Page 174 - MODELING OF ASPHALT CONCRETE

152 Cha pte r S i x

The interconversion in Eq. (6-27) provides a good relation between the response
functions when the material exhibits mostly elastic behavior with minimal viscoelasticity.

Power-Law-Based Interrelationship
An LVE material can be approximately represented by simple power law for small
ranges in their transition zone. Representing E(t) and D(t) in pure power-law form
[Eqs. (6-28) and (6-29), respectively], interrelationship Eq. (6-30) is obtained.
()
Et = E t − n (6-28)
1
()
Dt = D t n (6-29)
1
sin nπ
Et D t () = (6-30)
()
nπ
where E , D , and n are all positive constants. Equation (6-30) was first given by
1 1
Leaderman (1958). It is accurate in regions in which E(t) and D(t) are represented
approximately by straight lines on log-log scales, with the exponent n being the absolute
value of the slope of these lines. When n approaches zero, that is, for an elastic material,
the right-hand side of Eq. (6-30) becomes unity and Eq. (6-27) becomes exact. In equation
form, n is represented as

dlog R t)
(
n = H (6-31)
dlog t
where R (t) is the unit response function of interest.
H
Interrelationship by Christensen
Christensen (1982) developed an approximate interconversion method between E(t) and
D(t) using approximate relationships between the real and imaginary parts of a complex
material function, and between the transient function and the real part of the complex
material function: If E(t) is known then D(t) can be determined according to Eq. (6-32):

Et()
Dt() ≅
π 22 dE t() ⎫ 2 (6-32)
t ⎧
Et() + ⎨ ⎬
2
4 ⎩ dt ⎭
Equation (6-32) is also applicable when D(t) and E(t) are interchanged. In terms of
n, Eq. (6-32) can be expressed as follows:
1
()
Et D t() ≅ n π 2 (6-33)
2
1 +
4
Interrelationship by Denby
An approximation similar to that presented above was proposed by Denby (1975),
yielding the following interrelationship:

1
()
Et D t() ≅ n π 2 (6-34)
2
1 +
6