# /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) (6) (7) (8) (9) (10)
/MAT/LAW75/mat_ID/unit_ID or /MAT/POROUS/mat_ID/unit_ID
mat_title
${\rho }_{i}$
E $\upsilon$
mat_IDs Iflag1 Iflag2 itemax
PE Ps n
tol

## Definition

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 for porous material

(Real)

$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$
E Young's modulus

(Real)

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

(Real)

mat_IDs Material identifier of the solid (fully compacted) material.

(Integer)

Iflag1 Pressure formulation flag.
= 1 (Default)
Herrmann. 1
= 2
Modified Herrmann. 2

(Integer)

Iflag2 Deviatoric stresses formulation flag.
= 1 (Default)
Hydrodynamic
= 2
Elastic

(Integer)

itemax Maximum number of iterations on a calculation.

Default = 5 (Integer)

PE Elastic compact pressure (elastic limit). 3

(Real)

$\left[\text{Pa}\right]$
Ps Solid (matrix) compact pressure. 3

(Real)

$\left[\text{Pa}\right]$
n Exponent used for fitting the experiment data. 3

Default = 2 (Real)

tol Convergence tolerance on a calculation.

$\frac{|\text{Δ}\alpha |}{\alpha }

Default = 10-8 (Real)

## Example (Porous Soil)

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. LOCAL_UNIT_SYSTEM:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
#              MUNIT               LUNIT               TUNIT
g                  cm                 mus
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  1. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW75/1/1
porous soil
#              RHO_I
1.7
#                 E                   NU
3                  .3
#  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              epsmax              sigmax
10000                   0                   0                   0                   0
#               Pmin
0
#                  C           EPS_DOT_0                   M               Tmelt                Tmax
0                   0                   0                   0                   0
#              RHOCP                                                       Troom
0                                                           0
/EOS/POLYNOMIAL/2/1
EOS for soil
#                 C0                  C1                  C2                  C3
0          2.81999993                   2               -1.37
#                 C4                  C5                  E0                 Psh               RHO_0
1.53999996          1.53999996                   0                   0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

1. The porosity α is defined as:(1)
$\alpha =\frac{{\rho }_{s}}{\rho }$

Note that α ≥ 1 ( ${\rho }_{s}\ge \rho$ )

Where,
${\rho }_{s}$
Density of the solid (full compacted matrix) material
$\rho$
Density of the porous material
2. If the EOS of the solid (matrix) material is:(2)
$P=\mathrm{f}\left({\rho }_{s},e\right)$
Then the EOS of the porous material is:
$P=\mathrm{f}\left(\alpha \rho ,e\right)$
for Herrmann formulation
$P=\frac{1}{\alpha }\mathrm{f}\left(\alpha \rho ,e\right)$
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.

3. 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)
$\alpha =1+\left({\alpha }_{P}-1\right){\left[\frac{{P}_{S}-P}{{P}_{S}-{P}_{E}}\right]}^{n}$
Where,
${\alpha }_{P}$
Porosity where pressure reach the elastic compact pressure ${P}_{E}$
$\alpha =1$
Pressure reaches the solid (matrix) compact pressure ${P}_{S}$
${\alpha }_{0}$
Iinitial porosity
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