688
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
2 SIMULATION APPROACHES
2.1
Soil modelling in SPH framework
In the SPH method, motion of a continuum is modeled using a
set of moving particles (interpolation points); each assigned a
constant mass and “carries” field variables at the corresponding
location. The continuous fields and their spatial derivatives are
taken to be interpolated from the surrounding particles by a
weighted summation, in which the weights diminish with
distance according to an assumed kernel function. Details of the
interpolation procedure and its application to soil can be found
in Bui et al. (2008). The motion of a continuum can be
described through the following equation,
ext
f g σ u
(1)
where
u
is the displacement; a raised dot denotes the time
derivative;
is the density;
is the total stress tensor, where
negative is assumed for compression;
g
is the acceleration due
to gravity; and
f
ext
is the additional external force(s). The total
stress tensor of soil is normally composed of the effective stress
(
´
) and the pore-water pressure (
p
w
), and follows Terzaghi’s
concept of effective stress. Because the pore-water pressure is
not considered, the total stress tensor and the effective stress are
identical throughout this paper and can be computed using a
constitutive model.
In the SPH framework, Equation (1) is often discretized
using the following form,
N
b
a ext
a
ab a
ab
b
b
a
a
b
a
f
g W C
m u
1
2
2
(2)
where
and
denote Cartesian components
x
,
y
,
z
with the
Einstein convention applied to repeated indices;
a
indicates the
particle under consideration;
a
and
b
are the densities of
particles
a
and
b
respectively;
N
is the number of “neighbouring
particles”,
i.e.
those in the support domain of particle
a
;
m
b
is
the mass of particle
b
;
C
is the stabilization term employed to
remove the stress fluctuation and tensile instability (Bui et al.,
2011);
W
is the kernel function which is taken to be the cubic
Spline function (Monaghan & Lattanzio 2005); and
f
ext
a
is the
unit external force acting on particle
a
.
The stress tensor of soil particles in Equation (2) can be
computed using any soil constitutive model developed in the
literature. For the purpose of soil-structure interaction, the
Drucker-Prager model has been chosen with non-associated
flow rule, which was implemented in the SPH framework by
Bui et al. (2008) and shown to be a useful soil model for
simulating large deformation and post-failure behaviour of
aluminum rods used in the current paper as model ground. The
stress-strain relation of this soil model is given by,
)
(:
p
e
ε ε Dσ
(3)
where
D
e
is the elastic constitutive tensor; is the strain rate
tensor; and
is its plastic component. An additive
decomposition of the strain rate tensor has been assumed into
elastic and plastic components. The plastic component can be
calculated using the plastic flow rule,
σ
ε
p
p
g
(4)
where is the rate of change of plastic multiplier, and
g
p
is
the plastic potential function. The plastic deformation occurs
only if the stress state reaches the yield surface. Accordingly,
plastic deformation will occur only if the following yield
criterion is satisfied,
0
2
1
c
k J I
f
(5)
where
I
1
and
J
2
are the first and second invariants of the
stress tensor; and
and
k
c
are Drucker-Prager constants that
are calculated from the Coulomb material constants
c
(cohesion)
and
(internal friction). In plane strain, the Drucker-Prager
constants are computed by,
2
tan 12 9
tan
and
2
tan 12 9
3
c
k
c
(6)
The non-associated plastic flow rule specifies the plastic
potential function by,
constant
2
1
J I
g
p
(7)
where
is a dilatancy factor that can be related to the
dilatancy angle
in a fashion similar to that between
and
friction angle
. Substituting Equation (7) into Equation (4) in
association with the consistency condition, that is the stress
state must be always located on the yield surface
f
during the
plastic loading, the stress-strain relation of the current soil
model at particle
a
can be written as,
a a
a
a a
a
a aa
aa
a
s J G
K
K eG
dt
d
)
/ (
2
3
2
(8)
where
e
is the deviatoric strain-rate tensor;
s
is the
deviatoric shear stress tensor; and is the rate of change of
plastic multiplier of particle
a
, which in SPH is specified by
a
a a a
a a a
a
aa a
a
G K
s J G K
9
3
2
)
/ (
(9)
where the strain-rate tensor is computed by
a
a
u
u
2
1
(10)
When considering a large deformation problem, a stress rate
that is invariant with respect to rigid-body rotation must be
employed for the constitutive relations. In the current study, the
Jaumann stress rate is adopted:
a a
a a
a
a
ˆ
(11)
where is the spin-rate tensor computed by
a
a
u
u
2
1
(12)
As a result, the final form of the stress-strain relationship for
the current soil model is modified to
a a
a a
a
a a a
aa
a a
a a
a
s J G
K
K eG
dt
d
)
/ (
3
2
2
(13)
Equations (2) and (13) are finally integrated using Leapfrog
algorithm to describe the motion of soil medium. Validation of
this soil model with SPH has been extensively documented in
