Page 234 - Mechanics of Asphalt Microstructure and Micromechanics

226 Ch a p t e r Sev e n

Extended Model
σ
σ
σ
ε (, , ) = ε σ, ) + ε (, , ) + ε (, , ) + ε ( ,) + ε ( , ,) (7-65)
σ
t
T
T
t
T
(
T
t
t
T
t
,
T
ges el ve vp th tr
·
e tr (t, s, T) is tertiary strain rate.
The above formulations can be extended to 3D cases.
Elastic
E μ
σ = H ⋅[ ε + H ε ⋅ el δ ⋅ ] (7-66)
el
ij 1 − μ ij 1 −⋅ μ ij
2
H H
Viscoelastic
σ = σ + σ (7-67)
KH KN
σ = E (, ) ε t, σ T) (7-68)
σ T ⋅
,
(
KH K ve
σ = η σ T ⋅ ε ( t, σ T) (7-69)
)
,
(,
KN K ve
The stress s KH is the part of the overall stress s giving rise to the viscoelastic strain
at time t. In contrast, the stress s KN constitutes the part of the overall stress s responsible
for the increase in the viscoelastic strain at time t.
Viscoplastic
For viscoplastic strain the corresponding evolution equation already exists in the form
of Equation 7-70 below.
σ
,
ε (, σ T) = (7-70)
t
vp ησ T(, )
N
Tertiary Part
σ = σ + σ
DH DN
D (,
σ T)
σ =− 1 ⋅ ε t(, σ T, )
DH tr
σ T)
D (,
2
σ = 1 ⋅ ε t(, σ T, )
DN D (, tr (7-71)
σ T)
2
σ = σ + σ
HH HN
σ T)
H (,
σ =− 1 ε ⋅ t (, σ T, )
HH tr
σ T)
H (,
2
σ = − 1 ⋅ε t (, , T)
−
σ
σ
HN H (, T) tr (7-72)
2
The stresses s DH and s HH are coupled to the tertiary strain via the corresponding
ratios of the parameters D 1 and D 2 or H 1 and H 2 .