Porous materials such as soils or polymer foams display nonlinear elastic behavior caused by the effect of voids on the bulk modulus of the material during hydrostatic compression.
Compared to the solid portion of the material, voids are relatively compressible, and the effect on the bulk modulus varies according to void proportions. During hydrostatic loading, voids compress or dilate; however, the solid portion of the material remains comparatively stiff, exhibiting little volumetric deformation. As the void ratio of the material changes, the bulk modulus also changes.
The rate form of the elastic stress-strain relationship is:
where 
is the linearized elastic stiffness tensor, which can be given as a
function of the bulk and shear moduli:
where 
is the bulk modulus, 
is the shear modulus, 
is the second-order identity tensor, and 
is the fourth-order deviatoric projection tensor.
For elastic loading, the change in void ratio is proportional to the change in logarithmic pressure:
where 
is the swell index, 
is the elastic void ratio and 
is the pressure. If the material has a non-zero elastic tensile strength,
the relationship is:
 | (4–1) |
The elastic volumetric strain is related to the elastic void ratio:
 | (4–2) |
where 
is the elastic volume ratio. Integrating the porous elasticity
relationship from Equation 4–1, substituting into Equation 4–2 and rearranging gives the pressure as a function of the
elastic volumetric strain:
 | (4–3) |
where 
is the initial pressure. If the initial pressure is zero, then the model
requires a non-zero elastic tensile strength.
The bulk modulus depends on the pressure and the elastic volumetric strain:
Caution: Ensure small volumetric strain increments to prevent significant errors in the bulk modulus value used in the incremental stress update. This safeguard can be especially important when defining an initial stress that causes large volumetric deformation in the first solution substep.
The shear modulus does not depend on the void ratio or pressure directly, but Poisson's ratio is constant, resulting in the following relationship for the shear modulus:
where 
is the Poisson's ratio.
To define the porous elasticity model:
Define the material data table (TB,PELAS,,,,POISSON).
Input the appropriate constants (TBDATA).
Define the initial stress state (
) (INISTATE).
Table 4.1: Porous Elasticity Model Constants
| Constant | Meaning | Property | Unit | Range |
|---|---|---|---|---|
| C1 |
| Swell index | - |
|
| C2 |
| Elastic limit of tensile strength | Force/length2 |
|
| C3 |
| Poisson's ratio | - |
|
| C4 |
| Initial void ratio | - |
|
Example 4.1: Defining the Porous Elasticity Model
/prep7 ! Porous elasticity Kappa = 0.0024 NU0 = 0.279 pt_el = 5.835 E0 = 0.34 p0 = 69 TB,PELAS,1,,,POISSON TBDATA,1, Kappa, pt_el, NU0, E0 /solu !define initial stress state INISTATE,set,dtyp,stre INISTATE,defi,all,,,,-p0,-p0,-p0,0,0,0