Page 202 - Mechanics of Asphalt Microstructure and Micromechanics

194 Ch a p t e r S i x

In theory, according to the objectivity principle, the modulus, initial yielding stress,
the hardening laws, the ultimate yielding stress, the rate of softening, and the residual
strength are all related to strain rate and should carry it as an independent variable. A
review indicates that most of the viscoplasticity models used in AC do not include
strain rate in all the equations. However, many of them include the strain rate effect in
the yielding criteria.

6.4.2 Perzyna Models
Under the title of viscoplasticity, many models developed in AC do not have adequate
consistency. Most of the viscoplascticity models are based on Perzyna (1966). There is
no ambiguity in viscoelasticity and elastoplastcity. However, there is no unique defini-
tion for viscoplasticity. The two major sub-branch definitions, according to Perzyna, are
elastic-viscoplastic and elastic/viscoplastic; the former demonstrates viscous proper-
ties in both elastic and plastic regions; the latter shows viscous properties in the plastic
region only. Considering the following decomposition:
ε = ε + ε + ε p (6-148)
v
e
ij ij ij ij

Therefore the elastic-viscoplastic model can be considered as:
ε = ε + ε vp (6-149)
ve
ij ij ij
The elastic/viscoplastic relationship can be considered as:
ε = ε + ε vp (6-150)
e
ij ij ij
The rate (time) equation is:
ε = ε + ε + ε p (6-151)
v
e
ij ij ij ij
Therefore for the elastic-viscoplastic model, the rate equation can be represented
as:
ε = ε + ε vp (6-152)
ve
ij ij ij
The elastic/viscoplastic relationship can be represented as:
ε = ε + ε (time rate should be valid, please verify it) (6-153)
vp
e
ij ij ij
Where the elastic component is not time-dependent.
6.4.2.1 Elastic-viscoplastic Model
One example may involve the use of the general visecoelastic constitutive relation to
represent:
t ∂ σ ξ() t ∂ σξ()
2 ∫
ε () = δ a σ () + ∫ ϕ t − ξ) kk ]+ b σ () + ϕ ( t ξ) ij dξ (6-154)
−
ve
t
b
[
t
t
(
ij ij 0 kk 1 ξ ∂ 0 ij ξ ∂
0 0
For the plastic part, the viscoplastic yielding will be dependent not only on the
stress state and the plastic deformation but also on the viscoelastic strain, which is time
dependent. Therefore, even within the elastic region, depending on the stress path it