/MAT/LAW75 (POROUS)

Block Format Keyword Describes the P-α porous material model. This material describes ductile Porous material with Herrmann model. It only works with 8-node brick element and is not compatible with ALE.

Format

(1)	(2)	(3)	(4)	(5)
/MAT/LAW75/`mat_ID`/`unit_ID` or /MAT/POROUS/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
`E`		$υ$
`mat_ID`_s	`Iflag`₁	`Iflag`₂	`ite`_max
`P`_E		`P`_s		`n`
`tol`

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 for porous material (Real)	$[\frac{kg}{m^{3}}]$
`E`	Young's modulus (Real)	$[Pa]$
$υ$	Poisson's ratio (Real)
`mat_ID`_s	Material identifier of the solid (fully compacted) material. (Integer)
`Iflag`₁	Pressure formulation flag. = 1 (Default) Herrmann. ¹ = 2 Modified Herrmann. ² (Integer)
`Iflag`₂	Deviatoric stresses formulation flag. = 1 (Default) Hydrodynamic = 2 Elastic (Integer)
`ite`_max	Maximum number of iterations on a calculation. Default = 5 (Integer)
`P`_E	Elastic compact pressure (elastic limit). 3 (Real)	$[Pa]$
`P`_s	Solid (matrix) compact pressure. 3 (Real)	$[Pa]$
`n`	Exponent used for fitting the experiment data. 3 Default = 2 (Real)
`tol`	Convergence tolerance on a calculation. $\frac{\| Δ α \|}{α} < t o l$ Default = 10^-8 (Real)

Example (Porous Soil)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                   g                  cm                 mus
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW75/1/1
porous soil
#              RHO_I
                1.70
#                  E                  NU
                   3                .300
#  MAT_IDs    IFLAG1    IFLAG2    ITEMAX
         2         1         2         0
#                 PE                  PS                   N
                 .01                 .05                   0
#                TOL
                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/HYD_JCOOK/2/1
soil
#              RHO_I               RHO_O
          1.76000004                   0
#                  E                  nu
           3.5999999          .300000012
#                  a                   b                   n             eps_max           sigma_max
               10000                   0                   0                   0                   0
#                 C0                  C1                  C2                  C3
                   0                   0                   0                   0
#               Pmin                 Psh
                   0                   0
#                 C4                  C5                  E0
                   0                   0                   0
#                  c               EPS_0                   m               Tmelt               Tmax
                   0                   0                   0                   0                   0
#              RHOCP
                   0
/EOS/POLYNOMIAL/2/1
EOS for soil
#                 C0                  C1                  C2                  C3
                   0          2.81999993                   2               -1.37
#                 C4                  C5                  E0                 Psh
          1.53999996          1.53999996 
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|		  
#ENDDATAS
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

The porosity α is defined as:(1)
$α = \frac{ρ_{s}}{ρ}$

Note that α ≥ 1 ( $ρ_{s} \geq ρ$ )

Where,

$ρ_{s}$

Density of the solid (full compacted matrix) material

$ρ$

Density of the porous material
If the EOS of the solid (matrix) material is:(2)
$P = f (ρ_{s}, e)$

Then the EOS of the porous material is:

$P = f (α ρ, e)$

for Herrmann formulation

$P = \frac{1}{α} f (α ρ, e)$

for modified Herrmann formulation

Where, ( $e$ ) is the internal energy per unit mass. It is same in porous material and in the solid (matrix) material.
If $P < P_{E}$ the behavior is elastic, and if $P > P_{E}$ describes plastic region.
In the elastic region, the change of porosity α with pressure $P$ is reversible.

In the plastic region, the porosity α is assumed to depend on pressure as described below:(3)
$α = 1 + (α_{P} - 1) {[\frac{P_{S} - P}{P_{S} - P_{E}}]}^{n}$

Where,

$α_{P}$

Porosity where pressure reach the elastic compact pressure $P_{E}$

$α = 1$

Pressure reaches the solid (matrix) compact pressure $P_{S}$

$α_{0}$

Iinitial porosity

Figure 1.

¹ "Constitutive equation for the dynamic compaction of ductile porous materials", Herrmann W., J. Applied Physics 40, 1969

² "Static and dynamic pore collapse relations for ductile porous materials", Carroll M.M., Holt A.C., J. Applied Physics 43, 1972