Porous Elasticity

Porous Elasticity

The logarithmic porous elasticity model is valid for small elastic strains (normally less than 5%). It is a nonlinear, isotropic elasticity model in which the pressure stress varies as an exponential function of volumetric strain. The model allows a zero or nonzero elastic tensile stress limit.

Often, the elastic part of the volumetric behavior of porous materials is modeled accurately by assuming that the elastic part of the change in volume of the material is proportional to the logarithm of the pressure stress

where $\kappa$ is the logarithmic bulk modulus, $e_0$ is the initial void ratio, $p=-\frac{1}{3}trace\left(\sigma \right)$ is the equivalent pressure stress, $p_0$ is the initial value of the equivalent pressure stress, $J^{el}$ is the elastic part of the volume ratio between the current and reference configurations, and $p_t^{el}$ is the elastic tensile strength of the material in the sense that $J^{el}\to \infty$ as $p\to {-p}_t^{el}$ .

The deviatoric elastic behavior of a porous material can be defined by either defining the shear modulus, $G$ , or Poisson's ratio, $\nu$ :

• Shear Modulus ( $G$ ): The deviatoric stress, $S=\sigma +pI$ , is then related to the deviatoric part of the total elastic strain, ${\epsilon }^{el}$ , by $S=2G{\epsilon }^{el}$ ;
• Poisson's Ratio ( $\nu$ ): The instantaneous shear modulus is then defined from the instantaneous bulk modulus and Poisson's ratio as
  $$G=\frac{3(1-2\nu )(1+{\epsilon }_0)}{2(1+\nu )\kappa }(p+p_t^{el})\exp \left({\epsilon }_{vol}^{el}\right),$$
  where ${\epsilon }_{vol}^{el}=\ln \left(J^{el}\right)$ is the logarithmic measure of the elastic volume change.

Table 1. Shear Type=G

Input Data	Description
Log Bulk Modulus	Logarithmic bulk modulus, $\kappa$ . (Dimensionless.)
Shear Modulus	Shear modulus, $G$ .
Tensile Limit	Elastic tensile limit, $p_t^{el}$ . This value must be nonnegative.
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Table 2. Shear Type=Poisson

Input Data	Description
Log Bulk Modulus	Logarithmic bulk modulus, $\kappa$ . (Dimensionless.)
Poisson's Ratio	Poisson's ratio, $\nu$ .
Tensile Limit	Elastic tensile limit, $p_t^{el}$ . This value must be nonnegative.
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.