Technical information

3-12
polymers (Plazek, 1996), and for a limited range of temperatures,
T
a tend to follow the
Williams-Landel-Ferry equation (Williams et al., 1955):
)(
)(
)log(
02
01
TTc
TTc
a
T
−+
−⋅−
=
................................................................................. (3.3.6)
where
1
c and
2
c are both positive constants (
1
c is unitless and
2
c has units of
temperature). This equation was found applicable to HMA mixtures (e.g., Di Benedetto
et al., 2007).
As suggested in Levenberg and Shah (2008), a mathematical expression for the
relaxation spectrum ),(
0
Th
τ
of the following form is assumed:
(
)
2
321
)]ln()[ln(exp)( aaah −⋅−⋅= ττ ............................................................ (3.3.7)
where
1
a ,
2
a and
3
a are all temperature dependent positive constants. The variable
1
a
has units of modulus,
2
a is unitless, and
3
a has units of time (similar to
τ
). This
equation, along with equations 3.3.4, 3.3.5 and 3.3.6 were used to fit the complex
modulus test data given in Chapter 2 by Tables 2.3.5 to 2.3.8 (separate analysis for each
case). In this process six parameters needed evaluation, namely:
1
a ,
2
a ,
3
a (equation
3.3.7),
∞
E (equation 3.3.4),
1
c and
2
c (equation 3.3.6). Their numerical value obtained
simultaneously using a nonlinear minimization algorithm where the goodness of fit was
defined based on relative errors. The chosen reference temperature was 15.5ºC. A
summary of their derived values, for each of the four mix types, is given in Table 3.3.2.
The test data and corresponding master curves are shown in Figures 3.3.2 to
3.3.5. In these figures the dynamic modulus is depicted on the left ordinate and the
phase angle on the right ordinate. The abscissa represents reduced frequency
r
f defined
by the expression:
rr
f
⋅
⋅
=
π
ω
2 . In Figures 3.3.6 and 3.3.7 the master curves are
superimposed for graphical comparison. The four dynamic modulus muster curves are
all plotted in Figure 3.3.6, and the four phase angle master curves are plotted in Figure
3.3.7. It may be seen that mixes 1 and 3, and separately mixes 2 and 4, have very
similar master curves for all practical purposes.