Page 90 - Mechanics of Asphalt Microstructure and Micromechanics

Microstructure Characterization 83

3.4.2.7 Tensor Analysis
Fabric tensor is often used as a means to describe the spatial arrangement of particulate
systems. The orientational distribution of various microstructural quantities such as
contact normals, branch vectors, and long-axis orientations of particles can be described
by fabric tensors. A method to determine fabric tensors was developed by Hilliard
(1962, 1967) based on stereological principles. Kanatani (1984a, 1984b, 1985) developed
Cartesian tensor formulations to represent the distribution function f(n) of any fabric
quantities. An approximate formulation using only the second-order fabric tensor is
presented as follows:
nn ⎤
fn() = C ⎡ 1 + φ ij i ⎦ (3-15)
⎣
4π j
where n i is the unit vector; f(n) is the distribution function representing the orienta-
tional distribution of any fabric quantities; f ij is a second-order fabric tensor; and de-
notes the spherical domain of integration.
There are six unknown terms with a second order symmetric fabric tensor. The ob-
served distributions for the 12 directions, f(n) ob , listed in Table 3.3 for contact normal
vectors, branch vectors, and particle orientations can be used to obtain the six unknowns
for each of the distributions by minimizing the square differences between the observed
and predicted distributions (Equation 3-16). The unit vectors, n i , representing the 12
regions are also listed in Table 3.3. The C value for contact normal and branch vectors is
102 and for particle orientation is 52. These values can be substituted into Equation 3-16
to generate six linear equations. Then, the six unknown terms of the fabric tensor can be
solved using linear algebra.
∂ ⎛ ⎜∑ ⎛ fn() − C + φ nn ⎤ ⎞ 2 ⎞
⎣
∂φ ⎜ ⎜ ⎝ ob 4 π ⎡ 1 ij i ⎦⎟ ⎠ ⎟ ⎟ = 0 (3-16)
⎟
j
⎠
ij ⎝

The contact normal vector tensor, the branch vector tensor, and the particle orienta-
tion tensor for the specimen studied are calculated as follows:
⎡ 0.398 -0.359 -0.206 ⎤
⎢
φ = -0.359 0.296 -0.265 ⎥
ij ⎢ ⎥
Contact Normal Vector ⎢ ⎣ -0.206 -0.265 -0.437 ⎥ ⎦

⎡ -0.008 0.005 -0.120 ⎤ ⎡ 0.009 0.449 -0.266⎤
0
⎢
⎢
φ = 0.005 0.452 0.335 ⎥ φ = 0.449 0.510 -0.106 ⎥
ij ⎢ ⎥ ij ⎢ ⎥
Branch Vector ⎢ ⎣ -0.120 0 0.335 -0.235 ⎥ ⎦ Long Axis Orientation ⎢ ⎣ -0.266 -0.106 -0.296 ⎥ ⎦
It should be noted that the predicted distributions and the observed distributions
have large differences (Table 3.4), indicating that higher order fabric tensors might be
needed to represent the actual distributions. Although it is not rational to conclude that
fabric quantities cannot be accurately represented by second-order tensors based on this
dataset alone, similar doubts were cast by Tozeren and Skalak (1989) and Wang et al.
(2001a). A much larger dataset should be used, ensuring a statistically valid conclusion.

3.4.2.8 Limitation of 2D Techniques
None of the three fabric quantities, the contact normal, the particle orientation, or the
branch vector for irregular particles, can be quantified using rational mechanisms with a