Failure in accretionary wedges with the maximum strength theorem: numerical algorithm and 2D validation

(1)

HAL Id: hal-00584137

https://hal-mines-paristech.archives-ouvertes.fr/hal-00584137

Submitted on 11 Apr 2011

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Failure in accretionary wedges with the maximum strength theorem: numerical algorithm and 2D

validation

P. Souloumiac, K. Krabbenhøft, Y. M. Leroy, B. Maillot

To cite this version:

P. Souloumiac, K. Krabbenhøft, Y. M. Leroy, B. Maillot. Failure in accretionary wedges with the

maximum strength theorem: numerical algorithm and 2D validation. Computational Geosciences,

Springer Verlag, 2010, 14 (4), pp.793-811. �10.1007/s10596-010-9184-4�. �hal-00584137�

(2)

with the maximum strength theorem:

numerial algorithm and 2D validation.

P. Souloumia

Laboratoire MSS-Mat, CNRS ,

Eole Centrale Paris, Chatenay Malabry, Frane,

K.Krabbenhøft

Centre for Geotehnial and Materials Modelling,

University of Newastle, NSW 2308, Australia,

Y.M.Leroy

Laboratoirede Géologie,CNRS ,

Éole Normale Supérieure, Paris, Frane,

and

B. Maillot

Département Géosienes et Environnement,

Université de Cergy-Pontoise, Frane.

Abstrat

Theobjetiveistoapturethe3Dspatialvariationinthefailuremodeouringinaretionarywedges,

andtheiranalogueexperimentsinthelaboratory,fromthesoleknowledgeofthematerialstrengthandthe

struturegeometry. Theproposedmethodologyreliesonthemaximumstrengththeoremwhihisinherited

from the kinematis approah ofthe lassiallimit analysis. It selets the optimum virtual veloity eld

whih minimizes the tetoni fore. These elds are onstruted by interpolation thanks to the spatial

disretization onduted with ten-nodedtetrahedra in 3D, and six-nodedtriangles in2D. Theresulting,

disrete optimization problem is rst presented emphasizing the dual formalism found most appropriate

in the presene of non-linearstrength riteria, suh as the Druker-Prager riterion used in all reported

examples.

Thenumerialshemeisrstappliedtoaperfetly-triangular2Dwedge. Itisknownthatfailureoursto

thebak,fortopographislopesmallerthan,andtothefrontforslopelargerthan,aritialslope,dening

sub-ritialand super-ritial slopestability onditions,respetively. Thefailuremodeisharaterizedby

theativationofaramp,itsonjugatebakthrustandthepartialorompleteativationofthedéollement.

It is shownthat the ritial slope isaptured preiselybythe proposed numerial sheme, therampand

the bakthrustorrespondingto regionsofloaliz edvirtualstrain. Theinueneofthebak-wallfrition

onthisritialslopeisexplored. Itisfoundthatthefailuremehanismisharaterizedbyathrustrooting

at thebase ofthe bakwall andthe abseneofbakthrust,for small enoughvaluesofthe fritionangle.

This inueneis wellexplained by theMohronstrutionandfurther validated withexperimentalresults

withsand,onsideredasananaloguematerial. 3Dappliationsofthesamemethodologyarepresentedina

ompanionpaper.

Otober26,2009

Submittedforpubliation.

(3)

The objetive is to determine the 3D failure mode whih haraterizes the onset of thrusting

orfoldinginfold-and-thrustbelts and inaretionary wedges. The numerialmethodwhihis

proposedhasitsrootinthekinematisapproahoflimitanalysisalthoughonlytheknowledgeof

the materialstrength isrequired. The numerialalgorithmand its2Dvalidationare presented

in this ontribution, the 3D appliations ina ompanionpaper (Souloumiaet al.,2009).

Thekinematisof 2Dfoldsand thrustshas been studiedatlengthand isnow wellaptured

by geometrialonstrutionsinspiredby theseminalworkofSuppe(1983). Theabsene ofany

onept of mehanis, suh as material strength and mehanial equilibrium render however

impossiblethe omparison between two geometrialonstrutionsneessary toselet the most

relevant. The meritof theseonstrutionsis however learinviewof theirsimpliityand their

potentialappliationintheoilindustry,oneompletedbytheomputationofthetemperature

evolution(Zoetemeijer and Sassi, 1992, Siamannaet al.,2004).

Thelineofworkwhihhas been followed by theauthorstries totakethe mostadvantage of

the2Dgeometrialonstrutionwhileaountingformaterialstrengthand mehanialequilib-

rium. The priniple of minimum dissipation was applied by Maillot and Leroy (2003) in their

study of a simple fault-bend fold, with either brittle or dutile material response, to nd the

optimum orientationofthe bak thrust. A morerigorous framework isnow adopted, basedon

the maximumstrength theorem forfritionaland ohesivematerials(Salençon, 1974,2002). It

was appliedtothe evolution of akink-foldby Maillotand Leroy (2006)proposing that,at any

stage of the struture development, its main geometrial attributes, suh as the kink dip and

width,ouldbefoundbyminimizingtheupperboundtothe appliedtetoni fore. Cubasand

al. (2008) extended this argumenttostudy sequen es of thrustswithinanaretionary wedge.

Souloumia etal. (2008) proved that the optimum stress stateould be alulatedat any step

of the thrusting sequen e development, based onthe statiapproah of the limitanalysis.

Thereisadenitedesiretopropose 3Donstrutionsoffoldingand thrustingwhihisoften

inhibited by the lak of intuition for parameterizing simplythe failure mehanism (e.g. ramp

andbakthrustsystem)attheonsetandduringthedevelopmentofthefold. Itisthusneessary

to develop a systemati proedure to study the failure mode of 3D geologialstrutures. For

the onset, the kinematisapproah of limit analysis ouldprovide a rst insight onthe failure

mode. It is the subjet of the present ontributionand it ishoped that the results ould help

in onstrutingthe 3D kinematis of the evolving strutures.

The proposed method,referred toas the maximum strength theorem, is based onthe kine-

matis approah of lassial limit analysis. It is emphasized that a omplete plastiity theory

is not required and the provisionfor the ohesive and fritionalroks of interest of astrength

domain, onvex in the stress spae, sues to obtain an upper bound to the applied tetoni

fore. Overtheyears,anumberofdierentnumerialformulationsofthemaximumstrength(or

upper bound) theorem have been proposed. Early formulations, fousing on two-dimensional

problems (Anderheggen and Knöpfel, 1972; Pastor, 1978; Bottero et al., 1980; Sloan, 1989),

typiallyinvolvedalinearizationofthestrengthdomainandmadeuse ofthesimplexmethodor

one ofitsderivativestosolvetheresultinglinear programs. Inspired by theprogress ingeneral

onvex programming, these linear programming formulations have reently been replaed by

more general non-linear formulations avoiding the need to linearize (Lyamin and Sloan, 2002;

KrabbenhøftandDamkilde,2003). Themostreentdevelopmentonthisfronthasbeen theap-

pliationsoftheso-alledoniprogrammingalgorithmstosolvetypiallimitanalysisproblems

suhasthe onesonsideredhereaswellasarangeofotherplastiityproblems(Krabbenhøftet

al., 2007; Krabbenhøft etal., 2008). These algorithms are partiularly suited for dealing with

non-smooth strength domains suh as those typially haraterizing the strength of ohesive,

(4)

Initsprimal formthe maximum strength theorem isformulated intermsof kinemativari-

ables,thevirtualveloities. Theirdistributionisonstrutedbyinterpolationthankstoaspae

disretization. This primal form with disretization leads to a onvex minimization problem.

Alternatively, it is possible to work diretly with the dual form of the theorem whih leads

to a maximizationproblem reminisent of the stati approah leading to lower bounds to the

tetoni fore. The dual variables of the veloities (of its symmetri gradient to be more pre-

ise) in the sense of power are regarded as stresses after appropriate saling, although they

do not onstitute statially admissible elds (these dual variables do not satisfy equilibrium).

From a numerial point of view, this alternative, dual approah has a number of advantages.

For example, it is possible to impose ompletelygeneral strength riteria ina straightforward

mannerwhereasaprimalupperbound formulationwould requirethespeiation oftheorre-

sponding support funtion. This funtion denes the maxium power whih ould be provided

for a given veloity and strength domain. Its analytial expression is ertainly non-trivial to

derive and the resulting onstraints diult to aount for in a lassial optimization ode.

Furthermore,followingthe approahproposedby(Krabbenhøftetal.,2005),theinorporation

of kinematially admissible veloity disontinuities is straightforward and will be proposed in

this paper forthe generalthree-dimensional ase for the rst time.

The paper ontents are as follows. The next setion is devoted to the presentation of the

numerialalgorithm. The 2Dsetting ismost suitedfor suhpresentation forsakeof simpliity

and the extension to 3D is postponed to Appendix B. The onstrution of the dual problem

is highlighted with the help of the primal-dual algorithm of linear programming summarized

in Appendix A. Appendix C presents the link between these strength domains, typial of soil

mehanis,and the oniprogrammingalgorithmsadopted inMosek (2008),whihis used for

allexamplesreportedhere. Setion3isonernedwith2D appliationstoaretionary wedges

of perfet triangularshape. Failure inthe bulk ours either to the bak orto the front, with

the omplete ativation of the weak déollement at the base, depending on the topographi

slope. The transition from sub-ritial (failure to the bak) to super-ritial (failure to the

front) is aptured exatly, validating the numerial proedure. It is shown that the frition

angle onthe bak wallinuenes the failure mode for sub-ritial onditions. For small values

of the frition angle, a single ramp roots to the base of the bak wall whereas a ramp and

bak thrust ours for larger values. The transition infailure mode ours for a frition angle

deteted numerially whih is exatlythe one predited by the Mohr's onstrution. It is also

shown that thesetwomodes of failureare reproduedinthe laboratoryexperimentswith sand

by seleting the appropriate frition onditions atthe bak wall ontat.

2 The maximum strength theorem with spatial disretization

Theobjetiveofthissetionistopresentinthreestepsthetheoryappliedinthenextsetionfor

2Dwedgesandfor3Dexamplesintheompanionpaper. Therststepisthepresentationofthe

upper bound theorem of lassial limit analysis, as it is found in Salençon (2002) and Maillot

and Leroy (2006). It is proposed here to approximate the strength domain externally by a

seriesofhyper-plane,intheappropriatestressspae,tofailitatetheset upoftheoptimization

problem. Theseondstepisthedisretizationofthespaeandtheonstrutionofinterpolations

for the virtual veloities as well as for the virtual salars assoiated to these hyper-planes.

The third step onsists in the dualization of the upper bound problem after disretization,

resultinginamaximizationproblemwherethe basiunknownsare saledtohave dimensionof

stress. Thisdual formulationisusedinallexamplesbut shouldnotbeonfusedwiththe lower

(5)

disussed.

2.1 Summary of the upper bound theorem of limit analysis

The upper bound theorem of limit analysis is alled here the maximum strength theorem to

emphasize that only the onept of strength is required. This theorem is now presented in

details.

The starting point is the theorem of virtual power whih states the equality between the

internal and the external powers for any kinematially admissible(KA) veloity eld. The set

S

û ôf ^KA êlds ômprises âny êld

U ˆ

^whihîs ^zero ôver ^part ôf ^the ^boundary

∂ Ω

u ^where ^the

displaements are presribed. Elements of

S

û âre îdentied ^by â^superposed ^hat. ^The êxternal

power, dened by

P

^ext

( ˆ U) = Z

Ω

ρg · UdV ˆ + α Z

∂Ω_T

T

^o

· UdS , ˆ

⁽¹⁾

is due to the power of the veloity over the body fore

g

^,

ρ

îs ^the ^material ^density, ând ôf

the foreapplied onpart of the boundary

∂Ω

T^. ^This âpplied ^foreîs âssumed ^to^be ^known ⁱⁿ

distribution

T

ô ^but ^not ⁱⁿ îts întensity ^dened ^by ^the ^salar

α

^whih îs ^the ûnknown ôf ^the

problem and for whih we seek the best upper bound. Note that in (1) and in what follows,

vetors and subsequently tensors, are identied with bold haraters. The internal power is

given by

P

^int

( ˆ U) = Z

Ω

σ : d( ˆ U) dV ,

⁽²⁾

where

σ

and

d( ˆ U)

âre ^the ^Cauhy ^stress ^tensor ând ^the ^virtual ^rate ôf deformation tensor (also denoted

d ˆ

⁾ ^based ^on

U ˆ

^, respetively. The double dot produt in (2) between these two tensors results in

σ

ij

d ˆ

ji ⁱⁿ ^terms ôf ^their ômponents ⁱⁿ ânorthonormal basis. The expression (2)fortheinternalpowerdoesnotaountforpotentialdisontinuitiesintheveloityeldsand

bulk deformation is the only soure of dissipation. Expliit aount of disontinuities, whih

orientationsare partof theunknowns ofthe problem, istypialofanalytialdevelopments but

is not neessary in the numerialformulation onsidered in this paper. However, pre-dened,

physialdisontinuitiesthusofknowngeometryareapproahedaszonesofbulkmaterialhaving

a zero thikness. Their ativation is marked by a loalized deformation within these narrow

zones. The onventional nite-elementformulationsannot ope with the limitof zero length

in one diretion for an element beause of the resulting ill-onditioning of the stiness array

(see e.g. Day andPotts, 1994). To the ontrary,the formulationadopted inthe followingdoes

not involve suh ompliation. Indeed, as it willbe disussed in the last part of this setion,

it is entirely possible to inlude pathes of elements with a thikness identially set to zero.

Thisapproahwasrstsuggested byKrabbenhøftetal. (2005)intheontextoflinearveloity

elements and is extended here to quadrativeloity elementsin 2D and further generalizedto

3D.

Coming bak to the internal power (2), note that the stress eld is unknown and its elim-

ination is desired. For that purpose, we take advantage of the material maximum strength.

The stress is required to remain within the strength domain denoted

G( σ )

^. ^The ^strength ^of

ohesive,fritionalfaultsisusuallydesribed intermsoftheCoulombriterionandforpristine,

bulk materials the strength domain is

G( σ ) = { σ | σ

I

− σ

III

+ (σ

I

+ σ

III

) sin φ − C cos φ ≤ 0 } ,

⁽³⁾

(6)

yy

)

²

+ 4σ

_xy²

, P = (σ

xx

+ σ

yy

)/2 ,

⁽⁴⁾

in whih

σ

e ^and

P

âre ^referred ^to âs ^the êquivalent ^shear ^stress ând ^the în-plane ^mean ^stress,

respetively. The determination of the intermediate stress diretion beomes a burden in 3D

appliationsanditismoreonvenienttoonsiderthestrengthdomainbounded bytheDruker-

Prager riterion:

G

DP

( σ ) = { σ | α

DP

I

1

+ p

J

2

− C

DP

≤ 0 } ,

⁽⁵⁾

δ

the seond-order identity tensor in (5). The twomaterial parametersin (5) are the frition oeient and the ohesion for the

Druker-Prager riterion and they are onviniently dened as

α

DP

= tan φ

p 9 + 12 tan

²

φ , C

DP

= 3C

p 9 + 12 tan

²

φ ,

⁽⁶⁾

so that the domainboundaries desribed by (3) and (5) oinidefor 2D plane-strain problems

(see e.g. Davis and Selvadurai,2002, for further details).

Mostif not allstrength domainsonsidered in the literature are onvex. Consequently, the

maximum power

σ : ˆ d

îs ^bounded ând ^given ^for â ^given ^veloity

U ˆ

^by ^the ^support ^funtion

π(ˆ d)

^. Ît ^depends ôn ^the ^geometry ôf ^the ^strength ^domain ^boundary ând ôf ôurse ôn ^the

veloity eld. A graphial method to onstrut this funtion is presented in Figure 1 where

stress tensorsare represente d as vetors. The strength domainhas an arbitrary,albeitonvex,

boundary in the stress spae. Superpose in this stress spae the virtual rate of deformation

d ˆ

despite the dierene in dimension. This virtual rate of deformation is normal to the hyper-

plane represented with a dashed line. Translate this plane towards the strength domain, as

illustrated by the dotted urve, and the point of ontat, denoted

σ

^∗, is the stress providing the maximum poweraording tolassial onvex analysis. Consequently:

π(ˆ d ) = σ

^∗

: ˆ d

^and

the seletionof

σ

^∗ is indeeda funtionof the orientationof

d ˆ

ând ôf ^the ^shape ôf ^the^strength

domain boundary.

The analysis of the 2D results insetion 3 willbe failitated with the expliit expression of

the support funtion. It reads

case 1 : tr(ˆ d) > ( | ˆ d

1

| + | d ˆ

2

| ) sin φ , π(ˆ d) = C

tan φ tr(ˆ d) ,

⁽⁷⁾

case 2 : tr(ˆ d) = ( | ˆ d

1

| + | d ˆ

2

| ) sin φ , π(ˆ d) = C cos φ( | d ˆ

1

| + | d ˆ

2

| ) , case 3 : tr(ˆ d) < ( | ˆ d

1

| + | d ˆ

2

| ) sin φ , π(ˆ d) = + ∞ ,

for bulk materials having the strength limit dened by the Coulomb riterion (3) in 2D

(Salençon,2002). In(7),

d ˆ

1 ^and

d ˆ

2 âre^the^2D^prinipal^valuesôf^the^virtual^rateôfdeformation

(7)

of the rate of deformation. More speially, the trae of the virtual rate of deformation has

to be positive, for the bound to be nite, implying a virtual dilation whih we will not try to

interpret physially. This is due tothe innite resistane in pure ompressionassumed for the

Coulomb riterion.

*

L

ij

σ ij *

σ

d

σ ij

ij

Α ijb

Α ijc c

k _b

k _c b

Figure1:Thegraphialmethodtoonstrutthesupportfuntionforaonvexstrengthdomain. Thelinearized

strength domainbounds thesupport funtion externallyand isrepresentedby fourhyper-planes (twodashed

andtwosolidlines).

Itisfoundonvenientforwhatfollowstoapproximateexternallythestrengthdomainbound-

ary witha series of

n

hyper-planes inthe stress spae. Eahplane bounds ahalf-spaedened by

A

_a

: σ − k

a

≤ 0 , a = 1, ..., n ,

⁽⁸⁾

in whih

A

_a ^and

k

a ^are ^the ^normal ^(symmetri seond-order tensor) to the hyper-plane and the referene stress (ohesion-like) for the

a

^th ^plane, respetively. Suh an approximation is presented in Figure 1 with four hyper-planes, two dashed and two solid lines, the latter two

labeled

b

^and

c

^. Ît îs âlsoônvenient ⁱⁿ^what ^follows ^toîntrodue ^the ^new ^variables

s

a ^(slak

variable)whihdenethe distanebetween thestress pointand theboundaryofthe linearized

where

λ ˆ

a ^are ^the non-negative virtual deformation omponents. In the example of Figure 1, the

λ ˆ

a âssoiated ^to ^the ^dashed ^lines âre ^zero ând ^the ônly ^stritly ^positive ^salars âre ^related

to the planes

b

^and

c

^. ^Furthermore, the support funtion of the linearized riterion has the followingproperties

π

L

(ˆ d) =

n

X

a=1

A

_a

ˆ λ

a

: σ

^L^∗

=

n

X

a=1

λ ˆ

a

k

a

≥ π(ˆ d) ,

⁽¹¹⁾

the seond equality being a onsequene of

σ

^L^∗ belongingto eah ativated hyper-plane (non zero

λ ˆ

a⁾ ând ônâount ôf ^(8), ^whihîs ^then ânêquality.

The onept of support funtion is now used to derive the upper bound to the loading

salar

α

^. ^The înternal ^work ^dened ⁱⁿ ⁽²⁾ îs ^bounded ^by âbove^with

P

^int

( ˆ U) ≤

Z

Ω

π

L

(ˆ d)dV ,

⁽¹²⁾

so that the theorem of virtual powerprovides

α Z

∂Ω_T

T

^o

· UdS ˆ ≤ Z

Ω

π

L

(ˆ d)dV − Z

Ω

ρg · UdV , ˆ ∀ U ˆ KA .

⁽¹³⁾

The right-hand side providesthe upper bound

α

U^, ^after^propernormalizationin the left-hand side. The upper bound theorem, referred here as the maximum strength theorem, is thus

summarized asthe minimizationproblemwith respet to the veloity elds

minimize

α

U

= Z

Ω

n X

T

^o

· UdS ˆ = 1 ,

λ ˆ

a

≥ 0 ∀ x ∈ Ω ,

U ˆ ∈ S

^u

= { U ˆ | U ˆ = 0 ∀ x ∈ ∂ Ω

u

} .

(14)

2.2 Spatial disretization and interpolation of the veloity eld

The spatial disretization and the interpolation of the veloity eld as well as of the virtual

deformationomponents

λ ˆ

a ^are ^now^introdued.

Thedomainofinterest

Ω

^isapproximatedbythedomain

Ω

^h^where^the^boundary^orresponds

to a series of straight segments or planar surfaes, as illustrated for the 2D ase in Figure 2a.

The rest of this setion presents the 2D element, the generalization to 3D is postponed to

Appendix B. The interior of

Ω

^h ^is partitioned in

q

^six-noded ^triangles ⁽

q = 11

ⁱⁿ ^Figure ^2b).

Note that the mid-side nodes are at the same distane from the two nodes at the adjaent

3

U ˆ

₆

] ,

⁽¹⁵⁾