Page 177 - MODELING OF ASPHALT CONCRETE

Interr elationships among Asphalt Concr ete Stif fnesses 155

Approximate Analytical Methods
Numerous approximate analytical interconversion methods with different bases,
simplifications, assumptions, and accuracies have been proposed by Tschoegl (1989),
Schapery (1962), and Park et al. (1996), and others. Methods of primary interest are
those that have been widely used by asphalt researchers to relate between the time,
frequency, and Laplace domain representations of modulus and compliance.
Schapery (1962) presented two approximate analytical methods for interconversion

between the uniaxial relaxation modulus E(t) and the operational modulus Es( ), defined
as the Carson transform or the s-multiplied Laplace transform of E(t); refer to Eq. (6-53).

∞ ∞

∫
∫
−
−
()
st
()
st
t
Es ( ) ≡ s E t e d(ln t) = s E t te d(lnt) (6-53)
0 −∞
The first relationship is given by

Et() ≅ E s) or Es () ≅ E t ( ) (6-54)
(
s=α t t=α s

where Es() ≡ sE s(), Es() is the Laplace transform of E(t); and a = e , where C is Euler’s
−C
constant, thus yielding a ≅ 0.56.

The second relationship is given as follows:

Es () ≅ E t ( ) (6-55)
t=β s
n
where β = {(Γ 1 − )} − 1 n , Γ(.) is the gamma function, and n is the local log-log slope of the
(
source function given either as n ≡ dlog E t) or n ≡ dlog E s) . If the moduli in Eqs. (6-54)
(
dlog t dlog s ∼
and (6-55) are replaced by compliances, analogous relationships between D(t) and D(s)
are obtained.
Christensen (1982) proposed an approximate interconversion between E(t) and the
storage modulus E′(w) as follows:
E w)
Et() ≅ ′ ( or ′ Ew() ≅ E t( ) (6-56)
w=2 π t = t 2 π w
Analogous relationships hold for compliance functions when E′s in Eq. (6-56) are
replaced by D′s.
Staverman and Schwarzl (1955) gave the following approximate conversion from
storage modulus E′(w) to the loss modulus E″(w):
π dE ()
′ w
Ew ≅ (6-57)
′′()
2 dln( w)
Booij and Thoone (1982) proposed the following conversion from E″(w) to E′(w):
π w dE [ ′′( )/ ]
ww
′ E ()w ≅ E − , or (6-58)
e 2 dlnw
π ⎛ dln E ′′ ⎞
()
′′()
′ Ew ≅ E e + ⎜ 1 − ⎟ Ew (6-59)
2 ⎝ dln w ⎠
where E is the equilibrium modulus. Equations (6-57) and (6-59) also apply to compliance
e
components when E′(w) and E″(w) are replaced by D′(w) and −D″(w), respectively.