Page 139 - Mechanics of Asphalt Microstructure and Micromechanics

Mixture T heor y and Micromechanics Applications 131

Balance of linear momentum
..
∇•σ + ρ b + p = ρ x α (5-6)
α α α α α
r a = the linear momentum supply to the ath constituent
s a = the partial stress of the ath constituent
b a = the body force on the ath constituent

Balance of angular momentum
m 1 = σ 23 − σ T (5-7a)
α α α 32
m = σ 31 − σ T (5-7b)
α 2 α α 13
m = σ 12 − σ T (5-7c)
α 3 α α 21
m , m , m = components of the angular momentum in the x, y, and z directions.
a 1 a 2 a 3
Balance of energy
ρ e = tr t L − ∇ • q + ρ r + e (5-8)
T
)
(
αα α α α α α α
e = internal energy
a
q = heat flux
a
r = heat supply
a

e = energy supply
a
Unlike many of the traditional mixture theories, no entropy inequality is proposed
for each of the constituents. The second law of thermodynamics is proposed for the
mixture only.
By defining the corresponding mixture quantities as follows, balance equations for
the mixture can be obtained.
N
α ∑
ρ = ∑ φ γ = N ρ α (5-9)
α
α=1 α=1
. N .
x = ∑ φ α xα (5-10)
α =1
N
N
Μ= ∑ φγ x . α = ∑ ρ x . α (5-11)
α
α
α
α =1 α =1
N
σ = ∑ σ α (5-12)
α=1
N
ε = ∑ φ ε (5-13)
α α
α=1