Page 141 - Mechanics of Asphalt Microstructure and Micromechanics

Mixture T heor y and Micromechanics Applications 133

Balance of energy
N
N
N
αα ∑
∑ ρ e = N tr t L ) − ∇ • ∑ q α ∑ ρ r α (5-20)
+
T
(
α
α
α α
α=1 α=1 α=1 α=1
The second law of thermodynamics
N
∑ ∂ . α + q α + ρ r
αα
)
[( ρη +∇ •( ρη x ) θ θ ] ≥ 0 (5-21)
α
α
α
α
α=1 ∂t α α
Linear momentum interaction
The linear momentum supply p a in Equation 5-6 represents the interaction between any
two of the constituents. This interaction may also be affected by other constituents in
the mixture to a secondary degree (Truesdell, 1969). Following Bowen (1976), this sup-
ply can be represented as:
.
N
β ∑
p =− ∑ ξ αβ gradρ − N ζ αβ xβ (5-22)
α
β=1 β=1

N
N
with ∑ ξ αβ = 0 , ∑ ζ αβ = 0 , x ab and z ab are functions of dispersed densities of the
β=1 β=1
constituents and the temperature.
For a homogeneous material, it can be proved that: gradρ = γ gradφ .
β β β
5.1.3 Two-constituent Case
The following refers to the two-constituent mixture of a solid (i.e., aggregate coated
with asphalt binder) and air voids. The local solid volume fraction is f. The symbols
used follow those in the previous sections and will not be repetitively explained. The
objective of this section is to illustrate how mixture theory predicts the mechanical
properties. Due to this consideration, the main focus will be on the balance equation
of linear momentum.
Balance of linear momentum
For the solids:
..
∇•σ + ρ b + p = ρ x s (5-23)
s s s s s
For the air:
..
∇•σ + ρ b + p = ρ x (5-24)
a a a a a a
Constitutive relations
Assume that the solids are the aggregates coated with asphalt binder (with density
equal to g -the maximum density). The mixture therefore consists of only two constitu-
ents: the coated aggregate (effective aggregate) and air void. This application is to eval-
uate the mechanical properties, especially the initial stress distribution (not the creep-
ing behavior). It is more applicable to the situation of low-temperature conditions.