Page 294 - Mechanics of Asphalt Microstructure and Micromechanics

286 Ch a p t e r N i n e

Obviously, there are quite a few problems with these assumptions in relation to re-
alistic granular materials and their behavior.

1. Particles are not circular or spherical. They have complicated shapes, angularity,
and textures. As shape, angularity, and texture affect the contact properties sig-
nificantly, the original DEM has some limitations in quantitative predicting the
behavior of granular materials.
2. A lack of rules guides the relationship between slip and residual forces. In other
words, neither hardening nor softening is considered.
3. The linear contact model is not valid when stress level is high.
Since the pioneering development of DEM by Cundall, numerous publications and
improvements have been developed in the areas of soil mechanics or granular materi-
als. Improvements include the use of polygons and ellipsoids to represent more compli-
cated shapes, particle fluids interactions, and more complicated contacts such as Hertz
contact and viscoelastic contact. Two major issues involving the use of DEM for model-
ing AC are 1) quantifying the effects due to surrounding materials such as mastics or
binder; and 2) microscopic parameter characterization. Most current practice uses back-
calculation to obtain the microscopic parameters. Future research to quantify the effects
due to the surrounding medium is necessary.

9.2.1 Force and Displacement Analysis
For a set of particles, the core of DEM is represented by Equation 9-1:

.. .
Δ
Δ +
Δ +
MX C X S X = Δ R (9-1)
C = damping matrix
M = mass stiffness matrix
S = stiffness matrix
ΔR = incremental force referring to a dynamically equilibrium status
ΔX = incremental displacement
The mass should be generically understood as mass and mass momentum. The
displacement could be translational and rotational. The forces may include body forc-
es, surface tractions, distributed momentums (i.e., due to electromagnetic forces), and
other force-induced momentums. Figure 9.1 illustrates a 2D case and Equations 9-2a,
b, and c represent the generic force vector, the mass matrix, and the displacement vec-
tor in 2D cases.
A
A
A
A T
A
ΔR = ( ΔF , ΔF , ΔM / r ) (9-2a)
1 2
⎛ ⎞
⎜ M 0 0 ⎟
M = ⎜ 0 M 0 ⎟ (9-2b)
⎜ ⎟
⎜ 0 0 I ⎟ ⎠
⎝
r
ΔX = ( ΔX , ΔX , )ω T (9-2c)
1 2
The major procedures for DEM focus on defining and updating the individual terms
such as ΔR and ΔX. Determination of ΔR is mainly addressed by the determination of