Page 99 - Mechanics of Asphalt Microstructure and Micromechanics

92 Ch a p t e r Th r e e

has six independent variables, the area fractions of voids and/or cracks in six different
orientations are evaluated and fitted into the following equation to obtain the tensorial
parameters.
ϕ n = f (3-39)
ij j i
Where j ij is the damage tensor, n j is the orientational cosine, and f i is the observed
area fractions in six orientations. The six orientations used in this development are
(1,0,0), (0,1,0), (0,0,1), (1,1,1), (1,-1,1), and (-1,-1,1). By this method, the 3D dataset is re-
sampled and the cross-section images are interpolated. Figure 3.23 presents a section in
a certain orientation and the circular part that was used for the area fraction evaluation.
This section was virtually cut from the 3D volume rendering.

3.6.1 Modeling Applications of the Damage Parameters
Damage tensors and damage parameters are widely used in damage mechanics. The
following sections present some of these applications.

3.6.1.1 Specific Damaged Surface Area
Theoretically, the overall specific damaged surface area should be a pertinent quantity to
represent the severity of the damage. However, the same total damaged surface can be
contributed by a smaller number of large voids/cracks or a larger number of small
voids/cracks. In addition, the volume average of the damaged surface area smears the
information about the crack distribution and therefore the specific damaged surface area
may not accurately represent the damage status. For brittle materials the specific dam-
aged surface area is related to the energy required to generate the fractured surface.
Using the Griffith energy criterion, the following relationship shall hold.

d
+
( U − F W) = 0 (3-40)
dS
d dW
or ( FU)− = (3-41)
dS dS

FIGURE 3.23 A section in
the specimen and the
circular area for area fraction
evaluation.