Page 94 - Mechanics of Asphalt Microstructure and Micromechanics

Microstructure Characterization 87

The general methodology for the quantification of the distribution function is to
estimate F ij through observed N(m) in different orientations by using Equation 3-26 and
then assessing the distribution function through Equation 3-27. If the distribution can
be approximated by a second-order fabric tensor, the following equation results.
C
Nm() = [ + Fm m ] (3-28)
1
4π ij i j
If a cross-section with a unit normal I is revealed, the following integral over a circle
C(I) on the cross-section may be evaluated.
MI () = ∫ N m dm (3-29)
( )
CI()
MI () = ∫ m m N m dm (3-30)
( )
ij i j
CI ()
If experimental observations are made on three orthogonal planes with basis vec-
tors e 1 = (1,0,0), e 2 = (0,1,0), and e 3 = (0,0,1), it can be shown that:

C 1
1
Me() = ( − F ) (not summed on i) (3-31)
i ii
2 2
C
Me() = F (3-32)
ij k 8 ij
2
)
(
(
(
)
C = [ M e + M e + M e )] (3-33)
1
2
3
3
With these relations, F ij , and therefore f(n), can be obtained if M(e i ) and M ij (e k ) are
known. M(e i ) and M ij (e k ) can be evaluated by the following method:
1. Obtain three orthogonal sections (physically or virtually) from a specimen.
2. On each of the three cross-sections, a grid of parallel test lines in orientation q m
is laid and the quantities Q L (q m ) over the circle are evaluated. In the case of a
specific damaged surface area, this quantity is the number of intercepts per unit
length of test lines. In the case of the mean solid path, this quantity is the mean
solid path measured in q m orientation.
N−1 Q (θ )
Me() = 2π ∑ L m e k (3-34)
k N
m=0
⎛ π m⎞
2
Q (θ ) Sin
N−1 L m e k ⎜ ⎝ N ⎠ ⎟
Me() = π ∑ (3-35)
ij k N
m=0
As pointed out in the previous section, many quantities can be used to describe the
damage status of a material, which may include the specific damaged surface area, the
average spacing among the damaged surfaces, the area fraction of the damaged sur-
faces, and the size of the defects. However, most of these quantities are varying with
orientations, therefore tensorial representations are required.