Page 184 - Mechanics of Asphalt Microstructure and Micromechanics

176 Ch a p t e r S i x

The Tresca criterion actually states that the material will yield when the maximum
shear stress reaches a certain value.
Perfectly plastic: Y = constant
Linear strain hardening: Y ε ( ) = Y + hε p
p
0
ε
Power-law hardening: Y = Y + h( ) 1/ m
p
0
One can have various combinations for the yielding conditions and the hardening
laws.
6.2.2.2 General von Mises Yield Function
f (σε p ) σ − Y ε ( ) (6-59)
=
p
,
ij e
3
σ = SS (6-60)
e 2 ij ij
1
S = σ − σ δ (6-61)
ij ij 3 kk ij
The von Mises yielding criterion that includes a kinematic component (the back
stress) can be expressed as:

3
f (σα ij ,ε p ) = S ( ij − α ij )( S − α ij ) − Y(ε p ) = 0 (6-62)
,
ij
ij
2
The physical meaning of the kinematic terms is the center of the yielding surface is
also moving in the stress space. The direct implication of the back stress is the change of
the reference state (an offset of the zero stress state). There are different rules that gov-
ern the update of the back stress. The simplest approach is:
dα = 2 cdε p (6-63)
ij 3 ij
A more sophisticated approach is:

dα = 2 cdε − γα dε p (6-64)
p
ij 3 ij ij

6.2.2.3 Mohr-Coulomb Criterion
Neither the Tresca criterion nor the von Mises criterion is dependent on the mean stress
or the confining stress (the first invariant). The Mohr-Coulomb criterion presents one
that is dependent on the normal stress (therefore, the mean stress). For a specific plane
with an applied normal stress s, when its shear stress reaches:
f () =− tan( ) ϕ (6-65)
σ
c σ

The material will yield. Where c and j are the cohesion and the friction angle cor-
respondingly.