Page 92 - Mechanics of Asphalt Microstructure and Micromechanics

Microstructure Characterization 85

Strain-Fabric Tensor Relation
In continuum mechanics, the deformation rate tenor (D ij ) and the spin tensor (W ij ) are
used to represent the strain characteristics. Mehrabadi and Nemat-Nasser (1983) also
derived the relations relating D ij and W ij to contact normal tensor J ij and local strain
rates E ij .
1
D = ( E J + E J ) (3-19)
ij ik kj jk ki
2
1
W = ( E J − E J ) (3-20)
ik kj
jk ki
ij
2
Kanatani (1984a, 1985) offers relationships between strains and fabric evolution as
well.
Other Relationships
Tobita (1988), through introducing the modified stress tensor, derived a new yielding
function. Tobita and Yanagisawa (1992) systematically introduced the modified stress
tensor that is explicitly related to the fabric tensors. Cowin (1985, 1986) introduced the
relationships between fabric tensors, elasticity tensor, and strength.
Oda and Nakayama (1989), through modifying the first and second stress invari-
ants by the following equations, related the stress invariants with the particle orienta-
tion tensor R ij defined with 2D datasets.

−
I = ( δ + a R )σ (3-21)
a
1
1 ij 2 ij ij
−
J = 2( b δδ + 4 b R δ ) S S (3-22)
2
6 ik jl 7 ik lj ij kl
Where a 1 ,a 2 and b 6 ,b 7 are functions of the orientation tensor constant of R ij . S ij is the
deviatoric stress tensor.
The 3D non-destructive quantification of these fabric tensors provides a viable tool
to verify these relations.
Significances
The quantification methods presented offer potential applications in characterizing the
parameters needed in constitutive models that incorporate material microstructural
quantities. They can also provide applications in correlating microstrutural quantities
with anisotropic behavior of granular materials (or weakly bonded materials in discrete
senses). More importantly, these methods place a foundation to better understand the
deformation and strength mechanisms of granular materials on the basis of microme-
chanics, microstructure, and statistics. A rational elasto-plasticity constitutive model
based on the fabric description can be conceived to include 1) the anisotropic elasticity
base on the anisotropic fabric distribution; 2) anisotropic yielding strength that is based
on material properties and fabric distributions; and 3) non-associated flow.

3.5 Microstructural Quantities in View of Damaged Continuum
Another set of parameters is related to the general concept of damage or weakness in a
broader sense. For example, from a pure mechanics point of view, voids will weaken
the material due to the stress concentration. The interface between aggregates and