Page 161 - Mechanics of Asphalt Microstructure and Micromechanics

Mixture T heor y and Micromechanics Applications 153

K 1 = bulk modulus of aggregate
K 2 = bulk modulus of asphalt binder
K 3 = bulk modulus of air
K * = bulk modulus of AC
G 1 = bulk modulus of aggregate
G 2 = bulk modulus of asphalt binder
G 3 = shear modulus of air
G * = shear modulus of AC
c 1 = volumetric concentration of aggregate
c 2 = volumetric concentration of asphalt binder
c 3 = volumetric concentration of air

Luo compared the model predictions with experimental observations. The results
indicated that the model was reasonably accurate.

Shu and Huang (2008) applied the concept of equivalent medium (Eshelby, 1957;
Christensen and Lo, 1979) to first obtain the effective elastic modulus of a two-constitu-
ent (asphalt binder and aggregates) solid. Through the Laplace transform they obtained
the modulus in the transformed domain:
Es 12 )( − n)
()( −
v
1
Es a) = 1 0
(,
0 9 En 1 ( − v ) 2 (5-165)
x − 2 1
1 4(112− v )( 1− n E s) + 4E x
)
(
2 1 2 2
The equivalent complex modulus at a certain frequency for a fixed size of particle
(graduation) is then obtained.
ω
E ()( − v )( − n)
*
12
1
ω
E (, a) = 1 0
0 9 En 1( − v ) 2 (5-166)
x − 2 1
ω
1 4(112− v )( 1− n E ( ) + 4E x
*
)
2 1 2 2
The complex modulus of HMA mixtures that includes gradation information
(P(a i ) − P(a i + 1)) is expressed as:
1 N +1
ω
ω
ω
E ( ) = ∑ E [ ( , ) + E ( , a )][ P a ) − P a )] (5-167)
*
*
*
(
a
(
+
0 0 i 0 i 1 i i++1
2 i =1
Following the Prony series expression of the relaxation modulus, they obtained
both the real and imaginary part of the modulus and the phase angle:
e ∑
Et () = E + m E e − (/ρ i ) (5-168)
1
i
i=1
Using a two-step procedure with Step 1 to obtain the effective modulus of asphalt
binder and aggregates, and Step 2 to treat air voids as a medium of zero modulus, they