COMSOL BLOGs.docx

COMSOL BLOGs.docx

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COMSOLBLOGs

1YieldSurfacesandPlasticFlowRulesinGeomechanics

FabioBocchi |October16,2014

Inordertoensuresafegeotechnicalbuildingmethods,specificapplicationsrequirecertainfoundationsandstructurereinforcements.Testsarequiteexpensivetocarryout,sosimulationcanbereallyusefulandevenessential.Manynumericalmodelshavebeendevelopedtogiveadeepinsightintosoilbehavior.Here,weintroducethemostwidespreadmodelsforsoilsavailableinCOMSOLMultiphysicsandanalyzeatunnelexcavationexample.

QuickNoteonGeotechnicalEngineering

Thereseemstobeageneraltrendwhenitcomestoconstruction.Offshorestructuresareconstructedindeeperanddeeperwaters;buildingsareconstructedincreasinglyclosetoeachother;offshorewindturbinesaredevelopedindeepwatersfarawayfromthecoasts,whicharelikelytoexperienceextremeloadingconditions.Therefore,inrecentdecades,geotechnicalengineershavedevelopednumericalsimulationstocopewiththisconstructiontrendandensuresafebuildingmethods.

“ParisMetroconstruction03300288-3″.LicensedunderPublicdomainviaWikimediaCommons.

PlasticityandGeomaterials

Materialsforwhichstrainsorstressesarenotreleaseduponunloadingaresaidtobehave plastically.Severalmaterialsbehaveinsuchamanner,includingmetals,soils,rocks,andconcrete,forexample.Thesegiverisetoanelasticbehavioruptoacertainlevelofstress,the yield stress,atwhichplasticdeformationstartstooccur.

Theelastic-plasticbehaviorispath-dependentandthestressdependsonthehistoryofdeformation.Therefore,theplasticitymodelsareusuallywrittenconnectingthe rates ofstress,ratherthanstress,andtheplasticstrain.Themostwidespreadandwell-knownplasticitymodelthroughouttheindustryisbasedonthevonMisesyieldsurfaceforwhichplasticflowisnotalteredbypressure.Therefore,theyieldconditionandtheplasticflowareonlybasedonthedeviatoricpartofthestresstensor.

However,thismodelisnolongervalidforsoilmaterialssincefrictionalanddilatationeffectsneedtobetakenintoaccount.Let’sseehowthiscanbeworkedoutandbrieflyexplainthedifferentsoilplasticitymodelsavailableintheCOMSOLMultiphysics®simulationsoftware.

PlasticityofSoilsandRocks

Formaterialssuchassoilsandrocks,thefrictionalanddilatationaleffectscannotbeneglected.Thiswholeclassofmaterialsiswellknowntobesensitivetopressure,leadingtodifferenttensileandcompressivebehaviors.ThevonMisesmodelpresentedaboveisthusnotsuitableforthesetypesofmaterials.Instead,yieldfunctionshavebeenworkedouttotakeintoaccountthebehavioroffrictionalmaterials.

Let’sillustratethefrictionalbehaviorandplasticflowforthesematerialsbyconsideringtheblockshownhere:

Theblockisloadedasshownbyanormalload 

 andatangentialload 

.Assumingthattheblockrestsonasurfacewithacoefficientofstaticfriction 

accordingtoCoulomb’slaw,themaximumforcethattheblockcanwithstandbeforeslidingisgivenby 

.Therefore,theonsetofslidingoccurswhenthefollowingconditionisreached:

（1）

Thedirectionofslidingishorizontal.Fortangentialloadssuchas 

theblockwillnotslide,butassoonas 

theblockwillslideinthedirectionoftheappliedload 

.The Mohr-Coulomb criterion—thefirstsoilplasticitymodeleverdeveloped—isageneralizationofthisapproachtocontinuousmaterialsandamultiaxialstateofstress.Itisdefinedsuchthatyieldingandevenruptureoccurwhenacriticalconditionthatcombinestheshearstressandthemeannormalstressisreachedonanyplane.Thisconditionisstatedasbelow:

（2）

Here, 

 istheshearstress, 

 isthenormalstress, 

 isthecohesionrepresentingtheshearstrengthunderzeronormalstress,and 

 isthecoefficientofinternalfrictioncomingfromthewell-knownCoulombmodeloffriction.ThisequationrepresentstwostraightlinesintheMohrplane.AstateofstressissafeifallthreeMohr’scirclesliebetweenthoselines,whileitisacriticalstate（onsetofyielding）ifoneofthethreecirclesistangenttothelines.

Mohr-Coulombyieldbehavior.TheMohrcirclesarebasedontheprincipalstresses 

and 

.Asyoucansee,oneofthecirclesistangentialtotheyieldsurface,andsotheonsetofyieldingisoccurring.

Accordingtothefigureabove,thestressstateisgivenby 

 and 

.TheyieldcriterionandEquation2canthereforebere-writteninageneralizedformasfollows:

（3）

ItcanevenbeseenasaparticularcaseofamoregeneralfamilyofcriteriabasedonCoulombfrictionandwrittenbyequationsbasedon invariantsofthestresstensor:

（4）

RepresentationoftheMohr-Coulombyieldfunction.

The Mohr-Coulomb criteriondefinesahexagonalpyramidinthespaceofprincipalstresses,whichmakesitstraightforwardforthiscriteriontobetreatedanalytically.But,theconstitutiveequationsaredifficulttohandlefromanumericalpointofviewbecauseofthesharpcorners（forinstance,thenormalofthisyieldsurfaceisundefinedatthecorners）.

Inordertoavoidtheissueassociatedwiththesharpcorners,anotheryieldcriterionofthisfamily,the Drucker-Prager yieldcriterion,hasbeendevelopedbymodifyingthevonMisesyieldcriteriontotakeintoaccounttheCoulombfriction,i.e.,incorporateahydrostaticpressuredependency:

（5）

Thisrepresentsasmoothcircularconeintheplaneofprincipalstress,ratherthanahexagonalpyramid.Ifthecoefficients 

 and 

 arechosensuchthattheymatchthecoefficientsinthe Mohr-Coulomb criterion,asfollows:

（6）

the Drucker-Prager yieldsurfacepassesthroughtheinnerorouterapexesoftheMohr-Coulombpyramid,dependingonwhetherthesymbol 

 ispositiveornegative.The plasticflow directionistakenfromtheso-called“plasticpotential”,whichcanbeeitherthesame,associativeplasticity,ordifferent,non-associativeplasticity,thantheonsetofyielding（theyieldfunction）.Manydifferentnon-associativeflowrulescanbedeveloped.

UsinganassociativelawfortheDrucker-Pragermodelleadsthevolumetricplasticflowtobenonzero.Therefore,thereisachangeinvolumeundercompression.However,thisiscontradictorytothebehaviorofmanysoilmaterials,particularlygranularmaterials.Instead,anon-associativeflowrulecanbeusedsuchthattheplasticbehaviorisisochoric（volumepreserving）—amuchbetterreflectionoftheplasticbehaviorofgranularmaterials.

RepresentationoftheDrucker-Prageryieldfunction.

Non-AssociativeLawforSoilPlasticityinCOMSOLMultiphysics

Next,Iwillshowyouhowtouseanon-associativelawforsoilplasticityinCOMSOLMultiphysics.Non-associativeplasticlawscanbeusedregardlessoftheplasticitymodelusedinthesoftware.

Ifyou’reusingtheMohr-Coulombmodel,therearebasicallytwodifferentapproachestohandlingnon-associativeplasticity.TheplasticpotentialcaneitherbetakenfromtheDrucker-Pragermodelorbethesameasthe Mohr-Coulomb yieldfunctionbutwithadifferentslopewithrespecttothehydrostaticaxis,i.e.,theangleoffrictionisreplacedbythe dilatationangle（seescreenshotbelow）.

Moreover,whenusingtheDrucker-Pragermatchedtoa Mohr-Coulombcriterion,itiseasytoadaptthedilatationangletomatchwiththenon-associativelawthatyouwanttouse.Forinstance,thenon-associativelawpresentedabovecanbeworkedoutbytakingthedilatationanglenull.

Lastbutnotleast,ausefulfeaturecalled ellipticcap hasbeendevelopedtoavoidunphysicalbehaviorofthematerialbeyondacertainlevelofpressure.Indeed,real-lifematerialcannotwithstandinfinitepressureandstilldeformelastically.Therefore,tocopewiththis,wecanusetheellipticcapfeatureavailableinCOMSOLMultiphysics.

SoilPlasticityfeaturesettingswindow.

Let’strytoputintopracticeeverythingwe’velearnedsofarbyanalyzingtheexampleofatunnelexcavation.Thiswillalsobeanopportunitytofigureoutwhattheeffectsofthedifferentfeatureswementionedaboveare.

ExampleofaTunnelExcavation

Thesimulationofatunnelexcavationprocessisespeciallyimportantinpredictingthenecessaryreinforcementsthattheworkersneedtousetoavoidthecollapseoftheconstruction.

Thefollowingmodelaimstosimulatethesoilbehaviorduringatunnelexcavation.Thesurfacesettlement（i.e.,theverticaldisplacementalongthefreegroundsurface）andtheplasticregionarecomputedandcomparedbetweenthedifferentsoilmodelsusedtocarryoutthissimulation.Thegeometrywe’lluseispresentedinthefigurebelow.Tomakeourmodelrealistic,infiniteelementshavebeenusedtoenlargethesoildomain,whilekeepingthecomputationaldomainsmallenoughtogetthesolutioninarelativelyshorttime.

Thegeometryconsistsofasoillayerthatis100metersdeepand100meterswideplus20metersofinfiniteelements.Atunnel10metersindiameterisplaced10metersawayfromthesymmetryaxisand20metersbelowthesurface.

Firstofall,weneedtoaddthein-situstressesinthesoilbeforetheexcavationofthetunnel.Then,wecancomputetheelastoplasticbehavioroncethesoilcorrespondingtothetunnelisremoved.Thein-situstressesmustbeincorporatedinthissecondstep.ThisisfairlystraightforwardtosetupinCOMSOLMultiphysics.

Wecanbeginbyaddingastationarystepwherethein-situstresseswillbecomputed.Then,inasecondstepbutstillwithinthesamestudy,weaddasoilplasticityfeature.Finally,wecomputethesolution.Inordertogetthepre-stressesincorporatedintothesecondstep,weshouldaddanInitialStressandStr