Page 276 - Mechanics of Asphalt Microstructure and Micromechanics

268 Ch a p t e r E i gh t

τ
b
Where b = max( ),0 ≤ ≤ t (8-76)
τ

1
i
b = (: i 2 (8-77)
εζ ε:)
τ
And
ζ = λ δ δ + μ δ δ + δ δ )
v
v
(
ijkl ij kl ik jl il jk (8-78)
z ijkl is a Rank 4 isotropic tensor, d ij is the Kroneker delta. t is real time. l , m , r, n are
v
v
temperature-dependent material constants. Under a “thermorheologically simple” as-
sumption, the viscosity at different temperatures are related by the following relation:
(
CT − 0 T )
T
ζ = ζ e 0 TT (8-79)
0
T 0 = reference temperature at which material constants are characterized; T-current
temperature; C T = material constant.
Plasticity Component
The classical rate independent Mises plasticity model with associated flow rule and
linear isotropic and kinematic hardening was adopted for the plasticity component.
The yield function:
α
σ
α
f (, q, ) = η − 2 K( ) (8-80)
3
. . η
p
The fl ow law: ε = γ η (8-81)

The evolution of the equivalent plastic strain:

. 2 . .
α = ε : ε p (8-82)
p
3
The evolution of back stresses (kinematic hardening):
. 2 η
α
q = H ()
'
3 η (8-83)

The other equations were also presented as follows:
t .
α = ∫ 2 ε τ d τ (8-84)
p
()
0 3
η = dev[]
σ − q
(8-85)
tr q[] = 0
(8-86)