Page 228 - Mechanics of Asphalt Microstructure and Micromechanics

220 Ch a p t e r Sev e n

The virgin extension curve is:
E ε
+
ε
=
f () σ = ii (7-29)
i i i E ε
1 + ii
+
s
i
Replacing S i with S i one can obtain the unloading curve:
–
+
E ε
−
ε
=
f () σ = ii (7-30)
i i i E ε
1 + ii
−
s
i
S −
Denoting k = one can obtain the following relation (this applies to the magnitude
S +
of the strains):
ε
ε
=
− + i
f () = kf ( ) σ i
i
i
i
k (7-31)
For small strains, the linear viscoelasticity applies and the modulus can be repre-
sented as:
⎛ 1 n 1 ⎞ −1
ω
E * DBN (, T) = ⎜ + ∑ E + jωη T ()⎠ ⎟ (7-32)
⎝ E 0 i=1 i i
2
where j = complex number, j = –1
w = radian frequency (w = 2p f, f = frequency)
T = temperature
E i , h i can be obtained through a minimization process with a curve fitting the ex-
perimental results. By introudcing a function t = t 0 a T (T), where t 0 is a material constant
and a T (T) is the shift function, a complex modulus expression can be obtained.
Assuming that the WLF (William, Landel and Ferry) equation holds:
CT −T s )
1 (
−
τ = τ aT() = τ 10 C 2 T (7-33)
+−T s
0 T 0
The model can consider large strains, brittle failure at low temperatures, and be
extended to modeling damage, thixotropy, and healing (Benedetto and Olard, 2009).
Damage
(
d D )/ dN = function (ε , D )
iN E iN iN (7-34)
d Dη )/ dN = function (ε ,η )
(
iN η iN iN (7-35)
Thixotropy
Dh iNthixo = function thixo (dissipated energy, dissipated energy N , e iN ) (7-36)
Healing
d(Dh iN )/dt = function healing (loading parameters, D iN , Dh iN ) (7-37)