Page 175 - MODELING OF ASPHALT CONCRETE

Interr elationships among Asphalt Concr ete Stif fnesses 153

Park et al. (1996) found that Eqs. (6-30), (6-33), and (6-34) yield comparable and
accurate relationships for interconversion. This holds true as long as both functions E(t)
and D(t) exhibit broadband and smooth variations on log-log scales. On that same
premise, the following interconversion scheme has also been proposed:

Dt() = 1 , and (6-35)
α
Et()

Et () = ⎛ 1 t ⎞ (6-36)
D⎜ ⎟
⎝ α ⎠

where

1
⎛ sinn π ⎞ n (6-37)
α = ⎜ ⎝ n π ⎠ ⎟

Interconversion between E* and D*
The significance of using response functions in their complex domain in characterizing
the LVE behavior of asphalt materials has been highlighted here as well as in other
chapters. On that premise, it thus becomes important to establish a relationship between
*
*
the material functions in the complex domain, specifically between E and D . Deriving
such a relationship is more convenient when E(t) and D(t) are represented in the Laplace
domain as follows (Tschoegl 1989):
∞

∫
Es () ≡ s E t e dt
−
( )
st
(6-38)
0
∞

∫
Ds () ≡ s D t e dt
−
( )
st
(6-39)
0
where the integrals in Eqs. (6-38) and (6-39) are the Laplace transforms of E(t) and D(t),
respectively. Incorporating the above equations with Eq. (6-21) yields the following
relationship between the relaxation and creep compliance in the Laplace domain:

()
Es D s() = 1 (6-40)
Complex material functions arise from the response to steady-state sinusoidal
loading with angular frequency w, and are related to the Laplace-transformed functions
according to the following relationship proposed by Tschoegl (1989):

Ew() = E s) (6-41)
(
*
s→ iw

Dw() = D s) (6-42)
*
(
s→ iw
*
Hence, the relationship between E and D can be deduced from Eqs. (6-40) to (6-42):
*
*
*
()
Ew D w = 1 (6-43)
()