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

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