Geoscience Reference
In-Depth Information
with
s
c
= s − (r
h
− t
h
r
c
)p
· F · M
[9.24]
s
h
and
r
h
, which give the position and size of the field at the time of change in
the loading direction, are discontinuous historic parameters introducing a kinematic
hardening in the model. They are given by:
s
h
p
h
· F(p
h
,p
c
(ε
vh
)) · M
r
h
=
[9.25]
t
h
=
s
h
[9.26]
q
h
We calculate them at each change of direction. As it can be seen, writing
q
c
in the
form given in relationship [9.23] takes into account the Masing rule [MAS 26] and
Bauschinger effect [LEM 09].
8) The monotonic and cyclic yield functions having been described, we now
proceed to see how they can evolve, giving the evolution laws of the hardening
variables:
a) it is assumed that the deviatoric plastic strain rate follows an associated law.
Thus, we have:
∂σ
= λ
p
3s
= λ
p
Ψ = λ
p
∂f
p
ε
[9.27]
2q
where
λ
p
is the plastic multiplier which can be obtained from the consistency condition
(
f
s
=0
),
b) the evolution of
ε
v
, which is given by the dilatancy rule based on the relation
proposed by Roscoe [ROS 58]:
ε
v
= λ
p
Ψ
v
[9.28]
with:
M
(ψ)−
q
p
r(
p
)
Ψ
v
= −ξ
[9.29]
where
ψ
is the dilatancy angle and
ξ
is a positive coefficient. This angle is in fact the
friction angle, which corresponds to the state of null volumetric strain rate or phase
transformation. This relation reveals that the behavior can be divided into two domains
in the (
p
− q
) plane. Under the dilatancy line, where
M
−
p
=0
,the material is
contractant (
ε
v
> 0
). Above, it is dilatant (
ε
v
p
< 0
) and, on this line, the dilatancy is
zero. Parameter
ξ
takes into account the fact that, under drained conditions, the volume
change is zero until a threshold shear (
hys
) and that, under undrained conditions, the
pore pressure evolution (especially during the cycles) depends on the level of strain. It
Search WWH ::
Custom Search