Zhao Plasticity Model (LAW48)

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

(1)

σ = (A + B ε_{p}^{n}) + (C - D ε_{p}^{m}) . In \frac{\ddot{ε}}{{\dot{ε}}_{0}} + E {\dot{ε}}^{k}

Where,

$ε_{p}$: Plastic strain
$\dot{ε}$: 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:(2)

σ_{n + 1} = σ_{n} (\frac{ε_{t 2} - ε_{1}}{ε_{t 2} - ε {}_{t 1}})

Where,

$ε_{1}$: Largest principal strain
$ε_{t 1}$ and $ε_{t 2}$: Rupture strain limits

If $ε_{1} > ε_{f 1}$ , the stress is reduced by Equation 2. When $ε_{1} > ε_{t 2}$ 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.