1452
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
1.4
Plastic flow rule
The plastic potential
=
+ P" ⋅
is decomposed into its
deviatoric part which is associative,
′ =
(8)
and its dilationnal part which is non-associative
P" =
⋅
and
η =
⋅:
(9)
where
̅
is a material parameter that takes into account the
phase transformation line (Ishihara, et al., 1975). This parameter
rules the dilationnal behaviour and separates the p’-q plane into
two zones. Stress ratios (
) lower than
η
imply a plastic
contractive behaviour under shear loading while the other zone
depicts a dilative plastic behaviour.
1.5
Hardening rule
The hardening rule is purely kinematic. During loading, active
surface moves up to contact the next one. All surfaces inside the
active one stay tangential at the current stress state. The
relationship between plastic function and kinematic hardening is
determined through the consistency condition (Prevost, 1985)
and leads to
p
⋅
=
:
⋅ 〈L〉 ⋅
(10)
where
is a tensor defining the direction of translation of the
active surface in the deviatoric space. At this step, any direction
of translation could be used. The only requirement is that the
outermost activated surface has to be at most tangential to the
next one, at the end of a given step. The overlapping of surfaces
is then avoided. In the original paper, the Mroz rule was chosen.
This choice enforces an explicit integration of the law.
=
⋅ −
⋅
− −
⋅
(11)
1.6
Refinements
Dependency on stiffness (bulk, shear and plastic moduli) for the
mean effective stress is taken into account through (Prevost,
1985)
X(p
) = X
⋅ p
p
= , ,
(12)
where
is a reference pressure and
is the corresponding
stiffness at
. Typically n equals 0.5 for sands (Prevost, 1985) .
Other shapes of surfaces can be considered. The Lode-angle
dependency is taken into account through the use of Van
Eekelen surfaces (Yang, et al., 2008; Zerfa, et al., 2003).
2 COMPARISON WITH LABORATORY TESTS
2.1
Calibration
A series of tests on Nevada sand, available in the scope of
VELACS project (Arulmoli, et al., 1992), was used as
experimental background. It provides a large number of triaxial
monotonic and cyclic tests at several densities.
Elastic parameters for modelling are available in (Popescu,
et al., 1993). Monotonic triaxial tests in both compression and
extension are used to calibrate the basic plastic parameters of
the model (Zerfa, et al., 2003). For a triaxial test, Equation (7)
describing the i-th surface depends only on 2 scalars parameters
(
et
) and is transformed into
f ≡ q − α
∙ p
− (m
)
∙ (p
)
= 0
(13)
Fig. 1. Comparison between experimental p’-constant drained triaxial
tests (Arulmoli, et al., 1992) and numerical modelling for Nevada Sand
(Dr=40%) : deviatoric stress vs. vertical deformation. Tests are
available for 3 different initial mean effective pressures (40-80-160 kPa)
in compression and extension.
Parameters related to the surfaces are obtained from drained
tests:
1. An experimental q-
ϵ
y
curve (deviatoric stress vs. vertical
deformation) in compression is delimited into linear
segments along which plastic moduli are constant. The
number of segments corresponds to the number of
surfaces. Transitions from a surface to another give initial
upper bounds of surfaces.
2. The procedure is repeated for an experimental extension
curve at the same initial mean effective pressure but
plastic modulus are already known. The lower bound of
each surface is then found. Aperture (M
i
) and initial
position (
α
i
) of the centre of each surface are then
computed.
Figure 2. Comparison between experimental drained triaxial tests
(Arulmoli, et al., 1992) and numerical modelling for Nevada Sand
(Dr=40%) volumetric vs. vertical deformation. Tests are available for 3
different initial mean effective pressures (40-80-160 kPa).
The p’=80kPa curve was adopted as a reference curve for
calibration. The model is then applied to other initial mean
pressures. Results in the q-
ϵ
y
plane are given in Fig. 1.
Numerical and experimental curves fit relatively well. However,
regarding the volumetric deformation, the numerical curves
don’t match the tendency of experimental ones. The only
parameter
̅
that rules the plastic potential is the cause of those
discrepancies. The phase transformation line is accurately
defined in the p’-q plane but the amount of contractancy or
dilatancy is not adjustable.
