Page 42 - Mechanics of Asphalt Microstructure and Micromechanics

Mechanical Proper ties of Constituents 35

If n = 1, the PLM model degrades into the linear viscosity model. When n > 1, it de-
scribes the shear thinning behavior while n < 1 describes the shear thickening behavior.
By including the temperature dependence (Arrhenius Equation) in the Equation
2-2, the following equation can be obtained:
ε ⎛ σ ⎞ n p ⎛ − Q ⎞
p
ε = ⎜ ⎝ σ op ⎠ ⎟ exp ⎜ ⎝ RT ⎟ ⎠ (2-3)
op
Where Q p is the thermal activation energy, R is the universal gas constant, and T is
the temperature.
The power law model is applied at relatively higher stress levels. It is confirmed by
numerous experimental studies at stress levels of 100 kPa to 1 MPa, where bitumen was
found to behave as a power-law material.

2.1.5 Williams-Landel-Ferry Shift Function
The viscosity of binder at different temperatures varies. Experimental results indicate
that the viscosity at two temperatures T 1 and T 2 can be related by:
η() = η( ( ,T ) ) (2-4)
A
T
T
T
2 1 2 1
Where A is given by:
CT − T )
(
log[ ( ,T )] =− 1 2 1 (2-5)
AT
1 2 C + (T − T )
2 2 1
This is called the Williams-Landel-Ferry (WLF) shift function. C 1 and C 2 are mate-
rial constants.
2.1.6 The Modified Cross Model (MCM)
The MCM is to model the transition from linear behavior to power law behavior. It is
expressed as:
η − η
η = η + 0 (2-6)
1 + αγ · n c
–
Where h is the shear viscosity,
·
g is the shear strain rate,
·
η is the shear viscosity when γ → ,
·
η is the shear viscosity when γ → 0 , and a is constant.
0
For bitumen η is usually small compared with η and is rarely observable (except
0
for non-residual bitumens). Ignoring η , Equation 2-6 can be re-written for tensile de-

formation as:
η
η = oT (2-7)
1 + βε n c
Where h is the tensile viscosity,
·
e is the tensile strain rate,
η is the limiting viscosity when γ → 0,
oT
b is a constant.