Page 143 - Mechanics of Asphalt Microstructure and Micromechanics

Mixture T heor y and Micromechanics Applications 135

air voids), the stress in the air voids may be neglected. The boundary conditions may
be written as:
σ n = t (5-34)
s s s
t = /φ (5-35)
t
s
5.1.4 Analytic Solution of Two Simple Cases
Equations 5-24 through 5-31 and 5-34 through 5-35 plus the strain-compatible condi-
tions constitute a whole set of equations for solving mechanical boundary value prob-
lems. Obviously, unlike the classic continuum theory, field variables f and gradf are
needed for the solution of the set of equations. It should be noted, if gradf = 0, the set of
the equations becomes the same as those for a single continuum. Therefore, it may be
deduced that the spatial gradients of the local volume fractions are important in affect-
ing the mixture properties. The following will look into two simple cases: a 2D case and
a 1-D case. In both cases, a static problem is assumed and the momentum supply from
the air and the body force are considered negligible. With these simplifications, only the
stress component in the solid exists.

Two-dimensional case
Equation 5-23 becomes:
∂σ ∂σ
11 + 12 = 0 (5-36)
∂x ∂x
1 2
∂σ ∂σ
12 + 22 = 0 (5-37)
∂x 1 ∂x 2
or
∂(φτ ) ∂(φτ )
11 + 12 = 0 (5-38)
∂x ∂x
1 2
∂(φτ ) ∂(φτ )
12 + 22 = 0 (5-39)
∂x ∂x
1 2
A simple way to solve the problem is to solve the partial stress Equations 5-36 and
5-37. This way, the physical implication is not so explicit. However, the solution to the
material Equations 5-38 and 5-39 will be interesting. The above two equations can be
rewritten as:

φ
∂( τ ) ∂() ∂( τ ) φ ∂
φ 11 + τ + φ 12 + τ = 0 (5-40)
∂x 11 ∂x ∂x 12 ∂x
1 1 2 2
φ
∂( τ ) ∂() ∂( τ ) φ ∂
φ 12 + τ + φ 22 + τ = 0 (5-41)
∂x 12 ∂x ∂x 22 ∂x
1 1 2 2
φ
φ
The terms τ ∂() + τ φ ∂ and τ ∂() + τ φ ∂ play the roles of driving forces in
11 12 12 22
∂x 1 ∂x 2 ∂x 1 ∂x 2
the above two equations. These terms are similar to the materials forces (Maugin, 1993)
and will be called inhomogeneity-induced material forces.