In the case of material without strain rate effect, the hardening curve given by Equation 1 is identical to those of Johnson-Cook. However, Zhao law allows a better approximation of strain rate dependent materials by introducing a nonlinear dependency.

As described for Johnson-Cook law, a strain rate filtering can be introduced to smooth the results. The plastic flow with isotropic or kinematic hardening can be modeled as described in Cowper-Symonds Plasticity Model (LAW44). The material failure happens when the plastic strain reaches a maximum value as in Johnson-Cook model. However, two tensile strain limits are defined to reduce stress when rupture starts:

- ${\epsilon}_{1}$
- Largest principal strain
- ${\epsilon}_{t1}$ and ${\epsilon}_{t2}$
- Rupture strain limits

If ${\epsilon}_{1}>{\epsilon}_{f1}$ , the stress is reduced by Equation 2. When ${\epsilon}_{1}>{\epsilon}_{t2}$ the stress is reduced to zero.

^{1}Zhao Han,

A Constitutive Model for Metals over a Large Range of Strain Rates, Materials Science & Engineering, A230, 1997.

^{2}Zhao Han and Gerard Gary,

The Testing and Behavior Modelling of Sheet Metals at Strain Rates from 10.e-4 to 10e+4 s-1, Materials Science & Engineering" A207, 1996.