/MAT/LAW52 (GURSON)
Block Format Keyword This law is based on the Gurson constitutive law, which is used to model viscoelasticplastic strain rate dependent porous metals.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW52/mat_ID/unit_ID or /MAT/GURSON/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
E  ${\nu}_{12}$  Iflag  F_{smooth}  F_{cut}_{}  I_{yield}  
A  B  N  c  p  
${q}_{1}$  ${q}_{2}$  ${q}_{3}$  S_{N}  ${\epsilon}_{N}$  
f_{I}  f_{N}  f_{c}  f_{F} 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

Tab_ID  XFAC  YFAC 
Definitions
Field  Contents  SI Unit Example 

mat_ID  Material identifier. (Integer, maximum 10 digits) 

Unit Identifier  Unit Identifier. (Integer, maximum 10 digits) 

mat_title  Material title. (Character, maximum 100 characters) 

${\rho}_{i}$  Initial density. (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{3}}\right]$ 
E  Young's modulus. (Real) 
$\left[\text{Pa}\right]$ 
${\nu}_{12}$  Poisson's ratio. (Real) 

Iflag  Viscoplastic flow flag. 1
(Integer) 

F_{smooth}  Smooth strain rate are computed.
(Integer) 

F_{cut}  Cutoff frequency for strain rate
filtering. Default = 10^{30} (Real) 
$\text{[Hz]}$ 
I_{yield}  Flag for computing the Yield stress.
3
(Integer) 

A  Yield stress. (Real) 
$\left[\text{Pa}\right]$ 
B  Hardening
parameter. (Real) 
$\left[\text{Pa}\right]$ 
N  Hardening exponent. (Real) 

c  Strain rate coefficient in
CowperSymond's law. (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
p  Strain rate exponent in CowperSymond's
law. (Real) 

${q}_{1}$ , ${q}_{2}$ , ${q}_{3}$  Damage material
parameters. (Real) 

S_{N}  Gaussian standard
deviation. (Real) 
$\left[\text{Pa}\right]$ 
${\epsilon}_{N}$  Nucleated effective plastic
strain. (Real) 

f_{I}  Initial void volume fraction. 2 (Real) 

f_{N}  Nucleated void volume
fraction. (Real) 

f_{c}  Critical void volume fraction at
coalescence. 2 (Real) 

f_{F}  Critical void volume fraction at ductile
fracture. 2 (Real) 

Tab_ID  Yield stress table identifier
(stressstrain functions with correspond strain rate). (Integer) 

XFAC  Scale factor for the first entry
(plastic strain) in function which used for Tab_ID. Default = 1.0 (Real) 

YFAC  Scale factor for ordinate (Yield stress)
in function which used for Tab_ID. Default = 1.0 (Real) 
Example (with parameter input)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
g mm ms
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/LAW52/1/1
Steel
# RHO_I
.0078
# E NU_12 Iflag Fsmooth Fcut Iyield
200000 .3 0 0 0 0
# A B N c p
200 533 1 802 3.585
# q_1 q_2 q_3 SN EpsN
1.25 1 2.25 .1 .2
# Fi FN Fc FF
.01 .04 .12 .2
#12345678910
#ENDDATA
/END
#12345678910
Example (with function input)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
g mm ms
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/LAW52/1/1
Steel
# RHO_I
.0078
# E NU_12 Iflag Fsmooth Fcut Iyield
200000 .3 0 0 0 1
# A B N c p
200 533 1 802 3.585
# q_1 q_2 q_3 SN EpsN
1.25 1 2.25 .1 .2
# Fi FN Fc FF
.01 .04 .12 .2
# Tab_ID XFAC YFAC
1000 0 0
#12345678910
/TABLE/1/1000
curve_list with strain rates
2
# function stain rate
10010 1.0e4
10020 1.0
#12345678910
/FUNCT/10010
plastic strain vs yield stress funct dt=1.0e4
# plastic strain yield stress
0.0000 200.
1.0000 733.
#12345678910
/FUNCT/10020
plastic strain vs yield stress funct dt=1.0
# plastic strain yield stress
0.0000 250.
1.0000 783.
#12345678910
#ENDDATA
/END
#12345678910
Comments
 The von
Mises criteria for viscoplastic flow:If Iflag = 0:
(1) $${\mathrm{\Omega}}_{vm}={\sigma}_{eq}{\sigma}_{M}\sqrt{1+{q}_{3}{f}^{*2}2{q}_{1}{f}^{*2}\mathrm{cosh}(\frac{3{q}_{2}{\sigma}_{m}}{2{\sigma}_{M}})}$$If Iflag = 1:(2) $${\mathrm{\Omega}}_{vm}=\frac{{\sigma}_{eq}^{2}}{{\sigma}_{M}^{2}}+2\text{}{q}_{1}{f}^{*}\mathrm{cosh}\left(\frac{3}{2}{q}_{2}\frac{{\sigma}_{m}}{{\sigma}_{M}}\right)\left(1+{q}_{3}{f}^{\ast 2}\right)\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$$if ${\sigma}_{m}>0$
(3) $${\mathrm{\Omega}}_{vm}=\frac{{\sigma}_{eq}^{2}}{{\sigma}_{M}^{2}}+2{q}_{1}{f}^{\ast}\left(1+{q}_{3}{f}^{{\ast}^{2}}\right)\text{}\text{}$$if ${\sigma}_{m}\le 0$
Where, ${\sigma}_{M}$ is the admissible stress, ${\sigma}_{m}$ is the trace[ $\sigma $ ] (hydrostatic stress), ${\sigma}_{eq}$ is the von Mises stress, and ${q}_{1}$ , ${q}_{2}$ , and ${q}_{3}$ are the material parameter for Gurson Law,
${q}_{3}={q}_{1}^{2}$
${f}^{*}$ is the specific coalescence function.
${f}^{*}=f$ if $f\le {f}_{c}$
${f}^{*}={f}_{c}+\frac{{f}_{u}{f}_{c}}{{f}_{F}{f}_{c}}(f{f}_{c})$ if $f>{f}_{c}$
with
${f}_{u}=\frac{1}{{q}_{1}}$ corresponding to the coalescence function ${f}_{u}={\mathrm{f}}^{*}({f}_{F})$
 The void volume fraction parameters must be entered so that, ${f}_{I}<{f}_{c}<{f}_{F}$ .
 If one integration point reaches ${f}^{*}\ge {f}_{F}$ , the element is deleted.
 If the
I_{yield} flag is not activated
(I_{yield}=0), the yield
stress is computed using CowperSymond's law:
(4) $${\sigma}_{M}=(A+B{\epsilon}_{M}{}^{N})\left(1+{\left(\frac{\dot{\epsilon}}{c}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$p$}\right.}\right)$$If the I_{yield} flag is activated (I_{yield}=1), the yield stress is computed directly from the Yield stress curves (Tab_ID).
 This law is available for shell and solid elements.
 In plot files (/TH/SHEL, /TH/SH3N and /TH/BRICK) or animation files
(/ANIM), the following variables are available:
 USR1: plastic strain ${\epsilon}_{M}$
 USR2: ${f}^{*}$
 USR3: admissible stress ${\sigma}_{M}$
 USR4: f
 USR5: $\text{\epsilon}$