Page 209 - Mechanics of Asphalt Microstructure and Micromechanics

Fundamentals of Phenomenological Models 201

will have 50% pores or voids. This is equivalent to about 45% volume increase (consider-
ing the original air void content being around 5%). However, experimental observations
do not indicate such big volume changes. Therefore, it may be concluded that damage
effects cannot be completely represented by the damage effect thus defined. In other
words, the fatigue phenomena cannot be explained by CDM only.
Lamaitre (1996) also presented summaries on the development of plasticity and
viscoplasticty theories that include damage and coupling. These theories have not been
applied to AC, and are not presented here due to limited space.
6.5.8 Damage Concept Extended to 3D Cases
In 3D, the degree of damage on surfaces of different orientations may vary. The formu-
lations for the 1-D or homogenous damage may not be valid for the 3D cases.
Murakami (1988) introduced a symmetric fourth-rank tensor named damage effect ten-
sor M. M will link the effective stress and the stress in the undamaged material by the
following formulation:
σ = M σ (6-183)
ij ijkl kl

The matrix format of the damage effect tensor, according to Murakami (1988), is
linked to the second-rank damage tensor ö by the following relationship:
−1
M = ( I - ö) −1 = det( G G T (6-184)
)

Where G is a fictitious deformation gradient given by:
x ∂
G = i where x and x are the coordinates in the damaged state and the ficti-
–
ij x ∂
j
tious effective undamaged state, respectively. Details regarding the fictitious effective
undamaged state can be found in Vojiadjis and Kattan (1999).
6.6 Fracture Mechanics (Anderson, 1995)
6.6.1 General Concept
Fracture mechanics has been widely used in the modeling of fracture of AC, especially
fatigue cracking. This section presents only the most fundamental concepts essential to
the understanding of the research in binder, mastic, and AC. Fracture due to cyclic load-
ing is an important topic in this section as well. An excellent textbook reference is An-
derson (1994).
6.6.2 Stress Concentration-Macroscopic View
Figure 6.13 illustrates an elliptic hole embedded in a large plate (width w 2a and its
height h 2b). An elastic analysis for the problem with its boundary subjected a uni-
form stress s indicates the following relationship:
⎛ 2 a⎞
σ = σ 1 + (6-185)
A ⎜ ⎝ b ⎠ ⎟
2
In terms of the radius of curvature r (r = b for the ellipsis) at Point A, the above
equation can be represented as: a
⎛ ⎞
+
σ = σ 12 a ⎟ (6-186)
⎜
A ⎝ ρ ⎠