Page 185 - Mechanics of Asphalt Microstructure and Micromechanics

Fundamentals of Phenomenological Models 177

In 3D cases, the above criterion transfers to:
f ( ) (σ= − σ ) (σ+ + σ )sinϕ − 2 ccosϕ (6-66)
σ
1 3 1 3
6.2.2.4 Druker Prager Criterion
Druker and Prager (1952) proposed a criterion that combines the Mises criterion with
the Mohr-Coulomb criterion, which has been known as the Druker-Prager criterion.
ϕ
fs I ) = J + I tan / 6 − Y(ε p ) (6-67)
(
,
1 2 1
It may be interpreted as the Mohr-Coulomb criterion applied to the octahedral
plane.
6.2.3 Plastic Flow Directions
Another function that governs the magnitude and directions of the plastic flow is named
the plasticity potential function, typically represented as g = g(s ij ,q) and:
g ∂
dε = λ (6-68)
p
ij ∂ σ
ij
Where l is a scalar called plastic multiplier. l can be combined into g so that
g ∂
dε = . While this presents a simpler format it requires more criteria in selecting the
p
ij ∂ σ
ij
potential function. If g and f are the same function, it is named associated flow; other-
wise it is non-associated flow.
6.2.3.1 von Mises Criteria and Associated Flow
The following makes use of the von Mises criteria and associated flow as an example to
illustrate how one can calculate the plastic strain in increments.

Isotropic Hardening f ∂ S
dε = dε p = dε p 3 ij
p
ij ∂ σ 2 Y
ij (6-69)
where
3
Y ε
f (σε p ) = S S − ( )
p
,
ij ij ij
2
1
S = σ − σ δ
ij ij kk ij
3
f ∂ f ∂
f (σ + dσ ,ε + dε p ) = f (σ ,ε p ) + dσ + ε d p = 0
p
ε
ij ij ij ∂ σ ij ∂ε p
ij (6-70)
∂f ∂Y
⇒ dσ − dε p = 0
∂σ ij ∂ε p
ij (6-71)
1 ∂f 1 3 Sdσ
⇒ dε p = dσ = ij ij
h ∂σ ij h 2 Y
ij (6-72)
dY
h =
dε p