Page 145 - Mechanics of Asphalt Microstructure and Micromechanics

Mixture T heor y and Micromechanics Applications 137

distributions. As deformations take place in the materials (solids in this case), the stress
in the material actually counts. Both permanent deformation and fatigue cracking should
be related to the stresses in the material rather than the nominal stress.
5.1.5 Characterization of the Filed Variable e and grade
A procedure is presented in Chapter 3. There are other methods to define the local vol-
ume fractions such as the material cell and the influence cell concepts.
5.1.6 Air-void Reduction Simulation Using Mixture Theory
Krishnan and Rao (2000) developed a mixture theory and applied it to the modeling of
the air void reduction process. His theory is briefly summarized as follows.
Partial tractions and partial stresses
3
t = ∑ α (5-45)
t
α =1
3
σ = ∑ σ α (5-46)
α=1
t = σ T n (5-47)
s

Conservation of angular momentum
3
∑ M = 0 (5-48)
α
α=1
Momentum interaction terms
Asphalt-aggregate interaction
() γ u
φ
2
f = τ () 1 gradφ − 1 1 ( agg−− asp) (5-49)
( asp−− agg) ij 1 κ κ

φ
() γ u
2
f = τ () 2 gradφ − 2 2 ( agg−− asp) (5-50)
( agg−− asp) ij 2 κ κ

u = v − v is relative seepage speed.
( agg−− asp) 2 1
Asphalt-air voids interaction

f = τ () 3 gradφ (5-51)
−
( asp air) ij 3
f = τ () 2 gradφ (5-52)
−
( air asp) ij 2