Page 231 - Mechanics of Asphalt Microstructure and Micromechanics

Models for Asphalt Concrete 223

The modified yield function reduces to the classical Drucker–Prager function for
the special case of isotropic and intact (no damage) material. Assuming a power law for
the viscoplastic yield function, it becomes:
⎧ , 0 No Viscoplastic Strain , J − αI − κ ≤ 0
e
e
⎪
φ() f = ⎨ e e e 2 1 e e (7-47)
κ
⎩ ⎪ [ J −α I −κ ] Viscoplastic Strain Occurs J −α I −κ > 0
1
2
2
1

7.6 Temperature Dependent Viscoplastic Hierarchical
Single Surface (HiSS) Model
Huang et al. (2002) extended the HiSS model by Desai et al. (1986) to include temperature
dependence of the model parameters. The major structure follows the Perzyna model.
Similar work in this category includes the Delft model (Erkens et al., 2003) and the Uni-
versity of Maryland (UOM) model (Gibson et al., 2003; Gibson, 2006). The UOM model is
more inclusive but similar to the VEPCD model presented in Section 7.2. For the purpose
of conciseness, neither the delft model nor the UOM model is presented in detail.

Strain Components (rate equation)
eθ
e = e + e vpθ (7-48)
ij ij ij
Elastic Component
The elastic component follows the linear elasticity.

eθ
eθ
eθ
pθ
δ
dσ = C de = C ( de − de − α dθδ ) (7-49)
ij ijkl kl ijkl kl kl T kl
Viscoplastic Component
Yield function:
⎪
J ⎧ ⎪ J J n ⎪ ⎫
α
θ
γθ
F = ( 2 D 2 ) − ⎨ () ⋅( 1 2 − () ⋅( 1 ) ⎬ ⋅[1 − ( )βθ ⋅ ]S r − .05 (7-50)
)
p ⎩ ⎪ p p a ⎭ ⎪
a a
J 1 is the first invariant of the stress tensor; J 2D is the second invariant of the devia-
toric stress tensor; P a is atmospheric pressure; a (q ) is the hardening function; and b (q )
is material parameters. S r is stress ratio and q is temperature.
For associated isotropic hardening material, the thermoplastic strain increment
flow rule is:
F ∂
de = λ (7-51)
pθ
ij ∂ σ
ij
Where l is the proportionality factor.
Considering the time dependency, the total strain rate is:
e = e + e vpθ (7-52)
eθ
ij ij ij