Page 193 - MODELING OF ASPHALT CONCRETE

VEPCD Modeling of Asphalt Concr ete with Gr owing Damage 171

complex hysteretic behavior of asphalt concrete much easier. The results from
experimental verification of the correspondence principle have been documented by
Kim and Little (1990), Kim et al. (1995), and Lee and Kim (1998a) using uniaxial
monotonic and cyclic data of asphaltic materials under a wide range of test conditions.
The material property needed to calculate the pseudostrain in Eq. (7-6) is the relaxation
modulus. Typically, the relaxation modulus test is not easy to perform due to the large
amount of stress that develops at the beginning of the test from the step input of
displacement. Therefore, the relaxation modulus is determined from the complex modulus
using the theoretical inversion process. This approach is explained in detail in Chap. 9.
Because the complex modulus tests are performed on a separate set of representative
specimens for the mixture in question, the relaxation modulus determined from the
complex modulus tests may not be the same as the one for specimens used in damage
testing due to sample-to-sample variability. In that case, the initial pseudostiffness in
Fig. 7-4(b) may not follow the line of equality (LOE). In order to minimize the effect of
the sample-to-sample variability, the initial secant pseudostiffness I is introduced. Thus,
the governing constitutive equation in uniaxial mode becomes

σ = I ε R (7-7)
where
ξ ε ∂
−
ε = 1 ∫ E( ξ τ) τ d (7-8)
R
E τ ∂
R o
In most cases, the I value remains between 0.9 and 1.1. When the I value is
significantly out of this range, a reexamination of the data (both the relaxation modulus
and test results) is necessary.
Pseudostrain Calculation
The definition of pseudostrain shown in Eq. (7-8) naturally yields a solution through a
linear piecewise technique, as shown below:
t 1 ⎡ t 2 t n ⎤
1 ε d ε d dε
ε = ⎢ ∫ Et − τ) 1 τ d + ∫ E t − τ) 2 τ d +
+ ∫ Et − ) n dτ ⎥ (7-9)
τ

(
(
(
R
E R ⎢ 0 ⎣ τ d τ d dτ ⎥ ⎦
−1
t n
t 1
Such a technique, though fundamentally sound, is profoundly inefficient when
analyzing large amounts of data. The source of the inefficiency lies in the need to analyze
all the time steps that precede the time step of interest, thus resulting in exponentially
increasing analysis time for increasing data amounts. To overcome this shortfall, a
method commonly used in computational mechanics, the state variable approach, is
utilized.
The goal of the state variable approach is to transform the process of convolution
into an algebraic operation. Theoretical details of state variable techniques can be found
in the literature (Simo and Hughes 1998). In a physical sense, though, the state variable
approach assigns a variable to each Maxwell element in the Prony representation of the
relaxation modulus, as shown below:
t − ρ
m
Et() = E + ∑ E e i (7-10)
∞
i
i=1