Page 171 - MODELING OF ASPHALT CONCRETE

Interr elationships among Asphalt Concr ete Stif fnesses 149

the Prony series representation of both the source and target response functions. Mead
(1994) presented a numerical interconversion method based on constrained linear
regression with regularization, while Ramkumar et al. (1997) proposed a regularization
technique that used a quadratic programming technique. The aforementioned methods
as well as others not mentioned have been covered extensively by Schwarzl and Struik
(1967), Ferry (1980), and Tschoegl (1989). This chapter sheds light on some of the mostly
used techniques in converting between LVE response functions for asphalt materials.
Both the E(t) and D(t) are essential elements for characterizing viscoelastic responses.
They have been incorporated in several constitutive models that have been developed
and used over the years including some of those presented in this book (Schapery 1961
and 1974; Ferry 1980; Christensen 1982; Tschoegl 1989; Kim et al. 1990 and 1997; Lee and
Kim 1998; Uzan 1996; Bahia et al. 2000; Daniel and Kim 2002; Roque et al. 2002;
*
Chehab et al. 2003). As for E , its acceptance as an indicator of the AC mix behavior has
been on the rise especially after it was chosen as a Superpave simple performance test
for mix design, a QC/QA tool, in addition to an indicator of stiffness in the M-E design
guide. It is also used for depicting the material’s response under sinusoidal loading.
Thus, with the simplicity and wide use of the complex modulus test, using interconversion
*
methods to obtain E(t) and D(t) from E becomes more favorable than conducting
additional experimental tests to obtain them.
Interconversion is not always simple and straightforward. Using exact solutions
may require integration over an infinite range, a task that is often complicated whether
done analytically or numerically. In addition, experimental data required for the
interconversion may only be available for a limited range of the required time or
frequency domain. To overcome such difficulties, adopting approximate analytical and
numerical techniques becomes necessary. In what follows, both approximate and
analytical methods of interconversion are presented.

Approximate Numerical Methods
Several numerical methods have been used in converting between LVE response
functions. According to Taylor et al. (1970) and Park and Schapery (1999), those methods
are especially useful when the response functions are expressed in the Prony series
form. In what follows, t and w are used as symbols for time and radial frequency,
respectively, or as their corresponding temperature-reduced quantities.
Interconversion between E(t) and D(t)
The exact relationship between relaxation modulus E(t) and creep compliance D(t) in
time domain is given by the following integral:
t dD()
τ
∫ Et −τ ) d = 1
τ
(
dτ (6-21)
0
Taking the Laplace transform of Eq. (6-21) yields
Es D s() = 1 (for t > 0) (6-22)
()
s 2
where fs() ≡ ∫ ∞ f t e dt denotes the Laplace transform of f(t) and s is the transform
−
st
(
)
0
parameter. In solving the integral, a typical numerical approach requires that the
integral be divided into a large number of time segments. This can be easily achieved
when the functions are presented in their Prony series form. This being the case, when