Modeling Material Microstructure - Continuum Mechanics of Anisotropic Materials - page 188

This derivation has demonstrated that the hydraulic permeability depends upon
the viscosity of the fluid and the geometry of the pores. The intrinsic permeability

is defined as the hydraulic permeability times the viscosity of the fluid in the pores,
K eff

m

n c p r H eff

8 . Since n c , the total number of axially aligned cylindrical
cavities per unit area has the dimension of one over length squared, the intrinsic
permeability K eff

33 ¼ m 33 ¼

n c pr

8 is of dimension length squared. The intrinsic permeabil-
ity is independent of the type of fluid in the pores and dependent only upon the size
and geometrical arrangement of the pores in the medium. The comments above
concerning the connection between, and the relative properties of, the intrinsic
permeability and the hydraulic permeability are general and not tied to the particu-
lar model used here to calculate the effective permeability H eff

33 ¼

33 , thus K eff

H eff in ¼ m

general.

7.6 Structural Gradients

A material containing a structural gradient, such as increasing/decreasing porosity
is said to be a gradient material. Figure 7.4 is an illustration of an example material
with a layered structural gradient. Spheres of varying diameters and one material
type are layered in a matrix material of another type. As a special case, the spheres
may be voids. Figure 7.5 is an illustration of a material with a structural gradient
that is not formed by layering. Spheres of varying diameters and one material type
are graded in a size distribution in a material of another type. Again, as a special
case, the spheres may be voids. Gradient materials may be man-made, but they also
occur in nature. Examples of natural materials with structural gradients include
cancellous bone and the growth rings of trees.
The RVE plays an important role in determining the relationship between the
structural gradient and the material symmetry. In a material with a structural
gradient, if it is not possible to select an RVE so that it is large enough to adequately

Fig. 7.4 An illustration
of a material with a layered
structural gradient. Spheres
of varying diameters and one
material type are layered in
a material of another type.
As a special case, the spheres
may be voids. From
Cowin ( 2002 )