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238 Ch a p t e r Sev e n

heat transfer and thermal stress. Ullidtz (2005) proposed a damage model based on
Miner’s law with a nonlinear relation between damage and the number of load applica-
tions. Damage was defined as a decrease in the modulus of material or as an increase in
permanent deformation. The model was based on experiment data. Ker et al. (2008)
used an LTPP database to develop an improved fatigue cracking prediction model. The
generalized additive model (GAM) extended from the generalized linear model along
with assumption of the Poisson distribution and adoption of the quasi-likelihood esti-
mation method. The proposed model included variables such as KESALs, pavement
age, annual precipitation, annual temperature, critical tensile strain under the AC sur-
face layer, and freeze-thaw cycle for the prediction of fatigue cracking.

7.13 Non-Local Theory
Traditional elasticity relates the stress of a point to the strain at that point only. This type
of theory is usually named local theory. A more complicated theory could associate the
stress at a point with the average strain surrounding that point. In a simple case, one
may assume that the elastic strain at r (vector) surrounding a position x (vector) de-
pends linearly on the relative distance between r and x, which occurs in a small but fi-
–
nite material volume V surrounding x. The average strain e can be obtained by the
volume average of the local strain distribution within the representative volume ele-
ment RVE as follows:
ε = 1 ∫ ε xr dV (7-110)
− )
(
V
V Re f
Taylor’s series expansion limited to the second-order term of the function (x-r)
around x gives:

ε = ε() + ∇ 2 ε (7-111)
2
l
x
c
Where l c = characteristic length scale of the material microstructure. e is the strain
2
gradient around point x. The following gradient elasticity model is obtained when the
above average strain is substituted in Hooke’s law:
2
σ = E ε + l E ∇ 2 ε (7-112)
c
Where s and e = stress and strain tensors, respectively, and E = fourth-order elastic-
ity tensor.
Dessouky et al. (2006) applied the gradient elasticity theory and implemented an
FEM code for modeling the AC behavior. The microstructure of AC is also incorporated
into their modeling work.