Page 76 - Mechanics of Asphalt Microstructure and Micromechanics

Microstructure Characterization 69

The spatial distribution of the voids may be characterized by the local volume frac-
tion of voids and their spatial gradients. Equations 3-7 and 3-8 present the definitions of
these two parameters:
V a
γ = sd (3-7)
V
sd
Δ γ
gradφ = (3-8)
Δ d
γ = the local volume fraction of voids
V sd = the volume of any sub-domain in the mixture
V sd = the void volume in the sub-domain
a
gradφ = the gradient of the local volume fraction of voids
Δγ = the difference of the local volume fraction of voids of two points
Δd = the distance between the two points
There are several theories including mixture theory, material force mechanics, and
micro mechanics that relate the local volume fractions of the constituents and the me-
chanical properties of the mixture. These theories show that local volume fraction and
the gradient of local volume fraction are related to the stress concentration or strain lo-
calization within the mixture. The stress concentration and strain localization are relat-
ed to fatigue cracking and permanent deformation properties of the mixture, and there-
fore are related to the performance of the mixture. For example, in the continuum
theory for granular materials (Goodman and Cowin, 1972) the stress in the mixture
(granular particles and voids) under the applied strain D ij can be expressed in the fol-
lowing equation:
T = (β − β v +α v v + α2 vv )δ − α v v + λ D δ + 2 μ D
2
2
ij 0 k , k , , kk ij i , j , k kk ij ij (3-9)
In Equation 3-9, a, b 0 and b are material constants, v represents the local volume
v ∂
fraction of solids, and v = is the gradient of the local volume fraction of solids.
k , x ∂
k
Compared with classic linear elasticity, T = λ D δ + 2μ D , additional terms contribute
ij kk ij ij
to the stress tensor.
The second order terms of the gradients, v ,k v ,k , vv ,kk , and v ,i v ,j , are usually very small.
If these terms are neglected, and D kk and D ij (macro variables) are considered as con-
stants, then the stress difference between two points in the space can be expressed in the
following equation:
T − T = ⎡β − β v − β( − β v ⎤δ) (3-10)
2
1
2
2
⎣
1 ⎦
ij ij 0 2 0 ij
These two equations indicate that under the same macro-strain tensor D ij , the stress-
es at two different locations would be different if their local volume fractions (local
solid volume fractions) are different. This conclusion has an important implication to
heterogeneous materials such as granular soil and AC: two specimens that have the
same total void ratio (void content) may not have the same properties as the non-uni-
form void distribution will introduce additional stresses (inhomogeneity induced ma-
terial force) in the material.
3.4.1.1 Local Volume Fraction Determinations
Local void volume fraction of granular soil was quantified by Frost and Kuo (1996)
based on serial section and stereology principles. Wang et al. (2002) presented a method