Page 160 - Mechanics of Asphalt Microstructure and Micromechanics

152 Ch a p t e r Fiv e

(d) Alternative version
2 ⎡
2
'
+
+
+
Ec = Va pEa VmpEm +( Va s Vms Vvs) ⎢ Va s + ( Vms + VVvs) ⎤ ⎥ 1 −
'
'
⎣ Ea VmsEm ⎦
⎡ Va' ( Vm Vv) ⎤ −1
+
2
Ec = Pc Va Ea VmEm +(1 − Pc)⎢ + ⎥ (5-157)
+
(
)
'
⎣ ⎣ Ea VmEm ⎦
⎛ VFM Em⎞ P 1
×
⎜ ⎝ P + ⎟ ⎠
0
Pc = VMA'
⎛ VFM Em⎞ P P 1
×
P + ⎜ ⎟
2 ⎝ VMA' ⎠
Simple three-phase system of aggregate, asphalt binder, and air voids:
⎡ Va ( Vb Vv) ⎤ ⎤ −1
+
2
Ec = Pc VaEa VbEb +(1 − Pc)⎢ + ⎥ (5-158)
+
(
)
⎣ Ea VbEb ⎦
Incorporate film thickness:
⎡ Va ( Vb Vv) ⎤ −1
+
2
Ec = Pc VaEa VbEb +(1 − Pc)⎢ + ⎥ (5-159)
+
(
')
⎣ ⎣ Ea VbEb' ⎦

Effective binder modulus:
tEbE
Eb' = F g
t ( − t E + t Eb (5-160)
)
F T g T
Comparison between model predictions and experimental results indicates that the
parallel version of model AC properties is better.
Luo and Lytton (2009) made use of the Hashin and Shtrikman lower bound and
extended it into general coupled self-consistent formulations.
n
∑ c i K − K * * = 0 (5-161)
i
K +
i=1 3 i G
*
n
(
∑ cG − G ) = 0 (5-162)
i
i
*
i=1 * 9 K + 8 G *
G * * + 6 G i
K + 2 G
For the three-constituent asphalt mixture, the above formulations become:
cK − K ) cK − K ) cK − K )
*
*
*
(
(
(
1 1 + 2 2 + 3 3 = 0 (5-163)
3 K + G * 3 K + G * 3 K + G *
1 2 3 3
cG − G ) cG − G ) cG − G )
*
*
*
(
(
(
1 1 + 2 2 + 3 3 = 0 (5-164)
+
*
*
9 K + 8 G * 9 K + 8G * 9K * + 8G *
G * * * + 6 G 1 G * * * + 6G 2 G * * * + 6G 3 3
K + 2 G K + 2G K + 2G