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Numerical Formulation for solving Soil/Tool interaction problem
involving Large deformation.
Nicolas Renon, Pierre Montmitonnet, Patrick Laborde.
Key words: Soil Mechanics, Ploughing, Finite Element, Compressible Plasticity.
Abstract
Soil ploughing as used for military breaching induces very large deformation (flow) of a generally
compressible plastic medium with large structural rearrangement and mechanical properties evolution.
So modelling such a process requires to handle numerically the non-linearities induced by the material
and by unilateral contact and friction between the rigid structure and the soil. Moreover, the different
sizes of tines and blade make it a two-scale process, thus requiring large numbers of d.o.f’s in FE
simulations. Reaching a steady state takes a displacement several times the typical dimension of the
ploughing system, so that between a few hundred and many thousand time steps are necessary. We
present the description of the global model and the implementation of a one-phase, critical state
compressible non-associated plastic law [4]. The numerical platform is the large deformation FE
software Forge3®, initially devoted to metal forming. Then we focus on simulations of a single tine
ploughing different kinds of soil. Results obtained for complex tools involving several tines and/or a
blade will be presented in another paper.
2
1 - Introduction
The aim of this work is to provide a global 3D FE model devoted to the modelling of superficial soil
ploughing in the large deformation range and for a vast class of soil treatment tools. One of the most
complex soil ploughing tool, for example, is involved in military breaching. This process consists in
scraping off a ca. 30 cm superficial layer of soil (for cohesive soils) or combing mines off it (for non-
cohesive frictional soils, e.g. sand). A plough is used for this purpose, consisting in a V-blade with
sharp tines, pushed by a caterpillar vehicle. Especially in this case, more generally in ploughing
studies, the question is: what is the force necessary to push the plough, and, for military breaching, is
the soil / caterpillar contact able to transmit it ? What is the soil response depending on soil nature and
tool geometry ?
Modelling such processes faces several crucial points to determine accurately soil reaction and
deformation fields :
1. The interacting system consists of two materials, soil and metal, with complex interface
geometry. The heterogeneous and stratified nature of the soil adds further complexity.
2. The cutting process creates a flow of the soil and large deformations, hence a permanent
geometric evolution of the unconstrained boundary (free surface).
3. The behavior of surface soils under combined low stress state (free boundary) and large
deformations is poorly known, all the more as vegetation (roots) has to be considered.
4. And so is the soil / metal interface friction.
5. This is typically a two-scale problem, where deformation occurs at the tine tip vicinity (a
centimetre scale) and at the scale of the blade (metre scale) which pushes a heap of extracted
material.
Numerical simulation of such a problem can go back to the soil structure, namely its particulate
character, using a granular model like the DEM (distinct element method), now widely spread in geo-
mechanics. It is clearly adequate for purely frictional soils (sands); some kind of cohesion can be
3
introduced, although not in a straightforward way, so that this remains a debated matter. Numerical
simulation of the problem under investigation has been carried out with a DEM for a single tine in 2D
and 3D [25]. Quite intensive computations were necessary to examine the characteristics of this “small
geometric scale”, which makes intractable the two scale problem involved in blade-tine interacting
system: either the particles will be too many, or they will be too big for the small-scale part of the
system (tine tip vicinity).
The same difficulties arise for the alternative, continuum FEM modelling, but the mesh size can here
be locally adapted. Literature does provide FEM studies of soil ploughing [12],[22],[23]. Usually, they
address agricultural purposes, tools consist of either a single tine or a smooth blade. Simulations are
often 2D and flow patterns are prescribed, with adjunction of joint element at the tip of the tool for
instance [22]; large deformations, in terms of mechanical formulation as well as free surface evolution,
are not really dealt with. In most continuum approaches, a soil is idealised as an elastic-plastic body,
using classical soil mechanics models such as Mohr-Coulomb or Drücker-Prager [22]. More recent
alternatives include hypo-plasticity [17],[35]. In this paper, the elastic-plastic CJS model (which stands
for Cambou-Jafari-Sidoroff model [5]) has been chosen: based on the critical state concept, it can
handle the large deformation response of soils under tri-axial solicitations, namely, hardening
(densification) and/or softening (dilatation).
Our developments for the numerical resolution of the evolution problem corresponding to the plowing
process modelling, is based on a 3D FE software devoted initially to metal forming, namely Forge3®.
This software can take into account large mesh distortion, consequence of large displacement tool,
thanks to an automatic re-meshing routine. Crucial non linear points in the plowing modelling exposed
above, can be sum up at the numerical stage as : unilateral frictional contact, non associated plasticity
and large deformation in terms of change in free surface and stress strain measures. The numerical
resolution is performed through an incremental time decomposition with the Updated Lagrangean
procedure. In our case, penalization of contact conditions and mesh distortion point out a rather low
4
upper bound for time step size, and consequently the incremental tool displacement cannot exceed
1mm. Practically, if we compare this incremental displacement to the military breaching plough
dimension, about 5 meters in width for the blade, and 3 cm wide 30 cm deep for the connected tines,
time step can be considered as small. As our aim is to perform a robust 3D numerical model in the
context of highly intensive computations, able to model plowing application for different soil
treatment tools, we are lead to make proper relaxation of a part of the non linearities involved in the
numerical problem.
Hence the first part of the paper is devoted to the presentation of the global mechanical problem and
large deformation, followed with attention on linearisations made in order to provide eventually the
numerical model. We present also the space and time discretisation, frictional contact modelling,
general elasto-plastic law and resolution algorithm. The validation of CJS implementation is presented
by comparison with results from Flac2D©, 2D explicit FE software, on triaxial experiments with a set
of previously identified model parameters. The second part of the paper is devoted to the investigation
of numerical simulation for a single tine. Some influent parameters are discussed as well as soil nature
influence. Finally, the conclusion evokes results for multi-tine tool simulations, to be presented in
details in another paper.
5
2 - Global Model Description
2.1 Strain and stress measures
Let [ ] 3RI,0: →×Ω Tψ be the deformational motion of the continuum body with reference
configuration Ω , time interval of interest [ ]T,0 and particles labeled by Ω∈X . For a time
[ ]Tt ,0∈ , the current configuration of the domain occupied by the soil, theoretically semi-infinite, is
defined by ( ) ( )tt ,Ω=Ω ψ with current particles ( ) ( )ttXx Ω∈= ,ψ .
To describe the inelastic response in large deformation Lee and Liu [18] introduce the multiplicative
split of the deformation gradient, ( ) ( ) XtXtXF ∂∂= /,, ψ into elastic part EF and plastic part
PF :
( ) ( ) ( )tXFtXFtXF PE ,,, =
A suitable definition of strain rate can then be introduced by considering the spatial velocity gradient
lwhich is expressed as
( ) ( ) 1111 −−−−− +=+== EPEEEPPEEE FlFlFFFFFFFFl &&& 1
If we assume that the elastic transformations (strain and rotation) are negligible compared to the
plastic one, it appears reasonable to consider the elastic stretch tensor EV to be given by
EE IV ε+= where Eε is the infinitesimal elastic strain tensor and
EE VF = . Following Lee and Liu
[18] and neglecting higher order infinitesimal quantities, one can arrive at the following approximation
[ ] PE DDDl +≅=sym (1)
Assumption above made on the amount of elastic strain compared to plastic strain, is usual in metal
plasticity. Especially in forming processes the material undergoes large plastic transformation as well
as in metal powder compaction where the material is compressible and often idealised though classical
soil mechanics models. Hence strong numerical as well as experimental arguments make reasonable to
classify the process of soil plowing in this context. Moreover, the kinematics of the plowing process
6
can be closely related, in another scale range, to the cutting process for metals [11] or the scratching of
organic glasses [4], where elastic transformations are small compared to plastic transformation. Hence
in the following, we will make use of the additive decomposition eq.(1), coupled with an hypo-elastic
stress rate – strain rate constitutive equation.
As a stress measure we consider the Kirchhoff stress tensor στ J= where FJ det= and σ is the
Cauchy true stress tensor. In the case of incompressible plasticity it is common to approximate
Kirchhoff stress by Cauchy stress ( ) ( )( )1detdet == EFF . But, in our plastically compressible case
this approximation cannot be made. We remark that Kirchhoff stress tensor is an energy conjugate
stress measure associated with the rate of strain D (Simo J.C. [30]).
As we choose a hypoelastic law, we have to take into account an objective rate of stress, which leads
to the following relation between the elastic part of the strain rate and an objective rate of Kirchhoff
stress:
∇= τLD E
(2)
with L elastic compliance tensor. The superscript ∇ above a symbol refers to an objective time-
derivative involving rotation terms given by the spin tensor (see for example Ponthot J.P. [26] ).
The properly speaking plastic constitutive equations given below describe the yield criterion and the
general flow rule, with complementarity (Kuhn-Tucker) conditions for the elastic-plastic transition (N
is the number of plasticity surfaces):
( )∑= ∂
∂=
N
i
ii
P Rg
D1
,τσλ&
( ) ( ) NiRfRf iiii ,,1;0,;0,;0 K&& =≡≤≥ τλτλ
(3)
Yield surfaces if may differ from their plastic potentials ig . Finally, the evolution law of internal
variables ( )qiiRR ≤≤=
1 can express isotropic hardening as well as softening in a condensed form such
as :
7
( )DRR ,,τΨ=& (4)
A precise description of constitutive equation of CJS model implemented will be given in section 3.
2.2 Mechanical problem
Whether the displacements or strains are large or small the equilibrium conditions between external
and internal forces have to be satisfied. Thus, the equilibrium equation can be written on the current
configuration ( )tΩ , in a standard form as:
ργρσ =+ gdiv (5)
where σ is the total stress, ρ (scalar) soil density, γ the acceleration and g the gravity. The last two
terms have to be maintained as their order of magnitude is similar to the constitutive stresses. The
analysis is restricted to the soil, i.e. the tool is supposed rigid. Subscripts T and n denote tangential and
normal components, n is the external normal to the boundary ( )tΩ∂ .
Crucial boundary conditions mainly include unilateral contact and friction. For the former, Signorini’s
condition is written in terms of velocities on the contact surface ( ) ( )ttC Ω∂⊂Γ :
0).(
;0
;0).(
≡−
≤
≤−
n
n
tool
n
n
tool
vv
vv
σ
σ (6)
with n⋅= nn σσ the normal stress component and vtool
the prescribed velocity of the rigid tool.
In the present model, the singularities in inequalities above are relaxed by the penalty method :
( ) +⋅−−= ntool
n vvθσ ( ) on tC
Γ (7)
brackets <…>+ means that the repulsion stress in (7) is applied only if penetration occurs :
( ) 0<⋅− ntoolvv .Then θ is the “large” penalty parameter.
8
The displacement formulation of Signorini’s condition is the most widely spread in literature for quasi-
static approach in solid mechanics [37]. Velocity formulation appears to be more suitable in evolution
problems, at least from a numerical point of view. However in terms of incremental time steps, both
are equivalent.
The friction threshold law is replaced by a one-to-one regular relation:
( ) on 2
t
vv
vvC
tool
TTF
tool
TTTT Γ
−+
−−=
βπσ
(8)
where ( )TT nσσ = is the tangential stress and Tπ is the friction stress: nT µσπ = for Coulomb’s law
and kmT =π for the friction factor model (k is the shear yield stress). Parameter Fβ is a “small”
regularisation term.
Finally the velocity field is equal to zero on ( ) ( )ttD Ω∂⊂Γ , in order to reduce the semi-infinite
domain to a finite one. On ( ) ( )ttN Ω∂⊂Γ , the complementary part of ( ) ( )tt CD Γ∪Γ , normal stresses
are equal to zero (free surface) :
( )tn NΓ= on 0σ (9)
9
2.3 Variational formulation
To derive the variational formulation of equilibrium equation, at time t, we multiply terms of (5) by a
kinematically admissible virtual velocity field, and integrate over the current configuration domain.
We finally obtain, thanks to Green’s formula, the following variational formulation of the possible
dynamic equilibrium taking into account frictional contact
( ) ( ) ( ) ( ) ( )∫∫∫∫∫
ΩΓΓΩΩ
⋅=⋅−⋅−+⋅
ttttt
dVwgdSwdSwdVwwdV
C
TT
C
nn ρσσεσγρ )(: ( )tVw 0∈∀
(10)
where n
σ and T
σ are defined in equations (7) and (8), ( )tV0 is the set of velocity field vanishing
on ( )tD
Γ . At this stage stress and strain-rate measures are Eulerian and σ can be interpreted as the
Cauchy stress tensor. Applying the symmetric gradient operator )(⋅ε to the velocity field w of the
current configuration, we have ( ) *Dw =ε , which can be interpreted as a virtual strain rate.
A variational formulation of the isotropic non associated plasticity set of equations (1)-(4) can be
written as follows:
( )RE ,τπ ∈∀ ( ) ( )( )
( )
Ψ=
≥−
−∈ ∫
Ω
∇
DRR
dVDLRP
t
,,
0 : ;
τ
τπττ
&
(11)
Where plasticity convex ( )RP and set of the iso-potential surfaces ( )RE ,σ , are defined below:
( ) ( ) NiRfRP i ,,1;0,: K=≤= ππ
( ) ( ) ( ) ( ) 0,such that for ; ,,:, =≤= RfiRgRgRE iii ττππτ
The presentation of the set of variational equation in an Eulerian form have the interest to provide a
quite simple and synthetic formulation of the plowing problem involving unilateral frictional contact,
10
non associated multisurface plasticity and large deformation. Nevertheless, our aim is to perform a 3D
numerical model in the context of highly intensive computations, able to model plowing applications
for different soil treatment tools, variable in size and geometry. To that end we will adopt a step by
step approach based on time increments decomposition, i.e. [ ]ttt ∆+, , in order to numerically handle
this evolution problem. In this context, we developed a formulation in the Updated Lagrangean
framework with the configuration at the beginning of time step as the reference configuration. This
approach need to re-write the governing equations of the mechanical problem in the Lagrangean
frame. Moreover, such Lagrangean formulation lead, among other, to provide a proper and consistent
formulation of the hypo-elasoplastic constitutive relation which add non linearity to the problem (see
for example [7],[36]). On the other hand, at numerical point of view, the Updated Lagrangrean
description allows to relax a part of the non-linearity due to large deformation, typically in the choice
of lagrangean strain and stress measures (Piola-Kirchoff stress tensors and Green Lagrange strain
tensor [6]). Hence, in the present paper and for the sake of simplicity we focus on the approximations
associated to this approach and their justifications, in order to eventually provide the numerical model
actually solved.
2.4 Updated Lagrangean and Approximations
Approximation and linearisation that might be done, are related to time step size, but have to guarantee
that the kinematics underlying large transformation is eventually addressed. Hence significant
differences may appear in numerical simulation with Total or Updated Lagrangean formulations, see
for example [15]. In our case, independently from material large deformations, the time step size is
principally governed by (i) penalized contact conditions, which imply a rather low upper bound on
time step size, and (ii) by mesh distortion. Practically the corresponding incremental displacement of
the plowing tool cannot exceed 1 mm. This justifies the following approximation made on stress and
strain measures. Hence over the time step [ ]ttt ∆+, we assume that the current and reference
configuration are close enough (the change in volume remains small) and use the Cauchy stress tensor
11
as the stress measure in the constitutive equations as well as in the equilibrium equation.
Correspondingly the strain measure is the infinitesimal strain tensor. However small time steps do not
make the objective derivation equivalent to material derivatives ; such an approximation would lead to
erroneous numerical results, especially in the direction of stresses after numerous increments. In our
incremental formulation, see equation (12) below, the stress .approxσ is first computed with a non-
objective, material time-derivative, and a correction is performed at the updating stage, taking
WW approxT .σσ = , with W the total (infinitesimal) rotation tensor (in the following the mention
approx. will be omitted). Finally the Updated Lagrangean procedure lead to integrate on the reference
configuration denoted tΩ . These considerations allow us to describe the set of equation that are
actually numerically solved.
2.5 Incremental Formulation
As we approximate the problem through time steps, we have to apply an integration scheme to the set
of governing equations, especially for the constitutive law. To that end a Forward Euler scheme is
applied for the geometry actualisation and a Backward Euler scheme for the constituive equations. We
recall that hypoelastic relation involves simply a material time-derivative of stress (the rotation terms
being corrected for only at the updating stage, see above) and so we can derive the incremental
variational formulation of both equilibrium and constitutive equation. With the notation
ttt yyy −=∆ ∆+, the global incremental problem consists in finding the velocity field
ttt Vv 0∈∆+, the
stress field tt ∆+σ and the internal variables
ttR
∆+ satisfying the variational system (12)-(13).
12
∫
∫∫∫∫
Ω
ΩΩ
⋅=
Γ
⋅−
Γ
⋅−+⋅∆∆ ∆+∆+∆+
t
t
C
T
tt
T
t
C
n
tt
n
t
tt
t
dVwg
dSwdSwdVwwdVtv
ρ
σσεσρ
)(:
tVw 0∈∀
(12)
( ) ( )( ) ( )
( )( )
Ψ∆=∆
≥−∆−−∈
∆+∆+∆+
∆+∆+−∆+∆+∆+ ∫Ω
tttttt
t
ttttttttttt
vRtR
dVvtLLRP
εσ
σπεσσσ
,,
0:1 ;
( )tttt RE ∆+∆+∈∀ ,σπ (13)
with tV0 the set of velocity field vanishing on
t
DΓ , and the normal and tangential stresses given by
+∆+∆+ −−= δθσ tttt
n d and 2
tool
T
tt
TF
tool
T
tt
Tt
T
tt
T
vv
vv
−+
−−=
∆+
∆+∆+
βπσ
with the incremental distance function given by ( ) ttoolttttt nvvtdd ⋅−∆+= ∆+∆+and the smoothing
corner parameter δ allowing a (small) penetration. Repulsion stress is applied if the distance inside the
tool is larger than δ.
Hence contact conditions (non-penetration) are written at the end of the time step, but the normal
involved in the computation of d is taken at the beginning of the current time step. Similarly, in the
case of Coulomb’s friction model, we use the normal stress at the beginning of the current time step,
tnσ . These two approximations avoid the computation of complicated, dissymmetry-enhancing
derivatives for the stiffness matrix,. This is a further reason for taking small time-steps; otherwise the
neglected stiffness terms (e.g. v∂∂ /n ) become significant and convergence may be degraded.
13
2.6 Mixed Formulation
The solution of the inequation (13) is locally equivalent to the projection of an elastic predictor on the
plasticity convex :
( ) ( ) ttt
RP
tt vtLtt
∆+−∆+ ∆+= ∆+ εσσ 1
Proj
This projection operator is identity when the elastic predictor remains inside the convex of plasticity.
Directions of projection depend on the potential gradient and the elastic inner product. The main non
linearity of this operator is that projection depend also on the solutions, namely tt ∆+σ and
ttR
∆+.
Thanks to this local projection operator we can build a mixed velocity/pressure formulation of the
problem. We first introduce the following notations for hydrostatic pressure (spherical stress)
( )σtrp3
1−= ; deviatoric stress I1ps += σ ; and deviatoric strain rate ( ) ( ) ( )( ) I13
1 vtrvve εε −= .
Using the definition of pressure and thanks to the additive decomposition of strain rate we have:
( ) ( )PtttKtrtrKpp εε ∆+∆−=∆+
(14)
K is the bulk elastic modulus and ( )Ptr ε∆ the volumetric plastic deformation at the end of time step.
14
To perform velocity/pressure formulation we enforce weakly equation (14), which leads to the
following system :
tVw 0∈∀
Qq∈∀
( )
( ) ( )( )
=∆−∆+−
⋅=
⋅−−−+⋅∆∆
∈
∫
∫
∫∫∫∫∫
Ω
Ω
ΓΓΩΩΩ
∆+
∆+∆+∆+∆+
∆+
0
)()(:
;
0
t
Pttt
t
tC
T
tt
T
tC
n
tt
n
t
tt
t
tt
t
ttt
dVqtrtrKpp
dVwg
dSwdSwdVwtrpdVweswdVt
v
Vv
εε
ρ
σσερ
(15)
with ( ) ( ) [ ]dttt
RP
tt vtLs tt
∆+−∆+ ∆+= ∆+ εσ 1
Proj ; [ ]d refers to the deviatoric part and Q is the set of
sufficiently regular virtual pressures.
15
2.7 Spatial Discretisation
We briefly present the spatial discretisation performed with velocity/pressure finite element P1+/P1
[1],[3]. Linear shape functions for velocity are enriched with a bubble function at the centre of the
tetrahedron. Bubble shape functions are linear over each of the 4 sub-tetrahedra constructed with the
centre of the element.
Let hε be a triangulation of tΩ into tetrahedra, the velocity field is additively decomposed as follows:
hhhhhh wvbvw =⊕∈+= Β (16)
with interpolation spaces:
( )( ) ( )( )
∈∀∈∈= hhh
hhhhhh PvCvvv εϖϖϖ ,1andΩ:
330
( )( ) ( )( )
=∈∈∀∂=∈=Β 4,..,1,1,,on0,Ω:
330 iPbbCbb i
hh
hhhhhhhhh ϖϖϖϖ ε
We note ( )( )30 Ω hC the space of continuous functions on 3RΙ⊂Ωh , ( )( )3
1 hP ϖ the space of linear
functions on element 3RΙ⊂hϖ , hΒ the bubble space included in 4
th order polynomial space,
4,..,1, =ii
hϖ representing a sub-tetrahedron of the element hϖ . Bubble functions are continuous and
piecewise differentiable.
We describe now the resolution algorithm of system (15) once spatially discretised. Due to the strong
non linearity of the local projection operator the resolution of such a system involves two steps
(Algorithm 1). The first step is devoted to constitutive equation integration for a given incremental
deformation, that is to perform projection. This stage provides total stress and plastic strain, as well as
internal variable. The second step consists in solving, in terms of velocity and pressure, equation (15)
updated with only deviatoric stress and volumetric plastic deformation, provided by the previous step.
Note that the pressure ttp ∆+ is a basic unknown of the velocity / pressure formulation and as such is
16
discretized and has to be solved for with global eq.(14), not at elemental level. After convergence, the
continuous pressure field computed from the spatially discretised system (15) is equal to the one given
by the projection, at the element level.
The system (14) is still strongly non linear and we use the Newton-Raphson method with line-search to
solve it.
Algorithm 1: Mixed Incremental Resolution
1. Input Data : ε&,,,,, tttt Rspvt∆ (ε& is given)
2. Constitutive equations integration : ( ) εσσ &1
Proj −∆+ ∆+= ∆+ tLt
RP
tttt
3. Solve equation (15) in ( )tttt pv ∆+∆+ , , with Ptts ε∆∆+ , known, with Newton-Raphson
4. Data updating, return 1
We note that the stiffness matrix resulting from Newton-Raphson method is non-symmetric due to
both non-associativity and compressibility of the general hypo-elasto-plastic law. This is why the
Minimal Residual solver of Forge3® has been replaced here by the Bi-cgstab algorithm [34]. The
improvement as compared to using a symmetric solver with a symmetrized stiffness matrix has been
previously quantified [28].
In the present application, extensive use has been made of the fully automatic topological mesher and
remesher of the software [9]. The remeshing criteria depend on the geometric quality of elements and
a local refinement in the vicinity of the tool is specified by a set of element sizes. Each set is
associated with a prescribed volume, and in our case such parallelepiped “boxes” move with the tool.
After remeshing the former velocity, pressure field and state variables (stress and internal variables)
are projected into the new configuration.
17
3 - Integration of constitutive equations
Constitutive model have been derived from the general CJS Model [5]. This elastic-plastic model has
been designed as a phenomenological approach for geo-materials.
3.1 Compressible CJS constitutive equation
This model involves two yield surfaces, similar to Cap Models: a spherical mechanism represented by
a plane closing the elastic domain on the pressure axis (equation (18)) and a deviatoric mechanism
represented by equation (19). In the following sss II := is the second, ( ( )sdet the third) stress
invariant. This model accounts for hardening and/or softening responses exhibited by soils for
monotonous loadings, associated with densification and/or dilatations. The concepts of characteristic
state (separating contractive from dilatative stress states) and critical state (deformation at constant
volume, reached after large strain) are also included.
The elastic law is non linear and hypo-elastic :
Ι+= 132
1
K
ps
G
E &&&ε (17)
with ( ) ( ) ee n
a
nppGG
−= 0 and ( ) ( ) ee n
a
ne ppKK−= 0 pa is a reference pressure equal to 100 kPa, G0,
eK0 , ne are model parameters.
The yield surface of the spherical mechanism is written:
( ) 0,,1 =−= QpQpsf (18)
with ( ) ( ) 1011 λ&& n
a
np ppKQ−= .
18
Note that 1λ& >0 (spherical plastic multiplier); thus, only hardening (increase of Q) is possible in this
model. The corresponding flow rule is associated, p
0K is a constant hardening modulus, n1 a
parameter.
The yield surface of the deviatoric mechanism is written:
( ) 03)(,,2 =−−= HRpshsRpsf II (19)
with
6/1
3
)det(541)(
−=
IIs
ssh γ , γ is a parameter of the model allowing the cross section of the yield
surface to evolve between a circle and a rounded triangle approximating the Mohr-Coulomb non
regular hexagon. H is the (constant) cohesion, R the average slope of the yield surface, the evolution of
which is given below in equations (21) and (22). The model is non associated :
( )Rpsg
ij
P ,,222
σλε
∂
∂= && (20)
with
∂
∂−
∂
∂=
∂
∂ijkl
klijij
ffg ηησσσ
222 and 3'
)('
2 +
−=
β
δβη ijIIij
ij
ss, ( )c
IIII ss+−= 1' ββ . Tensor η is
a weighted average of the deviatoric stress direction (sij) and the pressure axis direction (δij), and
parameter β is a constant (β = 0 means plastic incompressibility). Superscript c stands for “in the
characteristic state”. Through (20), the direction of the strain rate (dilatation or contraction) depends
on the position of the current stress state (sII) with respect to the characteristic surface ( ) 0, =psf c ,
according to:
( )( ) dilatationpRshspsf
ncontractiopRshspsf
cII
c
cII
c
⇒>−=
⇒<−=
0)(,
0)(,
This surface is not a yield surface, it has a different nature from 1f and 2f .
19
The hardening-softening variable R evolves by :
m
m
RpA
pARR
+= (21)
with ( )ppRR ccm /lnµ+= and )exp(0 ϑcpp cc −= , where ϑ is the relative volume change; A,µ,
c(>0) are model parameters, p an internal state variable related to pressure and plastic deformation
by:
( )RpsR
f
p
pp
a
,,2
2/3
2
∂
∂
−=
−
λ&& (22)
Rm is the radius of the fracture surface, which represents the maximum shearing experienced by the
material. Rm converge to the critical state parameter Rc, here equal to the characteristic surface radius.
As seen in figure 1, the evolution of R may display a peak or not, depending on initial material density
(through 0cp ) : loose sand will monotonically “harden”, dense sand will first “harden” then “soften”.
As mRRp →+∞→ , , and cm RR → .
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0 10 20 30 40
axial def.%
Rm 800 L
R 800 L
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0 5 10 15 20 25 30 35 40
axial def. %
Rm 800 D
R 800 D
Figure 1: Maximal shear surface radius Rm and yield surface evolution R in a tri-axial test with
confinement pressure 800 kPa, Rc = 0.265 (dashed line). Dense sand (D, left), loose sand (L, right).
20
3.2 Implicit scheme
For a given incremental strain, we have to perform the incremental projection operator for the set of
evolution equation of CJS Model during the time step [ ]ttt ∆+, :
( ) εσσ ∆+= −∆+∆+∆+
1
, Proj Lt
RQP
tttttt
where ( ) ( ) ( ) 0,:0,:, 21 ≤∩≤= ∆+∆+∆+∆+ tttttttt RfQfRQP ττττ .
To provide such an operator, the implicit formulation of CJS constitutive equations is written below.
In the present work, we chose linear elasticity with non pressure dependent bulk and shear modulus.
Total incremental plastic deformation is split into its spherical and deviatoric parts.
In the case where both mechanisms are active, and according to the flow rules, the volumetric strain is
given by:
( ) ( )( ) ( )
∂∂
∂
∂∆+
∂∂
∂
∂∆=
∆+∆=∆
∆+∆+∆+∆+∆+∆+∆+∆+
σλ
σλ
εεε
pRps
p
gpQps
p
f tttttttttttttttt
PPP
,,tr,,tr
trtr
22
11
21
and shearing (deviatoric) strain by:
( )
∂∂
∂
∂∆+=
∆+∆=∆
∆+∆+∆+∆+
σλ sRps
s
g
eee
tttttttt
PPP
,,0 22
21
Thanks to the plane Cap, both mechanisms interfere only in the compressible part of plastic
deformation. Hence, the incremental stress equation is written in terms of spherical and deviatoric
parts :
21
( ) ( )
( )
∂∂
∂
∂∆−∆+=
∂
∂∆+∆−∆−=
∆+∆+∆+∆+∆+
∆+∆+∆+∆+∆+∆+
σλ
λλε
sRpss
gGeGss
Rpsp
gKKpp
ttttttttttt
ttttttttttttt
,,22
,,tr
22
221
(23)
The double consistency condition leads to:
( )
=−−=
=−=∆+∆+∆+∆+∆+∆+∆+
∆+∆+∆+∆+∆+
03),,(
0),,(
2
1
HpRshsRpsf
QpQpsf
tttttttt
II
tttttt
tttttttttt
(24)
The non linearity of system (23) in unknows p and s, is partially linked to the third invariant involved
in the deviatoric mechanism. More classical models such as Drucker Prager or Cam Clay, lead to a
simplified system (23), in which pressure and deviatoric stress at time t+∆t depend only on trial stress
and plastic multipliers. The latter appear in system (24) thanks to hardening/softening laws and stress
components. They are basic unknowns of the problem. Hence, the projection step consists in :
Find tttttttt ps ∆+∆+∆+∆+ ∆∆ 21 ,,, λλ satisfying the non linear system (23)-(24).
The numerical resolution of this system follows the general return mapping algorithm. It is well known
that numerical difficulties may occur according to, among others, the slope and non linearity of
evolution laws. To overcome these numerical difficulties at the local integration step, different
techniques are proposed in the literature [2],[10],[14],[19],[33]. Some authors ([2],[19]) replace a full
Newton method by a fixed-point like method, which results in a split-level resolution of equations in
system (23)-(24). We applied this method to solve the local projection step. However for the present
multi-surface plasticity model, we focused on problems where non linearity due to the third stress
invariant (function h) can be relaxed. Hence the projection step can be reduced to the resolution of the
non linear system (24).
22
Eventually in order to perform the global Newton Raphson method the consistent tangent operator,
[29],[31] has been derived.
3.3 Validation on Tri-axial Tests
To perform the validation of our implementation in Forge3®, we have compared our results with
experiments for loose and dense Hostun sand ; more precisely, we have compared the present
implementation of the CJS model with the implementation done previously in the soil mechanic FEM
software Flac2D© [13] (figures 2b and 3b), and recalled the comparison of Flac2D
© with experiments
detailed in [16] (figures 2a and 3a).
Parameter Loose sand Dense sand
K0e 50×103 kPa 80×103 kPa
G0 40×103 kPa 50×103 kPa ne 0 0
K0p 50×103 kPa 80×103 kPa
n1 0.6 0.6
β 2.2 2.2
A 0.11×10-3 0.4×10-3 γ 0.845 0.845
c 30 30
Pco 750 kPa 8×103 kPa µ 0.03 0.03
Table 1: CJS model parameters for loose and dense Hostun Sand.
The set of parameter listed in Table 1 has been used; it is the same set as used in [16], except for the
simplifications made in the present implementation (Forge3® is restricted to linear elasticity, so ne =
0).
The numerical domain is 1/8th of a cylinder (45 nodes and 102 tetrahedron). Convergence criterion on
the global Newton-Raphson’s Method is 6
NR 10−=ε , the iterative solver is Bi-cgstab. Convergence
criterion for local integration is 12
local 10−=ε .
23
Graphs on Figure 2 and Figure 3 represent results respectively for stress-strain and volumetric-axial
strain relations, for three confining pressures 100 kPa, 300kPa, 800 kPa. As a first observation CJS
correctly models the experimental stress-strain relation for both densities (figures 2a and 3a). The
concept of critical state, here clearly evidenced by experience, is correctly handled: whatever the sand
density is, the stress level eventually reached is the same, for each confining pressure. The description
of volume variation is not as satisfactory, pointing out severe difficulties met to model the whole range
of responses displayed by soils. Nevertheless, mechanisms related to the volumetric variation, namely
densification and dilatation, are well represented.
Comparisons of Flac2D© and Forge3® results are plotted on Figure 2 and Figure 3 (bottom). In terms
of stress-strain relation, the agreement between the two codes is very good. Slight differences appear,
mainly in the slope at the beginning of the curves for the 800 kPa confining pressure. These
differences are related to the treatment of elasticity, non linear in Flac2D©, linear in Forge3
®. In terms
of volume variation, more significant differences appear for extreme conditions (loose sand under high
pressure and dense sand under low pressure). It is thought that linear elasticity is the reason here also,
but how it influences the amount of plastic deformation at the re-projection step is more difficult to
understand. Moreover for the stress-strain relation, the concept of critical state will eventually provide,
for any elasticity modulus, roughly the same stress level, but the deformation mode will be different.
As a whole, the results of the two implementations are rather close to each other and to experiments.
As the two software are radically different in all respects (explicit vs implicit formulation, hexaedra vs
tetrahedra, …), this tends to prove that the present implementation has been done correctly.
24
0
0,5
1
1,5
2
2,5
0 10 20 30 40 50
axial de f. %
dev
iato
ric
stre
ss M
pa
Forge3 100L
FLAC2D 100L
Forge3 300L
FLAC2D 300L
Forge3 800L
FLAC2D 800L
Forge3 100D
FLAC2D 100D
Forge3 300D
FLAC2D 300D
Forge3 800D
FLAC2D 800D
Figure 2: stress-strain relations on tri-axial tests for three confining pressure:
100 kPa, 300 kPa, 800 kPa, for Dense (D) and Loose (L) sand.
(Top) comparison experience – Flac2D. (Bottom) comparison Flac2D© – Forge3
®.
25
-20
-15
-10
-5
0
5
10
0 10 20 30 40 50
axial def. %
vol.
def
%
Forge3 100L
FLAC2D 100L
Forge3 300L
FLAC2D 300L
Forge3 800L
FLAC2D 800L
Forge3 100D
FLAC2D 100D
Forge3 300D
FLAC2D 300D
Forge3 800D
FLAC2D 800D
Figure 3: volumic-axial strain relation on tri-axial tests for three confining pressure:
100 kPa, 300 kPa, 800 kPa, for Dense (D) and Loose (L) sand.
(Top) comparison experience – Flac2D. (Bottom) comparison Flac2D© – Forge3
®.
26
4 - A simple system: ploughing with a single tine
The ploughed superficial soil layer has a very complex behavior, in particular due to the large
heterogeneity of the soil structure, basically a granular solid influenced by pore water pressure,
maintained or not by a root network, more or less stratified … Most of this physical complexity is
overlooked in this work devoted to 3D simulation of the entire scarring process. A one-phase material
is assumed, whereby pore water suction is rendered approximately by a cohesion; water migration is
ignored in this large speed context. Stratification is neglected as a first approach, but could be
introduced in future developments; roots could then be re-introduced via an enhanced cohesion in a
layer of a prescribed thickness. Consequently we divide soils in two classes: purely frictional or
frictional-cohesive soils modelled as compressive pressure-dependent materials, and purely cohesive
saturated or quasi-saturated soils assumed incompressible.
For incompressible materials we use a Von Mises–based model with specific evolution law allowing
for softening / hardening response. Constitutive equations of the model are listed below:
0=−= Hsf II (25)
In this case, the flow rule is associated and cohesion H increases and/or decreases:
max
max
HA
AHH
+=
εε
, ∫=t
PP ee0
3
2: &&ε ,
2
cmc
b1
HHHH
)(max ε+
−+= (26)
A, b, Hc and Hm ≥ Hc are constant parameters of the model. Hc is the asymptotic cohesion:
cHH → when the cumulated plastic strain +∞→ε . If Hm = Hc, the peak disappears, H becomes a
monotonically increasing function of ε .
In subsequent sections a saturated clay and a silted sand will be involved. Elastic parameter are in both
case E = 8 MPa (Young’s modulus), ν = 0,3 (Poisson’s ratio). For clay, the moisture and density are w
= 17,67% and ρ = 1,66. Hardening parameters are A = 5 MPa, b =120, and asymptotic cohesion is Hc
= Hm = 71 kPa. For the silted sand, the moisture and density are w = 19,69% and ρ = 1,82. We use a
27
simplified version of the compressible CJS model : the deviatoric mechanism is restricted to an elastic-
perfectly-plastic one, i.e. R = Rc = 0.265 is a constant; the flow rule is kept non-associated, but only
dilatative plastic deformation is allowed (β = 0.2). The third stress invariant influence is neglected (γ
= 0). The spherical mechanism is activated with Q0 = 100 kPa and K0p = 60 Mpa, n1 = 0.6. A cohesion
H = 14.4 kPa has been chosen.
As for the friction, experimental studies, [16], resulted in using Tresca law (friction factor model) for
clay and Coulomb’s law for sand. Parameters are respectively : m = 1, and interface friction angle
equals to 2/3 of the soil friction angle.
Modes of deformation
Numerical experiments have been conducted with simplified tines; the parameter examined is the rake
angle, 15° and 45°; the response of incompressible clay is pictured in Figure 4. Two material flow
patterns clearly appear depending on the rake angle. For almost vertical tines, a chip forms in front of
the tool, like in a cutting process; only a small volume of soil is mobilised. For inclined tines,
deformation extends much farther and the material is pushed into well-developed lateral and frontal
bulges. In both cases, total plastic strain reaches a maximum value )30( ≈ε in the close vicinity of
the tool tip.
Such deformed configurations have been obtained after a tool displacement of 30 cm (10 times the
width of the tool). 28 hours of computation were necessary on a SUN E450 (processor Sun
UltraSparc-II, 400 Mhz), including about 130 remeshing operations (60% of the total CPU time). The
meshes are 6738 nodes, 23514 tetrahedra for the 15° tine and 5876 nodes, 18133 tetrahedra for the 45°
tine - intensive computations indeed.
28
Figure 4 : ploughing 30 cm deep with the 15° and 45°, 3 cm wide tine. Total tool displacement 31cm.
Influence of rake angle on forces
Another series of tines (closer in shape to real tools) has been used to study the effect of rake angle on
horizontal and vertical components of the ploughing force in saturated clay (Figure 5). Tine geometry
does not affect the steady-state horizontal thrust : the projection of the active surface of the tine in the
vertical plane almost remains the same. Moreover this cutting-like process leads to roughly the same
normal stress level whatever the angle is (about three times the asymptotic cohesion in this
incompressible case).
On the contrary, tine geometry influence is strong on the vertical force which gets closer to zero as
rake angle decreases.
29
Figure 5 shows also that both components increase strongly in the very first centimetres of tool
penetration, no matter the rake angle is. Then stabilisation occurs promptly for the vertical force,
whereas the horizontal one converges slowly to a steady state, without actually reaching it after a 15
cm tool displacement : in this high friction case, the characteristic dimension is not the depth of the
tine width, but the lateral surface of the tine in a vertical plane parallel to the ploughing direction; the
whole tine must have penetrated the block before the contact and the friction force come to saturation,
promoting steady state.
-6
-4
-2
0
2
4
6
0 2 4 6 8 10 12 14 16
Tool Displacement (cm)
Fo
rces
(kN
)
20
F-ver, 30°
F-hor, 30°
F-ver, 45°
F-hor, 45°
F-ver, 60°
F-hor, 60°
F-ver, 30°-60°
F-hor, 30°-60°
Figure 5: Vertical (<0) and horizontal (>0) force evolution during plowing for different rake angles:
30°, 45°, 60° ; the “30°-60° tine” has its lower half inclined by 60° , its upper half by 30°.
Compressible vs. Incompressible
Figure 6 compares the force evolution for the dilatative silted sand and an incompressible clay. The
striking point here is the “peak” in the dilatative case, after which the force drops dramatically to its
pre-peak level. The first, pre-peak part of the curve is similar to the incompressible case, this
corresponds to the tine abutment into the domain. As for the peak, explanation may lie in the dilatancy
30
of the compressive model under these shear-dominated conditions. Simulations for different domain
width and boundary conditions indicate us the following arguments. When lateral flow is limited,
which occurs as the domain is narrower, dilatancy is blocked and could bring about higher pressure, a
smooth curve but at the top-of-the-peak level. For a wider domain or when lateral boundary are free,
as the material displaced from the furrow can be pushed aside easily and reach the free surface,
conditions of blocked dilatancy no more take place. Once this new flow pattern is established, which
means the tine sufficiently penetrates the block, pressure and force drop down. Future work will try to
confirm this tentative explanation.
-4
-2
0
2
4
6
0 5 10 15Tool Displacement (cm)
Fo
rces
(kN
) F-ver.,Incomp.
F-hor., Incomp.
F-ver., Dilat.
F-hor., Dilat.
Figure 6: Comparison of force evolution for incompressible and compressible material.
5 - Conclusion
3D modelling of the ploughing process using FEM has been presented using the implicit FEM, large
deformation software Forge3®. A complex, non associated, two yield surface elasto-plastic law for
soils, named CJS, has been introduced in a mixed velocity/pressure formulation. It is able to predict
contraction and dilatation associated with the hardening and softening of the material. This elasto-
plastic law has been time-integrated by a generalized return mapping algorithm. Implementation has
31
been successfully validated on tri-axial tests by comparison with both experiments and another
software. The second part of this paper has been devoted to the numerical simulation of single tine
ploughing, showing the ability of the code to handle complex flow patterns, examining the influence
of the rake angle on flow pattern and forces as an example. Such numerical simulations of large tool
displacement are using intensively remeshing routines, which contributes significantly to the high
computing cost. In terms of soil nature influence, blocked dilatancy effects have been hypothesized to
explain a pronounced peak in force ploughing after tine abutment.
Not presented here are some applications of the model to more complex ploughing tools : single tine
plus a blade, multi-tine tool, three tines in V with a blade. For instance, for three tines aligned along an
oblique line, the computation was able to reproduce an experimentally observed effect, the central tine
experiencing a smaller horizontal force (by some 20%) than both external ones. Other, parametric
studies are in progress to better understand the deformation modes in ploughing.
Acknowledgement
This work was part of a multi-laboratory cooperative work, for which the financial support of the
French DGA (Délégation Générale à l’Armement) is gratefully acknowledged, as is the permission to
present this paper. Fruitful discussions must be acknowledged with Profs. B. Cambou, F. Sidoroff, C.
Bohatier, Drs. P. Gotteland, C. Bacconnet, A. De la Lance, C. Nouguier, P. Kolmayer, MM. O. Benoît,
M. Grima, M. Froumentin and A. Quibel.
32
References
[1] Arnold D. N., F. Brezzi, M. Fortin, A stable finite element for Stokes equations, Calcolo, 21,
(1984), 337-344.
[2] Borga R.I., Lee S.R., “Cam-Clay Plasticity, Part I : Implicit integration of elasto-plastic
constitutive relations” Comput. Methods Appl. Mech. Engrg., 78, pp. 49-72 (1990).
[3] Brezzi F., Fortin M., Mixed and Hybrid Finite Element Methods, Springer Series in
Computational Mathematics, 15, Springer-Verlag, (1991).
[4] Bucaille J.L., “ Simulation numérique de l’indentation de verres organiques”, Doctorate
Thesis, Ecole des Mines de Paris, (2001).
[5] Cambou B., Jafari, Sidoroff F., “Modèles de comportement des sols non cohérents”, Rev. Fr.
Géotech., 44, (1988), 43-55.
[6] Crisfield M.A., “Non-linear Finite Element Analysis of Solids and Structures”,Volume 1:
Essentials, Jhon Wiley & Sons, 1991.
[7] Chopra M. B., Dargush G.F., “Finite Element Analysis of time Dependent Large-Deformation
Problems ” Int. J. Numer. Anal. Meth. Geomech., 16, pp. 101-130, (1992).
[8] Chtourou H., Gakwaya A., Guillot M., “International Journal of Solids and Structures”, 39,
1077-1096, (2002).
[9] Coupez T., Génération de maillage et adaptation de maillage par optimisation locale, Rev.
Eur. Eléments Finis, 9, (4), (2000), 403-423.
[10] De Borst R., Heere O.M., A unified approach to the implicit integration of standard, non-
standard and viscous plasticity models, International Journal for Numerical and Analytical
Methods in Geomechanics, 26, (11), pp. 1059, (2002).
[11] Fourment L., Chenot J.L., Mocellin K., “Numerical formulations and algorithm for solving
contact problems in metal forming simulations”, Int. J. Num. Meth. Enging., 1435-1462, 1999.
[12] Gee-Clough D., J. Wang, W. Kanok-Nukulchai, “Deformation and Failure in Wet Clay Soil :
Part 3, Finite Element Analysis of Cutting of Wet Clay by Tines”, J. Agric. Engng. Res., 58,
(1994), 121-131
[13] ITASCA consulting, “Documentation de FLAC2D : strain-softening model”, User’s Manuel,
Minneapolis, 1994.
[14] Jeremie B., “Line search techniques for elasto-plastic finite element computations in
geomechanics”, Commun. Numer. Meth. Engng, 17, pp. 115-125 (2000).
33
[15] Khoei A.R., Lewis R.W., “Finite element simulation for dynamic large elastoplastic
deformation in metal powder forming”, Finite Element Analysis and Design, 30, 335-352,
1998.
[16] Kolmayer P., “Modélisation du comportement Mécanique des Sols de Surface et Identification
In Situ”, Doctorate Thesis, Ecole Centrale de Lyon, (2001).
[17] Kolymbas D., B. Cambou, C. Di-Prisco, Elements of hypoplasticity. Constitutive modelling of
Geomaterials. Revue française de génie civil, 4 (5), (2000), 85-107.
[18] Lee E.H., Liu D.T., “Finite strain elastic-pplastic theory with application to plane waves
analysis”, Journal of Applied Physics, 38, 19-27, 1967.
[19] Macari E.J., Weihe S., Arduino P., “Implicit integration in elastoplastic constitutive models for
frctional materials with highly non-linear hardening functions”, Mech. Cohes.-Frict. Mater., 2,
pp. 1-29, (1997).
[20] McMeeking R.M., Rice, J.R., “Finite-element formulations for problems of large elastic-
plastic deformation”, Int. J. Solids Structures, 11, 601-616, 1975.
[21] Meschke G., Liu W.N., “A re-formulation of the exponential algorithm for finite strain
plasticity in terms of cauchy stresses”, Comput. Methods Appl. Mech. Engrg., 173, 167-187,
(1999).
[22] Mouazen A. M., M. Nemeyi, “A review of the finite element modelling techniques of Soil
tillage”, Mathematics and Computers in Simulation, 48, (1998), 23-32.
[23] Mouazen A.M., H. Ramon, “A numerical-statistical modelling scheme for evaluation of
draught requirements of a subsoiler cutting a sandy loam soil, as affected by moisture content,
bulk density and depth”, Soil & Tillage Research, 63, (2002), 155-165.
[24] Mroueh H., Shahrour I., “Use of sparse iterative method for the resolution of three-
dimensional soil / structure interaction problems”, Int. J. Numer. Anal. Meth. Geomech., vol.
23, pp. 1961-1975, (1999).
[25] Nouguier C., “Simulation des Interactions Outil-Sol, Application aux Outils de Traitement des
Sols”, Doctorate thesis, Université de Montpellier II (1999).
[26] Ponthot J.P., “Unified stress update algorithms for the numerical simulation of large
deformation elasto-plastic and elasto-viscoplastic processes”, International Journal of
Plasticity,18, 91-126, 2002.
[27] Renon N., Sofonea M., Shillor M., “Un modèle Mathématique pour un problème de contact
elasto-plastique avec écrouissage. Application à la scarification”, 10ième Colloque Franco-
Polonais de Mécanique, Varsovie, Décembre 2002.
[28] Renon N., “Simulation numérique par éléments finis des grandes déformations des sols.
Application à la scarification”, Doctorate Thesis, Ecole des Mines de Paris, (2002).
34
[29] Runesson K., Samulesson A., “Aspects on Numerical Techjiques in Small Deformation
Plasticity”, Proceedings of NUMETA ’85 Numerical Methods in Engineering, Theory and
Application, Editors: J.Middleton and G.N. Pande, pp. 337-347, AA. Balkema. 1985.
[30] Simo J.C., “ Algorithms for static and dynamic multiplicative plasticity that preserve the
classical return mapping schemes of the infinitesimal theory”, Comput. Methods Appl. Mech.
Engrg., 99, 61-112, 1992.
[31] Simo J.C., Taylor R.L., “Consistent tangent operator for rate-independent elastoplasticity”
Comput. Methods Appl. Mech. Engrg., 48, pp. 101-118 (1985).
[32] Sofonea M., Renon N., Shillor M., “Stress formulation for frictionless contact of an elastic-
perfectly-plastic body”, to appear in Applicable Analysis
[33] Stolle D.F.E., Vermeer P.A., Bonnier P.G.,“ Time Integration of a constitutive law for soft
clays”, Commun. Numer. Meth. Engng, 15, pp. 603-609 (1999).
[34] Van der Vorst H.A., “BI-CGSTAB : A Fast and Smoothly Converging Variant of BI-CG for
the Solution of Non-Symmetric Linear Systems” J. Sci. Stat. Comput., 132, (1992), 631-644.
[35] Viggiani G., C. Tamagnini, “Ground movements around excavations in granular soils: A few
remarks on the influence of the constitutive assumptions on FE predictions”, Mech. Cohesive
frictional Mat., 5 (5), (2000), 399-423.
[36] Voyiadjis G.Z., Abu-Farsakh M.Y., “Coupled Theory of Mixtures for Clayey Soils”
computers and geotechnics, 20, pp. 195-222 (1997).
[37] Wriggers P., “Finite Element Algorithms for Contact Problems”, Archives of Computational
Methods in Engineering, State of the art reviews, 2, 4, pp. 1-49, (1995).