An interdisciplinary approach towards improved understanding of soil deformation during compaction

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Copyright

An interdisciplinary approach towards improved

understanding of soil deformation during compaction

Thomas Keller, Mathieu Lamandé, S. Peth, M. Berli, Jean-Yves Delenne, W.

Baumgarten, W. Rabbel, Farhang Radjai, Jean Rajchenbach, A. P. S.

Selvadurai, et al.

To cite this version:

Thomas Keller, Mathieu Lamandé, S. Peth, M. Berli, Jean-Yves Delenne, et al.. An interdisciplinary

approach towards improved understanding of soil deformation during compaction. Soil and Tillage

Research, Elsevier, 2013, 128, pp.61-80. �10.1016/j.still.2012.10.004�. �hal-00843268�

Review

An

interdisciplinary

approach

towards

improved

understanding

of

soil

deformation

during

compaction

T.

Keller

a,b,1,

*

,

M.

Lamande´

c,1

,

S.

Peth

d,e

,

M.

Berli

f

,

J.-Y.

Delenne

g

,

W.

Baumgarten

d

,

W.

Rabbel

h

_,

_F.

_Radjaı¨

g

_,

_J.

_Rajchenbach

i

_,

_A.P.S.

_Selvadurai

j

_,

_D.

_Or

k

a

AgroscopeReckenholz-Ta¨nikonResearchStationART,DepartmentofNaturalResourcesandAgriculture,Reckenholzstrasse191,CH-8046Zu¨rich,Switzerland

b

SwedishUniversityofAgriculturalSciences,DepartmentofSoil&Environment,Box7014,SE-75007Uppsala,Sweden

c

AarhusUniversity,DepartmentofAgroecology&Environment,ResearchCentreFoulum,P.O.Box50,DK-8830Tjele,Denmark

d_{Christian-Albrechts-UniversityKiel,InstituteforPlantNutritionandSoilScience,Hermann-Rodewaldstrasse2,D-24118Kiel,Germany} e

UniversityofKassel,Faculty11,OrganicAgriculturalSciences,GroupofSoilScience,Nordbahnhofstrasse1a,D-37213Witzenhausen,Germany

f

DesertResearchInstitute,DivisionofHydrologicSciences,755E.FlamingoRoad,LasVegas,NV,USA

g

Universite´ Montpellier2,PhysicsandMechanicsofGranularMedia,Cc048,F-34095MontpellierCedex5,France

h

Christian-Albrechts-UniversityKiel,InstituteofGeosciences,Otto-Hahn-Platz1,D-24118Kiel,Germany

i

Universite´ deNice,LaboratoiredePhysiquedelaMatie`reCondense´e,28AvenueValrose,F-06108NiceCedex2,France

j_{McGillUniversity,DepartmentofCivilEngineeringandAppliedMechanics,817SherbrookeStreetWest,Montre´al,QC,H3A2K6,Canada} k_{SwissFederalInstituteofTechnology,SoilandTerrestrialEnvironmentalPhysics,Universita¨tstrasse16,CH-8092Zu¨rich,Switzerland}

Contents

1. Introduction... 62

2. Deformationofporousmedia:theory,approachesandapplicationsofdifferentresearchfields... 63

2.1. Soilphysicsandsoilmechanics... 63

2.1.1. Soilphysicsandsoilmechanics–similarsubject,differentapproaches ... 63

Keywords: Soilcompaction Continuummechanics Granularmedia Seismicmethods Modelling

X-raycomputedtomography

Soilcompactionnotonlyreducesavailableporevolumeinwhichfluidsarestored,butitaltersthe arrangementofsoilconstituentsandporegeometry,therebyadverselyimpactingfluidtransportanda rangeofsoilecologicalfunctions.Quantitativeunderstandingofstresstransmissionanddeformation processesinarablesoilsremainslimited.Yetsuchknowledgeisessentialforbetterpredictionsofeffects ofsoilmanagementpracticessuchasagriculturalfieldtrafficonsoilfunctioning.Conceptsandtheory usedinagriculturalsoilmechanics(soilcompactionandsoiltillage)areoftenadoptedfromconventional soil mechanics (e.g. foundation engineering). However, in contrast with standard geotechnical applications,undesired stressesappliedbyagriculturaltyres/tracksarehighlydynamicandlastfor veryshorttimes.Moreover,arablesoilsaretypicallyunsaturatedandcontainimportantsecondary structures(e.g.aggregates),factorsimportantforaffectingtheirsoilmechanicalbehaviour.Mechanical processesinporousmediaarenotonlyofconcerninsoilmechanics,butalsoinotherfieldsincluding geophysicsandgranularmaterialscience.Despitesimilarityofbasicmechanicalprocesses,theoretical frameworks often differ and reflect disciplinary focus. We review concepts from different but complementary fields concerned with porous media mechanics and highlight opportunities for synergisticadvancesinunderstandingdeformationandcompactionofarablesoils.Wehighlightthe important roleoftechnologicaladvances innon-destructive measurementmethods atpore(X-ray tomography)andsoilprofile(seismic)scalesthatnotonlyoffernewinsightsintosoilarchitectureand enable visualization of soil deformation, but are becoming instrumental in thedevelopment and validationofnewsoilcompactionmodels.Theintegrationofconceptsunderlyingdynamicprocesses that modify soil pore spaces and bulk properties will improve the understanding of how soil managementaffectvitalsoilmechanical,hydraulicandecologicalfunctionssupportingplantgrowth.

* Correspondingauthor.Tel.:+41443777605;fax:+41443777201. E-mailaddresses:Thomas.Keller@slu.se,thomas.keller@art.admin.ch(T.Keller).

1

2.1.2. Mechanicsofagriculturalandforestsoils... 63

2.1.3. Soilrheology... 63

2.2. Geomechanics... 64

2.2.1. Mechanicalbehaviourofanisotropicelasto-plasticsaturatedmaterial... 65

2.2.2. Poroelasto-plasticbehaviourofaone-dimensionalcolumn... 65

2.3. Geophysics... 66

2.3.1. Electricalconductivity... 67

2.3.2. Electricalpermittivity... 67

2.3.3. Seismicmethods... 68

2.4. Physicsofgranularmedia ... 68

3. Modellingapproaches... 69

3.1. Analyticalsolutionsforstresstransmission ... 69

3.1.1. Influenceofgeomaterialinhomogeneityonstresstransmission... 70

3.2. Thefiniteelementmethod... 70

3.2.1. EssentialingredientsoftheFEM... 70

3.2.2. PotentialandlimitationsofmodellingsoilcompactionwithFEM... 71

3.3. Thediscreteelementmethod... 71

4. Non-destructivemeasurementtechniquesforsoilstructureanddeformation ... 72

4.1. Computedtomography ... 72

4.1.1. Background ... 72

4.1.2. PotentialandlimitationsofCTandmCTinsoilcompactionresearch... 73

4.2. Scanningelectronmicroscopy... 73

4.3. Seismicmethods... 74

5. Synthesis:potentialsandchallenges ... 74

5.1. Scale-dependentsoilstructuralorganizationandhowitinfluencessoilstrength,stresstransmissionandsoildeformation... 74

5.2. Mechanicaldeformationasatime(rate)-dependentprocess... 75

5.3. Visualizationofsoilstructureandsoildeformation... 76

5.4. Linkingseismicmeasurementstosoilmechanicalproperties... 76

6. Conclusions... 77

Acknowledgements... 77

References... 77

1. Introduction

Soil compaction(i.e.reductionof soilporosity)isoneof the mainthreatstosustainingsoilqualityinEurope(COM,2006).A rangeof importantecologicalfunctionsis affectedwhensoil is compacted(e.g.van Ouwerkerkand Soane,1995; Alaouiet al., 2011):compactionreducessaturatedhydraulicconductivityand thus triggers surface runoff and soil erosion by water; it may induce preferential flow in macropores, and hence facilitates transportofcolloid-adsorbednutrientsand pesticidestodeeper horizonsandwaterbodies;compactionreducessoilaeration,and hence reduces root growth and induces loss of nitrogen and production of greenhouse gases through denitrification by anaerobicprocesses.Consequently,soilcompactionisoneofthe causesofa number of environmental and agronomicproblems (flooding,erosion,leachingofchemicalstowaterbodies,cropyield losses)resultinginsignificanteconomicdamagetothesocietyand agriculture.

Our understanding ofdeformationprocessesinarable soilis stilllimited.Onereasonmightbethatinsoilcompactionresearch, themajorfocushasbeenontheagronomic(and,morerecently, environmental)impacts of compaction, rather than on thesoil deformation process itself. That is, the externally applied mechanical stress(e.g. by agricultural machinery)is related to (physical)soilfunctionsorcropresponse.Whilesuchstudiesare undoubtedlyvaluable,theydonotdirectlyaddtotheknowledgeof the soil compaction process itself. In order to understand the impactsofsoilcompactiononsoilfunctions(reaction),knowledge ofthesoildeformationprocess(cause)isneeded.

Soildeformationistheresponseofasoiltoanappliedstress, which could be either mechanical or hydraulic. Stress and deformationarecoupledprocesses:soildeformationisafunction ofsoilstressandsoilstrength,andthepropagationofstressinsoil isafunctionofsoilstrengthandsoildeformation.

Abettermechanisticunderstandingofstresstransmissionin structuredsoilandthedeformationbehaviourofunsaturatedsoil will improve models for prediction of soil compaction. Such modelsareneededtodevelop strategies andguidelinesfor the prevention of soil compaction. Furthermore, improved under-standingof the soildeformation processes willpromote better predictionsoftheimpactofsoilmanagementpracticesonphysical soilfunctionsandtheireffectonprocesseslistedabove.

Conceptsandtheoryusedinagriculturalsoilmechanics(soil compactionandsoiltillage)aregenerallyadoptedfrom conven-tionalsoilmechanics(e.g.foundationengineering).Examplesare analytical solutions for stresspropagation in soil based on the worksofBoussinesq(1885)andFro¨hlich(1934),soilcompressive strengthcharacterizedbytheprecompressionstressthatisbased onCasagrande(1936),ormodelsforpredictingdraughtforceof tillageimplements that arebased on early studiesof Coulomb (1776)andRankine(1858).

Nevertheless, agricultural soilmechanics differs from geo-technical applications in a range of aspects, most notably:(i) stressesappliedbyrunningtyres,tracks,andtillageimplements are dynamic and the loading time is very short (1s); (ii) agriculturalsoilistypicallyunsaturated,whichmakes interac-tionsbetweenhydraulicsandmechanicsveryimportant,butalso complex(Hornetal.,1998;RichardsandPeth,2009);and(iii) arablesoilisstructuredwithcompoundparticlesofdifferentsizes andshapesandanetworkofvoidswithvariousmorphologiesand orientation (e.g. biopores, shear fractures, desiccation cracks). Consequently, models for calculating stress transmission and deformation in arable soil that are based on foundation engineeringconceptssufferfrominsufficientknowledgeofsoil structure(fabric)andsoilmoistureeffectsonstresstransmission, andpoorcharacterizationofsoilmechanicalpropertiesrelevant for short-termdynamic loading occurringonagriculturalsoils (KellerandLamande´,2010).

Stresstransmissionanddeformationprocessesinporousmedia arenotonlyatopicinsoilmechanicsandgeomechanics,butalsoin other research fields including geophysics, granular material science, and snow/avalanche research. Although dealing with similar processes, the theoretical frameworks used are often research-fieldspecific.Webelievethatacombinationofdifferent approaches could advance our understanding of deformation processesinarablesoils.AnInternationalExploratoryWorkshop heldduringtheautumnof2010attheAgroscopeResearchStation ARTin Zu¨rich(Switzerland)broughttogetherscientists dealing withporousmediamechanicsfromdifferentperspectivesranging fromclassicalsoilphysicsincludingsoilmechanicsto geomecha-nics,geophysics,andphysicsofgranularmixtures.Theaimofthe workshop was to introduce and jointly discuss theoretical frameworksandmodellingapproachesappliedindifferentfields to porous media mechanics, and to explore possible new approachestoquantitativedescriptionofsoilcompaction.

Theprimaryobjectivesofthispaperwereto:(i)reviewtheory, modellingapproachesandnon-destructivemeasurement techni-quesappliedindifferentresearchfieldsthatdealwithdeformation ofporousmedia,and(ii)delineatenewapproachesandpathways forimprovingthetheoreticalandexperimentalbasisformodern soilcompactionresearch.

2. Deformationofporousmedia:theory,approachesand applicationsofdifferentresearchfields

Several disciplines (such as soil mechanics, geotechnics, geophysics,granularmaterialscience,etc.)dealwithdeformation processesinporousmedia.However,thetheoreticalapproaches that describe the physical nature of deformation and the frameworkstosolveapplicationsdifferbetweendisciplines.This can beattributed todifferences in material properties, loading characteristicsand boundaryconditions oftherelevant specific processes.Nevertheless,someofthedistinctionsmaysimplybe duetohistoricalreasons.

2.1. Soilphysicsandsoilmechanics

2.1.1. Soilphysicsandsoilmechanics–similarsubject,different approaches

Inoneoftheearliesttextbooksonsoilphysics,Baver(1940)

definessoilphysicalpropertiesas:‘‘themechanicalbehaviourof thesoilmass’’.Aroundthesametime,Terzaghi(1943)definessoil mechanics as: ‘‘the application of the laws of mechanics and hydraulicstoengineeringproblemsdealingwithsedimentsand otherunconsolidatedaccumulationsofsolidparticlesproducedby themechanicaland chemicaldisintegration ofrocks’’. Although thetwodefinitionsaddresscloselyrelatedtopics,soilphysicsand mechanicsevolvedparallelwithlittleinteractionsforagoodpart ofthe20thcentury.Whathappened?

Soilphysicswasdevelopedwithinthedisciplinesofagronomy andforestry,drivenbytheneedforlandreclamation,irrigation, and drainage as well as related topics such as groundwater recharge, salinization and flood control. Soil physics primarily focused on quantifying hydraulic properties and processes of partiallysaturatedsoils.Incontrast,startingfromtheearlyworkof

Coulomb (1776), Rankine (1857) and Boussinesq (1885), soil mechanicswasaddressingfoundationandslopestabilityproblems related to fortification, road and dam construction. As an engineeringdiscipline,soilmechanicsdevelopedstronglyinthe early20thcenturywithFillunger(1913),Terzaghi(1923,1925)

and Fro¨hlich (1934). Withthe development of more advanced mechanicalmodelsforsoilsuchasCriticalStateSoilMechanics (CSSM) (Roscoe et al., 1958; Schofield and Wroth, 1968) and extendingfromsaturatedtounsaturatedsoils(e.g.Bishop,1959;

Bailey andVandenBerg, 1968; Alonsoet al.,1990), frameworks becameavailablethatexplicitlyconsiderchangesinvoidratio(or porosity)asafunctionofappliedstressaswellastheimpactof moistureconditionsonthemechanicalbehaviourofsoil.

However,itisclearthatfluidtransportthroughsoil(primary focus of soil physics) and stability and deformation processes (main focus of soil mechanics) cannot be separated fromeach other, because any deformation results in a change in fluid transport properties,whileanyhydraulicprocessinfluencesthe deformation behaviour of soil. The emerging need to solve environmentalissuesrelatedtodeformablesoils(e.g.contaminant transportthroughengineeredclayliners,geologicsequestrationof greenhouse gases, impact of vehicle traffic on soil hydraulic properties)finallybroughtsoilphysicsandsoilmechanicscloser together(Vullietetal.,2002). Theprotectionofagriculturaland forestsoilsfromcompactionisoneoftheseneeds.

2.1.2. Mechanicsofagriculturalandforestsoils

Since its beginning in the 19th century, the mechanics of agricultural and forest soils was linkedto problems related to trafficabilityandsoil–vehicleinteractions.Soilcompactiondidnot receive muchattentionbefore increasedmechanization ofpost WorldWarIIagriculturesparkedconcernsaboutdecreasingsoil fertilityduetovehicle-inducedcompaction.Althoughthefocusof attention moved away from the problem of pure trafficability towardspreservingsoilfertility,militaryneedsremainedthemain driver to study soil–vehicle interaction until the late 1960s (Bekker,1956,1969).

Todescribeandpredictsoilcompactionduetovehicletraffic, theworkbySo¨hne(1951,1953,1958)wascertainlypioneering andaddressedthemainissuesofmoderndaycompactionresearch suchasstressdistributionatthetyre–soilinterface,stresstransfer intothesoilaswellasimpactofthesestressesonthedegreeofsoil compaction.BothBekker(1956)andSo¨hne(1951,1953)employed mechanical concepts developed by Rankine (1857), Boussinesq (1885),Terzaghi(1925,1943)andFro¨hlich(1934)toderivetheir stress–strainmodelsforsoilsundervehiculartraffic.Thestrength ofthesemodelslayintheirsimplicity,whichmakesthemhugely populartothisdayforapplicationsinagricultureandforestry.

WiththeintroductionofconceptslikeCSSMandunsaturated soilmechanics,moreadvancedframeworksbecameavailableto describesoildeformation.Althoughsomeofthesemodelswere adoptedtodescribethemechanicsofagriculturalsoilsasearlyas in the1960s (e.g.Baileyand VandenBerg,1968)andhavebeen furtherdevelopedeversince(e.g.Hettiaratchi,1987;Gra¨sleetal., 1995;Kirby,1994;O’SullivanandRobertson,1996),no compre-hensive mechanical theory for agricultural and forest soils is currentlyavailable.

Onereasonisthatagriculturalandforestsoilsfeaturevarious formsofsecondarystructurethataffectthemechanicalaswellas hydraulic propertiesof thesoil (e.g.Hartgeand Sommer,1980; Horn,1993)butarehardtoquantify.Onlyrecently,advancesin imaging using X-ray (micro-) tomography have allowed non-destructivevisualizationofthesoilstructure,akeysteptowards quantifying the impact of structure on soil hydraulic and mechanical properties. Driven by theincreasing possibilities of soil structure visualization, structure-based micro-scale soil mechanicsmodelsweredeveloped,whichallowtheconsideration oftheinteractionofmechanicsandhydraulicsofthesoilatthe porescale(e.g.OrandGhezzehei,2002;Eggersetal.,2006). 2.1.3. Soilrheology

The extent of soil compaction and subsequent structural recoveryaregreatlyinfluencedbyloadinganddeformationrates that in turn are determined by soil rheological properties. In contrastwithmotionandmechanicsofrigidbodies,rheologydeals

withrelative motions ofthe parts ofa body relativelyto each another(Reiner,1960).Under theactionofaforce, abodymay deformelastically(deformationisfullyrecoverablewhentheforce isremoved),plastically(permanentdeformationevenwhen the force is removed), or the material may flow at a certain rate (continuousdeformationwithoutlimitundertheactionofaforce).

Reiner(1960) defines ‘‘rheology in thenarrower sense’’ as the sciencethatdealswiththedeformationandflowofmaterialsthat areclassifiedbetween theextremesof solid(Euclid-solid: rigid body;andHooke-solid:elasticbody)andliquid(Newtonianliquid: idealviscousliquid;andPascalianliquid:inviscidliquid).

We may distinguish macro-rheology from micro-rheology (Reiner,1960):macro-rheology considers materialsas homoge-neous, whereas micro-rheology considers material structure. Macro-rheologyis partlytreated within‘‘classical’’(continuum) soil mechanics, dealing primarily with strain and stress (e.g. stress–strain curves obtained from uniaxial compression tests, triaxialtestsorsheartests).Nevertheless,mechanicalbehaviourof soilis alsodependentupon strainand stressrates,which have receivedlittleattentioninclassicalsoilmechanics(perhapsdueto preoccupation with foundations and static structures). Micro-rheologyhastwomaintasks(Reiner,1960):itaimsateither(i) obtainingapictureofthestructureofthematerialgivingriseto measured flow curves, or (ii) at explaining the rheological behaviour of a complex material from the known rheological behaviourofitsconstituents.

Inpractice,soilrheologydealswithdynamicsoildeformation processesthatconsidersoilstructure,ratesofloadapplicationand resultingdeformation.AccordingtoOr(1996),traditionalstress– strainapproachesfailtocapturesalientfeaturesofsoilstructure dynamicsbecausetheseapproaches(i)arebasedonequilibrium statewhiledeformationinagriculturalsoilsaredynamicprocesses thatrarelyreachequilibrium,and(ii)oftendescribebulkvolume changesonlybutcannotdescribetheevolutionofthesoilstructure attheporescalethatiscrucialforflowandtransportprocesses centraltohydrologicalandagronomicapplications.

Recent developmentsin soilrheology addressthetwo main tasksofmicro-rheologyasdescribedbyReiner(1960).Theworkof Markgraf,Hornandco-workers(seeMarkgrafetal.,2006)aimsat characterizingthestructureofsoilfromobservedrheologicaltest curves.Orandco-workers(seeOrandGhezzehei,2002)developed modelsfor soil structure dynamics at the porescale based on rheologicalproperties.

Ghezzehei and Or (2000, 2001) developed models for soil aggregate deformation and coalescence due to wetting–drying cyclesaswellasduetoexternalstaticandcyclicstresses.These modelswerefurtherextendedtoincludeclosureofisolatedpores (Ghezzeheiand Or,2003;BerliandOr,2006;Berlietal.,2006), whichcanthenbeused,forexample,toinvestigatetheevolutionof hydraulicconductivityuponcompression(Eggersetal.,2006;Berli et al., 2008). The approaches are based on (i) geometrical representationsofaggregates,porespaceandliquidmenisci,(ii) energyconsiderations,and(iii)rheologicalpropertiesof unsatu-ratedsoil.

Itwasshownthatsoilundersteadystatestressbehavesasa viscoplasticmaterial that canbe described as a Bingham body (Vyalov, 1986; Ghezzehei and Or, 2000, 2001)as illustrated in

Fig.1a,whichingeneraltermscanbegivenas(Reiner,1960;Or andGhezzehei,2002): ˙

g

¼ _ðtt0

t

<

t

y y Þ/_h pl

t



t

y  (1) where

t

istheshearstress,

t

yistheyieldstress,

g

˙ isthestrainrate,

and

h

pl is the plastic viscosity. It is seen fromFig. 1 that the

rheologicalproperties

t

yand

h

pl(theinverseslopeofthestraight

lineof theBinghammodel)arefunctionsofsoilwater content: both

t

yand

h

pldecreasewithincreasingwatercontent.

Energy generally appears under three forms in rheological phenomena(Reiner,1960):kinematicenergy,elastic(potential) energy,anddissipated(thermal)energy.Elasticenergyisstored (conserved)duringdeformationandfullyrecoveredupon stress release, while energy is dissipated during viscous flow is permanent (ifelasticenergy is not permanentlyconserved but dissipates with time, this is referred to as dissipation by relaxation.)Hence,totalstraincanbedividedintoanelasticand a viscous component. The ratio of elastic to viscous strain is dependentonthestressrateandtheloadingtime,asshownby

GhezzeheiandOr(2001):theshortertheloadingrate,thelarger the elastic strain and the smaller the viscous (permanent) component of the total strain. Consequently, storage of elastic energy(and thereforeelasticstrain)is ofimportanceespecially duringdeformationundercyclicortransientstressesasoccure.g. duringthepassageofagriculturalmachinery.Thebehaviourofthe soilcanthenbedescribedas‘‘visco-elastic’’(Ghezzeheiand Or, 2001).Itisconvenienttoexpressthevisco-elasticpropertiesina complexplanesystem.Theshearstress,

t,

andstrain,

g,

canbe relatedby(GhezzeheiandOr,2001):

t

¼G

g

(2)

whereG*isthecomplexshearmodulus;similarly,

t

andstrainrate, ˙

g,

canberelatedby

t

¼

h



_g

_{˙ (3)}

where

h*

isthecomplexviscosity.TherealcomponentsofG*and

h*

indicate the storage (elastic) shear modulus, G0_{, and elastic} viscosity,

h

0_{, respectively, while the imaginary components of} G*and

h*

indicatetheloss(viscous)shearmodulus,G00_,andloss viscosity,

h

00_{,respectively(GhezzeheiandOr,2001).Therelative} proportionoftheelasticandviscouscomponentcanbeobtained fromthephaseshiftangle(alsotermedthemechanicallossangle),

d,

thatdescribesthedelayinstrain(peak)duetoanappliedstress (peak)asaresultofthetimedependenceofviscousstrain(d=0for idealelasticmaterial;

d

=

p/2

foridealviscousmaterial)( Ghezze-heiandOr,2001).

Markgrafetal.(2006)measuredthestress–strainrate relation-shipinarotationalrheometerinordertoderiveG0_,G00_{andthe‘‘loss} factor’’ tan

d

(=G00_/G0_{), as well as the linear visco-elastic (LVE)} deformationrangeandthe deformationlimitingvalue,

g

L.Three

phasesofmaterialbehaviourcanbeidentified(Markgrafetal.,2006; Markgraf,2011),asillustratedinFig.2.InphaseI(initialorplateau phase)G0_>G00_{,aquasi-elasticbehaviourcanbeobserved.The} quasi-elastic stage is defined by the LVE range and the deriving deformationlimit

g

L.Atthisstageoflowstrain,afullrecoveryof

themicrostructurecanbeassumed.InphaseII.1,astageof pre-yieldingoccursduetohigherstrain;soilparticlesarere-oriented, microstructuralstabilityisgiven,butdecreasing.Attheendofphase II.2, an intersection of G0 _{and G}00 _{indicates the yield-point. In} comparison,the intersection oftan

d

withthe tan

d

=1-linealso indicates a complete loss of stability of the soil microstructure (MarkgrafandHorn,2009).Bycalculatingtheintegralz(Fig.2b),the structuralstrength,whichincludesquasi-elasticityandpre-yielding behaviour,canbequantified.Hence,phaseIIIdefinesthefinalstage ofstructuralcollapse,G0_<G00_{;aviscouscharacterpredominates,and} substancesarecreepingorflowing.

2.2. Geomechanics

Applicationsofclassicalcontinuummechanicsrangefromthe use of elasticity theory (e.g. Davis and Selvadurai, 1996) to micromechanical approaches based on particulate mechanics

(e.g.MisraandHuang,2009;seealsoSection2.4).Thesubjectof geomechanicshasmadeimportantadvancesintermsof describ-ing the mechanics of porous geomaterials by taking into considerationtheirmulti-phasenature,especiallypertainingto thecoupledbehaviourinvolvingfluidflowthroughtheporespace, mechanicaldeformations(bothreversibleandirreversible)and thermal deformations of the separate phases (e.g. Desai and Siriwardane,1984;Selvadurai andNguyen,1995;Pietruszczak, 2010).

TheobjectiveofSection2.2istoexaminetwoproblems,which demonstrate the ability of advanced theories of continuum poromechanics(continuummechanicsthatstudiesthebehaviour offluid-saturatedporousmedia)toprovideexplanationsofsoil compaction phenomena. The compaction of an array of soil aggregates(cf.Fig.3)isamorecomplexproblemthatcannotbe obtainedconvenientlywithcontinuummechanicsasdiscussedin Sections2.1.3and5.1.

2.2.1. Mechanicalbehaviourofanisotropicelasto-plasticsaturated material

Themechanicalbehaviourofafluid-saturatedporousmedium undergoinginfinitesimalelasticstrainswasfirstdevelopedbyBiot (1941),takingintoconsiderationDarcy’slawtodescribetheflow ofthefluidthroughtheporespaceandHookeanelasticbehaviour of the porous skeleton to describe deformations. The basic constitutive equations governing the mechanical and fluid transportbehaviourofanisotropicporoelasticmediumconsisting of non-deformable solid matter, which is saturated with an

incompressiblefluid,canbewrittenintheforms

v

f¼

Ks

m

r p (4)

s

¼GðruþurÞþðlruÞIþpI (5) where

s

isthetotalstresstensor,uisthedisplacementvectorof theskeletalphase,vfreferstothevelocitiesofthefluid,pisthe

fluidpressureintheporespace,Ksisthesaturatedpermeability

matrix,Gand

l

areLame´ elasticconstantsoftheporousskeleton,

m

isthedynamicviscosityofwater,5isthegradientoperator,andI istheunitmatrix.InEq.(4)thevelocityoftheporousmatrixis neglected.

Inextendingthestudiestoincludeporoelasto-plasticityeffects, we need to select an appropriate constitutive response for saturatedclay-type materials.A varietyof constitutiverelations have been proposed in the literature. For the purpose of illustration, an elasto-plastic skeletal response of theModified Cam Clay type (e.g. Desai and Siriwardane, 1984; Davis and Selvadurai,2002)ispresentedhere.Attentionisrestrictedtoan isotropicelasto-plasticmaterialdefinedbytheyieldfunction ð ˜

s

aÞ2þq_/M2_a2_{¼0 (6)}

whereqisthevonMisesstress,aistheradiusoftheyieldsurface, ˜

s

isthemeaneffectivestress,Mistheslopeofthecriticalstateline andthesearedefinedby

q¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi3˜sij˜sij/2; p ˜

s

¼s˜kk/3   ; ˜sij¼ ˜

s

ijþ ˜

sd

ij (7) Thehardeningruleisdefinedby

˜

s

¼ ˜

sðe

pl

kkÞ¼ ˜

s

0

cþ ˜

s

cðe_kkplÞ (8) andtheincrementalplasticstrainsaredefinedbyanassociated flowruleofthetype

deiplj ¼dl

@G

@

s

˜ij ;G¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˜

s



s

˜ o c 2  ˜

s

cðe_kkplÞ 2 !2 þ q M  2 v u u t ₍₉₎

wherethehardeningruletakestheform ˜

s

c¼ ˜

s

cðe_kkplÞ¼Hðe_kkplÞ (10) and H is a positive constant. In addition to the elasto-plastic constitutiveresponsefortheporousskeleton,itisassumedthat the fluidflow through theporous skeleton remains unchanged during yield andsubsequent hardening ofthe porous skeleton. However, itis recognizedthat deformationchangesthe perme-abilityofgeomaterials (e.g.Berliet al.,2008). Furthermore,the compactionprocesscaninducestratificationsatthemacro-levelof thefabricthatcangiverisetotransverselyisotropicpropertiesin allmechanicalandtransportphenomena.

2.2.2. Poroelasto-plasticbehaviourofaone-dimensionalcolumn Forthepurposeofillustration,weconsidertheproblemofa one-dimensionalcolumn,whichissaturatedwithan incompress-ible fluidand theporous skeletoncan possessanelasto-plastic constitutiveresponsedescribedbyEqs.(6–10).Thesurfaceofthe one-dimensional column is subjected to a quasi-static normal traction or in a cyclic loading–unloading mode in a time-dependentfashionasshowninFig.4a.Theuppersurfaceofthe columnismaintainedatzeroporefluidpressure.Theconstitutive models are implemented in a general purpose computational multiphysics codeand theinitialboundary valueproblems are analysedviaafiniteelementtechnique.

Fig. 4b shows the response of the poroelasto-plastic soil skeleton model.Unlike inporoelasticity (notshown),thecyclic

Fig.1.(a)Exampleofastrainrateversusstresscurve(siltloamattwodifferent watercontents)(fromOrandGhezzehei,2002).(b)Coefficientofplasticviscosity andyieldstressasfunctionsofsoilmatricpotentialforthesamesoil.

loadingresults in surface displacement patterns that exhibit a permanent deformation. Furthermore, the number of cycles of loadingcan influence the permanent deformation(Horn et al., 2003). The monotonicapplication of a peakload resultsin the greatestpermanentstrain.Theporepressurehistoryisinfluenced by the load cycling(Fig. 4c). The negative pore fluid pressure generationisgenerallysuppressedbytheelasto-plasticityeffects inthesoilskeleton.Itisevidentthatinaninitiallyunstressed fluid-saturated porous medium, there can be negative pore fluid pressure induced for a period after the unloading process. ConsideringTerzaghi’sprincipleofeffectivestresses,thenegative porefluid pressure can have a strengthening effect of the soil skeleton,leadingtoshort-termapparentstrengthsthatarehigher

than the innate strength of the medium when all pore fluid pressureshavedissipated.

2.3. Geophysics

Geophysicalexplorationmethodsaretoolsforinvestigatingthe interior of the earth from the earth’s surface. They are non-destructive and rely on recordingand analysingphysical fields extendingorpropagatinginandaroundtheearth.Examplesare thegravityandmagneticfieldsandelectromagneticandseismic waves (‘‘earthquake waves’’). For prospecting both natural and artificiallygeneratedfieldsareuseddependingonthequestionsto besolved.Insoilsciencetheapplicationofgeophysicalmethodsis

Fig.2.Representativeresultsfromaconductedamplitudesweeptest(AST)withcontrolledsheardeformation.(a)PlotsofstorageandlossmodulusG0_(Pa);G00_(Pa)versus

deformationg;threephasesofstiffnessdegradationaregiven:PhaseI:quasi-elasticstage,PhaseII:pre-yielding(II.1),yield-point(endofII.2),PhaseIII:viscousstage, microstructuralbreakdown,yielding.(b)ResultsofanASTcanbeplottedastan(d)versusdeformationgbasedonthesamedatabase.Lossfactortand(G00_/G0_)versus

deformationg;iftand=1iscrossed,aviscouscharacterisgiven.Bycalculatingintegralz,stiffness(rigidity;elasticity)canbequantified. FromMarkgraf(2011).

Fig.3.Discretetocontinuumgenerationduringaggregatecoalescence(left)andcompaction(right).Left:coalescenceofsiltloamsoilaggregates(2–4mm)underwetting– dryingcycles;(a)drysoilaggregatesseparatedbymechanicalsieving,(b)clusterofaggregatesformedbycoalescenceduetowetting–dryingcycles,and(c)partlycrushed clusterofaggregates(fromGhezzeheiandOr,2000).Right:Fluidvelocityfieldwithin(A)verticaland(B)horizontalcrosssectionsthroughthecentreofacylindricalsampleof modellingclayspheresof5mmaveragediameterinoctahedralpackingasafunctionofverticalsamplestrainezof0,20,and40%.

attractivebecausetheyarecapableofcontinuouslymappinglarger areasintermsofphysicalsubsurfaceparameters(e.g.Allredetal., 2008).

Inordertodecidewhetherornotascientificproblemforasoil can be approached by geophysical prospecting, the following needstobeinvestigated:(i)iftheproblemunderconsiderationcan be translated into temporal or spatial changes of the physical propertiesofthesoil,and(ii)ifthesechangesexceedthedetection swell of the geophysical instruments. Examples are the clay contentthatmapsintonaturalradioactivity(e.g.FrickeandScho¨n, 1999;vanderKlosteretal.,2011),orthemoisturecontentthat influences the electrical conductivity and permittivity (e.g. Al Hagreyetal.,2004;Kirsch,2010).

Soilcompactionisconnectedwiththefollowingchangesinthe soil structure that are relevant for geophysical mapping: (i) a reductionofthe porosity(or,conversely,anincreaseindensity) leadingtoadecreaseofthevolumecontentsofsoilairandpore waterandanincreaseofthevolumefractionofclay,(ii)amechanical stiffeningofthesoilmatrix,and(iii)aspatialrearrangementofsoil grains and deformation of the soil fabric. These changes imply modificationsofsomesoilphysicalproperties,whichcanpotentially bemeasuredinthefieldwithcorrespondinggeophysicalmethods. 2.3.1. Electricalconductivity

Thecompaction-inducedchangesinthevolumeportionsofthe soil constituents have a strong influence on the electrical conductivity.Soilscontainingclayshowasignificantincreasein electrical conductivity (and conversely, decrease in electrical

resistivity) basically becausethe raisingvolume contentof the highlyconductiveclayusuallyover-compensatesthereductionof thealsoconductiveporefluid(e.g.Scho¨n,1996;Lu¨cketal.,2009). Fabricchangesthat leadtoamoreintenseconnectionbetween clayparticlesmayalsocontributetothiseffect.Bessonetal.(2004)

measuredadifferenceinelectricalresistivityof10

Vm

betweena compacted (30

Vm,

bulk density=1.59Mgm3_{) and a porous}

structure(40

Vm,

bulkdensity=1.41Mgm3_{)ofaloamysoil.A}

decrease of electrical conductivity with compaction can be expectedinpuresandonlyifthecompactionleadstoareduction inthevolumetricporewatercontent(whichisnotnecessarilythe case in the vadoze zone). The electrical conductivity can be measuredinthefieldbyDC-geoelectricalsounding, DC-geoelec-tricaltomographyandelectromagneticinductionmeasurements (EMI)(e.g.Corwin,2008).

2.3.2. Electricalpermittivity

Theelectricalpermittivityordielectricconstantisinfluenced bycompactioninasimilarwayastheelectricalconductivity.Itis basically sensitive to thegeneral contentof water in thesoil because it expresses the capability of the soil to become electrically polarized. The permittivity can be determined via the propagation velocity andreflection amplitudesof electro-magneticwavesintheradarfrequencybandbytheuseofground radarequipment.However,groundpenetratingradar(GPR)can hardly be applied in clay rich soil becauseits high electrical conductivityleadstoastrongabsorptionoftheradarwavesinthe ground(e.g.Kirsch,2010).

Fig.4.(a)Theproblemofaone-dimensionalporoelasto-plasticelementsubjectedtotime-dependentsurfaceloadings;andtheresponseoftheone-dimensional poro-elasto-plasticelementsubjectedtotime-dependentsurfaceloadings:(b)surfacedisplacement,and(c)porefluidpressures.

2.3.3. Seismicmethods

Seismic methods haveseldom been applied tosoil science research (Petersen et al., 2005) although these geophysical techniques offer an intuitive and direct approach to the characterizationofthemechanicalstateandpropertiesofsoils. For the available soil science applications, seismic methods relied primarily on the analysis of seismic wave propagation velocities.The velocity of seismic waves depends onthe soil elasticmoduli,thebulkdensity,andonthedegreeofsaturation ofsoillayers.Thevelocityofcompressionalwaves(P-waves),

v

P,

isintherangeofafewkms1_{inrock(‘‘earthquakewaves’’),and}

around750and2000ms1_{in consolidated dryandsaturated}

sand,respectively.Inunsaturatedsoil,

v

Pistypicallylowerthan

thespeedof soundwaves in air(335ms1_{)(e.g. Baker et al.,} 1999).

Onetheprimaryeffectsofcompactionisthestiffeningofthe soilandthealterationoftheelasticmoduli.Thebulkdensityisalso affected,but relativeto changes inthe soilelasticmoduli, this effecttendstobeminor.Soilcompaction(i.e.anincreaseinbulk density)thusresultsinanincreaseinseismicwavepropagation velocity.Uyanik(2010)reportedvaluesfor

v

Pof176and269ms1

fora clayeysandwitha bulkdensityof 1.41and1.62Mgm3_,

respectively.Inthefield,depthprofilesofseismicwavevelocity canbemeasuredwithavarietyofdifferentmethodsusedforthe exploration of deep targets, such as aquifers or hydrocarbon reservoirs.

Geophysical mapping offer excellent means for gathering informationonsoilchangesinspaceandtimewheretheemphasis ison‘‘change’’.Theabsolutevaluesofgeophysicalsoilproperties usuallydonottranslatedirectlyintocompositionalandstructural soilproperties.Thegeophysical measurementsneed calibration andvalidationbyclassicalsoilanalysisinordertoaccountforthe localfieldconditions,meaningthatpedo-transferfunctionshave tobesetup.Therefore,a combinedapplicationofmethodsfrom geophysicsand soilscienceappearsasapromisingapproach to investigatesoilcompaction.

2.4. Physicsofgranularmedia

Soiliscomposedofmineralandorganicprimaryandcompound particlesofdifferentshapesandsizes.Thegranularnatureofthe soilleadstoa verylargesurfacearea.Theinter-particlesurface areasarethelocationofphysicalinteractionsofcontact,friction, lubrication,capillary action or cementingcohesion.The macro-scopicbehaviourofsoilsemergesfromthecollectiveinter-particle interactions.

Theclassicalapproach(Section 2.2)is basedon phenomeno-logical relationships between macroscopic parameters, while a

granular approach starts from the particle properties and interactions at the grain scale with the aim of deducing the homogenizedbehaviourofthematerial.Amajoradvantageofthe latterapproachistheabilitytoaccountfortheinternalstructureat variousscalesandtopredicttherheologicalbehaviourfromthe localphysicsundercomplex loading(Fig.5).A number ofnew insightsintothestressandstraininhomogeneitiesofsoilshave been obtained using the granular approach. For example, the complexityofstresstransmissionthroughforcechainshasbeen extensivelystudiedandshowntobeorganizedintwowell-defined forcenetworks: the weak network (forces below average)that doesnotcontributetoshearstress,andthestrongnetworkthat supportsalltheshearstress(Fig.6)(Radjaietal.,1996,1998).Such a structureshows boththeweaknessesofa soilin responseto external excitations and the real mechanisms of its stability. Another example, which is not accessible from continuum modelling, is the micromechanical origin of shear strength. It wasrecentlyshownthat theinternalangleof frictiondoesnot depend on the particle size distribution because the largest

Fig.5.Evolutionofsolidfraction,r,ofapackingofgrainsasafunctionofshearstrain,e.(a)Comparisonbetweeninitiallylooseanddensesamples,wherethesamecritical solidfractionisreached;and(b)effectoflowamplitudecyclicdeformationontheevolutionofsolidfraction.

FromRadjaiandRoux(2004).

Fig.6.Exampleofheterogeneousforcetransmissioninagranularsamplesubmitted toaconfiningpressure.Themagnitudeofthecontactforcesisproportionaltothe widthoftheinter-centreconnectingsegments.

particles inthe bulk material capture mostof the strong force chains.TheCoulombcohesionis,however,foundtobedependent ontheparticlesizedistribution(Voivretetal.,2009).

Experimentalstudiesofmodelgranularmaterialshavebeen helpful in highlighting many phenomena inherent to their discrete structure. Besides the classical testing tools of soil mechanics,newexperimentalapproacheshavebeendeveloped for the measurement of individual particle displacements, contact forces, etc. For example, 2D photoelastic grains have beenused to visualizethe heterogeneity of localstressesin a granular material under compressive loading (Da Silva and Rajchenbach, 2000; Majmudar and Behringer, 2005). The methods based on imaging and Particle Image Velocimetry (PIV)havebeenusedtostudytheelasticpropertiesofgranular assemblies at the grain scale (El Hadji Bouya et al., 2011). Experimental devices for studying stacks of polydisperse cylinders were performed in order to analyse the shear behaviour in line with the movements and the localization phenomenasuchasshearbands(Calvettietal., 1997).Forthe studyofsaturatedmedia,iso-indexfluorescentliquidandgrains have been used to track the motion of particles inside the volumewithalasersheet(Tsaietal.,2003).Today,newimaging techniquessuchascomputedtomography(cf.Section4.1)open interesting perspectives for the granular approach and its applicationstosoils(Desrueset al., 2006).

Inparalleltoexperimentalmethods,discreteelementmethods (DEM) have been developed over the last 30 years for the computationoflargepackingsofparticleswithincreasinglymore complexinteractions(cf.Section3.3).Theavailabilityofpowerful computers has made it possible to investigate the mechanical behaviourofdiscretemodelsofsoils.Themechanicalbehaviourof granularmaterialsisinfluencedbyvariouslocalpropertiessuchas particlesizedistribution(Voivretetal.,2007),frictionalslidingand rolling(Estradaetal.,2008),particleshapes(Aze´maetal.,2009), andcohesion(Delenneetal.,2004),andtheseaspectshavebecome majorfieldsofresearch.

Statisticalhomogenizationconceptsarealsousedinassociation withDEMtoanalysethedisorderedlocalstructuresandtoupscale thelocalbehaviour tothemacroscopicscale (Rouxand Radjai, 2001).Simplerelationshipshavebeenshownbetweenthe micro-scaleparametersofcohesion,thestrengthpropertiesatthe macro-scale (internal friction angle, dilatancy angle, and Coulomb cohesion)andtheparametersdescribingtheshapeand polydis-persityoftheparticles.Theinternalangleoffrictionisfoundto resultfromdifferentcontributionsofgeometrical,kinematic(due todilation)andfrictionalorigins(Taboadaetal.,2006).

Granularphysicsisnowamaturefieldwiththepotentialfor applicabilitytoavarietyofcomplexsystemssuchaslandslidesand soil–rootinteractions(Taboadaetal.,2005;StaronandLajeunesse, 2009).Thisbroadscopecallsforamultidisciplinaryapproachusing aninnovative combination ofconcepts, experimental tools and numericalmethods.

3. Modellingapproaches

Modelsusedtosimulatethestresstransmissionand deforma-tionin geomaterialscanbedividedintotwo groups:analytical modelsandnumericalmodels.Analyticalmodelsassume elemen-tarymateriallaws(e.g.linearelasticity).Soilcompactionmodels basedon analytical approachessolve compaction in twosteps: first,stresstransfer,andsecond,deformationasafunctionofthe calculated stress. Numericalmodels solve thedeformation and stress simultaneously by satisfying the equilibrium equation. Numericalmodelsincludefiniteelementanddistinct(ordiscrete) elementcodes.Thesemethodsrequirecomputingpower,andwere

firstdevelopedinthe1940’s(finiteelementmethod)and1970’s (distinctelementmethod).

3.1. Analyticalsolutionsforstresstransmission

Stresstransmissionis atopicthat hasbeenofscientific and engineeringinterestsincethetimeofArchimedesandculminated intheconceptthatwasproposedbyAugustinLouisCauchy(see e.g.Truesdell,1961;DavisandSelvadurai,1996).Cauchysimply assumedthatthenatureofthereactiveforcesgeneratedduring transmissionofexternallyappliedloadstotheinteriorofasolidis nodifferentfromthetractionsthatareappliedtotheboundaryofa stressedsolid.SpecificvaluesforthetractionsTactingonaplane locatedattheinteriorofthebodyarerelatedtothestressstate

s

at apointthroughtherelationship

T¼

sn

(11)

wherenistheoutwardunitnormaltotheplane.Thisabilityto defineameasureforthestateofstresswithinthebodywithout considerationofthematerialpropertiesofthegeomaterialorits fabric was a major accomplishment in the development of mechanicsofmaterials.

The spatial transmission of stresses within the geologic materialrequiresknowledgeofthemannerinwhichtheinternal structure of thegeomaterialresponds totheexternallyapplied loadingthroughstrains.Thereisnouniquemodelofstress–strain behaviour that is universally applicable to geomaterials. The earliest application of analytical concepts dates back to the classicalworksofRankineandCoulombthatwerelargelybasedon thestress distributionsin geomaterialsas theyapproachedthe stateoffailure(seeDavisandSelvadurai,2002).

Theseminalanalyticalproblemdealswiththenormalloading ofthesurfaceofanisotropicelastichalfspacebyaconcentrated forcePB,maintainingtheunloadedregiontractionfree(Fig.7).In

addition,thetractionsfarfieldfromthepointofapplicationofthe concentratednormalloadshoulddecayinsuchafashionthatthe resultantofthefarfieldtractionsshouldbalancethenormalforce appliedattheboundary.Thesolutiontothisclassicalproblemis due to Boussinesq (1885). The static boundary value problem requiresthesolutionofthegoverningEqs.(12–14)(withzerobody forcesanddynamicterms):

s

¼

leI

þ2me (12)

e

¼1 2 ruþðruÞ T h i ;e¼tre (13) rsþf¼

r

@

2 u

@t

2 (14)

whererdenotesthegradientoperator,trdenotesthetrace,fisa bodyforcevectorand

r

isthemassdensity.Sincetheconcentrated

forceactsnormaltothesurfaceofthehalfspace,thereexistsastate ofsymmetryaboutthez-axis, which allowstheproblem tobe formulatedasanaxisymmetricproblemrelatedtothecylindrical polarcoordinatesystem(r,u,z),and thedisplacementandstress fields can be expressed as follows (see e.g. Timoshenko and Goodier,1970;DavisandSelvadurai,1996):

urðr;zÞ¼ PB 4pmR rz R2 ð12nÞr Rþz  uuðr;zÞ¼0 uzðr;zÞ¼ PB 4pmR 2ð1

nÞ

þ z2 R2  (15) and

s

rrðr;zÞ¼PB 2p 3r2_z R5  ð12nÞ RðRþzÞ 

s

uuðr;zÞ¼ PBð1 2nÞ 2p 1 RðRþzÞ z R3 

s

zzðr;zÞ¼ PB 2p 3z3 R5 

s

rzðr;zÞ¼ PB 2p 3rz2 R5 

s

ru¼

s

uz¼0 (16)

where R=(x2_+y2_+z2₎½_{. For relatively simple loadings (e.g.}

circularload withuniform stressintensity),expressions canbe developed to calculate the transmission of stress within the halfspaceregion due tothesurface loading(see e.g. Davis and Selvadurai,1996).

Empirical knowledgeof stress propagation is availablefrom comprehensivestudiesreportingstressmeasurementsin undis-turbed soil profiles, including investigations of the effects of contactareaandstressdistributionatthetyre–soilinterface(see e.g.KellerandLamande´,2010)andsoilconditions(e.g.Hornetal., 2003;Trautner and Arvidsson,2003; Lamande´ and Schjønning, 2011)onstresstransfer.Therateofdecayofthestresspredictedby theclassical theoryofelasticityis foundtobeatvariance with experimental observations, in particular the vertical stress distributionswithinthegeomaterialregions.TheworkofFro¨hlich (1934)isanempiricaldevelopmentthatadjuststheformofthe vertical stress

s

zz(r,z) derived from Boussinesq’s solution by

introducinga‘‘concentrationfactor’’,n,whichallowsthealteration ofthedecaypatterntosuitanexperimentallyobservedpattern. Fro¨hlich’smodeliswidelyusedinagriculturalsoilmechanics(e.g.

Keller and Lamande´, 2010). The mathematical basis for the introductionof this conceptis lacking, in thesense that there appearstobenoformallinearsolutionoftheequationsgoverning theclassicaltheoryofelasticity(i.e.Eqs.(12–14)),thatwillyielda solutiontoBoussinesq’sproblemfortheactionofaconcentrated forcePBnormaltothesurfaceofanelastichalfspaceintheform

s

zzðr;zÞ¼ nPBzn 2pRnþ2;R¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2_þz2 p (17) A criticalmathematicalexaminationofFro¨hlich’s solutionto Boussinesq’sproblemisprovidedbySelvadurai(2012).Analytical soilcompactionmodelshavetheadvantageinthattheyaresimple touse,requirefewinputparametersandarerobust.

3.1.1. Influenceofgeomaterialinhomogeneityonstresstransmission Geomaterialsareinherentlyinhomogeneousatanyscale.The generalapproachfordealingwithgeomaterialheterogeneityisto considereffective material parameters applicableto units con-sistingofvariablemechanicalproperties.

Theinhomogeneityinthedeformabilitycharacteristicsthatis of particular importance to soil tillage mechanics and stress

transmissionisthelayeredsoilprofilethatcandevelopthrough pedo-and/orgeogeneticprocessesorduetoperiodiccompaction and tillage of the soil. This type of soil inhomogeneity is characterized by mechanical properties that are assigned to distinct strata.Thelayered elasticsystemapproachtoestimate stress transmissionhas beenextensively studiedin connection with transportation infrastructure analysis (Poulos and Davis, 1975). Analytical solutions exist for the settlement of layered systemssuchastheproblemofanelasticlayerthatrestsonarigid base (see Davis and Selvadurai, 1996); however, these are of limitedinteresttosoilscience.

3.2. Thefiniteelementmethod

Thefiniteelementmethod(FEM)isanumericalapproachfor solving differential equations describing any kind of physical phenomenainacontinuousmediumforwhichanalyticalsolutions donotexist,areavailableonlyforverysimplecases,oraretoo complex.Theoriginaldifferentialequationsdescribingaspecific problem arereplacedbya finitesetof equationswhich canbe solvednumerically.Thisrequiressubdividingthecontinuumintoa numberofelementsthatareconnectedtoeach othervianodes (spatialdiscretization).TheFEMallowsthetreatmentofcomplex geometriesofheterogeneousproblems.FEMwasemployedforthe firsttime inthe1950s intheconstructionof airplanesand the designofbridges(BearandVerruijt,1987;ZienkiewiczandTaylor, 2000).TheconceptofFEMwaslatergeneralizedand,alongwith increasingcomputerpower,becameastandardtoolinalmostall areasofengineering,physicsandenvironmentalsciences.TheFEM hasalsobeensuccessfullyappliedtoawiderangeofsoilrelated problems with geotechnical, environmental or agricultural aspects. Examples include theanalysis of waterflow, chemical and heattransport in soilsand aquifers(e.g. Bear andVerruijt, 1987; Radcliff and Simunek,2010), soil-implementinteractions (e.g.Kushwahaetal.,1993;Abu-HamdehandReeder,2003),slope stabilityandfoundations(GriffithsandLane,1999;Zdravkovicand Carter,2008;DannoandKimura,2009),andsoilcompaction(see reviewbyDe´fossezandRichard,2002).FEMhasalsobeenapplied tosmallscaledeformationprocesses,forinstance,thoseassociated withlateralrootexpansion(RichardsandGreacen,1986)andits effectonhydraulic conductivity(Aravenaetal.,2011). Today,a variety of commercial FE programs are available (multiphysics programs suchas ABAQUS (Abaqus FEA, D S Simulia, Dassault Systems) and COMSOL(COMSOL Multiphysics, COMSOL AB) or codes designed for geotechnical purposes such as PLAXIS (Brinkgreve,2002)),as wellas non-commercial FEM developed forsoilcompactionresearch(e.g.Richards,1992;Gysietal.,2000). 3.2.1. EssentialingredientsoftheFEM

AcomprehensivetreatmentoftheFEMincludingitsapplication fornon-linearproblemsforplasticandviscoplasticbehaviouris providedinZienkiewiczandTaylor(1994a).Here,onlythebasic stepsandnotationsoftheFEMwillbeoutlined.

The first step in FE analysis is the discretization of the continuumintoafinitenumberofelements(in2D:trianglesor quadrilaterals),whichareconnectedvianodesbuildingaFEmesh as a substitute for the continuum. Each element consists of a numberofnodes.Nodesareassignednodalparametervalues(e.g. pressure, velocity, displacement, etc.). In deformation analysis mostFEprogramsuseadisplacementformulationwherethestrain within anelement is defined in terms of nodal displacements. Initialstrainsplusstrainsassociatedwiththenodaldisplacements definethestateofstresswithinanelementandonitsboundaries basedontheconstitutivepropertiesofthematerial(Zienkiewicz andTaylor,1994b).Thedisplacements uatanypointwithinan element(e)canbeapproximatedbyacolumnvectoruˆ ,whichis

obtained from prescribed functions of position Ni and nodal

displacementsuiforaparticularelement:

ˆ u¼½N1;N2;N3 u1 u2 u3 8 < : 9 = ; e ¼Nue ₍₁₈₎

whereNisthematrixofinterpolationfunctions(alsoreferredtoas shapefunction)andue_{isthedisplacementvectorofanelement.}

Withthenodaldisplacementsdeterminingthedistortion ofthe elementthespatialderivativeofthedisplacementfunctionsyields thestrainsatanypoint:

e

¼Bue ₍₁₉₎

whereBistheelementstrainmatrixand

e

thestrainvectorofany pointinanelement.Assuminglinearelasticmaterialbehaviour, thestressesareobtainedfromthestrainsandthecorresponding elasticmaterialproperties:

s

¼Dðe

e

0Þþ

s

0 (20)

whereDistheelasticstress(materialconstitutive)matrix,

e

0isthe

initialstrainand

s

0istheinitialresidualstress.Usingtheprinciple

ofvirtualwork,anodalforce–displacementrelationshipcanfinally beestablished(ZienkiewiczandTaylor,1994b):

qe_¼Ke ue_þfe

(21) whereqe_{isthenodalforcevector,K}e_{istheelasticstiffnessmatrix,}

andferepresentsforcesduetobodyforces,initialstrain,andinitial stress,respectively.

Fora two-dimensionalplanestresselasticproblemthetotal strain

e

inEq.(19)atanypointwithinanelementcanbedefinedby three strain components

e

x=

@

u/

@

x,

e

y=

@

v/

@

y, and

g

=(

@

u/

@

y+

@

v/

@

x)withu,

v

beingthedisplacementcomponentsparallel tothex-andy-axis,respectively.Assuminganisotropicmaterial and considering the constitutive stress–strain relation, the correspondingstresscomponentsaregivenby:

s

x¼ E 1

n

2ðexþ

ne

yÞ (22)

s

y¼ E 1

n

2ðeyþ

ne

xÞ (23)

t

¼ E 2ð1þ

nÞ

g

¼Gg (24)

whereEistheelasticmodulus,

n

isthePoisson’sratioandGthe shear modulus. Note that the elasticmoduli E,G and K (bulk modulus)arerelatedaccordingto:

G¼ E

2ð1þ

nÞ

(25a)

K¼ E

3ð12nÞ (25b)

Thematerialparameters(i.e.K,GandE)usedinthematerial constitutivematrixDandtheelasticstiffnessmatrixK,areoften highlynon-linearandhystereticforsoils,i.e.mechanicalmaterial propertiesarestress-statedependent.Therefore,itisnotpossible to provide general values for the above properties, but for unsaturated arable soil, E is typically of the order of 10MPa, andatypicalvalueforthePoisson’sratioofsoilis

n

=0.3.Usingthe modified‘‘hyperbolic’’formulaeofNelson (1970), stress-depen-dentmodulihavebeenformulatedbyRichards(1992):

K¼k1

s

nmþk2hmþk0 (26) G¼ðg1

s

mpþg2h r Þ 1

t

t

f  q  þg0 (27)

where

s

misthepreviousmaximumofthemeannormalstress,his

thesoilwatersuction,

t

istheshearstressand

t

ftheyieldstress

(e.g.Mohr–Coulombfailurecriterion).Theotherparametersk0...

k2,g0...g2,n,m,p,r,andqareempiricalmaterialconstants,which

can be determinedfrom mechanical soil test data (Peth et al., 2006).

3.2.2. PotentialandlimitationsofmodellingsoilcompactionwithFEM TheFEMhasbeensuccessfullyappliedtoarangeofsoiland environmentalproblems.Itdoes,however,requireagreatereffort to thoroughly characterize the mechanical material properties basedon soiltesting inordertoderiveparametersforthe non-linearstress–strainrelationshipencounteredinsoils.Traditionally, standard soiltestswithstaticloadingor constantdisplacement rates, such as direct shear tests, confined consolidation tests (oedometer)ortriaxialtests,areemployedtoobtainshear(G)and bulk moduli(K)forFEMsimulations.However, thestresspaths followedinsuch‘‘static’’soiltestsdonotrealisticallyrepresentthe short-term dynamic loading situation in the field (Keller and Lamande´,2010;Pethetal.,2010a).

AmongtheprimarylimitationsofFEMarenumericalchallenges to consider large strain rates, and limited representation of openingandclosureofporesandcracks.

Duringdynamicsoilloadingporewaterpressureevolutionhas a significant effecton thedeformation process.A FE-model for simulating the interaction between hydraulic and mechanical processesinsoilshasbeenpresentedbyGra¨sleetal.(1995),based on previous work of Richards (1992). Despite an advanced understandingofthetheoreticalprinciplesgoverningdeformation behaviourofunsaturatedsoils(FredlundandRahardjo,1993)and theavailabilityofFEMforsolvingcomplexcoupledprocesses,the majorobstacleinusingsuchmodelsseemstobethelackofan adequatequantitativedescriptionoftherelevantsoilpropertiesby meansofsuitableinsituandlaboratorytests.

TheFEMisbasedonacontinuumapproach,butforstructured soils in particular, this assumption can be problematic, for example to simulate the effect of deformation on transport functions(seealsoSection2.1.3).Instructuredsoilsnumerous macro-scalediscontinuitiesarepresent(cracks,largebiopores, etc.), which are of paramount importance for gas and water movement.Thishasleadtoatwo-domainordual-permeability approachinmodellingwaterandsolutetransportbyassuming multiplecontinua(e.g.Gerke,2006).However,therepresentation of the effective local (structural) geometry of the conducting pores,andtheeffectsofsoilstructuredynamicsontheporespace geometry andhence transport processes remains a challenge. RecentadvancesinvolvemultiscaleFEapproaches(e.g.Heand Ren,2009)andnon-invasivecharacterizationoftheporenetwork dynamics(cf.Section4.1).

3.3. Thediscreteelementmethod

The discrete element approach comprises a large class of numericalmethodsforthesimulationofacollectionofparticles interactingthroughfrictionalcontactsandcollisions(Radjai and Dubois, 2011). Different variants of these discrete element methods (DEM) have been developed for the simulation of granular packings and slow deformations (Radjai and Roux, 2002),dense granularflows(daCruz etal.,2005),andgranular gases(McNamaraandYoung,1992).TwomajorvariantsofDEM areMolecularDynamics(MD),alsosimplycalledDEMfollowing Cundall(CundallandStrack, 1979),and ContactDynamics(CD) initiatedbyMoreauandJean(Moreau,1994;Jean,1999).

Thediscreteelementapproachisbasedontheintegrationofthe equationsofmotionsimultaneouslyforallparticles,describedas rigidelements, byconsideringthecontactforcesas wellas the externalforcesactingontheparticles(RadjaiandDubois,2011). Given the boundary conditions, the mechanical response of a collectionofparticlestoexternalloadingleadstorelativeparticle motionsconstrainedbystericexclusionsinadensestateand/orby inelasticcollisionsinalooseordilutestate(Moreau,1994).

ThemaindifferencebetweenMDandCDisinthenumerical modellingoffrictionalcontact.InMDthecontactforceisdescribed asafunctionofalocalstraindefinedfromparticlepositions.These forcesareregularfunctions(twicedifferentiable)oftherelative positions

d

ofparticlesandtheirtimederivatives.Hence,thetime discretizationmust beadapted tocorrectlydescribe forcelaws acrossthecontact.IncontrasttotheMDmethod,whereanexplicit schemeisusedtosolvetheequationsofmotionbyintroducingstiff repulsivepotentialsandviscousdampingbetweenparticles,the CDmethodisbasedonanimplicitschemeinvolvinganiterative Gauss–Seidel algorithm yielding, simultaneously, the contact forcesand particledisplacements at theendof eachtime step. Thisiterativeprocessisdefinedsothatitsatisfiesthekinematic constraints related to mutual exclusions of particles and the Coulomb friction law. For this reason, the CD method is unconditionally stable and does not require elastic repulsive potentialbetweenparticles.Thisallowsforlargertimestepsthan inMD,inparticularinthelimitofhighlystiffparticlesorverylow confiningstresses.

AtypicalDEMcomputationloopproceedsinthreesteps: (i) Determinationofthelistofnearestneighboursandcontacts

betweenparticlesfromtheircurrentpositions;

(ii) Integrationoftheequationsofmotionofallparticlesovera singletimesteptocalculatethecontactforcesandvelocities; and

(iii) Updatingtheparticlepositions.

Three levels of complexity may be discerned in DEM algorithms. The first (basic) level is based on a minimalistic descriptionofthematerial withsphericalgrains interactingvia frictional contacts. This elementary algorithm is already rich enoughtoreproduceemergentbehavioursofgranularmaterials under complex loading.The second level involves an enriched descriptionofparticleshapesandsizedistributionsaswellastheir interactionsviavariousadhesionforces.Thethirdleveldealswith advancedprocessesandinteractionssuchasparticle fragmenta-tion, hydrodynamic interactions in the presence of a fluid or particlesembeddedinasolidmatrix.Coupledcontinuum-discrete algorithmsareoftenrequiredtotreatsuchproblems.Forexample, themethodsrelyingonsubparticlemeshingoftheparticles,such astheLatticeElementMethod(LEM),maybeusedtointroduce particle fragmentation (Topin et al., 2007). Another example concerns fluid–grain interactions in which the continuum me-chanics of fluids must be combined with discrete mechanics. Different approaches have been proposed to compute the evolutionofthefluidphasebetweenparticles,suchastheLattice Boltzmannmethod(Mansouri etal.,2009),or DirectNumerical Simulation(Wachs,2009).

During the last 30 years, many examples of experimental validation have established DEM as an efficient and effective methodologytoaddressphysicalandengineeringissuesdealing withawidevarietyofmaterialsincludingsoils,powderprocesses, gravitationalflowsandinstabilities,etc.Manystudiesreportedin theliteratureconcernhomogenousboundaryconditionswiththe aimofinvestigatingthegrain-scaleoriginsofeffectiveproperties suchasshearstrength,dilatancy,creep,etc.,andthe microstruc-ture.Three-dimensionalsimulationsoflargesystemsofparticles

withrealisticshapesandsizedistributionsarestill computation-allyintensiveandrequirespecialoptimizationstosavesimulation time; the scope of large-scale simulations for application to complexsystemsandprocessesisbroadandpromising.

Challenges in the application of DEM in soil compaction research include (see also Shmulevich, 2010): (i) physical interpretationanddeterminationoftheconstitutive micromecha-nicalmodelparametersthatrepresentthemechanicalproperties ofsoil,(ii)modellingofthereal(spanof)particlesizeandshape, and (iii)modellingoftheprocess ofbreakageandformation of compoundparticles(fragments,aggregates).

4. Non-destructivemeasurementtechniquesforsoilstructure anddeformation

Non-destructive(non-invasive)measuringtechniquessuchas computedtomographyofferanexcellentmeansfor characteriza-tionandquantificationofsoilstructurefromtheaggregatetothe soilcorescale(e.g.Tulleretal.,2010).Geophysicalmethodsare alsonon-invasive,andallowforinsitu measurementsoflarger scale structures at the field scale. The propagation of seismic waves,forexample,isafunctionofthestructure,themechanical propertiesandthemoisturestatus(distribution ofsolids,water and air)of thegeomaterial. Knowledgeof structural properties acrossscaleswilladvancethecharacterizationofheterogeneities incontinuum-basedapproaches,andhenceimprovethe predict-abilityofsoilcompactionprocessesincomplexporousmediasuch as soils. Furthermore, the non-invasive techniques provide an insightintothedynamicsofsoilstructure.

4.1. Computedtomography 4.1.1. Background

Computedtomography(CT)hasbeenatoolinsoilresearchsince thepioneeringworkofPetrovicetal.(1982).Analysingsoilandglass beadsampleswithamedicalscanner,theydeterminedbulkdensity variation at a spatial resolution of 1.251.252mm3_{, and}

highlightedthemethodas‘‘potentiallypromisingtoolforresearch intheareasofcompaction,soilmanagement,andcultivation’’.One ofthefirstpapersstudyingsoilcompactionusingCTwaspublished byVazetal.(1989),whodetectedthincompactedlayersatthe ploughingdepth.Otherworkwheresoilcompaction/deformation wasinvestigatedwithCT(including

g-ray

andX-raybasedsystems) treatedeffectsofcompactiononearthwormburrowing(Langmaack etal.,1999;Je´gouetal.,2002),differenttillagemanagementsystems (Wiermannetal.,2000;GantzerandAnderson,2002;Pedrottietal., 2005;Mikitaetal.,2009;Kimetal.,2010),trafficked sewage-sludge-treatedsoils(Piresetal.,2003),puddlinginducedcompactionin paddy-soils(Sanderetal.,2008), andsoilstructureregeneration aftercompactionbywettinganddrying(WernerandWerner,2001; Piresetal.,2005,2007).

Overthelast10years,highresolutiontubeand synchrotron-basedmicrotomographysystemshavebroughtsignificantquality improvementsintermsofacquisitiontime,resolution,andsignal tonoiseratios,allowingstudiesoflocaldeformationprocessesat themmto

mm-scale.

Examplesaretheinvestigationof compres-sionaroundroots(Aravenaetal.,2011)andearthwormburrows (Rogasik et al., 2003; Schrader et al., 2007), and the effect of hydraulicand mechanicalstresseson thedynamicsof soilpore system(Scha¨fferetal.,2008b;Pethetal.,2010b).CT-imagescanbe renderedintoa3Dvolumecontainingthespatialconfigurationof voids and solids, allowing quantitative morphological and topological characterization of the pore space (Vogel, 1997; Horgan,1998;Delerueetal.,1999;Pethetal.,2008).

X-raycomputedtomographysetupscanbeclassifiedintothree different types: (1) medical scanners (XCT or short CT), (2)

industrialtubebasedmicrotomographysystems(XCMTor

mCT)

and(3) synchrotron-basedmicrotomographysystems (SR-mCT). Thephysicalprincipleunderlyingallsystemsisthatanincident X-ray beam with a given energy is absorbed by the material componentsoftheradiatedobject(seee.g.Peth,2011).

4.1.2. PotentialandlimitationsofCTand

mCT

insoilcompaction research

ThepotentialofX-rayCTand

mCT

insoilcompactionresearchis that it is non-invasive, permitting the mapping of the spatial arrangementofsoilconstituentsandporeswithoutdisturbingthe sample structure. This is an important improvement over traditionalmethods,wherestructuralpropertiesofsoils(e.g.pore sizedistribution) are measured indirectlyfrom waterretention functions or directly by thin section microscopy, which risk undesiredmodificationofstructuralfeaturesduringmeasurement (byhydraulicstresses)orsamplepreparation(dehydrationusing acetoneandimpregnationwithresin).

Hydraulicstressesmaycausesignificantsoildeformationby inducingswellingandshrinkagethroughcapillaryforces.The non-rigidity of the pore space architecture results in unreliable estimatesof e.g. water retentionfunctions.This problem could be overcome by repeatedly scanning the same sample under different hydraulic stress states, and relating the morphology/ topologyofaporenetworktoitstransportfunctionsatvarious moisture conditions in order to derive hydraulic properties/ functionsasafunctionofsoilmatricpotential.Similarapproaches couldbeusedforquantifyingtheimpactofmechanicalstresseson soilfunctions,andultimatelyalsoforthecouplingofmechanical andhydraulicstresstoaccountfortheinfluenceofwateronthe mechanicalbehaviourofsoil.

Systematicnon-invasivestudiesofchangesinporespacedueto soil deformation by mechanical and hydraulic stresses could inspire/facilitate the development and validation of new approaches in soil compaction modelling (see also Section 5). Forexample,CTor

mCT

imageanalysistechniqueswereusedby

Scha¨fferetal.(2008a,b)tostudythestabilityanddeformationof artificiallygeneratedmacroporesunderuniaxialcompression,by

Kremeretal.(2002)toinvestigatethedeformationofmacropores undervaryingloadandsoilmoistureconditions,andbyBerlietal. (2008) to develop and test an analytical model describing the evolutionof aggregatecontactsizeswithcompressionasa key determinantofunsaturatedwaterflow.Moderntechniquessuch as digital image correlation have successfully been used in combinationwith

mCT

analysistolocalizesoilstrain(Pethetal., 2010b) demonstrating the highly heterogeneous and complex

deformationfieldinstructuredsoilsandtheinfluenceofwateron thedeformationprocess(Fig.8).

Despitethegreatpotentialof(m)CTasanon-invasivetoolfor investigatingsoildeformationprocesses,resolutionislimitedand doesnotcoverthecompleterangeofporesizesencounteredinsoils. ModernCT-scannersareabletoprovideavoxelresolutionofslightly betterthan1:1000ofthesamplediameter.Hence,thesmallerthe structureofinterestthesmallerthesamplehastobe,whichconflicts withthescale-dependentspatialheterogeneityofsoils.Coveringthe hierarchy of pore spaces fromlarge inter-aggregate cracks and continuous biopores tosmall intra-aggregate micropores is not easilydoneandrequirestheextractionandanalysisofsubsamples. Anotherchallengeistheextrapolationofporescaleeffectsofsoil deformationandrelatedchangesinsoilfunctionsmeasuredonsoil corestothesoilmanagementlevel(pedonandfieldscale). 4.2. Scanningelectronmicroscopy

Thefirstcommercialscanningelectronmicroscope(SEM)was constructedbyCambridgeScientificInstrumentsin1965,basedon theconceptofVladimirZworykin(ZworykinandRamberg,1941; Zworykinetal.,1942).Todaythereareseveraltypesofinstruments thatworkaccordingtothisprinciple.AdetaileddescriptionofSEM, microanalysisandtheirapplicationisgiveninGoldstein(2003).

SEMisamethodcommonlyusedinmaterialscience,medical research,andgeosciencesincludingappliedclayscienceandsoil micromorphology. SEMs deliver important data about particle properties,i.e.structure,shapeandsurfaceproperties,andallowa visualidentificationofphysicochemicalcompoundssuchas(clay) minerals, (hydr)oxides, organic matter, and fungal hyphae (MarkgrafandHorn,2006;Markgrafetal.,2012).

In soil science, SEM analysis is performed on (oven-dried) samplessuchasmicroaggregates(<250

mm)

orisolatedfeatures (e.g. oxides or clay minerals). SEM analyses require good conductivity, which is achieved by gold–palladium coating (sputtering) of thesample underhighvacuumconditions. SEM micrographs are obtained at 15KeV at a working distance of 15mm.Monochromephotographsaretakenwithareflexcamera, whichisintegratedinaSEMworkstationasanexternalunit.The latestgenerationofSEMchamberssupportfastandeasy3Dstereo imaging via PC-controlled beam deflection with optional 3D surfacemodellingandanalysis.Furthermore,theyprovideachoice ofdifferentchambersandstageswithahighresolutionanalytical geometryforEnergyDispersiveX-ray(EDX)analysis.Ifnohigh vacuumconditionsareapplicable,environmentalscanning elec-tronmicroscopy(ESEM)isasuitablemethod.

Fig.8.StrainlocalizationduringthedeformationofastructuredLoesssample(1=5cm,height=4cm)in(a)airdry,and(b)wet(6kPamatricsuction)conditionafter applyingincreasingverticalsurfaceloads.e33anddV/Vrefertotheverticalandvolumetricstrain,respectively.Negativevaluesindicateaxialshortening/compression,while

positivevaluesindicateaxialextension/loosening.FurtherdetailsontheexperimentalsetupanddataanalysiscanbefoundinPethet al.(2010b). ModifiedafterPethetal.(2012).