Page 47 - MODELING OF ASPHALT CONCRETE

Modeling of Asphalt Binder Rheology and Its Application to Modified Binders 25

Formulatifon of temperature and strain dependencies followed the WLF (Williams-
Landel-Ferry) equation (Williams et al. 1955) to express the temperature shift factor
( −
()
aT cT T )
log T =− 1 0 (2-4)
()
aT 0 c + (T − T 0 )
2
T
where T = reference temperature
0
c = constant
1
c = temperature constant
2
In view of the similarity of strain and temperature dependencies for constructing
single curve purpose, the WLF equation is utilized to express the strain shift factor:
a () γ d (γ − γ )
log γ =− 1 0 (2-5)
a γ 0 d + (γ − γ 0 )
()
γ
2
where g = reference strain
0
d = constant
1
d = strain constant
2
The reduced frequency in Eqs. (2-2) and (2-3) is defined as follows:
f
log f ′ = loga = loga + loga γ (2-6)
T
where a = overall shift factor
a = strain shift factor
T
a = strain shift factor
g
The reference temperature, T , and reference strain, g , in Eqs. (2-4) and (2-5) can be
0 0
arbitrarily chosen at convenience. At the reference values, the respective shift factors
are unity, or their logarithms are zero. The logarithms of both shift factors are the
amounts of shift to form a single curve; one unit represents a shift of one logarithmic
decade, positive to the direction of high frequency or low loading time.
Based on testing a relatively large number of binders and mixtures Zeng and Bahia
characterized the models in Eqs. (2-2) through (2-5). The reference temperature and
reference strain were chosen to be 52°C and 0 percent, respectively. The reference strain
is chosen assuming that material properties at zero strain represent linear properties,
which cannot be directly measured but can be projected from measurements at strains
∗
∗
greater than zero. For binders, constants G g = 1.0 GPa, G = 0, and m = 1 were assigned
e e
as it has been shown that G ≈ E /3 ≈ 3.0/3, GPa = 1.0 GPa and that asphalt binders are
g g
nearly a linear viscoelastic fluid; parameters k and f were estimated using a minimum
c
square of error fitting procedure. For mixtures, all five parameters were estimated to
give the best fit by minimizing the square error routine.
An example of the fitting results for binder and mixture is presented in Figs. 2-3 and
2-4. The binder in the example is the PG 76-22 modified by ethylene terpoly; the
aggregate in the mixture is crushed limestone with fine gradation.
The Asphalt Viscoelastic and Failure Properties Selected in the SHRP
Although there are many methods of characterizing viscoelastic properties, SHRP
researchers selected the dynamic (oscillatory) testing as the best technique to represent