Page 225 - Mechanics of Asphalt Microstructure and Micromechanics

Models for Asphalt Concrete 217

J 3D = third invariant of the deviatoric stress tensor
n = parameter associated with the phase change from contractive to
dilative response
g and b are associated with the characteristics of the ultimate yield surface
a
α = 1 is the hardening or growth function
ξ η 1
a 1 and h 1 are parameters to describe the hardening features.
Where:
ξ = ξ + ξ (7-22)
v D

1
ξ = ε p (7-23)
v ii
3
D ∫
p 12
p
ξ = ( dE • dE ) / (7-24)
ij ij
p
p
e ij , E ij and e ii are the total, devitoric, and volumetric strains.
p
The disturbance D follows the following evolution law:
D = D (1 − e − ξ z ) (7-25)
A D
u
Where D u , A, and Z are parameters to characterize the disturbance (could be heal-
ing) evolution. While the definition of D is quite clear, the association of D with the
microscopic damage or other macroscopic observable quantity is not an easy task. For
the rate independent plasticity, by plotting the plastic strain-loading process, one can
calculate x D and then fit the disturbance evolution law.
In the case of cyclic viscoplasticity, one may assume a power law to associate x D at
different loading cycles.
⎛ N ⎞ b
ξ N = ξ N ) ⎟ (7-26)
()
(
r ⎜
D D ⎝ N r ⎠

⎡ ⎪ ⎧ 1 ⎛ ⎞ ⎫ 1/ Z ⎤ 1/b
⎪
N = N ⎢ 1 ⎨ ln ⎜ D u ⎟ ⎬ ⎥ (7-27)
r ⎢ ⎢ ξ ( N ) A ⎝ D − D⎠ ⎪ ⎥
r ⎩ ⎪
⎭
⎣ D u ⎦
Through plotting the accumulative plastic strain-N relationship, one may obtain the
parameter b and therefore define the damage evolution law. The DSC model has a ratio-
nal core. More applications in using it to model the behavior of AC are necessary.
7.4 The Benedito Model
Another model that targets at the general deformation and failure of AC in a unified
format is the Di Benedito and Neifar (DBN) Model. In a series of publications (Bene-
detto et al., 2004a, b, 2007a, b, 2009), Benedito and his colleagues presented a general
model of viscoplasticity. The major features of the model lie in its general format to