Page 100 - Mechanics of Asphalt Microstructure and Micromechanics

Microstructure Characterization 93

Where U is the elastic energy contained in the volume; F is the work performed by
external force; W is the energy for crack formation, and S is the cracked surface.
d dW
−
( FU) is the energy release rate and is the crack resistance. If the material is
dS dS
truly brittle like glass, the energy for crack growth is the surface energy to form the new
dW
fractured surface, and γ − the surface energy for the material. If the volume
dS
change is further assumed to be negligible during the crack growth process, then:
dS = dS V (3-42)
v 0
Where S v is the specific damaged surface area and V 0 is the constant volume. There-
fore, the following evolution equation for S v holds.

d
( FU)− = V γ (3-43)
dS 0
v
3.6.1.2 Damage Tensor
The damage tensor s ij thus defined is widely used in CDM. The typical application can
be found in the following equation (Murakami, 1988):
−
σ = M σ (3-44)
ij
ijkl kl
Where s ij , s kl are the effective stress tensor and the Cauchy stress tensor respec-
tively, while M ijkl is a symmetric fourth rank tensor—the damage effect tensor. By using
the vector formats of the tensors, Murakami (1988) showed that:
–1
M = (I − j) (3-45)
Where I is the second-rank identity tensor, and j is the damage tensor.
Voyiadjis and Kattan (1999) derived an explicit relation between M and j. The di-
agonalized form involves the terms of 1/(1 − j i ), and 1/(1 − j i )(1 − j j ) (i,j=1,2,3, i ≠ j).
The averages of these terms and 1/(1 − j 1 )(1 − j 2 )(1 − j 3 ) may serve as empirical quan-
tity to represent the state of damage. Voyiadjis and Kattan (1999) also introduced a term
ϕ = ϕ + ϕ + ϕ to represent the overall damage. This term was used to represent the
2
2
2
1 2 3
damage states of the specimens obtained from the WesTrack project (Wang et al., 2001b).
3.6.1.3 Spacing-Size Ratio
The spacing between defects and the size of the defects are important variables in non-
local damage theories. For modeling applications, the size of cracks or voids and the
average spacing among the cracks offer two relevant parameters that are related to
performance. Chudnovsky et al. (1987a, 1987b) developed a simple model addressing
the interaction of two cracks, in which the two important model parameters are the
spacing between the two crack tips and the sizes of the cracks. Figure 3.24 presents the
configuration of two interacting cracks. The stress intensity (mode I) of the large crack
K eff
eff
is affected by the adjacent cracks. The ratio 1 (where K is the effective stress inten-
1
K 0 1
sity of a crack when subjected to the influence of the adjacent crack while K l is the stress
0
intensity without the influence of adjacent cracks) decreases with the increase of l/d,
which means, the larger the crack size and/or the smaller the spacing, the larger the