Page 195 - Mechanics of Asphalt Microstructure and Micromechanics

Fundamentals of Phenomenological Models 187

σ
FIGURE 6.11 Illustration of
the Burger model.

R ε
1 1

η ε
1 2
ε

σ σ
1 2

R η 2 ε
2 3

σ

For the constant-strain process, the generic governing equation can be expressed
as:

ε
σ + p 1 σ + p 2 σ = q 1 0 + q 2 0 δ dt() (6-105)
ε δ t()
dt
where
η η η η η ηη
p = 1 + 1 + 2 , p = 12 , q = η , q = 12 (6-106)
1 R R R 2 RR 1 1 2 R R
1 2 2 1 2 2
The Laplace transform of the above equation yields:

ˆ σ + ps ˆ σ + ps 2 ˆ σ = q ε + q ε s (6-107)
1 1 10 2 0
ε (q + q ) s
ˆ σ = 0 1 2 (6-108)
+
1 + ps p s 2
1 2
The inverse transform of the above equation yields:
ε
σ() = 0 [(q − q ) r e − 1 rt − (q − q r )e − 2 r t ]
t
A 1 2 1 1 2 2 (6-109)
where
2
,
r = ( p − A)/ 2 p r = ( p + A)/ 2 p A = p − 4 p
,
1 1 2 2 1 2 1 2