Page 186 - Mechanics of Asphalt Microstructure and Micromechanics

178 Ch a p t e r S i x

Linear Kinematic Hardening
ij (
f ∂ p 3 S − α ij)
dε = dε p = dε
p
ij ∂ σ 2 Y
ij (6-73)
Where the yield criterion is:
ij (
3
f (σα ) = S − α ij)( S − α ij) − Y
,
ij ij ij
2
1
S = σ − σ δ
ij ij kk ij
3
Where Y = constant:
f ∂ ∂ ∂f
f (σ + dσ ,α + dα ) = f (σ ,α + dσ + dα = 0 (6-74)
)
ij ij ij ij ij ij ∂ σ ij ∂α ij
ij ij
If the linear kinematic hardening law is followed, one has:
S − )
ij (
2 α
dα = cdε = cdε p ij (6-75)
p
ij 3 ij Y
And therefore:
ij (
13 S − α ij) dσ ij
dε = (6-76)
p
c 2 Y
Following the similar philosophy, one can obtain the solution for combined isotro-
pic hardening and kinematic hardening.
6.2.4 General Laws on Plastic Deformations
There are several general principles regarding plastic deformations. These include: the
loading, unloading conditions; normality and regularity; and stability conditions.

6.2.4.1 Unloading Condition
For 1-D situations, unloading is easy to judge. For multi-dimensions, it is much more
difficult.
For isotropic hardening and the unloading condition:
Sdσ < 0 (6-77)
ij ij
For kinematic hardening, it is:
ij (
S − ) dσ < 0
α
ij ij (6-78)
The physical meaning is that the incremental stress will point to the inside of the
yielding surface.

6.2.4.2 The Kuhn-Tucker Conditions
p
It can be observed that dε and ƒ obey the Kuhn-Tucker conditions:
dε p f = 0, dε ≥ 0, f ≤ 0 (6-79)
p