/MAT/LAW21 (DPRAG)
Block Format Keyword This law, based on DrückerPrager yield criteria, is used to model materials with internal friction such as rockconcrete. The plastic behavior of these materials is dependent on the pressure in the material.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW21/mat_ID/unit_ID or /MAT/DPRAG/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
E  $\nu $  
A_{0}  A_{1}  A_{2}  A_{max}  
fct_ID_{f}  K_{t}  Fscale_{P}  
$\text{\Delta}{P}_{\mathrm{min}}$  P_{ext}  
B  ${\mu}_{\mathrm{max}}$ 
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) 
$\left[\frac{\text{kg}}{{\text{m}}^{3}}\right]$ 
E  Young's
modulus (Real) 
$\left[\text{Pa}\right]$ 
$\nu $  Poisson's
ratio (Real) 

A_{0}  Material plasticity
coefficient. (Real) 
$\left[{\text{Pa}}^{\text{2}}\right]$ 
A_{1}  Material plasticity
coefficient. (Real) 
$\left[\text{Pa}\right]$ 
A_{2}  Material plasticity
coefficient. (Real) 

A_{max}  Limiting von Mises
stress. Default set to 10^{30} (Real) 
$\left[{\text{Pa}}^{\text{2}}\right]$ 
fct_ID_{f}  Function identifier
describing
$\mathrm{P}\left(\mu \right)$
. (Integer) 

K_{t}  Tensile bulk modulus.
3 (Real) 
$\left[\text{Pa}\right]$ 
Fscale_{P}  Pressure function scale
factor. Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
$\text{\Delta}{P}_{\mathrm{min}}$  Minimum
pressure. Default = 10^{30} (Real) 
$\left[\text{Pa}\right]$ 
P_{ext}  External pressure. 4 Default = 0 (Real) 

B  Unloading bulk modulus.
3 (Real) 
$\left[\text{Pa}\right]$ 
${\mu}_{\mathrm{max}}$  Maximum volumetric strain
in compression. 5 (Real) 
Example (Sand)
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
Mg mm s
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/DPRAG/1/1
Sand
# Init. dens.
1.6E9
# E Nu
100 .3
# A0 A1 A2 Amax
1E7 .001 1 0
# If Kt Fscale
2 1 0
# P_min
1.5E4
# B Mu_max
80 .4
#12345678910
# 3. FUNCTIONS:
#12345678910
/FUNCT/2
Sand
# X Y
1 0
0 0
.1 1000
.2 2500
.3 5000
.4 10000
#12345678910
#ENDDATA
#12345678910
Comments
 Hydrodynamic behavior is given by a
userdefined function
$P=\mathrm{f}\left(\mu \right)$
.Where,
 P
 Pressure in the material
 $\mu $
 Volumetric strain with $\mu =\frac{\rho}{{\rho}_{0}}1$
 DrückerPrager yield criteria uses a modified von Mises yield criteria to
incorporate the effects of pressure for massive structures:
(1) $$F={J}_{2}\left({A}_{0}+{A}_{1}P+{A}_{2}{P}^{2}\right)$$Where, ${J}_{2}$
 Second invariant of deviatoric stress, with ${\sigma}_{VM}=\sqrt{3{J}_{2}}$
 P
 Pressure, with $P=\frac{{I}_{1}}{3}$ ( ${I}_{1}$ is the first stress invariant)
 A_{0}, A_{1}, and A_{2}
 Material plasticity coefficients
 ${A}_{1}={A}_{2}=0$
 Yield criteria is von Mises ( ${\sigma}_{VM}=\sqrt{3{A}_{0}}$ )
 It is recommended to set Unloading Bulk modulus, B is equal to the initial slope of function describing $\mathrm{P}\left(\mu \right)$ and Tensile Bulk modulus K_{t} equal to 1/100 of Unloading Bulk modulus B and K_{t} must be positive.
 External pressure is needed in
case of relative pressure formulation. In this specific case, yield criteria and
energy integration require total pressure value. Radioss outputs a pressure which is relative to
${P}_{ext}$
. You can conclude the total pressure value
from:
(2) $$P={P}_{ext}+\text{\Delta}P$$Total pressure limit is concluded from:(3) $${P}_{\mathrm{min}}={P}_{ext}+\text{\Delta}{P}_{\mathrm{min}}$$If ${P}_{ext}=0$ , the output result is a total pressure:
$P=\text{\Delta}P$ and ${P}_{\mathrm{min}}=\text{\Delta}{P}_{\mathrm{min}}$
 B is
unloading bulk modulus. If B is defined, then it must be greater
than any slope
$\frac{dP}{d\mu}$
in
$\left[0,{\mu}_{\mathrm{max}}\right]$
.
 If $B=0$ and ${\mu}_{\mathrm{max}}=0$ , the unloading and loading paths are the same.
 If $B=0$ or ${\mu}_{\mathrm{max}}\ne 0$ , the default value for B is $B={\frac{dP}{d\mu}}_{{\mu}_{\mathrm{max}}}$ .
 If $B\ne 0$ or ${\mu}_{\mathrm{max}}=0$ , the default value for ${\mu}_{\mathrm{max}}$ is $B={\frac{dP}{d\mu}}_{{\mu}_{\mathit{max}}}$ .