Page 301 - Mechanics of Asphalt Microstructure and Micromechanics

Applications of Discrete Element Method 293

x = ( x · t ( +Δ t/ ) 2 − x · t ( −Δ t/ ) 2 )
1
¨
t ()
i t Δ i i (9-17)
·
·
ω t () = 1 ω ( · t ( +Δ t/ )2 − ω t ( −Δ t/ )2 )
i t Δ i i (9-18)
From the known accelerations and initial velocities, it is possible to solve for the
velocities.
⎛ t () ⎞
· t ( +Δ 2 · t ( −Δ 2 F i
t/ )
t/ )
x = x + ⎜ + g Δ t (9-19)
i⎟
i i ⎝ m ⎠

⎛ M ⎞
t ()
·
ω t ( +Δ t/ )2 = ω t ( −Δ t/ )2 + ⎜ i ⎟ t Δ (9-20)
i i ⎝ I ⎠

From the velocities, the position can be obtained.
t ( )
t/ )
x t ( +Δ t) = x + x · t ( +Δ 2 t Δ (9-21)
i i i
9.2.4 Contact Detection
Contact detection is a complicated procedure. As PFC3D uses only spheres, the contact
detecting becomes much simpler: as long as the distance between the centers of two
balls is smaller or equal to the sum of the radii of the two balls, the two balls are consid-
ered in contact (Figure 9.7).
Between balls:
−
D AB ≤ r + r (9-22)
A B
Between ball and wall:
−
D AW ≤ r (9-23)
A
There are more complicated contact-detecting mechanisms for complicated shapes
(Majiza, 2002). Research in this area is fundamental in nature and there is no commer-
cially available software; progress in this area is slow. The Clump mechanism (bonding
small balls together, Itasca Consulting Group, 2005) provided by PFC allows the use of
bonding small balls to represent more complicated shapes.

FIGURE 9.7 Contact
detecting algorithm.
r
B
r
A

D Ball Ball ≤ r + r B ?
−
A
D ≤ r ?
−
Ball Wall A