Page 89 - Mechanics of Asphalt Microstructure and Micromechanics

82 Ch a p t e r Th r e e

Contact
Normal Branch Long Axis
Regions n 1 , or x n 2 , or y n 3 , or z Vectors Vectors Orientation
1 0.2706 0.2706 0.9239 4 7 4
2 0.8536 0.3536 0.3827 8 8 3
3 0.3536 0.8536 0.3827 4 14 7
4 –0.2706 0.2706 0.9239 8 6 2
5 –0.3536 0.8536 0.3827 10 13 4
6 –0.8536 0.3536 0.3827 12 10 4
7 –0.2706 –0.2706 0.9239 6 9 3
8 –0.8536 –0.3536 0.3827 9 8 9
9 –0.3536 –0.8536 0.3827 12 7 5
10 0.2706 –0.2706 0.9239 4 5 4
11 0.3536 –0.8536 0.3827 13 8 6
12 0.8536 –0.3536 0.3827 12 7 1
Total NA NA NA 102 102 52
TABLE 3.3 Distribution of contact normal, branch vectors, and long axis orientation.

Considering the location of the regions (e.g., Regions 1, 4, 7, and 10 and are close to
the z axis), the numbers of contacts (Table 3.3) in different orientations are different,
indicating the anisotropic distribution (it may require a larger specimen).
3.4.2.5 Branch Vectors
When two particles are in contact, the unit vector connecting the two mass centers of
the particles is called the branch vector. Figure 3.16 also shows the branch vector of the
two contacting particles. Since the 3D Cartesian coordinates of the mass center of each
particle have already been identified, the branch vectors can be obtained conveniently.
The distribution of the branch vectors can be obtained in the same manner as the distri-
bution of contact normal vectors. Table 3.3 also shows the distribution of the branch
vectors for the specimen. By comparing the data presented in Table 3.3, the differences
of the two distributions should be noted. For spherical particles the two distributions
should be the same.

3.4.2.6 Particle Orientation
The particle orientation has been defined as the orientation of the longest axis orienta-
tion of a particle cross-section for many 2D studies. The orientation of the longest orien-
tation of a cross-section may be significantly different from the particle orientation. This
problem can be solved by using the 3D surface datasets. Since the Cartesian coordinates
of the surface of each particle are already determined, the real long-axis orientation of
each particle in 3D can be determined. The method presented is simple but inefficient.
It calculates the distance between any two points on the surface and selects the two
points that have the longest distance. The line connecting the two points represents the
orientation of the longest axis or the particle orientation. The orientational distribution
of the particle orientation can be obtained in the same manner as the contact normal
vector distribution was obtained. Table 3.3 also presents the distribution of the particle
orientations for the specimen. Considering the location of the regions, it can be seen that
more particles are oriented in the horizontal direction, consistent with common sense as
gravity is in the vertical direction.