Page 163 - MODELING OF ASPHALT CONCRETE

Interr elationships among Asphalt Concr ete Stif fnesses 141

asphalt concrete were purely elastic, then D(t) and E(t) would be the reciprocal of each
other. However, due to the viscoelastic nature of asphalt concrete this is only true in the
Laplace transform domain.
In equation form
ε t ()
Dt() = , and (6-5)
σ 0

σ t ()
Et() = (6-6)
ε
0
where D(t) and E(t) = creep compliance and relaxation modulus, respectively
s and e = constant input stress and strain, respectively
0 0
s(t) and e(t) = stress and strain response, respectively
For uniaxial loading, and nonaging, isothermal conditions, the linear viscoelastic
stress-strain relationships are represented by the Boltzmann convolution integral as
follows:
t ε d
∫
σ = Et − τ) τ d (6-7)
(
0 τ d
t σ
∫ − d
ε = Dt τ) τ d τ d (6-8)
(
0
where t is an integration variable. Substituting e in the right-hand side of Eq. (6-7) by its
equivalent from Eq. (6-8) results in the constitutive equation relating D(t) and E(t) as
follows:
t dD t()
∫
τ
1 = Et − ) dτ dτ (6-9)
(
0
Complex Modulus
∗
The complex modulus E is a response function that relates stresses and strains for a
linear viscoelastic material subjected to sinusoidal loading. It is composed of two
∗
components: dynamic modulus |E | and phase angle f, defined as follows:

σ
E = amp , and (6-10)
*
ε
amp

Δ
π
φ = 2 ft (6-11)

where s = stress amplitude
amp
e = strain amplitude
amp
f = loading frequency
Δt = time lag between stress and strain response