Page 81 - Mechanics of Asphalt Microstructure and Micromechanics

74 Ch a p t e r Th r e e

AC is similar to that in a granular material. By the skeleton view, important parameters
of the microstructure of a granular system include:

• The average number of contacts among the particles forming the particle
skeleton
• The contact normal distribution (number of contacts in different orientations)
• The branch vector distribution (average length in different orientations)
• The particle orientation distribution (number of particles with the longest
dimensions in different orientations)

These parameters have been shown to be related to the strength and deformation of
granular materials. They have been used in micromechanics (Christoffersen et al., 1981;
Chang, 1991; Rothenburg, 1992), doublet mechanics (Granik and Ferrari, 1993; Ferrari et
al., 1997), and Distinct Element Method (Cundall, 1979). Applications of granular me-
chanics theory with assumed idealized particles and packing have successfully made
qualitative predictions about granular behaviors such as dilatancy (Rowe, 1962) and
shear band thickness (Mauhlhaus and Vardolarkis, 1987). These idealized theories cannot
account for the effects of particle shape and surface texture, the contacting characteristics,
and the actual particle configurations including clustering, and usually fail to predict the
behavior of granular materials quantitatively. A general practice in granular mechanics is
to introduce various fabric terms such as the contact normal tensor (Oda et al., 1982), the
branch vector tensor (Mehrabadi et al., 1983), and the combined contact normal and
branch vector tensor (Nemat-Nasser,et al., 1984) to describe the spatial arrangement of
the particles and their inter-particle contacting characteristics.
Although significant efforts have been focused on the quantification of these fabric
quantities since Oda’s (1972a, 1972b, 1972c) study in the early 1970s, these studies were
limited to 2D sections and thus inherently not correct for some of the quantities. For
example, the number of contacts and the contact orientations are inaccessible parame-
ters, which means they cannot be quantified without assumptions from a small number
of sections. For two particles in contact, the chance for a random cross-section to cut
through the contacting point (a small area) is very small. Figure 3.11 illustrates this con-
cept, where the two particles pointed by arrows are not in contact, while they may actu-
ally contact each other at some other sections. Therefore, contact counts based on 2D
sectional information are usually misleading. In addition, particle orientation, defined
as the orientation of the longest axis of a particle, is also an inaccessible parameter. The
measurement of orientation by Oda (1972a, 1972b, and 1972c) used the angle between
the direction of the longest Feret diameter of a particle cross-section and the horizontal
direction. The orientation based on 2D sectional observation varies with sections and is
usually not unique and may not be coincident with the actual longest dimension in 3D
space. Figures 3.11a and 3.11b illustrate how the irregular shape of the particles affects
the orientation of the longest dimension on particle cross-sections. Generally, for irregu-
lar shapes, the orientation of the longest Feret diameter on a cross-section is not coinci-
dent with that of the longest dimension of the particle in 3D space.
The limitations of the current 2D methods have long been recognized; however, no
general methods have been developed to quantify the branch vector, the contact normal
distribution, or the particle orientation distribution in a true 3D scheme. A most recent
development (Wang et al., 2001a), by use of the relation between the whole space and
the sub-space probability distribution functions of random vectors, quantified the
branch vector distribution from the sub-space distributions of projected random branch