Page 198 - MODELING OF ASPHALT CONCRETE

176 Cha pte r Se v e n

microcracking phenomenon. Therefore, only one internal state variable (i.e., S) is used
to model the damage growth in tension.
The method used to solve the damage evolution law in Eq. (7-24) is a matter of
preference and, as such, two different solutions are hereby proposed. The first, proposed
by Park et al. (1996), transforms the original form of the equation to an integrated form,
ˆ
assumes a >> 1 and defines a new parameter S . Equation (7-25) presents, in discrete
form, the method proposed by Park et al.:

1
⎡ 1 ⎞ ⎤ + 1
1
⎛ ˆ
S = ⎢ S 1 + ⎟ ⎥ α (7-25)
⎜
⎣ ⎝ α ⎠ ⎦
ˆ
where S is given by Eq. (7-26)
ˆ
) t
S ˆ = S − 1 (C − C )(ε R 2 α 1 (7-26)
i+1 i 2 i i−1 i
Lee and Kim (1998a, b) also propose a solution that utilizes the chain rule and makes
no assumption regarding a. The solution of these researchers is presented in Eq. (7-27).
It is noted that both methods have been successfully applied in asphalt concrete research
(Park et al. 1996; Daniel and Kim 2002; Chehab et al. 2003).

α 1
+α
⎡ 1
1
)
S i+1 = S + − ( C − C )(ε i R 2 ⎤ ⎥ ⎦ t Δ 1 +α (7-27)
i−1
i
⎣
i
⎢ 2
To reconcile the approximations of these methods, an iterative refinement technique
is incorporated into this research. In short, this method assumes that the rate of change
in damage is constant over some discrete time step. This rate of change is determined at
a point near the current value of damage (S + dS) where the extrapolation error is
i
minimized.
The method begins with an initial calculation of S by either of the approximate
methods, both of which require results from constant crosshead rate tests for the stress-
pseudostrain relationship. The initial S values are plotted with the pseudostiffness values
C, calculated from the following relationship, which is obtained from Eq. (7-17):
σ
C = (7-28)
I × ε R
The C and initial S values are then fit to some analytical form, such as the one
presented in Eq. (7-29), where a and b are fitting parameters:

C = e aS b (7-29)
Returning to the damage evolution law, and noting that the increments of time are
generally small, one can write the rate of change in damage as
dS = Δ S (7-30)
dt t Δ