Page 325 - Mechanics of Asphalt Microstructure and Micromechanics

Applications of Discrete Element Method 317

orthogonal to each other. For the example aggregate particle, the principal moments of
inertia and principal axes are obtained as:
I 1 = 3.819 109, I 2 = 3.266 109, I 3 = 1.023 109
0 918 ⎤
0 239⎤
.
⎡ . ⎡ − . − ⎡ 0 317⎤
⎢
⎥
⎥
ω = ⎢ ⎢ 0 107 , ω = ⎢ ⎢ 06 . 119 ,and ω = − 0 778 ⎥ ⎥
.
.
⎥
⎥
⎢
1 2 3
.
.
.
3
⎣ − ⎢ 0 382 ⎥ ⎦ ⎢ ⎣ 0 748 ⎥ ⎦ ⎢ ⎣ 0543 ⎥ ⎦
9.6.2 Equivalent Ellipsoid
Due to the complexity and difficulty in reconstructing real particle shape, either sphere
or ellipsoid is used in DEM or other computational models. The use of sphere to repre-
sent an aggregate particle is relatively simple and widely used in DEM simulation.
However, a sphere does not reflect any shape accurately enough. An ellipsoid, however,
presents much more potential to represent more complicated shapes with reasonable
accuracy. The size and shape of an ellipsoid can be determined by its lengths of semi-
axes a, b, and c in the ellipsoid Equation 9-39. The principal moments of inertia of the
ellipsoid are determined by Equation 9-40 (Goldstein, 1950).
x 2 + y 2 + z 2 = 1 (9-39)
a 2 b 2 c 2
From Equation 9-40, the semi-axis lengths of the ellipsoid can be expressed as:
1 1 1
I = m b +( 2 c ), I = m c +( 2 a ), I = m a +( 2 b ) (9-40)
2
2
2
1 2 3
5 5 5
By substituting m with the mass converted from the scanned sectional images and
I 1 , I 2 , and I 3 values obtained from the eigenvalue solutions, the semi-axis lengths of the
ellipsoid can be conveniently determined. This means that the ellipsoid will have the
same mass momentums as those of the real particle. This ellipsoid is named as the
equivalent ellipsoid. Nevertheless, it should be noted that the volume of the ellipsoid
might be different from that of the real particle. The magnitude of the deviation is re-
lated to the shape of the particle. By this approach, the semi-axis lengths of the ellipsoid
for the example particle are obtained as:
a = 34.5, b = 63.2, and c = 123.9 (pixels)
Then the ellipsoid should be rotated based on the eigenvectors so that the orienta-
tion of the three principal axes of the ellipsoid is the same as those of the real particle.
The eigenvector for the I 3 , which is the smallest among the principal moments of inertia,
will point to the direction of longest axis of the particle simply because the moment of
inertia along the longest axis will be the smallest. The final ellipsoid configuration for
the aggregate reconstructed from scanned images in Figure 9-23a is illustrated in Figure
9-23b. The blue line in both Figure 9-23a and Figure 9-23b is the direction of w 3 that is
corresponding to the longest axis. In addition, the red line and the green line are the
directions of w 1 and w 2 that are corresponding to the shortest axis and the medium axis.
From Figures 9-23a and 9-23b, it can be noted that all three axes also match well with
the visually determined principal axes of the aggregate.