Page 183 - Mechanics of Asphalt Microstructure and Micromechanics

Fundamentals of Phenomenological Models 175

model to calculate the stress. However, as the unloading and reloading (up to the
stress where unloading starts) are elastic, one can calculate the stress using the follow-
ing equation:
−
p
e
σ = E ε = E( ε ε ) (6-54)

p
Where s is the stress, e is the total strain, while e and e are the elastic and plastic
e
strains respectively.
Clearly from Figure 6.1, to describe the plasticity behavior, one needs to determine
the initial yield stress (could be zero) where yielding starts, the magnitude of the plastic
strain and its directions, and the increase of the yielding stress with plastic strain or
plastic work.
e
p
The uniaxial strain decomposition e = e + e can be extended to the 3D cases
e ij = e ij + e ij and could be in an incremental format. The incremental format is more
e
p
useful in that the plastic strains are usually quite non-linear with stresses.
dε = dε + dε p (6-55)
e
ij ij ij
If it is linear elastic, the following equation will calculate the incremental stress.
dσ = C dε e (6-56)
ij ijkl kl
6.2.2 Yielding and Hardening
For 1-D cases, the initial yield stress can be conveniently expressed as s = s Y . For 3D
problems, it is much more complicated. In other words, one may have to express the
yielding as a yielding condition such as the von Mises condition, the Tresca condition,
and the Druck Pluger. For geomaterials or granular materials, the yielding stress is
very much affected by the effective confining stress I, the larger the I, the larger the
yielding stress.
The yielding conditions can then be generalized into a yielding function (including
a hardening component).
θ
For example, f = (σ ,ε p ,) = f I I I ,ε p ,) .
θ
(
,
,
f
ij 1 2 3
For objectivity considerations, the yielding function is often represented as a func-
tion of the stress invariants, which means that the function will not change during co-
ordinate transformation. Overall, the typical criteria can be classified into three types:
maximum tensile/compression stress, maximum shear stress, and shear energy. More
complicated yielding criteria may include the anisotropic characteristics (Hill, 1948;
Gotoh, 1977a,b).
6.2.2.1 General Tresca Yield Criterion
{
σ
f (σ ,ε p ) = max σ − σ , σ − σ , σ − } − Y ε ( ) = 0
p
ij 1 2 1 3 2 3 (6-57)
p
Where Y ε ( ) also carries a hardening component.
The effective plastic strain magnitude is defined as:
ε = 2 ε ε p ε = ∫ 2 dd p ij (6-58)
ε ε
p
p
p
p
ij ij
ij
3 3