/MAT/LAW52 (GURSON)

Block Format Keyword This law is based on the Gurson constitutive law, which is used to model visco-elastic-plastic strain rate dependent porous metals.

Format

(1)	(3)	(5)	(6)	(7)	(9)
/MAT/LAW52/`mat_ID`/`unit_ID` or /MAT/GURSON/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
`E`	$ν_{12}$	`Iflag`	`F`_smooth	`F`_cut	`I`_yield
`A`	`B`	`N`		`c`	`p`
$q_{1}$	$q_{2}$	$q_{3}$		`S`_N	$ε_{N}$
`f`_I	`f`_N	`f`_c		`f`_F

If I_yield > 0:

(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)
$ρ_{i}$	Initial density. (Real)	$[\frac{kg}{m^{3}}]$
`E`	Young's modulus. (Real)	$[Pa]$
$ν_{12}$	Poisson's ratio. (Real)
`Iflag`	Viscoplastic flow flag. 1 = 0 von Mises criteria. = 1 von Mises criteria. = 2 1 + void nucleation set to zero in compression = 3 0 + void nucleation set to zero in compression. (Integer)
`F`_smooth	Smooth strain rate are computed. = 0 (Default) No strain rate smoothing. = 1 Strain rate smoothing is active. (Integer)
`F`_cut	Cutoff frequency for strain rate filtering. Default = 10³⁰ (Real)	$[Hz]$
`I`_yield	Flag for computing the Yield stress. 3 = 0 Using Cowper-Symond's law. = 1 Using Yield stress table (Yield stress vs plastic strain). (Integer)
`A`	Yield stress. (Real)	$[Pa]$
`B`	Hardening parameter. (Real)	$[Pa]$
`N`	Hardening exponent. (Real)
`c`	Strain rate coefficient in Cowper-Symond's law. (Real)	$[\frac{1}{s}]$
`p`	Strain rate exponent in Cowper-Symond's law. (Real)
$q_{1}$ , $q_{2}$ , $q_{3}$	Damage material parameters. (Real)
`S`_N	Gaussian standard deviation. (Real)	$[Pa]$
$ε_{N}$	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 (stress-strain 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
#---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/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
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Example (with function input)

#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/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
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/TABLE/1/1000
curve_list with strain rates
         2
# function                    stain rate
     10010                        1.0e-4
     10020                           1.0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/10010
plastic strain vs yield stress funct dt=1.0e-4
#     plastic strain       yield stress    
              0.0000                200.
              1.0000                733.
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/10020
plastic strain vs yield stress funct dt=1.0
#     plastic strain       yield stress
              0.0000                250.
              1.0000                783.
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

The von Mises criteria for viscoplastic flow:
If Iflag = 0:(1)
$Ω_{v m} = σ_{e q} - σ_{M} \sqrt{1 + q_{3} f^{* 2} - 2 q_{1} f^{* 2} \cosh (\frac{3 q_{2} σ_{m}}{2 σ_{M}})}$

If Iflag = 1:(2)
$Ω_{v m} = \frac{σ_{e q}^{2}}{σ_{M}^{2}} + 2 q_{1} f^{*} \cosh (\frac{3}{2} q_{2} \frac{σ_{m}}{σ_{M}}) - (1 + q_{3} f^{* 2}) $

if $σ_{m} > 0$

(3)
$Ω_{v m} = \frac{σ_{e q}^{2}}{σ_{M}^{2}} + 2 q_{1} f^{*} - (1 + q_{3} f^{*^{2}}) $

if $σ_{m} \leq 0$

Where, $σ_{M}$ is the admissible stress, $σ_{m}$ is the trace[ $σ$ ] (hydrostatic stress), $σ_{e q}$ 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 $f \leq f_{c}$

$f^{*} = f_{c} + \frac{f_{u} - f_{c}}{f_{F} - f_{c}} (f - f_{c})$ if $f > f_{c}$

with

$f_{u} = \frac{1}{q_{1}}$ corresponding to the coalescence function $f_{u} = f^{*} (f_{F})$
The void volume fraction parameters must be entered so that, $f_{I} < f_{c} < f_{F}$ .
If one integration point reaches $f^{*} \geq f_{F}$ , the element is deleted.
If the I_yield flag is not activated (I_yield=0), the yield stress is computed using Cowper-Symond's law:
(4)
$σ_{M} = (A + B ε_{M}^{N}) (1 + {(\frac{\dot{ε}}{c})}^{\frac{1}{p}})$

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 $ε_{M}$
- USR2: $f^{*}$
- USR3: admissible stress $σ_{M}$
- USR4: f
- USR5: $ε$