Page 196 - Mechanics of Asphalt Microstructure and Micromechanics

188 Ch a p t e r S i x

6.3.6.4 Generalized Models
Generalized Maxwell and Kelvin models are comprised of a number of the basic ele-
ments in parallel or series. The solutions to these generalized models can be derived
following the Laplace transform approach. In some cases, they can be obtained through
simple linear addition. Generally, the governing equations of these models can be ex-
pressed as a generic format:
P( , , ,...) = Q( , , ,...)
εεε
σσσ
In operator format, it is expressed as:
Pσ = Qε

∂ a σ ∂ b σ
Pσ = p σ + p σ + p σ + ... + p = q ε + q ε + ε + ... q Q ε (6-110)
+
q
0 1 2 a t ∂ a 0 1 2 b ∂t b
Through Laplace transform, one can have:
+
+
s =
Ps σ s = p s p s + ... p s σ+ a ) ˆ () Q (s) ˆ( ) (ε s = q + q s q s + ... q s b ) ˆ()
ˆ
ε
+
() ˆ ( ) (p +
ˆ
2
2
s
)
0 1 2 a 0 1 2 b
ˆ
()
Qs = ˆ σ (6-111)
ˆ ( P s) ˆ ε
s

Through an inverse Laplace transform one can obtain the following equations:
q + q s q s + ... q s+ b
+
2
σ s = 0 1 2 b ] ˆ ε s () (6-112)
ˆ () [
+
2
1 + ps p s + ... p+ s a
1 2 a a
or
+
2
1 + ps p s + ... p s+ a
ε s = 1 2 2 a b ] ˆ σ s () (6-113)
ˆ() [
+
q + q s q s + ... q+ b b s
1
2
0
The above methods can be extended to the stress tensors and strain tensors in 3D
cases:
Pσ () = Q ε () (6-114)
t
t
1 ij 1 ij
It can be decomposed into the deviator stress-strain relationship and the volumetric
stress-strain relationship:
∂ ∂ 2 ∂ a ∂ ∂ 2 ∂ b
=
p
st
q
[ ′ + ′ p + ′ p +...+ ′ p ] ( ) [ ′ q + q′ + q′ +...+ q′ ] dt ( )
0 1 2 2 a a ij 0 1 t ∂ 2 t ∂ 2 b t ∂ b ij
∂t ∂t ∂t
(6-115)
Pσ () = Q ε ()
t
t
2 ij 2 ij
And the bulk stress and volumetric strain relationship:
∂ ∂ 2 ∂ a ∂ ∂ 2 ∂ b
σ
= q
p
[ ′′+ ′′ p + ′′ p +...+ ′′ p ] (t))[ ′′+ ′′ q + ′′ q +...+ ′′ q ]ε t()
0 1 2 2 a a ii 0 1 2 2 b b ii i
∂t ∂t ∂t ∂t ∂t ∂t
Pσ () = Q ε () (6-116)
t
t
3 ii 3 ii