# /MAT/LAW48 (ZHAO)

Block Format Keyword This law describes the Zhao material law used to model an elasto-plastic strain rate dependent materials. The law is applicable only for solids and shells.

The global plasticity option for shells (N=0 in shell property keyword) is not available in the actual version.

## Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW48/mat_ID/unit_ID or /MAT/ZHAO/mat_ID/unit_ID
mat_title
${\rho }_{i}$
E $\nu$
A B n Chard ${\sigma }_{\mathrm{max}}$
C D m EI k
${\stackrel{˙}{\epsilon }}_{0}$ Fcut
${\epsilon }_{p}^{max}$ ${\epsilon }_{t1}$ ${\epsilon }_{t2}$

## Definitions

Field Contents SI Unit Example
mat_ID Material identifier

(Integer, maximum 10 digits)

unit_ID Unit Identifier

(Integer, maximum 10 digits)

mat_title Material title

(Character, maximum 100 characters)

${\rho }_{i}$ Initial density

(Real)

$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$
E Young's modulus

(Real)

$\left[\text{Pa}\right]$
$\nu$ Poisson's ratio

(Real)

A Plasticity yield stress

(Real)

$\left[\text{Pa}\right]$
B Plasticity hardening parameter

(Real)

$\left[\text{Pa}\right]$
n Plasticity hardening exponent

Default = 1.0 (Real)

Chard Plasticity Iso-kinematic hardening factor.
= 0
Hardening is full isotropic model.
= 1
Hardening uses the kinematic Prager-Ziegler model.
= between 0 and 1
Hardening is interpolated between the two models.

Default = 0.0 (Real)

${\sigma }_{\mathrm{max}}$ Plasticity maximum stress.

Default = 1030 (Real)

$\left[\text{Pa}\right]$
C Relative strain rate coefficient.

Default = 1.0 (Real)

$\left[\text{Pa}\right]$
D Strain rate plasticity factor.

Default = 0.0 (Real)

m Relative strain rate exponent.

Default = 1.0 (Real)

EI Strain rate coefficient.

Default = 0.0 (Real)

$\left[\text{Pa}\right]$
k Strain rate exponent.

Default = 1.0 (Real)

${\stackrel{˙}{\epsilon }}_{0}$ Reference strain rate.

(Real)

$\left[\frac{\text{1}}{\text{s}}\right]$
Fcut Cutoff frequency for strain rate filtering.

Default = 0.0 (Real)

$\text{[Hz]}$
${\epsilon }_{p}^{max}$ Failure plastic strain.

Default = 1030 (Real)

${\epsilon }_{t1}$ Tensile failure strain 1.

Default = 1030 (Real)

${\epsilon }_{t2}$ Tensile failure strain 2.

Default = 1030 (Real)

## Example (Metal)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
g                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW48/1/1
metal
#              RHO_I
.008
#                  E                  nu
200000                  .3
#                  A                   B                   n               Chard             sig_max
145                 550                 .42                   1                   0
#                  C                   D                   m                  E1                   k
35                  47                  .3                 185                  .3
#         eps_rate_0                Fcut
.05                   0
#            eps_max              eps_t1              eps_t2
0                   0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

1. The stress-strain function is based on the formula published by Zhao:(1)
$\sigma =\left(A+B{\epsilon }_{p}{}^{n}\right)+\left(C-D{\epsilon }_{p}{}^{m}\right)\cdot \mathrm{ln}\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}+{E}_{1}{\stackrel{˙}{\epsilon }}^{k}$
Where,
${\epsilon }_{p}$
Plastic strain
$\stackrel{˙}{\epsilon }$
Strain rate
2. Except for the strain rate formulation, the plasticity curve is strictly identical to a Johnson-Cook model:

However, compared to Johnson-Cook, the Zhao law allows a better approximation of a nonlinear strain rate dependent behavior.

3. Yield stress should be strictly positive.
4. The hardening exponent n must be less than 1.
5. The iso-kinematic hardening parameter is defined as:
• If Chard = 0, hardening is a full isotropic model
• If Chard = 1, hardening uses the kinematic Prager-Ziegler model
• If 0 < Chard < 1, hardening is interpolated between the two models
6. If $\stackrel{˙}{\epsilon }\le {\stackrel{˙}{\epsilon }}_{0}$ , the term $\left(C-D{{\epsilon }_{p}}^{m}\right)\cdot \mathrm{ln}\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}=0$ , and Equation 1 becomes: (2)
$\sigma =\left(A+B{\epsilon }_{p}{}^{n}\right)+{E}_{1}{\stackrel{˙}{\epsilon }}^{k}$
7. The strain rate filtering is used to smooth strain rate. It is only available for shell and solid elements.
8. When ${\epsilon }_{p}$ reaches ${\epsilon }_{\mathrm{max}}$ in one integration point, then based on the element type:
• Shell elements: The corresponding shell element is deleted.
• Solid elements: The deviatoric stress of the corresponding integral point is permanently set to 0, however, the solid element is not deleted.
9. If ${\epsilon }_{1}>{\epsilon }_{t1}$ ( ${\epsilon }_{1}$ is the largest principal strain), the stress is reduced as:(3)
${\sigma }_{n+1}={\sigma }_{n}\left(\frac{{\epsilon }_{t2}-{\epsilon }_{1}}{{\epsilon }_{t2}-{\epsilon }_{t1}}\right)$
10. If ${\epsilon }_{1}>{\epsilon }_{t2}$ , the stress is reduced to 0 (but the element is not deleted).