802
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
n
p
p
BB
n
p
p
GG
1
1
1
1
(2)
Where,
n
= experimental parameter,
p
1
=reference normal stress,
G
1
and B
1
are also experimental parameters.
2.2. Yield function and plastic potential
The material model used in this paper is a generalized elasto-
plastic, pure kinematic hardening one. A simple hyperbolic
equation (Tatsuoka
et al
., 1993, Hossain
et al
., 2007) has been
used as the evolution function of yield surface. The yield
surface used is a generalized Drucker-Prager one given by;
0
22
pm pα s pα s f
ij
ij
ij
ij
n
(3)
, where
p
is the mean normal stress (i.e., hydrostatic stress
component) ,
is the desiatoric component of stress tensor,
is the kinematic deviatoric tensor defining the coordinates
of the yield surface center in deviatoric stress sub-space; m is
the material parameter defining the opening of the cone,
n
is the
number of yield surfaces.
ij
s
ij
A plastic potential function (
g
) is selected such that the
deviatoric plastic flow is associative. A non-associative plastic
flow rule is used for its dilatational component. So the
deviatoric component of plastic potential is defined same as the
yield function. The dilatational or plastic volumetric component
is defined by Rows’ dilatancy relationship given by
dKR
(4)
Where,
h
v
R
and
p
vd p
hd d
for loading and vice-
versa for unloading, K is the material constant.
2.3. Kinematic hardening rule
A pure kinematic hardening rule is formulated as follows:
ij
a
ij
p
(5)
Where,
= deviatoric component of tensor defining the
direction of translation of the yield surfaces. = amount of
translation determined through the consistency condition as
follows:
ij
a
If the yield function was isotropic, then it could be described
by eq. (6) and (7) and for kinematic surface, by eq. (8) & (9) -
0 ,
ij f
(6)
0
f
ij d
ij
f
d
f
ij d
ij
f
(7)
0 ,
ij ij f
(8)
0
ijd
ij
f
ij d
ij
f
(9)
f
ij d
ij
f
(10)
H
ij
ij
f
p
a
(11)
ij
ij
f
pH a
1
(12)
So
ij
ij ij
f
pH
ij
1
(13)
The yield surfaces are all self-similar conical surfaces in
general three-dimensional stress space. The yield surfaces are to
be translated by the current stress point upon contact. In order to
avoid the overlapping of the surfaces, the direction of
translation
of the active yield surface is chosen such that
ij
ij p ij s
ij p ij s
m
m
ij
(14)
Where
m
and
ij
are the plastic parameters associated with
the next outer surface of the nested yield surfaces.
3 RETURN MAPPING ALGORITHM
In this algorithm, elastic trial stress is returned to the current
yield surface, following the existing hardening law and flow
rule. In this way, the incremental elasto-plastic relation is
integrated in a robust way (Simo and Ortiz, 1986). In this
particular scheme, stress tensor is divided into two components,
deviatoric stress ( ) and mean stress (
p
). is designated as
angle of the center line of the concentric cone in the context of
pure kinematic hardening. Expanding the yield function into a
Taylor’s series gives Eq. (15)-
0
, ,
ij
ij
p sf
(15)
0
, ,
ijs
ij d
ij
f
dp
p
f
ijds
f
ij p ijsf
(16)
Now considering the followings facts:
ijs
g
G p dG ij ds
2
2
(17)
(18)
DK p
v
dK dp
As it has been defined that
pd
p
vd
D
and
D dD pdD
p
v
d
as it has been known that
for plane strain
situation.
d pd
Combining Eqs. (15), (16) and (17) and using Prager’s
kinematic hardening rule defined in Eq. (13) the plasticity
multiplier can be derived-
0
2
, ,
H KD
p
f
ij
s
g
G
ijs
f
ij p ijsf
(19)
H KD
p
f
ij
s
g
G
ij
s
f
ij
p
ij
sf
2
, ,
(20)
Using the trial stresses, following integrated elasto-plastic
stresses and kinematic hardening parameter are obtained as
shown in Eq. (19) and Eq. (20).
ij
s
ij
