Page 146 - Mechanics of Asphalt Microstructure and Micromechanics

138 Ch a p t e r Fiv e

Aggregate-air voids interaction
f ( agg−− air) = τ () 3 gradφ 3 (5-53)
ij

f ( air−− agg) = τ () 1 gradφ 1 (5-54)
ij

Here 1, 2, and 3 represent asphalt binder, aggregates, and voids, respectively.
Total force of interaction
f + f = f + f = f + f = 0 (5-55)
( asp−− agg) ( agg−− asp) ( air−− agg) ( agg−−aair) ( air−− asp) ( asp−− air)

In his model, aggregates are isotropic and linear elastic. Binder is modeled using the
Burger’s model.
By introducing the interaction forces into the governing equations of motion, and
mass conservation equation for each constituent, Krishnan established a complete set of
equations for the three-constituent asphalt mixture. Krishnan also applied his theory to
model the air reduction processes at different pressures and achieved rational results.

5.2 Micromechanics and Its Application
Micromechanics has many branches and it is difficult to give a rigorous and scientific
definition. The author would like to define micromechanics as a branch of mechanics
that predicts the mechanical behavior of a material by incorporating any distribution
and properties of constituents. The distribution information could include the total vol-
ume fractions of a representative volume, the local volume fractions of constituents and
their gradients, interfaces, and particle arrangements. It could be in the form of analyti-
cal solutions or through computational simulations. This section will mainly focus on
the introduction of Eshelby Mechanics. Computational techniques including the use of
the FEM, Boundary Element Method (BEM), and Discrete Element Method (DEM) will
be described in other chapters.

5.2.1 Eshelby Mechanics
5.2.1.1 Green Function
Eshelby (1952) developed a solution to predict the stress and strain field of a medium
that has an ellipsoid inclusion. Through the use of Green function (see Chapter 1), he
obtained a solution for the disturbed strain field. The Green function in an unbounded
elastic domain can be represented in Equation 5-56 (a concise derivation of the Green
function is given in Qu and Ckerkaoui, 2006).

∞
−1
(,
Gx y = 1 −∞ ∫ ∞ N () ξ D () ξ e i •− )ξ ( x y dξ (5-56)
)
ij (2π ) 3 ij
Where N and D are functions of the stiffness tensor.
∞
(,
By placing Gx y) in the equilibrium Equation 5-57, one can prove that the Green
ij
function is actually the displacement field created by a unit force (for an infinite domain).
It is also the basis for the BEM to be discussed in Chapter 8.
∞
∂ 2 G (,
x y)
−
)
L km +δδ ( xy = 0 (5-57)
ijkl ∂∂ im
xx
i j