/MAT/LAW102 (DPRAG2)

Block Format Keyword This law, based on extended Drücker-Prager yield criteria, is used to model materials with internal friction such as rock-concrete. The plastic behavior of these materials is dependent on the pressure in the material.

This law is similar to /MAT/LAW10 (DRAGP1); the only difference being that yield criterial is calculated from the Mohr-Coulomb parameters $c$ and $\varphi$ . The aim is to calculate the Drücker-Prager criteria that fit the Mohr-Coulomb criteria.

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 $\nu$
$c$ $\varphi$ Amax
Pmin

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)

$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$
Iform Formulation flag. 1
= 1 (Default)
Circumscribed criterion.
= 2
Middle criterion.
= 3
Inscribed criterion.

(Integer)

E Young's modulus.

(Real)

$\left[\text{Pa}\right]$
$\nu$ Poisson's ratio.

(Real)

$c$ Cohesion (Mohr-Coulomb parameter).

(Real)

$\left[\text{Pa}\right]$
$\varphi$ Internal friction angle (Mohr-Coulomb parameter).

(Real)

$\left[\mathrm{deg}\right]$
Amax Yield criteria limit.

Default = 1030 (Real)

$\left[\text{P}{\text{a}}^{2}\right]$
Pmin Minimum pressure (usually negative or zero, positive value for tension).

Default = -1030 (Real)

$\left[\text{Pa}\right]$

Example (Concrete)

#RADIOSS STARTER
/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/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
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

1. An extended Drücker-Prager yield criteria can be defined as:

The aim of this material law is to automatically compute A0, A1, A2 parameters from Mohr-Coulomb parameters $c$ (cohesion) and $\varphi$ (angle of internal friction).

The Mohr-Coulomb 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 Mohr-Coulomb parameters three different extended Drücker-Prager yield criteria can be calculated.

The following values are computed:

Where,
Criteria $k$ α
Circumscribed $k=\frac{6.c.\mathrm{cos}\left(\phi \right)}{\sqrt{{3}_{}}.\left(3-\mathrm{sin}\left(\phi \right)\right)}$ $\alpha =\frac{2.sin\left(\phi \right)}{\sqrt{{3}_{}}.\left(3-\mathrm{sin}\left(\phi \right)\right)}$
Middle $k=\frac{6.c.\mathrm{cos}\left(\phi \right)}{\sqrt{{3}_{}}.\left(3+\mathrm{sin}\left(\phi \right)\right)}$ $\alpha =\frac{2.sin\left(\phi \right)}{\sqrt{{3}_{}}.\left(3+\mathrm{sin}\left(\phi \right)\right)}$
Inscribed $k=\frac{3.c.\mathrm{cos}\left(\phi \right)}{\sqrt{9+3{\mathrm{sin}}^{2}\left(\phi \right)}}$ $\alpha =\frac{\mathrm{sin}\left(\phi \right)}{\sqrt{9+3{\mathrm{sin}}^{2}\left(\phi \right)}}$
2. 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.

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.

3. Drücker-Prager 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.

4. Pressure is always a total pressure. To model a relative pressure, the /EOS, Psh parameter must be used to shift the pressure output.
5. The yield maximum Amax is as the yield function becomes:(3)
${J}_{2}=\mathrm{min}\left({A}_{\mathrm{max}},\left({A}_{0}+{A}_{1}P+{A}_{2}{P}^{2}\right)\right)$