Page 138 - Mechanics of Asphalt Microstructure and Micromechanics

130 Ch a p t e r Fiv e

Masad et al., 2002). This technique can be used to quantify not only the local volume
fractions of the constituents but also the spatial gradients of the local volume fractions
(Wang et al., 2002; Chapter 3 of this book). With this technique, it is possible to perform
verification of mixture theory predictions by incorporating the required field vari-
ables, and to apply it to model the behavior of AC and/or other mixture materials.
Many mixture theories (i.e., Muller, 1968; Twiss and Eringen, 1971; Twiss, 1972;
Druheller and Bedford, 1980; Dobran, 1985) have been proposed in this century. Most
of these theories follow the general philosophy that was originally proposed by
Truesdell (1957). These theories defined the single mixture variables based on the
momentum equivalency, which results in some erroneous formulations (Hansen,
1989). Hansen’s formulations (volume-fraction-based) and Bowen’s formulations
(Bowen, 1976) for the general balance equations for the mixture will be followed in
this presentation.

5.1.2 General Framework of Mixture Theory
For volume-fraction-based mixture theories, important field variables include local vol-
ume fractions of the constituents and their gradients (Chapter 3). The local volume frac-
tions are usually assumed to be absolutely continuous spatially and differentiable to the
second order to their independent variables. For each of the constituents, x a = c (X a , t)
represents the motion of the particle belonging to the αth constituent (a = 1, N).

x = χ( X , t) (5-1)
α α
Where x a represents the coordinates of the deformed configuration while X a repre-
sents the coordinates in the reference configuration. The velocity, acceleration, and ve-
locity gradient of the particle can be represented as:

. ( χ ∂ Xt , )
xα = α (5-2)
t ∂
.. ∂ 2 ( χ Xt , )
xα = α (5-3)
t ∂ 2

.
∂ x i
L = x ∂ j (5-4)
α
For each constituent, the following balance equations should be satisfied.

Balance of mass
. .
ρ +∇ • ρ x α ) = c (5-5)
(
α α α
c a = the mass supply to the a th constituent
r a = dispersed density, related to the actual local density of the constituent by
ρ = φ γ
α α α N
f a = local volume fraction of the αth constituent and has a restriction of ∑ φ = 1.
α
α=1