668
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
3 ELASTIC PERFECTLY PLASTIC MODEL WITH
MOHR-COULOMB YIELD FUNCTION
Elastic-perfectly plastic abstractions are relatively simple and
despite their shortcomings very popular. The main derive for
their popularity is perhaps their similarity with limit equilibrium
and linear elastic solutions and simple abstractions. However,
some aspects of these models are still unexplored and
overlooked.
In this section, a linear elastic perfectly plastic Mohr-Coulomb
model with a linear elastic stiffness tensor following Hoek’s law
2
1 2
e
ijkl
ik jl
jk jl
v
C G
v
d d
d d
æ
ö÷
ç =
+
÷
ç
÷÷
çè
ø
-
,
(3)
is considered; where the shear modulus
G
and Poisson’s ratio,
v
, are elasticity parameters; and
ij
d
is the so called
Kronecker’s delta.
The stiffness degradation due to plasticity,
p
C
, is established
from the consistency condition in plasticity theory as
,
,
,
,
:
:
: :
e
e
p
e
g f
f
g
σ
σ
σ
σ
C
C
C
C
Ä
=-
,
(4)
wherein
f
is the yield function and
g
is the plastic potential
function,
,
x
x
σ
σ
= ¶ ¶
.
Here, the Mohr-Coulomb criterion is cast into a yield
function. The Mohr-Coulomb criterion can be written in terms
of stress invariants as
(
)
6sin
3 :
0
2
3 sin
p
p
f
p a
s s
q
j
j
j
= -
+ £
-
,
(5)
where
1
3
:
s σ σ δ
= -
, is the deviatoric stress tensor and
1
3
:
p
=
σ δ
is the mean normal stress. The peak friction angle,
p
j
, is a model parameter and
cot
p
a c
j
=
is called attraction
(Janbu 1973
a
), where
c
is cohesion.
The corresponding Mohr-Coulomb type plastic potential
function may be written as
max
max
3
2sin
:
:
2
3 sin
g
s s
σ δ
q
y
y
y
= -
-
,
(6)
where maximum dilatancy angle,
max
y
, is additional model
parameter.
The Lode angle dependent functions
q
j
and
q
y
can be found
from trigonometric considerations in Figure 2 as
(
)
,
,
1
2
,
3ω
2 ω sin
sin
j y
q
j y
j y
q
q
=
+
(7)
where
(
)
(
)
,
max
,
,
max
3 sin
ω
3 sin
p
p
j y
j y
j y
-
=
+
(8)
and
1
1,2
2
6
p
q
q
=
.
(9)
q
is the Lode angle defined here as
3
3/2
2
1
3 3
arcsin
3
2
J
J
q
æ
ö÷
ç
÷
ç
= - ÷
ç
÷÷
çè
ø
(10)
where
3
det
J
s
=
,
2 1
2 2
J tr
s
=
(det = determinant, tr = trace).
For triaxial compression and extension deformation modes, the
Lode angle,
q
, is
6
p
and
6
p
-
respectively.
p
=35
0
yield surface
plastic potential
surface
max
=5
0
1
2
o
C
A
B
Figure 2: One sector of the Mohr-Coulomb surface (normalized by
deviatoric stress at triaxial compression) in deviatoric plane in relation
to a corresponding Mohr-Coulomb plastic potential function. A:
Triaxial compression, B: Triaxial extension.
Stresses enclosed within the failure surface are elastic. On the
failure surface, all strains are plastic. Generally,
max
p
y j
£
.
when
max
p
y j
=
, associated flow rule is recovered and plastic
strain increments are oriented normal to the failure surface
given in Eq. (5).
It should be mentioned here that, although Eqs. (5) and (6) can
be used for implementation of the model, in this study they are
used for a short presentation only. The implementation is done
by sorting eigenvalues of the stress tensor such that the Mohr-
Coulomb criterion is established by choosing the major and the
minor principal stresses.
In Figure 3, pure deviatoric strains,
i.e
., isochoric condition, are
applied and stress paths are plotted in the normalized deviatoric
plane. The plots indicate that some stress paths deflect towards
triaxial extension and compression modes. The less the
maximum dilatancy angle the more is the deflection. At failure,
the stress paths are therefore different from the principal strain
rate increment directions, hence generally non-coaxial to the
strain increment direction. This non-coaxiality is inherent to the
geometry of the potential surface in the deviatoric plane and
hence distinguished here as geometric non-coaxiality. Often,
typical plots are illustrated for triaxial extension and
compression states. However, the stress paths in between do not
follow the trends of triaxial extension and compression stress
paths as shown in Figure 3. Reasons are
since the plastic potential function is a function of the
major and the minor principal stresses, the intermediate
stress state is not corrected for plasticity. Hence,
touching the Mohr-Coulomb line does not guarantee
that the stress state be a constant.
the geometry of the potential surface causes a drift,
since the normal to the surface is not necessarily coaxial
to the current stress path.
the drift may be amplified by deviatoric non-
associativity.
Considering one of the six sectors in the normalized
-plane
(Figure 2), two Mohr-Coulomb lines with friction angles
1
j
and
2
j
make an angle of
(
)(
)
2
2
2
1
1
1
2
2
2
2
1
1
2
2
4
4
4
4
3
arcsin
4
1
1
j
j
j
j
j
j
j
j
j
j
w w w w w w
a
w w w w
ì
ü
ï
ï
ï
ï
+ - - + -
ï
ïï
= í
ý
ï
ï
+ - + -
ï
ï
ï
ïï
î
þ
(11)
with each other. This angle introduces deviatoric non-
associativity when the yield function and the plastic potential
function assume different angles, which is the case for non-
associated flow. Deviatoric associativity may be achieved
simply by considering
q
q
j
y
=
.
