Page 46 - MODELING OF ASPHALT CONCRETE

24 Cha pte r T w o

More recently, during the SHRP, Christensen and Anderson (1992) proposed using
a function derived from the Weibel distribution to represent asphalt rheology. Following
that work, Marasteanu, working with Anderson (Marasteanu and Anderson 1999),
offered a modification of the original Christensen–Anderson model and introduced
what is called the CAM model. The CAM model was used in many studies and is
considered an effective phenomenological model for unmodified asphalt binders whose
properties are within the linear viscoelastic range.
As a result of studying the rheology of modified binders Zeng et al. 2001
introduced a generalized model to represent the complex behavior of modified
binders and mixtures, which allows for a shift for the nonlinearity (strain dependency)
and the plateau region at high temperatures or very long loading times (Zeng et al.
2001). The model reduces dynamic test data, measured at multiple temperatures
and strains, by constructing single complex modulus and phase angle curves. The
model was considered universal because it is used to reduce the test data for the
binders and mixtures with four formulations for complex modulus mastercurve,
phase angle mastercurve, temperature shift factor, and strain shift factor. Details of
the development of the model are found elsewhere (Zeng et al. 2001). The following
is a brief review:
The complex modulus mastercurve of both asphalt binder and mixture may be
expressed by the following equation:

G − G *
*
G = G + g e (2-2)
*
*
′
e [ + f ( f ) ] e /
k mk
1
c
∗
∗
where G = G (f → 0), equilibrium complex modulus
e
∗
∗
G = G (f → ∞), glass complex modulus
g
f = location parameter with dimensions of frequency
c
f´ = reduced frequency
k, m = dimensionless shape parameters
e
The phase angle mastercurve is represented by the following equation:
−md
′ ⎤ ⎪
⎧ ⎪ ⎡ log( f f ) 2 ⎫ 2
δ = 90I − 90I − δ ) ⎨1 + d ⎬ (2-3)
(
⎩ ⎪ ⎣ R d ⎦ ⎭ ⎪
⎪
m ⎢ ⎥
where d = phase angle constant
m
f´ = reduced frequency
f = location parameter with dimensions of frequency
d
R and m = dimensionless shape parameters
d d
I = 0 if f > f
d
I = 1 if f < f for binders
d
I = always 0 for mixtures
Equation (2-3) satisfies the requirement that the phase angle varies from 90 to 0° when
the frequency is increased from zero to infinity for asphalt binders. For asphalt mixtures,
this equation satisfies the requirement that the phase angle increases from 0° to a peak
value and returns to 0° when the frequency is increased from zero to infinity.