/MAT/LAW102 (DPRAG2)
Block Format Keyword This law, based on extended 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/LAW102/mat_ID/unit_ID or /MAT/DPRAG2/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
Iform  
E  $\nu $  
$c$  $\varphi $  A_{max}  
P_{min} 
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) 
$\left[\frac{\text{kg}}{{\text{m}}^{3}}\right]$ 
Iform  Formulation flag. 1
(Integer) 

E  Young's
modulus. (Real) 
$\left[\text{Pa}\right]$ 
$\nu $  Poisson's
ratio. (Real) 

$c$  Cohesion (MohrCoulomb
parameter). (Real) 
$\left[\text{Pa}\right]$ 
$\varphi $  Internal friction angle
(MohrCoulomb parameter). (Real) 
$\left[\mathrm{deg}\right]$ 
A_{max}  Yield criteria
limit. Default = 10^{30} (Real) 
$\left[{\text{Pa}}^{\text{2}}\right]$ 
P_{min}  Minimum pressure (usually
negative or zero, positive value for tension). Default = 10^{30} (Real) 
$\left[\text{Pa}\right]$ 
Example (Concrete)
#RADIOSS STARTER
/UNIT/1
unit for mat
g mm ms
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/DPRAG2/102/1
Concrete
# RHO_I
.0024
# Iform
2
# E NU
61000 .17
# c PHI AMAX
50 40 0.0
# P_min
0
/EOS/POLYNOMIAL/102/1
Concrete
# C0 C1 C2 C3
0 10000 0 0
# C4 C5 Psh Rho0
0 0 0 .0024
#12345678910
#ENDDATA
/END
#12345678910
Comments
 An extended
DrückerPrager yield criteria can be defined as:
The aim of this material law is to automatically compute A_{0}, A_{1}, A_{2} parameters from MohrCoulomb parameters $c$ (cohesion) and $\varphi $ (angle of internal friction).
The MohrCoulomb criteria is usually defined by:(1) $$\tau =c{\sigma}_{n}\mathrm{tan}\varphi $$Where, $\tau $
 Shear stress
 ${\sigma}_{n}$
 Normal stress
 $c$
 Cohesion
 $\varphi $
 Angle of internal friction
From the MohrCoulomb parameters three different extended DrückerPrager yield criteria can be calculated.The following values are computed: ${A}_{0}={k}^{2},\text{}{A}_{1}=6k\alpha ,\text{}{A}_{2}=9{\alpha}^{2}\text{}$
Where,Criteria $k$ α Circumscribed $k=\frac{6.c.\mathrm{cos}(\phi )}{\sqrt{{3}_{}}.(3\mathrm{sin}(\phi ))}$ $\alpha =\frac{2.sin(\phi )}{\sqrt{{3}_{}}.(3\mathrm{sin}(\phi ))}$ Middle $k=\frac{6.c.\mathrm{cos}(\phi )}{\sqrt{{3}_{}}.(3+\mathrm{sin}(\phi ))}$ $\alpha =\frac{2.sin(\phi )}{\sqrt{{3}_{}}.(3+\mathrm{sin}(\phi ))}$ Inscribed $k=\frac{3.c.\mathrm{cos}(\phi )}{\sqrt{9+3{\mathrm{sin}}^{2}(\phi )}}$ $\alpha =\frac{\mathrm{sin}(\phi )}{\sqrt{9+3{\mathrm{sin}}^{2}(\phi )}}$  Pressure
$P\left(\mu ,E\right)$
is defined through an equation of state
(/EOS).Where,
 $P$
 Pressure in material.
 $\mu $
 Volumetric strain with $\mu =\frac{\rho}{{\rho}_{0}}1$ .
 $E=\frac{{E}_{\mathrm{int}}}{V}$
 Energy density.
Unloading:
If $\mu \le {\mu}_{\mathrm{max}}$ , then unloading bulk modulus, $B$ is used for unloading/reloading path. For each $\mu $ over ${\mu}_{\mathrm{max}}$ , unloading path is the same as loading path.
 DrückerPrager yield
criteria is given by:
(2) $$F={J}_{2}\left({A}_{0}+{A}_{1}P+{A}_{2}{P}^{2}\right)$$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)
 ${A}_{1}={A}_{2}=0$
 Yield criteria is von Mises ( ${\sigma}_{VM}=\sqrt{3{A}_{0}}$ )
The polynomial expression should have at least one root and should be increasing.
 Pressure is always a total pressure. To model a relative pressure, the /EOS, P_{sh} parameter must be used to shift the pressure output.
 The yield maximum
A_{max} is as the yield
function becomes:
(3) $${J}_{2}=\mathrm{min}\left({A}_{\mathrm{max}},({A}_{0}+{A}_{1}P+{A}_{2}{P}^{2})\right)$$