# Zhao Plasticity Model (LAW48)

The elasto-plastic behavior of material with strain rate dependence is given by Zhao formula: 1 2(1)
$\sigma =\left(A+B{\epsilon }_{p}^{n}\right)+\left(C-D{\epsilon }_{p}^{m}\right)\text{\hspace{0.17em}}.\text{\hspace{0.17em}}\mathrm{In}\text{\hspace{0.17em}}\frac{\stackrel{¨}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}+E{\stackrel{˙}{\epsilon }}^{k}$
Where,
${\epsilon }_{p}$
Plastic strain
$\stackrel{˙}{\epsilon }$
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 式 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:(2)
${\sigma }_{n+1}={\sigma }_{n}\text{ }\text{\hspace{0.17em}}\left(\frac{{\epsilon }_{t2}-{\epsilon }_{1}}{{\epsilon }_{t2}-\epsilon {}_{t1}}\right)$
Where,
${\epsilon }_{1}$
Largest principal strain
${\epsilon }_{t1}$ and ${\epsilon }_{t2}$
Rupture strain limits

If ${\epsilon }_{1}>{\epsilon }_{f1}$ , the stress is reduced by 式 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.