/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)	(3)	(5)	(7)	(9)
/MAT/LAW48/`mat_ID`/`unit_ID` or /MAT/ZHAO/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
`E`	$ν$
`A`	`B`	`n`	`C`_hard	$σ_{\max}$
`C`	`D`	`m`	`E`_I	`k`
${\dot{ε}}_{0}$	`F`_cut
$ε_{p}^{m a x}$	$ε_{t 1}$	$ε_{t 2}$

Definition


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)
$ρ_{i}$	Initial density. (Real)	$[\frac{kg}{m^{3}}]$
`E`	Young's modulus. (Real)	$[Pa]$
$ν$	Poisson's ratio. (Real)
`A`	Plasticity yield stress. (Real)	$[Pa]$
`B`	Plasticity hardening parameter. (Real)	$[Pa]$
`n`	Plasticity hardening exponent. Default = 1.0 (Real)
`C`_hard	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)
$σ_{\max}$	Plasticity maximum stress. Default = 10³⁰ (Real)	$[Pa]$
`C`	Relative strain rate coefficient. Default = 1.0 (Real)	$[Pa]$
`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)	$[Pa]$
`k`	Strain rate exponent. Default = 1.0 (Real)
${\dot{ε}}_{0}$	Reference strain rate. (Real)	$[\frac{1}{s}]$
`F`_cut	Cutoff frequency for strain rate filtering. Default = 0.0 (Real)	$[Hz]$
$ε_{p}^{m a x}$	Failure plastic strain. Default = 10³⁰ (Real)
$ε_{t 1}$	Tensile failure strain 1. Default = 10³⁰ (Real)
$ε_{t 2}$	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----|

Comments

The stress-strain function is based on the formula published by Zhao:
$σ = (A + B ε_{p}^{n}) + (C - D ε_{p}^{m}) \cdot \ln \frac{\dot{ε}}{{\dot{ε}}_{0}} + E_{1} {\dot{ε}}^{k}$
Where,

$ε_{p}$

Plastic strain

$\dot{ε}$

Strain rate
Except for the strain rate formulation, the plasticity curve is strictly identical to a Johnson-Cook model:

Figure 1.

However, compared to Johnson-Cook, 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.
Figure 2.
The iso-kinematic hardening parameter is defined as:
- If C_hard = 0, hardening is a full isotropic model
- If C_hard = 1, hardening uses the kinematic Prager-Ziegler model
- If 0 < C_hard < 1, hardening is interpolated between the two models
If $\dot{ε} \leq {\dot{ε}}_{0}$ , the term $(C - D {ε_{p}}^{m}) \cdot \ln \frac{\dot{ε}}{{\dot{ε}}_{0}} = 0$ , and Equation 1 becomes:
$σ = (A + B ε_{p}^{n}) + E_{1} {\dot{ε}}^{k}$
The strain rate filtering is used to smooth strain rate. It is only available for shell and solid elements.
When $ε_{p}$ reaches $ε_{\max}$ 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 $ε_{1} > ε_{t 1}$ ( $ε_{1}$ is the largest principal strain), the stress is reduced as:
$σ_{n + 1} = σ_{n} (\frac{ε_{t 2} - ε_{1}}{ε_{t 2} - ε_{t 1}})$
If $ε_{1} > ε_{t 2}$ , the stress is reduced to 0 (but the element is not deleted).