Page 240 - MODELING OF ASPHALT CONCRETE

218 Cha pte r Ei g h t

Plasticity Models
In the general DSC equations, Eq. (8-1), C i represents the behavior of the RI material. It
~
can be characterized as elastic, nonlinear elastic, or elastoplastic hardening; for the
latter, the unified hierarchical single surface (HISS) plasticity model can be used. The
yield function F in the HISS model for isotropic hardening is given by Desai (1980) and
Desai et al. (1986) as follows:
F = J − −α J + γ J ) (1 − β S ) −0 5 . = 0 (8-3a)
2
(
n
2 D 1 1 r
⋅
= J − F F = 0 (8-3b)
2 D 1 2
2
where J = J / p = nondimensionalized second invariant (J ) of the deviatoric
2 D 2 D a 2D
stress tensor S
ij
p = atmospheric pressure constant
a
J = ( J + 3 R)/ p = nondimensionalized first invariant of the total stress
1 1 a tensor σ
ij
R = parameter proportional to cohesion
3
S = stress ratio J 3 D J ⋅ 2 − D 2
r
J = third invariant of S
3D ij
n = parameter associated with the phase change from contractive
to dilative response
g and b are associated with the ultimate yield surface, Fig. 8-7, and a is the hardening
or growth function given by
a
α = 1 (8-4)
ξ η 1
p
where ξ =∫(d ε ⋅ d ε ) / is the trajectory of total strains and a and η are the hardening
p 12
ij ij 1 1
parameters. x can be decomposed as
ξ = ξ + ξ
v D
1
ξ = ε p (8-5)
v ij
3
p 12/
ξ =∫ dE( p ⋅ dE )
D ij ij
p
ε , E , and ε are the total, deviatoric, and volumetric plastic strains, respectively.
p
p
ij ij ii
Schematics of F in various stress spaces are shown in Fig. 8-7.
The yield function F in Eq. (8-3) is found to be suitable for compression or tension
(yielding) response, which can be shown in the positive quadrant of J − J plot,
1 2 D
Fig. 8-7. For predominantly (yielding) compressive behavior, the possibility of tensile
condition can be accounted for approximately by using an ad hoc scheme such as the
stress transfer method when the material enters the tensile regime. Erkens et al. (2002)
have discussed such a need for asphalt concrete, and modified functions with an ad
hoc procedure have been proposed. For some materials, F in Eq. (8-3) may need to be