Page 382 - Mechanics of Asphalt Microstructure and Micromechanics

374 Ch a p t e r E l ev e n

binder modulus, is a function of temperature and loading frequency. In applying this
approach, the binder master curve or mastic master curve and the time-temperature
superposition principle may be used.
There are many other analytical micromechanics formulas (upper or lower bounds)
to estimate the mixture modulus from the components’ moduli and their volume frac-
tions. A simple one, the Mori-Tanaka method, may have its advantages due to it sim-
plicity (Wang and Lai, 1997). Typically, these methods predict only the isotropic modu-
lus. Formulas based on the Eshelby mechanics (Chapter 5), on the other hand, can
account for the anisotropic modulus resulted from ellipsoid inclusions. This approach
translates the predictions as:
E = f E ( a E , b , v v a b b c) (11-6)
,
, / , /
ijkl ijkl ijkl a b
E ijkl , E ijkl and E ijkl are the elasticity tensor for the mixture and component a and com-
b
a
ponent b, respectively; v a and v b are the volume fractions of the two components; a, b,
and c are axial lengths of the ellipsoid inclusions. By this formula, the moduli in dif-
ferent directions may not be the same when the particle shape and orientations are
different. Even with the same particles, when the particle orientations change, the
anisotropy will change as well. During the compaction, aggregates of different shapes
may change their orientations, resulting in the evolutions of the anisotropy. In sum-
mary, through this set of formulas, volume fractions, aggregate shapes and their ori-
entations, and interlocking are important to the stiffening of the mixtures (e.g., the
vertical modulus).
As particle shapes and their orientations are complicated in estimating the compos-
ite modulus, a powerful approach is the computational mechanics approach to calcu-
late the effective modulus of a mixture by including its microstructures in FEM simula-
tions (Zhang and Wang, 2007).
A final note is that the micromechanics method has been widely applied in pave-
ment design since 1978 (Shell). The Shell normograph, although based on tremendous
experimental data, is actually a micromechanics approach. It estimates the mixture stiff-
ness from the binder modulus, penetration grade, temperature, volume fractions of
binder and aggregates, and vehicle speeds.

11.2.3 Macro Thermodynamics Modeling
Krishnan and Rao (2000) developed a mixture-theory-based constitutive model (see
also Chapter 5) and performed simulation of the air-void reduction in the compaction
process. Krishnan and Rajgopal (2004) further developed a constitutive model that is
based on fundamental thermodynamics principles.
Koneru et al. (2008) developed a thermodynamics framework for a constitutive
model of AC and applied it to solving the gyratory compaction of asphalt concrete. An
analytical solution was derived for the 1D case (gyratory compaction). The work was
then extended to the 3D case using FEM code ABAQUS with UMAT implementation.

11.2.4 DEM Simulations
Analytical micromechanics estimations are usually only the upper or lower bounds of
the modulus and are static. Actual solutions typically require incorporating the micro-
structure and computational methods such as FEM. Considering the special properties
of AC in compaction, when binder is very soft and aggregate particles almost translate