Page 201 - Mechanics of Asphalt Microstructure and Micromechanics

Fundamentals of Phenomenological Models 193

Where l and G are constants describing the linear time-independent stress-strain
relationships, Y 1 and Y 2 are the stress relaxation functions for e kk and e ij .
λ 1
ε =− σδ + σ
ij kk ij ij
2 GG + 3 λ) 2 G
(
2
or
t ∂ σ ξ() t ∂ σξ()
ε t() = δ a σ () + ∫ ϕ t − ξ) kk ]+ b σ t() + ∫ ϕ t − ξ) ij ξ d (6-147)
σ
[
(
(
t
ij ij 0 kk 1 ξ ∂ 0 ij 2 ξ ∂
0 0
Where a 0 and b 0 are constants describing the linear time-independent strain-stress
relations, j 1 and j 2 are creep functions for s kk and s ij .
There are non-linear viscoelasticity theories. These theories may not be necessary
and can be replaced with viscoplastcity and therefore will not be discussed. Interested
readers may refer to Findley et al. (1989).
6.4 Viscoplasticity (Perzyna, 1966)
6.4.1 General Concept
For a set of tests at different strain rates, one may obtain a set of curves as illustrated in
Figure 6.12. In other words, the Young’s modulus, the initial yield stress, the hardening
parameters, the peak strength, and the residual strength are not only a function of the
strain and plastic strain, but also a function of the strain rate. This phenomenon can
then be modeled as viscoplastcity. There are many viscoplasticity theories. Typical ones
used in AC followed the Perzyna model (isotropic hardening) and the Chaboche model
(isotropic and kinematic hardening). For example, the SHRP permanent deformation
model has both isotropic and kinematic hardening.

ε
ε →•

σ
ε

ε → 0

ε

FIGURE 6.12 Typical stress-strain curves for metals and other materials.