/MAT/LAW48 (ZHAO)
Block Format Keyword This law describes the Zhao material law used to model an elastoplastic 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  C_{hard}  ${\sigma}_{\mathrm{max}}$  
C  D  m  E_{I}  k  
${\dot{\epsilon}}_{0}$  F_{cut}  
${\epsilon}_{p}^{max}$  ${\epsilon}_{t1}$  ${\epsilon}_{t2}$ 
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) 

C_{hard}  Plasticity Isokinematic hardening factor.
Default = 0.0 (Real) 

${\sigma}_{\mathrm{max}}$  Plasticity maximum stress. Default = 10^{30} (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) 

E_{I}  Strain rate coefficient. Default = 0.0 (Real) 
$\left[\text{Pa}\right]$ 
k  Strain rate exponent. Default = 1.0 (Real) 

${\dot{\epsilon}}_{0}$  Reference strain
rate. (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
F_{cut}  Cutoff frequency for strain rate
filtering. Default = 0.0 (Real) 
$\text{[Hz]}$ 
${\epsilon}_{p}^{max}$  Failure plastic strain. Default = 10^{30} (Real) 

${\epsilon}_{t1}$  Tensile failure strain 1. Default = 10^{30} (Real) 

${\epsilon}_{t2}$  Tensile failure strain 2. Default = 10^{30} (Real) 
Example (Metal)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
g mm ms
#12345678910
# 2. MATERIALS:
#12345678910
/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
#12345678910
#ENDDATA
/END
#12345678910
Comments
 The stressstrain function is based
on the formula published by Zhao:
(1) $$\sigma =\left(A+B{\epsilon}_{p}{}^{n}\right)+\left(CD{\epsilon}_{p}{}^{m}\right)\cdot \mathrm{ln}\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{0}}+{E}_{1}{\dot{\epsilon}}^{k}$$Where, ${\epsilon}_{p}$
 Plastic strain
 $\dot{\epsilon}$
 Strain rate
 Except for the strain rate
formulation, the plasticity curve is strictly identical to a JohnsonCook model:
However, compared to JohnsonCook, the Zhao law allows a better approximation of a nonlinear strain rate dependent behavior.
 Yield stress should be strictly positive.
 The hardening exponent n must be less than 1.
 The isokinematic hardening
parameter is defined as:
 If C_{hard} = 0, hardening is a full isotropic model
 If C_{hard} = 1, hardening uses the kinematic PragerZiegler model
 If 0 < C_{hard} < 1, hardening is interpolated between the two models
 If
$\dot{\epsilon}\le {\dot{\epsilon}}_{0}$
, the term
$\left(CD{{\epsilon}_{p}}^{m}\right)\cdot \mathrm{ln}\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{0}}=0$
, and Equation 1 becomes:
(2) $$\sigma =\left(A+B{\epsilon}_{p}{}^{n}\right)+{E}_{1}{\dot{\epsilon}}^{k}$$  The strain rate filtering is used to smooth strain rate. It is only available for shell and solid elements.
 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.
 If
${\epsilon}_{1}>{\epsilon}_{t1}$
(
${\epsilon}_{1}$
is the largest principal strain), the stress is reduced
as:
(3) $${\sigma}_{n+1}={\sigma}_{n}\left(\frac{{\epsilon}_{t2}{\epsilon}_{1}}{{\epsilon}_{t2}{\epsilon}_{t1}}\right)$$  If ${\epsilon}_{1}>{\epsilon}_{t2}$ , the stress is reduced to 0 (but the element is not deleted).