Page 278 - Mechanics of Asphalt Microstructure and Micromechanics

270 Ch a p t e r E i gh t

p
,
Where σ,,q α ε are the stress tensor, kinematic center, equivalent plastic strain,
and plastic strain tensor. are correspondingly the rate of the variables.
The unified implicit solution scheme could be represented in Equations (8-92),
(8-93), and (8-94).
p p
. .
Δε = Δt[(1 − θ ε k + θ ε k+ ] (8-92)
p
)
1
. .
Δq = Δt[(1 − ) q +θ q ] (8-93)
θ
k k+1
. .
Δα = Δt[(1 − θ α + θα +k ] (8-94)
)
k 1
• • •
p
If e k+1 , q k+1 and a k+1 are expressed as a Taylor series to the first order, the following
equations could be obtained:
. . .
. p . p ∂ε p ∂ε p ∂ε p
ε k+ ≈ ε k + : Δ σ + : q + • Δ α (8-95)
Δ
1
∂σ ∂q ∂α
. . .
. . ∂ q ∂ q ∂ q
Δ
Δ
q ≈ q + : σ + : Δ q + • α (8-96)
k+1 k ∂σ q ∂ ∂α

. . .
. . ∂α ∂α ∂α
α ≈ α + : Δ σ + : Δ q + • Δ α (8-97)
k+1 k ∂σ ∂q ∂α

Assume the stress increment could be approximated as:
p
Δσ = C :( Δε − Δε ) (8-98)
Then one would have the following equations:
. . .
∂ε p . ∂ε p ∂ε p
Δ
[C + Δ tθ ]: Δσ = Δε − t Δ ε p k − t Δ θ : q − Δtθ •Δ α
−1
t
∂σ q ∂ ∂α (8-99)
. . .
∂ q . ∂ q ∂ q
[I − Δ tθ ]: q = Δ tq + Δ tθ : Δσ + t Δ θ • Δα
Δ
Δ
k
q ∂ ∂σ ∂α (8-100)
. . .
∂α . ∂α ∂α
[1− Δtθ ]• Δα = Δtα k + Δtθ : Δσ + Δtθ : Δq
Δ
∂α . . ∂σ q ∂ (8-101)
.
∂ε p ∂ q ∂α
In the SHRP model, 0, 0, 0 and thus one has:
∂α ∂α ∂α
. .
∂ε p . ∂ε p
[C + Δ tθ ]: Δσ = Δε − t Δ ε p k − t Δ θ : q (8-102)
Δ
−1
∂σ ∂ q
. .
∂ q . ∂ q
Δ
[I − Δ tθ ]: q = Δ tq + Δ tθ : Δσ (8-103)
k
∂ q ∂σ
. .
. ∂α ∂α
Δ θ
Δα = Δ α + Δ θ : Δσ + t : Δq (8-104)
t
t
k ∂σ ∂q

Note: All the derivatives are evaluated at status k and Δe = eΔt.