/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

ϕ

. The aim is to calculate the Drücker-Prager criteria that fit the Mohr-Coulomb criteria.

Format

(1)	(3)	(5)
/MAT/LAW102/`mat_ID`/`unit_ID` or /MAT/DPRAG2/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
`Iform`
`E`	$ν$
$c$	$ϕ$	`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)
$ρ_{i}$	Initial density. (Real)	$[\frac{kg}{m^{3}}]$
`Iform`	Formulation flag. 1 = 1 (Default) Circumscribed criterion. = 2 Middle criterion. = 3 Inscribed criterion. (Integer)
`E`	Young's modulus. (Real)	$[Pa]$
$ν$	Poisson's ratio. (Real)
$c$	Cohesion (Mohr-Coulomb parameter). (Real)	$[Pa]$
$ϕ$	Internal friction angle (Mohr-Coulomb parameter). (Real)	$[\deg]$
`A`_max	Yield criteria limit. Default = 10³⁰ (Real)	$[{Pa}^{2}]$
`P`_min	Minimum pressure (usually negative or zero, positive value for tension). Default = -10³⁰ (Real)	$[Pa]$

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----|

Comments

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 $ϕ$ (angle of internal friction).

The Mohr-Coulomb criteria is usually defined by:(1)

τ = c - σ_{n} \tan ϕ

Where,

$τ$: Shear stress
$σ_{n}$: Normal stress
$c$: Cohesion
$ϕ$: Angle of internal friction

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

The following values are computed: $A_{0} = k^{2}, A_{1} = 6 k α, A_{2} = 9 α^{2}$

Where,

Criteria	$k$	α
Circumscribed	$k = \frac{6. c . \cos (φ)}{\sqrt{3_{}} . (3 - \sin (φ))}$	$α = \frac{2. s i n (φ)}{\sqrt{3_{}} . (3 - \sin (φ))}$
Middle	$k = \frac{6. c . \cos (φ)}{\sqrt{3_{}} . (3 + \sin (φ))}$	$α = \frac{2. s i n (φ)}{\sqrt{3_{}} . (3 + \sin (φ))}$
Inscribed	$k = \frac{3. c . \cos (φ)}{\sqrt{9 + 3 \sin^{2} (φ)}}$	$α = \frac{\sin (φ)}{\sqrt{9 + 3 \sin^{2} (φ)}}$

Pressure $P (μ, E)$ is defined through an equation of state (/EOS).
Where,

$P$

Pressure in material.

$μ$

Volumetric strain with $μ = \frac{ρ}{ρ_{0}} - 1$ .

$E = \frac{E_{int}}{V}$

Energy density.

Unloading:

If $μ \leq μ_{\max}$ , then unloading bulk modulus, $B$ is used for unloading/reloading path. For each $μ$ over $μ_{\max}$ , unloading path is the same as loading path.
Drücker-Prager yield criteria is given by:(2)
$F = J_{2} - (A_{0} + A_{1} P + A_{2} P^{2})$

Figure 3.

Where,

$J_{2}$

Second invariant of deviatoric stress, with $σ_{V M} = \sqrt{3 J_{2}}$

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 ( $σ_{V M} = \sqrt{3 A_{0}}$ )

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)
$J_{2} = \min (A_{\max}, (A_{0} + A_{1} P + A_{2} P^{2}))$