Page 296 - Mechanics of Asphalt Microstructure and Micromechanics

288 Ch a p t e r N i n e

Δ
Δ U = ( x A , x A )
Δ
A
1 2
f Δ AB = k ( Δ U • n) n X 2
A
n n B
n AB
Its component in x 1 and x 2 directions: 2
n AB
1
f Δ AB ( x = ( k Δ U • n) n = k ( n ) Δ U A A X 1
AB 2
A
)
n 1 n 1 n 1
f Δ n AB ( x = ( k Δ U • n) n = k ( n ) Δ U A
A
AB 2
)
2
n
n
2
2
Its components
X 2
For the cosines
in the t t B n in x 1 and x 2
directions
direction: AB
n
2 AB 2 A
k ( n ) Δ U
2
t
n (t ) = − n (n )
1 2 AB 2 A
n 2 (t ) = n 1 (n ) AB k ( n ) Δ U
1
t
n
1
A
f Δ AB = (k Δ U • n (t ))n (t )
A
t t
FIGURE 9.3 Force components for pair of particles in contact.
The derivation for the components (first term as an example) in the matrix can be
illustrated in Figure 9.3. This approach is named “granular element method” (Oda and
Iwashita, 1999). Other terms can be derived in a similar approach.
9.2.2 Contact Models
There are quite a few contact models that are widely used in DEM simulations. They
include the bilinear model, Hertz-Mindlin contact model, the parallel bond contact, and
the viscoelastic contact model. There are more complicated models that incorporate
rough contacts, however, these models are barely used in the DEM simulation for AC.
Nevertheless, models that can consider the rough contact will present better approxi-
mations to accurate solutions.
9.2.2.1 Bilinear Contact Model
For two particles in contact, a series contact model will have the following relation-
ships.
k Δ U = k Δ U = k (Δ U + Δ U )
A
[
[]
A
B]
A
B
B
n
n n n n n n