Page 140 - Mechanics of Asphalt Microstructure and Micromechanics

132 Ch a p t e r Fiv e

N
q = ∑ q α (5-14)
α =1
N
η = ∑ φ η α (5-15)
α
α=1
where r = density of the mixture
·
x = velocity of the mixture
M = linear momentum of the mixture
s = stress tensor of the mixture
e = strain tensor of the mixture
q = heat supply to the mixture
h = entropy of the mixture
h a = entropy of the constituent

Balance of mass for the mixture

∂ρ N .
=
+∇ • ∑ ρ x ] 0 (5-16)
[
α
∂t α =1 α
N
∑ c = 0 means that no net mass should disappear or produce.
α
α=1
By the definitions of the mixture quantities, the above equation can be rewritten
as:
∂ρ . N .
+∇ • ρx) +∇ •[ ∑ ρ u ] = 0 (5-17)
(
α
α
∂t α
=1
· · ·
Where u a = x a − x is the diffusion speed.
.
N
It is clear only when ∇•[ ∑ ρ u α ] = 0 , will the mixture formula have the same form
α
for a single constituent. α =1
Balance of linear momentum for the mixture
By summing the linear momentum equations of the constituents, the following equa-
tion is produced.
N
N
N
∇• ∑ σ + ∑ ρ b = ∑ ρ .. α x (5-18)
α αα α
α =1 α =1 α =1

The net momentum supply vanishes.
Balance of angular momentum for the mixture
σ = σ T (5-19)

It should be noted that the partial stress may not be symmetric, but the stress of the
mixture is symmetric.