Pro -Structur

Finite Element Modelling of Damage Processes in Ice-Structure Interaction

by

@Jing Xiao,a.Eng.

A thesis submitt edtothe Schoolof Graduate Studie s inpartia l l'ulfillmen tortherequire ments forthe degree of

Master of Engineering

Faculty of Engineering and Applied Science MemorialUniversityof Newfoundlan d

August, 1991

St.John's Ne wfoundla nd Canada

1+1

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Abstract

Two fieldprograms were carriedout onthe Hobson's Choice Ice Islandill April.

\989and May,1990using differentsizes of sphericaland11M indenters.Several coresof multiyearfieldice were recoveredfromthe iccislandandtranspor tedto MemorialUniversityto conduct compressive testsin thelaboratory.Both constant st rain-ratetests and constant stresstests were performed withuniaxial stress(0

investigate the deformationof multiyearice andcalibrate thematerialconst ant'!

fortheoretical modelling.

Thesphericalindentatio ntestsaremodelled using anaxisymmetrical finite element model. Theiceda mage process isrelatedtothe growingnetwork of micro- cracks andthe icecreep process isalso enhanced bythe existenceof cracks.The damage model is developedinFORTRANcode andimplementedas a user subrou- tinein the ABAQUSfiniteelementanalysisprogram. The analysis resultsshow thatmostCoftheice damage is closetothe contactsurface.ami that themaximum damage occurs atthe edge of the interface whereshear stress is concentrated. This isin agreementwith thetestresults.Themodelalso providesgood resultsonthe total load versus time history.

Acknowledgements

lamverygrateful to thefacultyof EngineeringandApplied Science that.

inmanyways, providedassistanceinthe completionofthis project. Without their activecollaborationand cooperationthis project wouldhaveremain ed an unfulfilleddrea m.Inparticular,Iamgreatly indebtedtoDr.IanJordean,my thesis superv isor,whonot onlyhelped at everystage ofthedevelopment,reading throughalloftheearlydrafts of the manuscripttimeand timeagain,butwhoalso had aprofound influenceonmy academictrainingthrough scholarship,andwhose constr uctivecommentsandmeticulouscriticism have been a constantsource of inspirationandencouragement duringthewrit ingofthisthesis.

Specialtha nksgo toDr.RichardMcKenn a whonotonlyprovided me with muchguida nceas wellas assist anceduringthe processof completing this project , but whoalso introduced me to manyrelevantreadings andmaterials.

Iwishtoespeciall y thankmy friends and colle agues.SanjaySingh,RickMeaney and Shawn Kenny fortheir consta ntencourageme nt,aswell as sincereconcern s, valuablesuggestionsandcrit icismsthroughout my research. Ialsowantto extend mygratit ude toMr. Barr y Stone.whose concretehelp andcontribution inmany of my laboratory experiments aregratefullyacknowledged.

Finally,aspecialword of thanks goes tomy wife, YangMeng,forher helpin designing theformat.Iwantto acknowledgegratefullyher keeneyefornoting discrepancies,andher patienceintypingandmaking changesandaddit ions.

iii

Contents

Listof Figures ListofTables Nomenclature 1 Int ro d u ct ion andScop e 2 Literatu reReview

2.1 Elasticity of Ice....

2.2 CreepofIce... .. .. ... ... 2.3 CrackingofIce

2.4 Damage~fechanicsand Damageof Ice 3 Exp u im ent a tion

3.1 U:1iaxialTests.

3.1.1 SpecimcmPreparation. . 3.1.2 Test Setu p ... 3.1.3 ResultsandDiscussion. • . 3.2 SphericalIndention Experiments•.

3.2.1 Experim entSetu p.... . . . . 3.2.2 Resu lts and Discussion.

" Cons t it u tive Modelling 4.1 Ice~Iodel . .. 4.2 Dilatati onofIce. 4.3 DamageEvolution Law.. 4.4 Crack Nucleat ion 4.5 CreepEnhancement ..

4.6 FiniteElementImplementatio n and~IodelVerification

ix

"

17

2 .

2;

2;

2' 29 3·\

3!) :)6 47

..,

03 .\6 .'59 6i 62

s Fini teElement Modellin g ofSpher icalIndentation Tests 72 .'}.1 TheElasticSolutions ofSpherica lIndentation . 72

.5.2 Finite Element Mcde!.. . 73

.')..1 ModellingtheSphericalIndentation Tests 74

.;.·1 EquivalentViscosityofDamagedIce 77

6 Conclusions 87

Refere nces 90

Ap p en dix:The Relat ion shipbet we en von Mises Stres sand Eq uiv-

ale nt Strain 96

:!·I '.tl

List of Figures

2.1 BurgersBod)";Eand JI areclastic modu lus;10,1viscositycoefficient,

respectively. . .

2.2 Applied slresshis~or}'and strain response ofconst antstress lest ..

2.3 Stress -sl raincurve ofcons tant strainret e lest ..

2.4 The threephasesofcreep lest: (I) prima ry;(II)secondary; (III) tertiar y.

2.5 Idealiza tion of ice sheet: planview of (a)phot ogra phi crepr esent ation showing progressof damage. and{h ] idcaliaaticnintothree zones [Jordaan andTimes.1985I.

2.6 TheIailure modesobservedin theiceshee tiudcntetion tests; (a) Crushi ngwith rad ial andcircumferenti alcracking:(b~Crushingwith radialcracking [Ttmec.1986). .... . . •.

2.7 Possiblemode of puh'crizationaheadofspherical indenter[Jord ean

and~lcKenna.198da). . ... . ... :!.'j

2.8 Abodywith anoverall section areaof.40 amiadamaged areaof_'1. :!5 J.I Phctcgrep h oficesamples: (a)beforelest;(h)aftertest, :18

3.2 Testsetup for measuring theaxialstrain. :!!)

3.3 Test setupfor measuringboth axialandlate ralstrain. :!!) 3.-1Stress-straincurveforuniaxialtestNo.I. -10 3.5 Stress-stra incurve [or uniaxialtestNo.2. ·10 3.6 Appliedstress histor yandstrain responseofcreep test No.3. ·11 3.7 Applied stresshistor yand strain responseofcreep test No. ·1. ·11 3.8 Applied st resshistoryandstrainresponse ofcreeptest No..J. ,12 3.9 Appliedstresshistoryand strain response ofcreeptestNo.6. ,12 3.10Ap pliedst reuhistory andstrainresponse ofcreeptest No.7. ,1:1 3.11 Stress-straincurveforuniaxialtestXo. S. ·13 3.\2Stress-straincurves of constant strainrate lesls on bothfreshwater

andmultiyeariceat st rainrateofto-~.'I-I. ·14 3.13Strainresponses ofintactand predamagedice forstress at0.75~IPa. 14

3.).1Cree p tests: (a )stressversuselasticstra in for intactand predam- agedice;(b)stressversustota lcreepstru mat20seconds after the applicationof the load for intactandpredamagedice. 4.5 :I.J!iSche ma tico(theact uator inde nter system [Frcder king eral., 1990a) . 45 :l.W Srhe rn ati c of thespherica l indente rand the locationsoflocal pres-

sure cells (Frcdcrki ugetal..1990a ). ·16

:J.17 Thetotal loadVt: fS UStim e recordsoftest:(al NRC1;(b) NRC2

(Fred e rkingctal.,1990a ), 46

4,] Stressversusstraincurvesfor thefourtest slistedinTable4.2 (Dor-

ris,1989). 67

·L:! St res sversusstra incurvesforthe four testslisted in Table4,2 (Dc r-

ris. 1!J89). 68

1.:1 Ratioofbulkst rai n rate to equivalent strainretevers us ratioofbulk stress to von Misesstress. Data point swerederived from Dorris (1989) Fig.4.2 and abest fitlineis shown(~Ic Kennaet al.1990). 69

·1..1 Eflectivc moduli:drycircular cracks;Gis the shearmodulus;Eis the Young'smodulus:1\isthe bulk modulusandIJis thePoisson's ratio (Budiansky and O'Connell, 1976 )... .. ... . . ... 70 4.5 Effectivemoduli versusthecrack-den sityparameterfor ind icat ed

value sofstressratio,v

=

0.3,wherejJisthe frictioncoefficient;P isthenorm..1stress to thecracksurface andTis the shear stress acrossthe crack surface[HoriiandNe rna t -Nasse r,1983). 70

·1.6 Com p arisonofcree ptestresults with modelresult s. il 1.7 Com p arison ofco nstantst rain ratetest resul tswith model results. 71

·

').1 finiteelementmesh for sphericalindentationtests. 79

·

5.2 Comparisonof theoreticalelasticsolutionof sphericalind e nt a t ion withfiniteelement solution . ... . .•. .. . . 79

·').3 Four differentmeshsizeshave been testedfor compa risons. 80 .').·1 Thetotal load\'5.limehistories:mode l resu ltsand testresults;(a)

lest NRCI;(b) test NRC2. 81

5.5 Distributionsof damage.Os,intheice adjacent totheindenter (..reaA, see Fig. 5.1).ON=0 forcontour level1;DN=0.5for conto u rlevel 10.theincrem entofONfor eachcontourlevel is 0.05.

(a) test NRC1; (b )lest NRC2.. 82

.5.6 Distri butionsof maximumprincipalstress between1MPa.and 2

~tPanearthe contactIeee:(a)testNRC1; (b) testNRC2. 83 5.7 Calcu latedpress u redistribu tionsfor tes t NRClon the contactface. 84

vii

5.$ Modelledequivale nt viscosity ofin.'as ,1function'Ifappliedshear

stress and damage state, $.1

5,9 Progression of equivalent viscosity adjacent to theccnt ectface: (a )

test:-;"RC1 ; (b)tes t :'iRC2, Sf)

.

".10Distri bu tion sof equiva lentviscositybetween '20~lPa.~[con t our level I)and 100 0~IPa.s(cont ourlevelIu)adja centto thecont actface :

(a)lest NRCI: (b)testNRC2. 86

viii

List of Tables

:1.1 Listofthe Test Series .. :1.2 List of StrainComponents.. .t.J ParametersUsedintheDamage~todel. -1.2 TestConditionsfor eachlee Sample (Dorris, 1989)

ix

30 33 66 66

Nomenclature

e, o

a N

stresstensor(~IPal strai nte nsor elastic modu lus (:-"I Pa) elasticstiffnessi nKelvinunit ela.stics tiffnessinMaxwell unit shea r modulus(~'IPa) bulkmodul us (:-.tPa) Poisson' sratio time(s)

viscositycoefficient(~I Pa's) viscosityinKelvin unit viscosit yinMaxwell unit forthorder co mp lia ncetensor elast icstraincomponents delayedelastic st raincomponents secondarycreepstraincomponents elasticstrain deviator delayedelastic strain deviato r seconda ry creep stra indeviator equivalentstrain volumetricstrain volumetric str ess(MPa) von Misesstres s (MPa) stress deviator(MPa)

criticalstressfor crack nucleatio n(MPa) average grainsize(m)

one halfofthe cracklengt h(m) crack density

ON damage parameter

/'10 damage constant

/1 creepenhanceme nt parameter

ig

^{delayedclasticstrainreferencerate}

'6 secondary creepstrain referencerat e

Ii'J delta function

m: damageexponent

creep expo nent

xi

Chapter 1

Introduction a n d Scope

Sinceearly19805,increased exploration for ccnvenricnalcncrgy sourcesin the arct ic and near-arcticoffshoreareas,has focusedattentionon theengineer in g problems of designingand buildinglarge structuresinice-covered waters.Theinte rac tion of ice withmarine structureshas been recognized asamajor design considera tion.

Therehas been a greatdeal of effortinrecent years,bothexperimenta llyand theoretically,to dete rminearelia ble approach for the estimatio nof both glob al andlocal ice loads all offshorestructures.These iceforces exertedon a structure may take manydifferentmodes,such as, crushing,fracturing,spalling, buckling ,or thecombinatio nsofthem.Thesearecomplex processesand involveseveral possible factors, suchas,loading rat e,ice type,tempe ratureandthe shapeofinterface.

Medium scale ice indent ationtests were conductedontheHobson's ChoiceIce Island in April,1989(Frederking et al., 1990a, b)and in May,1990, andalso earlier in1984,atPondInlet byMobil Oil Canada [Ccctech, 1985).In 1989 eleven tests wereperform ed withthreety pes ofindente rs,rigid spherical,flexible flatand rigid flat. There were six sphericalindent.ationtests with a speed rangefrom0.3 mm/ s to90rom/soInall tests,icecrushingwas observed in front ofthe indenter and

therh.ct-ness ofthecrushedlayerwasirregular. Therewasusually lesscracking inthe centerarea dueto high confinementand moredamage at the edgeof the interface (in the presentwork. damageis relatedto thedensity of microcracks).

Maximum pressuresmeasure d atthe center were ill therangeof 10 to 20~IPa.

Pressure meltinghas alsobeen reported (Gagnon and Sinha,1991).

Uniaxialtestshave beenperformedon multiyearfieldice which wasrecovered fromthe iceisland.Thecomparison of the testresultsonboth intact and predam- agediceshowclear evidenceof anenhancement ofthecree p straindueto crack anddamage.The elasticmodulusof the ice was calculatedfrom thetestresults.

An isotropicdamage model,whichouhaes a power-lawrelation betweencracknu- cleation rate and stress,isused inthe present work.Thisrelationship isbased onratetheory. Additionally, dilatationoficeundercompressionis modelledas a functionof theratio ofvolumetricstress to theequivalentvon Mises stress. To verifythis model.direct comparisons totheuniaxialtestshave been performed.

Finiteelementanalysisprogramshave beendeveloped.to simulatethe spherical indentationtests atlower rates,withthe ice damage modelcalibrated from uni- axialtests.The modelpredictionsshow good agreement wit hthe test resultson totalload versustime historiesand pressure distributionsandprogressions.The modelleddamagedistribution and progressioncan be utilized,toacerta indegree, to characterize thelayerof crushed ice. Thecalculationof ice viscosityis a simple approach,whichis proposedtoinclude the influenceof confining pressurein further studies.

Finiteelement analysiswithdamage mechanicsis a newand uniqueapproach,in modelling icebehaviourunderboth uniaxialloading and mediumscaleindentation

testing conditions.

As outlined above, thescopeof this wc rkmdYbecategorizedasfollows:

1.Litera turereviewof recent theoriesonice mechanics.includingicemechanical anddamage models,aswe11as experimentalobs~r\'ation,onicecracking behaviour.

2.Conduct inguniaxial testsinthelaboratoryonthe multiyearicctocalibrate the materialconstants (ortheoretical modelling;descript ionof iceindenta- tiontests carriedout on Hobson 's Choice lee Island (1989 )andsome ma jor observations.

3.Constitu tive modelling oficeda ma ge process, includingcreep enhancement dueto theexistence ofcracks anddamage;finiteelement impl ement ationand modelverification.

4.Finit e elementmodellingof sphe ricalindentationtestsandcompa rison with the experimentalresults.

5.Conclusionsandreccm rnendatic n s forIurthc rstudies.

Chapter 2

Literature Review

Ice in natureis a pclycrysralline materialcomposedof a largenumberofsing le crysta ls usua lly in differentorientations. Michel(1979)providedadeta ildescript ion cf the structure andclassification of ice (see also Cammaer t and Muggeri dge (1988) onsea ice). Typ ically, therearctwo mainkindsof polycrystallineicefoun d in

L.Gra nular ice , which is randomlyorientedpoiycrystallineice, canbe found inicefeat ur es; such as. glaciers.icebergs,lake ice andseaice.Thegrain sizeis classifiedas fineto medium. ln thelaboratory,this typeof ice canbe obtainedby freezing wat er seeded wit h fullmould ofran domly orientedfineice cryst als,an d it canbe treate d as ast at isticallyisotr opicmaterial.

2. Column arice is formedwiththe grains growing paralleltothe hea tflowand wit hc-axisperpend iculartothe colu mn lengt h.Thistypeofice isreferred to as $2 iccwhich ca n befou nd inlake, riverandarcticsea ice.The mechanical properties ofcolumnarice are orthotropic,or more usually, transverse isot ropic.

Iccis characterizedas a viscoelasticmaterialwith itsdeformation respo nse dependent upon theloadingrate;it is alsovery brittleunderhigh loadi ngrate.A

spring-dashpotmodel,called Burger'smodel. is oftenusee! forpolycrystalllneicc.

Thismodel is a combinationofitKelvinunit anda~Iax\\'cll'mit,as showninFig.

2.1.

Themechan icalpropertiesof ice can be divided into twopart s\5 all(lcrson , IU88):

L Continuumbehav iou r.This includes elasticaut! ductile rrccp deformation.

which canbe extendedto include thc uniformly Jislrihulc(l mirrocrnckingami damageprocesses.

2. Fracture behaviour.This includescrack propagation and brittlefailure . Thecont inuu mbehavio urof granu lar ice is markedly similar10 tha tofcclum ner ice, butwithsom edifferencesdue to orthotropyoranisotropy(Sinha,1989a),

2.1 Ela sticit y of Ice

In engineeringapplicati ons, the elasticityofgranular icc is typicallytreatedas isot ropicandcanbe characterized by twoconstants.the elastic modulus,E,and Poisson 'sratio,/I. Whe na const a ntloadorstressois appliedat timetoand releasedat time!..a strai nversus timecurveas shown sche ma ticall yinFig. 2.2 isprodu ce d . Following Hoo ke's law, the elastic strainof ice is given as

(2. 1)

wher e theelasticmodulu s oficeis the stiffnessof themainspringintheMaxwell unitandhence the elast ic strain corr esponds to theddorm at ion of the mainspring (see Fig.2, 1).

Theelastic modulusandPoisson's ratioare depende ntontheice tem pera t ure andporosity.Thevariationsof£and10'ontempera tureweregivenby Sinha(1989a)

forbot hgranular and columnaricc.This work shows thattemperat ure doesnot haveastrongeffectontheseconsranrs. ThevalueofEchanges from9CPato 10.16 GPa andIJfrom

o.aoa

^to0.3605 in the temperaturerange ··jO°C to0·C.Thereare twomet hods for determiningthe elasticmodulus ofice:staticand dynamic.The couvontionalstatic tests include uniaxial compression, uniaxial tension andbeam bending.Thetwomost commontests areuniaxialcompressiontests under either const antload (or stress) orconstant displacement rate (orstrainrat e).The former testgives a strainversustime curveas showninFig.2.2, andthe elasticmodulus E=u/e.,fort=O;thelattertest givesa stress versus straincurve asshownin Fig.2.3, and theelasticmodulusE=_~,whene=O.Sothecalculatedvalue ofclast ic modulusis stronglydependenton the accuracy ofthetest. However, ice isnot purelyelastic,it creeps at allstresseswit ha time-dependentrate,so readings of theinitial tangentmodulus from a stress-stra incurvewill notbe very accurat e. Dyna mic test ing techniques areconsideredto bemore accurate sincethey minimizethetime-dependent effects.Forpolycrystallineice of lowporosit y, the clastic modulusgiven byhighfrequencydynamic measurementsisapproximately

!Jto9.5 CPain thetemperatu rerange-5DCto·10DC(Mellor,1983). Thisis astand ardrangeofvaluesofelasticmodulusforlow porositypolycryst allineice.

Thecommonlyaccepted rangefor Poisson'sratio is 0.3 to 0.33.

Theelastic modulusof sea ice has been investigatedbyMellor(1983) based onprevious research,whichshows thatelasticmodulus ofseaice variesfrom10 CPa(pure ice) toICPain theporosityrange0 to0.3.Theoreticalmodels which calculatetheelasticmodulus asa function ofbrinevolumewerealso proposedby Weeks andAssur(1967),Schwarzand Weeks(1977).

2.2 Creep of Ice

Aconstantst resscreep test011polycrys rallin e ice givesiIcc nv..ntionatere'cPcu rv e as shown in Fig.2.4. Theidealized creepcurve cnn be divided luto th ree phas es:

prim ary,secondaryand tertiarycreep. TheToleofeach individualphnseincreep deform ationhasnet beenfully understood. In general.the deform ationof icc includ es several kindsof processes or mechanism s,and the influenceofeachin d i- vid ualprocessor phasemight be maximizedor minimizeddepending onthe typeof ice, tem perat ure andloading condition.Eachofthesethreephaseseoutd domi n at e thecreepstrainunder certaincircumstances.Sin h a(19i8)de velopedaviscoelast ic const itu tive equatio n(orcolumnarice under uniaxialcompressi on.Thetolalst ra ln e is consideredas the sumof threecomponentsas shownin Fig. 2.2,i.e.

(1.:q

wheret'istb...instantaneouselasticcomponent;(Jis thedelayed elas tic compo- nent,or recover ableprimary creep.and(~isthe permanent viscous component ,or secondary creepstrai n. Sinha' smodelislimitedto the firs t twophase s ofcree p and doesnotaddress thetertia rycreepphase.

Sinha(19i8)gives anexpressionfor delayed elastic strainunder cons ta ntstress

4

~~ (' )'I 'il

t:(t ) = 7 E l-exp{-(aTl) , 12.3)

whereC1,5,b andaTareall constantsdependingonthe tem p erature and thegra in size ,d; where E

=

9.5CPa;CI

=

9, isaconstan tcorrespon d ing tothe unitgra in sized,(dl

=

0.001);"

=

I;b

=

0.34:aT

=

2.5 xIO-~3-1(T=263(j.Thedelaye d elast icstraincorresponds tothedeforma tionofthe Kelvin un itinFig. 2.1.

;\ nonlinea rdes hpctbasedKelvinunit wasproposed hyJorclaanand~lcKenna (198Sb) to model delayedelastic strain,inwhichthe viscosityPkis a function of stress(J~in thedashpot. The str aininthe Kelvin unit isthen

(2.4)

whereE kis theelast icmod ulus ofthe spring, andPkis theviscosity of the dashpot inthe Kelvinunit.Itwas assumed thatthe dashpol follows apower-lawrelation withstre ss

(2.5)

whereh'~is theviscosity parameter: n isaconstant, Usingthe equationof equi- Jibrium for thedementoftheKelvin unit,i.e.

then it is found

Ij~

=

(n - l jE" I+/lIrll' (2.6)

Substitutin gEq. (2,6)intoEq .(2..1),the delayed elast ic strain isthengivenby

(2.7)

where,w=EIJ1J~O.PItOis the viscosityat time1=O.

Theseconda ry creepst ra indescribes theeffect of theviscous flow and dido- cationmovement withinthe grains, andappearsto be independent ofthegrain size(Cole.1986), Thiscreepstr aincorresponds tothe creepdeformation inthe

dashpotof the~Iaxwellunit(sec Fig.2.1).Forpolycrystallineice under uniaxial compressionor tension,a.power-lawrelation of strain rateandstresswassuggested by Glen(1955) ofthe form

i^e

=

Au"

wheren is a consta ntandAisa.functionoftemperatureinrho form (2.8)

whereR

=

$.3HJmol-l K-l.isthe universal gasconstant:Tisthetemperat ure in degreesKelvin:Q istheactivation energy andBis a materialconstant,bothQ andBare depend entof the ice type.

In Sinha'sexpressionforcolumnarice, the creepstrain ratewas given by a similar relationship:

(2.9) where11 ::::3,andf~=l.iBxIO-~.s -'(1'=2631\) ,is theviscous strainratefor unit stress uo, (0-0

=

^{IMPa ),}

Thetertia rycreepwasconsidereddue to theeffectofmicrocracking(Gold, 1970), but itwas Iaundthatcrackingisnot essential for the occurrenceoftertiary creepinpolycrysta llineice. even duringthe rreusluonfromprimaryto terti ary creep(Mellorand Cole,1982). The real precess is notwellunderstood.Inthis study,secon dary creep includesenhancementassociated with thecurrentlevelof crack density.Thiscrack-enhanced creepis used to model thepermanentcreep strain.

10 The viscosityofseaiceisdifTerentfrom thatofpure ice,sincesea ice contai ns many airbubbles. brine POCkt'lSiW Jimpuritlcswhichcauses localstressdiscontinu- it!l!sand concentrations:hence.sea ice is softe ned and has lower viscosity(Pounder, J!J6.5),(Wang,I!}T9a, b,198 1),{wee ks and Assur,(967).Arat ionali zed creep rate expression of seaice was givenby Sanderson(1988) as

where

(2.10)

andIIiis brinevolu meor porosityof theice andVois a normalizingconstant.

Subst ituti ngEq.(2.8)into Eq.(2.10),itis foundthat

;1'=.4_ _1_ _. (1-

;;;:;;;o?

For multiaxialstress states.the elasticrespo nse ofice can bewritten as (2.11)

(2.12) where^C;Jklisthefourth ordercompliancetensor:U,jis thesecondorderstress tenser.

The delayedelas ticstrain ra te.orrecoverablecreep strain ratewas generalized by Ohnoetul.(1985) to multiaxialstressstatesin theform

(2.13) wherea.1\andbatematerialconstants. which arepossiblyfunct ionsofst rain and stress;sisthevonMisesequivalentstressandS;jis thedeviato ric stress,Ifb:::: 1, the dashpotin the Kelvi nunitis linearand the aboveequat ion becomes

II

This equatio nwas adopt edby[\ <\HandChoi(19S9 ).

Thegeneralizationoftheseconda ry creepstrainTatefor iurompressib!r-be- haviourof iccwas givenby AshbyandDuval(1!l8!i)as

(:.!.15)

whe re K' isaviscosityconst ant.Sothe totalstrain rateoficc is then

(2.16)

wherefijisthe secondorder straintensor.

2.3 Crackin g of Ice

Crack nucleat ionin iceisa com plexprocess associat ed withthe transi tionfrom ductiletobrittl ebehav io r.Themechanismofnucleat ionde pends onthe loadlevel andloading rate. Gold(19i 2)firstdescribed thefa ilur eof columnar-grained ice interms of mlcrocrackiugduring com pressive creeptests, with specia l attent ion to the crack initia tiontime,strain and crackdensitydevelopment.Based on sta- tistica l analysis.lIVOtypesofcrackdistributionswere foun d. Strain-dependent crack distribu tionswereproposed to be the resultof a dislocation pileup mecha- nism.Strain-independe ntcrack distributi onsappeared to berelated to the elastic anisotropywhich causes stressconcentrati onsatgrain boundaries.

For fracture of ice intension.the appliedload mustbesufficient to nucleate microcracks,and the load mustbe increaseduntilthe crack beginsto propagate.

Cracknucleationis likelyto be associatedwith criticaltensilestrain{Seng-Kicng

12

andShyamSunder.19M }orcrit ical delayedelasti c st rain as proposedbySinha.

(1982).

Foriceofgrainsize 1\'5$thanI mm.nucleat ion ofcracks may occurat astress or aboutI toI.:!:\IPaAnd thepropa&ation stressis about1.2to 2 :\IP;Lfrom test data.ohtaincdat strain retes10-",,-1 to 1O-3_{J -}1bySchulson et.1.1.(1984. 1989), Schulson(1987.1989),andCurrierclal.(l9S2l.tensile cracknucleation occurs at acritica lst resswhichcan be expressedas

(2.17)

whereI1Dis 0.6 i\.lPaandkis0.02~IPamIllanddis the grainsize. The criterion fortensilecrackpropagationis givenby

(2.18)

whereKtcistheerilinlstressintensity factorformodeIload ing,ais haJfof the craclt lengthand )'isag~m e l rica lparameter. Tensilecracks and fract uresurfaces are a.Iwaysperpendicularto thetensile stress axis.

Incomp ression,thecrack nucleationprocessismore:compl icated and h.ighly rat esensit ive.Seng-Kiongand ShyamSunder(19M ), Hallam(1986) proposed that cracknucleationoccurswhen theassociatedlater al tensilestraininduced by the Poisso n expansion reechesacrilicalvalue.The requ iredcompressive nucleation srreseshouldbeabout3 tim eshigherthan thAtfor tension.

Sinha(1984) usedthetestres ultsof Cold (1972) torela tecrack nucleation toILcriticaldelayedelastic strainassociate d withgrain boundarysliding,Le.,the delayedela.sticstrain(~giveninEq.(2.3)is equaltothe straininducedbythegrain

1:1

boundarysliding{gb ..and whenf1b'reaches a critical valuet~b^••cracknucleation occurs. Thecriticalvalueof grain boundaryslidingt~b'isrelatedtothecritical stressneededto producedocrackat theend ofa slidinginterface.

Thedislocation pileup mechanismwas adopted by Schulsonct11.1.(198·1)• Cole (1986)and Kamaetal.(J9S9). Thismechanism is based on theconcept lhat dislocationpileupat grainboundariesmay providea highstress cnncent ratjon which can induce crack nucleationwhen thestressreachesa.critical level.

More testswere carriedout recently by Sinha (1988) on columnaricc,Hallam et al. (1987) on granular ice at constant!oad,and byCole(1986) on granularice at constant strain rates.Kalifa eta!' (1989) performedaseries oftriaxialcompression tests withstrainratesvaryingbetween2.5 xIO-s.'l-Iand10-³5-1andconfining pressureranging from 0 MParo 10 MPa.

From theirworksome conclusionscan be summarized:

{lJ Cracksusually startfromthe grain boundarieswherehighstressconcen- tretio nsexistand are arrested at grainboundari es.Cracknucleat ionoccurs atthe largergrainsfirst .The planeofcrackshas a strong tendencyto be parallelto the axis of com pressive stress.

(2) Fortheconsta nt st resstests(Cole. 1986), thecrack densityincreaseswith gra insize and stress,andthecrackingratedecreasesas the crack densityapproaches onecrackpergrain.

(3)The numberof intergranular(between thegrains) and int ragranular(within thegrains)cracksare aboutthesa me.but cracks are preferentiallyintergranula r at high strainrates.

(4)The average cracksizeis about 0.65 timestheaverage grain size,andthe

14

maximum crack size is typically 2to 3 times theaverage grainsize. The crack sizeisnot affecte dby thestress.The grainsize was calculated, accord ing toCole (D86),u.~ing(1

=

(6{1ftV)l/l,whereNis the numberofgains perunitarea.It was foundthat the peak stressdecreases with increasing grain size.

(.S) Althoughthefinal crack densityis very high,the microcracksdonotappear toint er act ; that is,the nucleat ion of onecrack does not triggertheothercrack nearby.

(6) No "wingcracks" wereobserved byCole(1986) and Kalifa et al(1989).

Onthe other hand,a few wing crackswereobservedbyHallam et al.(1987) and Schulson(198i).but dama geinice is mainly dueto thenucleatio nofnew grain- sized cra cks.rat her thanthepropagationofthose which havealready nucleate d.

(7)AccordingtoKalifaet al. (1989), the stressandstrain levelsforcrack nucleationIncreasedwiththe confining pressure, and so didthe st andar d deviation orthedistributionof crackorientation. Thesizeofcracks didnot change with pressure andthestrainratehas no significanteffectoncracknucleation. An equat ionofthe critical stressat the first cracks was given as

(2.19)

wherecrlis axialstress andcrJis confiningstress. Bothcrtandcr3arenegative in compression.

Therearefour stages in the failure processofice duringcompression exp eri- merus ofst rainrate at1O-³J-l(Cole,1989):

(I)Inthefirst stage,stress-stra in relatio n isbasically elasticbutwithslightly nonlinea rbehaviorand novisible microcra cking observed.

15

(2)Microcrack sbegin to nucleate inthe secondstagewithstress between 3.29 MPa to3.95MPa.The crackdensityincreasedveryquicklywithincreasing stresses up to5,65~IPa.

(3)Cracknucleation stopped.the increasingstress causes110more visiblemi- crost ructuraldamageand theexistingcracks appear stable.

(4)The finalstageis thespecimencompletely damagedwithpossiblysudden brit tle failure.

Infact , onlywhen thestra inrate is relat ivelyhigh,about IO-J s-".does the ice become britt le andcomplete fract ure failureoccurs. In this case cracks extendto thefreesurfaceorcracks interact to formalarger crackOfshear fract ure surface.

Ifthe loadi ng rete islow,the stress-straincurve eventually reachesaplateauand icecreeps withoutsuddenfa ilure.

Theelastic anisotropymechanism has also been applied to iceby Cole(1988) andShyamSunderandWu (1990). Theirrecentwork showedthatclast icanisotropy of the ice latti ce isan effecti vesourceof stressconcentration and can betakenas analtern ativefor crack nucleation whendeformatio n rete istoo high to allow dislocat ions topileup.Thesemodelsgave good agreementwithtestresults.

Microcr acking andfractu reof ice is verycommonin ice indent at ion andthere is much work onice intera ctingwit h Bat.cylindricaland sphericalindenters. As addressedin theworkof Jordaan andTimco(1988). Timcc(1986),Tominet al.

(1986) and Jordaan and McKenna (1988,1.),whenan ice sheetinte racts witha fiatindenter, a layerofcrushedice isformed in frontoftheindent erandthe microcracksaredeveloped along the maximu mshea rstress,as shown inFig.2.5a.

Theice isidealized intothr ee zones, as shown in Fig.2.5b, undamaged virginice;

16

partlydamagedicewithrelati vely high density ofcracks and reduced stiffness;and crus hed ice which eventuallywillhe ext ruded(,ut butthis leecan carrycom pressive loadsducto itsfrictionalpropertie s,i.e.the compressive strengt h of crushedice is not zero.

Several differ ent failuremodes ofan icesheetwere observed(Timeo1986)de- pending ontheloading rateandtheratio ofthe indenterwidt htoicethickness.

Generally,allowspeed. thereismainlycru shingand microcrackinginthe icewit h someshortcracks less than afewcent imetersin length (Fig.2.5).At high speed, there is crushingand spallingrightin front of theindenter,but the failureof ice is mainlyduetothe occurrence of theradialand circumferen tialcracks andmaybe buckling(Fig. 2.6a). ln somecases thereare mainly45⁰-60⁰radial cracks ex- tendingfromthecorners and the crackswould be acoupleof meterslong (Fig.

2.6b).Moretestshave beencarriedout recentlyin the icetank in theInstitute for MarineDynamics,Canada,which providessimilarevidenceof ice crackingin interaction(Finn,1991).Inthe caseof the cylindrical indention,crushing,microc- racking, radial and circumferentialcrackscan alsobe observed similarlyto theflat indention(Halla m, 1986).A possible crushing and damage mode ofthe spherical ind enta t ion tests,as discussedin theworkof Jordaan and McKenna(198811.),is illustrated inFig.2.7.Alaye r ofcrushediceis underthe indenterand the ice beyondthe crushed zone is partl ycracked .The density ofthe crushedice isless th an the intactice. Radialcrack s could alsoform andreachthe surface.solarge flakeswouldspallaway.More detailsofthis kind of testswillbediscussed later.

2.4 Dama ge Mecha n ics and Damage of Ice

Thedeformat ionprocessof engineeringmaterials under loading often resultsin changing the structure of the material. This change. to a large degree,will de- [lend on the combined effects of geometry,loading, and the most important, the growthof micro-defectsin the structure.The accumulationof micro-defects is of- tentermed"theprocess of damage" which is always associated with the change ofthemechanicalbehaviorofthe materialandthe dissipationof strain energy.

Mostof the early work of damagemechanicswas basedonthe originalideathat the damage ofa structure can be measuredby a scalar factor [Kachanov,1958), whichis equalto the ratio of the area of voids and the whole crosssection,orthe density of microcracksand voidswhich would permanently affect eithertheelastic modulus,Eor shear modulus. G. Thiswas the guidelinefor most of theearlywork.

Theimportance of thiskindof damagemodels is the establishmentofa rational damag elaw which defines the rate of damage accumulationin ter msofthecurrent valuesof statevariablesand internalvariables.

Based onKechancv'smodel(1986), a body withanoverall sect ionarea110and fract ured(damaged)areaA,is shownin Fig.2.8.In the case ofuniaxialloading Pwithout damage, stress in the body is given as

(2.20)

Withisotropic damage ,the damage variable0 can bedefined as

D = ~;

^05D

s

1 andthe effective stressd.is int rod ucedas

(2.21)

p p a oe» ~=.1₀(1-0)

=1=0'

18

(2.22)

Itisassumed that thest rain respon se of thebodyis modifiedby damage only through the effective stress, so thestress-s train relationofthe damagedmat erial is

e:=~=_Eo --"-=:!-_{£0(1-Dj E} (2.23)

whereEois theelast ic modulusof virginmaterialand£=Ea(1-DJcanbecalled the"effective" modulus.Sothebehaviourof damaged material ca n beconsidered to beequivalent to the behaviour of undamaged material, providedthat the origina l elastic mod ul us Eo isreplacedby

E=E,(I- D). (2.24)

Theevoluti on of damagegenerally rela tes thepresent strain,str ess anddamage.

The kineticor evolutlona lequationcanbe introducedin the generalform

D

=

!(e,e,d,q,D , ....b,

wheree. a andDarestrain,stress anddamage respectively.

(2.25)

Acontinuumdamage model was proposed byResende and Marti n(1983,1984) forrock-like materialswhichdefinesthe elastic strain-stressrelationof thematerial

and

e =

e·+~p

(2.26)

(2.27)

wheres is the stressinvariant:(,'0istheinitial shear modulus:0is thedamage measurement;e isthe totalshearst raininvarian"., and e"anae^Pare theelast icand damage part ofc, respectively. So the rate form of Eq.(2.2·1)wasgivenas

.s^=Go(l- OW

-o.eo .

for loading(D>0), and .s=Go(l-DW ;for unloading(0

=

0).

The damage evolutionlaw was definedas

(2.28)

(2,29)

(2.30)

whereAand8arematerial constants and dependenton loadingsituation,The invariant volumetr ic strain rate E.was also assumed as

(2.3 1)

where

i:

^andi~arethe elastic component and inelastic damage componen trcspcc- tively. They are alsofunctionsof strain.stress anddamage. See thereferences [ordetails, Other references on damage mechanicsinclude Krnjcinovic(1983);

Krajcinnovi cand F'onseka(l981)iLeckie (19iS); Schapery(l981,19801 and1988).

Damagemechanicshas beenintrodu cedto icebyCormeau cr al.(1986),McKenna etal.(1989), Jordaanand McKenna (1988), Karr(1985), Karrand Choi (1989), Sjclind (1987),etc,An isotropicdamage modelwith asingle scalarda magemea- surehas beendevelopedin somepapers.Someoftherecent workhas focused on the relatio n ofthe extent ofdamage and thegrowing networkof microcrackswhich is oftenassumed to beuniformly(isot ropically) distributedand randomlyoriented,

20

As describ ed in the previous section, Sinha'smodel predicts cracknucleat ion when(,60:=:(~~.,and thE" forma t ion of subsequentcracks was given in the form (Sinha, 198,1, 1988, 19891»

N

=

Nc[¢exp(i -xc)-I] (2.32)

whereNcis thecrack density for the first cracks;I/Jis a const ant;.fis theaverage grainboundarysliding(gbs) displacement; :toiscritical (gbs) displacement.

Cree p strainrete was also foundto beinfluencedbythe forma tio nofcracks.

FollowingWeert ma n (1969),the enhancemen tofcracks oncreep wasgivenas (Sinha,198B,1989b)

(2.33)

whereNisthenumber of cracksper unit areaanda is halfof thecrack lengt h.

t\rateexpress ionof crack form ation wasalso proposed byMcKenna etel.

(1989. 1990), Jord aan andMcKCl'\Oa(1989) basedonratetheory intheform

i"

⁼IVc[exp

(u :Ou c)

_-I),

andalso

(2.34)

(2.35)

where

N

=0,if^U~^{Uc , d,}is the criticalstress,Uoisaconst ant( unib ofstress) andflcis a referencerate.Theisotr opicdamageparameterDN•afterBudiansk y andO'Connell(1976), wasdefined as

(2.36)

whereaisthesameasabove and,Vistheden sityofcracks/I1-J, Whenthecrackdensity is high.thectlcct of(racks01\the creep ratewas esti mated,basedonthe work of Weertman l [!)o!)l.byintroduci ngan exponential form

i::::toexp(J N )

where

a

is a constan t.

(:1.:17)

Some specially designeduniaxialtest swere conduc tedonbot hintactand predam- aged iceto investigatethe influenceofthe presence of cracksonthedeformationof ice,Stonectill.(1989),Jcrdaan and~lc Kell na.(1989)andJc rdeanet ...1.(1990a., b).These testswere alsousedto verifythetheoreticaldamagemodel andwillbe discussedin moredet aillater,

"

I<elvin unit Max wellunit

Figure2.1:Burgers Body;EandIJareelast icmodulusand viscositycoefficient, respectively.

Figure2.2:Applied stresshistory andstrai n respon se ofCODstantstresstest.

:?J

0.01 0.02

Suaia<")

Figure 2.3:Stress-strai ncurve ofconstantstrain ratetest.

Tillie

Figure 2.4:The tbreephesee ofcreep te!lt:(I) prim&rJ i (II)aeeondaryj(lit)terti&rJ .

2'

Figure2.5:Idealizationof ice sheet;planview of (a)photographic representation showing progressof damage.and (b) idealizationintothreetones (Jordaan and Timco,(988).

(a) (b)

Figure2.6:The failuremodesobservedinthe icesheetindentat ion tests: (a.) Crushing wit h radialand circumferen tial cracking;(b) Crushing withradial crack- ing(Timco,1986).

MOVlm.nt01 pulvlrlzotlon Iro"t N.w loyn boundar)'

Figure2.7: Possible mode ofpulverizationaheadofsphericalindenter (.Iorda.a.n andMcKenna,1988&).

~ ^{. o 0}

ⁱⁱ

A

@

Figure 2.8:A bodywith an overall sectionarea of Ao and&dam aged areaofA.

Chapter 3 Experiment a tion

III April. 1989 andMay,1990.two fieldprogram swerecarriedout onHobson's Choice IccIsland Research Stationby MemorialUniversity, theNat ionalResearch CouncilofCanada(~RC ) .Canad ianCoast Guard(e e G)and SandwellSwan Wooster(55 \V).A hydrauli c indentat ion systemwasutilized withdifferentsizes of sphericaland Hatindente rs.The ice islandisa 2.5kilomet erwide, 8 kilometer long, ·I,'}meter thick floatingblockof icethat brokeaway from theWardHuntIce Shelf.Ellesmere Island,in1982.Itisprimarilycomposedof freshwatershelf ice, withalarge amount of thick,up to 10 meters, multiyear ice surroundingtheshelf ice core.Thetestsitewas in the area of multiyear ice(Kennedy,1990;Frederking ctal.,1990a, b). These programswere designedto determine anaccurateand reliable methodologyforthepred iction orice Icrcesonoffshore struc tures.

Severalcoresofmult iyearfieldicewererecovered (romthe ice islandand trans - portedtoMemoria l Universityfor compressiontesting in thelaborato ry. These tests weredesignedtoinvestigate the deformationofmultiyear ice, and toobtain the relevantmaterialconstants fortheoretical modelling.Theinfluenceofcracks an d damageonthe creepresponse wasinvestigat ed.As previouslymention ed,sim-

26

:?i

ilartests onfreshwa t erice werecarried out'bySlone etill. (1989),Jordaan and McKenna. (1989) andJcrdaanctal.(I!)90 a , b].

3.1 UniaxialTests 3.1. 1 SpecimemPreparation

The blockof lee-islandice was cut withabandsaw into theshape ofrecrangulnr prismon theorde rofi5 x i,5x200 mm^J•Cylindricalsamples of desireddiameter we re machined from therectangularsamples on alathe . Then thecylindrical sampleswereheld ona pr ecisionV·blockjigwith its axisparalleltothelong it ud inal axis ofthe lat heand perpendicularto the cross head⁵¹⁾that thesamplecoul dhe cuttoth e desir edsize with two endsparalleland perpe ndi culartotileaxisof the cylinder.The fina lspecime nswere54±O.05mmin dia me te rand135±0.:l5 rom in le ngth .Thesizesofthespecimenswere determinedbythesetup of thetest system.

Thespe ci menswere st or e d in a freezerat atem pe ratureof_30°C un t iltheteat.

Asrep o rtedbyFrederklng et aI.,(1990a)andSinha(1990),the structureof theice was basicallyfrazil with a small portionofcolumnaricc, and compriseda significan tnumberofairbubblesand brinepocke ts.The percentage of airpocket volum e wasaveuud2 to5%. Thegrain sizewas about2 to7mm.Thesalinity ofthe ice varied from0 to 0.4%depend ingonthe loca tion,amithedensit y wa.'\

about0.8 75 to0.886 g/cm³.So mephotographs ofthespecime ns , beforeant!after thetest.ing,Mesh own in Fig.3. 1.Itwasfou nd tha t the rewer e abou t4-8big bu bbles,ontheorderoC2.5to!;mm in each specimen,plusgro upsof bu bbleswith thesize ofIto2 mmandrandomlydistributedsmallbu bb leswit h thesizeofO.:l to 0.7 mm.

28

3.1.2 Test Setup

A~ITSSystemsCorporatio nMedel905 Structure TestingSystemwasused for alltests. TwoLVD T ' swere mounted directlyonthe specimen as showninFig . :1.2,overagauge lengthof approximately 8·) mm. The two LVDToutputs were averagedto providethein-sit u measure of axial st rainaswelt as a closed-loop feedbackcont rolsignal to the MTS servo-valve. Lateral strainwasalso measured

011thefirsttwoteststo tryto find evidencesof dilatation duringthedeforma tion.

Forthis purpose,twoLVDT' s were mounted onthetwosides of thespeci men and approximatelyon aline as shownin Fig.3.3,but the results werenotvery satisfactoryclue to theirregular deformationofthe specimeninthelateral directio n, which isevide nt in Fig. 3.1.

Severalhoursbefore eachtest,the specimen wasplaced in thecoldroomto allowtemperatureequalization. During thetest,thetemperat ures atthetop and botto m of the specimenweremeasured.The temperatureat the botto m wasusually sligluly higher than thatatthe top. This was attributed to the hyd raulic fluidbeing suppliedfromoutsidethecoldroom.Amaximum bottom temper atureof·9.6·C wasmeasur ed for -IO'Ctest s.

A1\the test data including load, stroke,displacement and time, were recorded

01\amicrocom put er viaa multifunctiondata acquisition board. Accord ingto Stone etal.(1989),an acquisition rale of

is

sample/sec/channel wasfound to be adequate forloa ding andunloadi ngphasesof thetestsat strain ra tesoC10-·s -¹ to10-'\'.'1 -¹.forthe higherstrainrate of1O-³s-¹,an acquisition rateof 175sam- ples/ sec/channelwas recommen ded. Betweeneach unload ing andloading, i.e., during the period of relaxat ion , wherethedeformationrateis verylow, theecqui-

29 sition rate is set to 10-100 times lower than that on loading,sothal morestorage spacein thecomputerca n be saved.The dataacquisitionrntefor thecreeptl'st~

was 245 samples/se c/channelduringtileloading.Theacquisitionboard providesa measurementaccuracy andresolutionof ±O.02%in the rangeof±IOV.Loadand stroke were also plottedby an X-Y plotterduringthetests.

3.1.3 Resultsand Discu ssion

AslistedinTable 4.1.three uniaxial const antstrainrate testsand five constnut load testswereca rried out onthe multiyear icc describedin section3.1.1.Atthe temperatureof_lOo

e,

test Xo. Iwas a constantstrain-ratelest subjected to a loadingrate of 5x10-5 s-Ito a maximumstrain of 2% as showninFig.:lAoTest No.2was similar buthadtwoloadings with different rates.Thespecimenwas first loadedat a strainrateof·')< 10-⁵3-1to astrainof 2%,followed byunloadlug andabou t 100seconds of relaxation,followedagainbyreloading al a strainrateof 2.5x 10-⁵05-1to a total strainof4%,as shown illFig.3.';.TestNo.3,eondueted on an intactice specimen,was a constantstress creeptest ,which. wasa.series of creep tests. Eachcreeptestconsisted of a 20 second load pulse followed by atominuterelaxationperiod,rollowed againbyreloading, and soon(see Fig.

3.6 ).Thereasonforloading only 20 secondsis thatthesetests weredesignedto investigatethe short-timeresponses orice, suchas, elast ic and delayed elastic strain components.TestNo. 4 was a creep testconducted on a predamagedspecimen.

The speci menwaspredamagedby subjectingto aconstantstrain rate loading of 10-⁴$-1to a strainor 2% as shownin Fig. 3.7.At thetemperature of ·20°C,a creep test,No..j,onan intact specimenwas carried out withthe sameloadings

30

Table 3.1:ListoftheTest Series

tes t strainrate ice temp.

No. type orstress type 'C rem ar ks

I const antstrai n 5x10-s- intact .io-c rete(C.S.R.) (see Fig.3..1) ice

2 co nstan tst rain 5x 10-$-&. intact ·IOOC sample was reloaded rat e 2.5 :oc IO-ⁱ.s-^I ice at"~rainoC2%

( seeFig. :1.5)

3 const antstress 0.25~IPa- int act ·'O·C

(C.S.) 1.5 MPa ice

(seefi.3.6)

,

constantstress 0,25~I Pa - p.d. .re-c predamaged;

2.0:vlPa ice sam plewas predamaged

(secFig.3.7) underC.S.R.to 2%strain

5 constant stress 0.25~IPa- intact <woe

:.0MPa ice

(seeFig.3.8)

6 constantstress 0.25~I Pa p.d . ·20·C sample was predam aged

2.0MPa. ice underC.S.R. to 2%strain

(seeFig. 3.9)

7 const an tslrleU 0.25MPa - p.d. ·200C sam ple waspredam aged

2.0MPa ice under C.S.R. to2% strain

(seeFig.3.10)

s

co ns t an tstrain 5x10-s-& intact ·20·C sample wasreloaded

rale 10-3 ,,-1 ice ll.l slrll.inof2%stnin

(seeFig.3.11)

as thatoftestNo.3(Fig.3.8).TestNo.6andNo.1weretwocreep testson predamagedspecimens,i.e.•theicewas loaded toatotalstrain of2%at constant

"t rain rate of10-⁴,,- 1.AIshowninFig.3.9 and Fig.3.10.respect ively,TestNo.

Swas a constant strainratetest,asshownin Fig.3.11,toastrainof2%witha stra inrateof5x10-4,,-1,and reloade d again to astrain of,,%withastrainrete oflO- Jj-l.

Fig.3.12shows thestress-strai n curves ofsomeconstant strai n-ra te testson

:11

bothfreshwater andmultiyearice. Based on theunloadings of thetests onboth types of ice,it was found that the recoverablest rain.i.e. elastic strain plus delayed elastic strain. was less than 10% of till' total strain. Wlwn the tot al strainis more than 2%, the stress reaches a plateau . The recoverable stealn "

+

^{tJfrom the}

test of freshwate rice is only 2.05 x 10-³,1.82X10-³and 2.84x 10-³upon first.

secondand thir d unloading,respectively.Notice thatthe first value is largerthan the second, becausethe stressis higherat firstunloading.Theicc specimenwas not totally relaxed duringthefirst two unloadings: the relaxationperiodswere both about 10 minutes.Thespecimenrelaxedfor aboutone hour after thethird unloading.therefore thethird value of recoverablestrainis thelargest. Butthe secondaryorperma nent creepstrainis still thepredominantstraincom ponent.

The ratiosof recoverable strainto total strainare 10.5%, .).2%and S.5%forthe three unloedings .Thiswas also found inthe tests of multiyear field ice.

Comparing the constant strainrate testsof mult iyearice tofreshwaterice (Fig.

3.12)shows that the peakstresses ofthe multiyeariccarc much lower.Onereason forthelower strength must be the defects. such as the airpockets, in the multiyear ice.These defectscan be considered as damage, which wouldsignificantlysoften the ice.Anotherreason is that the structureof the multiyeariccis acombin ation of fraziland columnar ice with the grain sizerangingfrom 2 mmto7 mm. The laborat ory.madefreshwaterice is granulariccwith a grainsize of:I10m .Itis expect edthatthe twokindsof ice wouldhave differentresponses underthesame loading. Themulti year ice,as addressedin the previous chapter,must have a lowerviscosity duetodefect s and crystalstructu re.As shown in Fig.3.12,the stressesstartto build upalmost linearly with str ainsandallthe curves arcclose,

32 which mea nsthattheelasticmod ulus ofthe twoty pesof icehave sim ilarvalues, sincemost ofthe lolalstrain iselastic strain althebeginningof loa ding(Sinh a

1981).Inthestra inrange of 0.190to0.5%,thereisit.significantdifferencein the stresses.The peakstress oftheIreshwatericeis almost double thatofthemultiyea r ice. Assum ingthat thestressincreaseslinearlytothe peakstress,itis possibleto estimate theport ionofeachstrain component at the peak stress.i.e.

("=(//£

Here ais assumedtcbe

(J=~t

I,

where

u ,

^{isthe peak}stress at timel,.So it isfoundtha t

wheren

=

3is assumed. Herethepeakstress" are5.1 MPaand 3 MPa (or freshwaterand multiyearice. respectively,andt,is 27.9seconds. Assumingthe twotypes ofice havethe same elastic modulus,theviscosityparameteroffreshwater ice is 1,;6x IO-T(Sinha,1981), andtheestimation ofporosityofthemultiyearice is about 5%.With Eq.(2.1t),the viscosityparamet eriscalculatedtobeabout 2.05x10-⁶•The straincomponentsatpeakstressatelistedinTable 4.2. So, forfreshwater ice at peak stress,morethanhairor the totalstrainiselast ic, bu t formultiyear ice, the delayedelasticstrain isthelargest component,and both(II

:13

Table :1.2: List of StrainComponents

( x l O - 3 (tX10-3 (J.x10-³ «x10-:¹ !..~% ~% C%

I

^~reshwater

1.395 O.iSS 0.3S:] .')6.5%

tee O.:!:!-l :!i ..5% 16%

1~ultiyear

1.395 o.u: 0..591 0.:3$7 :10%

ice -I:!..I% 27.6%

andcare largerthan thatof freshwaterice.So it is concludedthatthemultiyear ice must haveless stiffnessandlowerviscosity in the Kelvin unit(see Fig.3.1), due to thedefects and crystal struct ure. Bothdelayedclasticstrainand secondary creep st rainare enhanced by thedefects (t hiswill bediscussed later).Afteratotal strainofabout 1.5%to :!%.the strain-stresscu rves beginto convergeagain,and as mentionedpreviously,the recoverable strainis less than10% ofthe total strain whenthestress reachesaplateau,most ofthe strain is secondarycreepstrain.i.c.

(" ::::: fifurthermore, the stress(J'is almost constant,so.i! :::::i,1:::::0,thisyields (3.1)

(3.2)

where superscript primemeans damaged materialpropertiesandthesubscripts fandi stand forfreshwater ice and multiyearice,respectively. The equation shows that thetwotypes of;··e havesimilarviscosities.Thissuggests that inthe strain range of 0.1%to0.5%,asmentioned above, the freshwaterice suffers more damagethanthat of the multiyeariccdue to the much higherstress.Therefore thefreshwater icc has been givena greatdealofstrain softeningandviscosity reduction .So theviscosityof ice,which wasmeasuredfrom the consta ntstrain rate testafterthe plateau,is actually thatof damagedice, notintactice,

34

Thecreep tests provided additionalinformationontheelast icity and viscosity of theice.Fig. 3.13showsthe strai nresponses of inta ctand predamaged iceat a stressofO.i5 :vtPa(~.and(,. aretheelasticstrains;fd+«andltd+l'Care delayed elastic strainsplus secondarycreep strai nsfor intactand predamagedice, respectiv ely.3yfocussingon the strainversus timecurvesdose to time

=

0, the instantaneous elasticstrainscan beest ima ted ). Theelasticstrainsand totalcreep strainsat20secondsare plottedinFig . 3.14. The staticelasticmodulus of intact multiyearice is estimatedas 8000~IPaat ·20 "C,andthemodulusof predam aged icc is about6000 MPaat -20 'C.These valuesarc takenfrom the slopes of the stress-st raincurvesin fig. 3.t-\a,and theseelasticresponses ofthe intact multiyear icc showthatthesmallrepeat ed loadingshave not addedsignifican tdamagein the ice.Comparing thecreep responses (delayedelastic strainplus secondarycreep strain ) ofthetestsof intactto predamagedicein Fig. 3.14b(see alsoFig. 3.13), shows that the presenceofcracks anddamage significantlyinfluence creepstrain.

The creepst rainof predamagediceis about1)tototimes that ofintactice.As showninfig.3.13,the stra inresponseof the intactice has mostlyrecovered, andthe permanent viscousstrainis closeto zero. This suggeststhat theelastic anddelayed elastic componentsofstrain dominateforshortload times.The strain response of predam agedicehas a significantpercentageof secondarycreep,andthe delayedelastic strainrate (the slopeofthest rainversus timecurvein Fig. 3.13) ismuchhigherthanthat of intact ice.This impliesthatcracking anddamage can significantlyenhancethecreepresponseof ice(seealso StoneetaI.,1989;Jord aan etaI.,1990a,b).

3.2 Spherical Indention Exper im ents

3.2.1 Experim en tSetup

lnApril,1989, a lolal ofeleventestswerepreformed on theHobson's ChoiceIce Island.Six ofthem werespherical indentation tests with speedrangingfrom0.:1 mm/s to90mm/s.Theinsitu icetemperaturewas about·1.\°C.Theles ts were carrie dout inanareaof 8 m thick multiyearice which was attached to theedge ofthe ice island.A trench 3 m wide.-tm deep and 100 m longwas excavatedto conduct the tests. The walls of thetrench were roughlysmoothedwithachain sawand the test areas were speciallymachined with a verticallymounted circula r saw.The wa ll oppositethe testface was also machined and maul" parallel to the test face.

The iceindentat ionsystem consisted of a hydraulicactuatormountedupon a largemobileskidof beamand strutconstruction(rig. 3.15). The actuat or was poweredby a bankof pressureaccumulatorsand controlled by a servo-control systemwhichprovidedaconstant displacementrate(with exception of110servo- controlsystembeingused for thefirst test, NRCl).The indentation speeds could beassumedconstantoverthe whole testperiod.Seven100 mm diamete rpressure cellsformeasuringlocal pressureswere mounted to rhe frontofthe indente r.The locationof pressurecells is shown inrig.3.16.A flat backplate wasat tached to the rearendofthe actu atorto supportthesystem. This back plate hadalarger contact areathan theindenter to force crushingfailureon the indentation face only.

36 3.2 .2 Results and Discu ssion

As discussedin the workof Frederking etal. (1990a, b), bothlarge scale and local crushingunderthe indenterfacety pically accompaniedthe indentation tests.

low speed testsallowed sufficienttimeforcreep deformationand microcracksto extendinto theice, andthe totalloadversustimecurves were relat ively smooth, whilehighspeedtests appearedto producelocalizedfailure nearthe indent er and dynamicice forcesonthe indenterwererecorded. Analysisofcrus hedlayer profiles during spherical indenta tion tests showedthat thelayer thicknesswas irregular.

Thethicknessof thecrushed layerwas about20 mm to 50mmfortest No.7, and themaximumthickness observedwasabout 320 mm atthe centeroCthe contact areaduring testNRC5. Therewas a clear boundarybetween the parenticeand crushedice. For the low speedtests, the ice under theindenterwaspartlydamaged withshortcracks.In thepresentwork,attention is focussed on twooCthe spherical indentationtests, these are denoted asNRCt and NRC2, respect ively.Theload- timeresultsof thetwo tests areshownin Fig.3.17a, b.The loa dingrate Cortest NRCI was0.3 mm/s .Theindenter came in contact withthe iceat pointA andthe systemstoppedat pointC.Itwas observed that a verylargepieceofice spalled offduringthe test ,whenlarge cracksextend ed6·9 mon either sideoCtheindenter towardsthe top icesurface.[tisbelieved thatthe spall occurredatpointB.andso the test results afterpointB have not beenusedCormodelling purposes. A similar situationexistedCortestNRC2conducted at2.5 mm/s, in whichtwo big spalls probablyoccurred.Here againtheportionofthe test aCterpoint B was neglected.

From thepressuremeasurements, themaximum pressureswererecordedat the centerof the indenterandthe average pressures were intherange of.5 MPa to20

37

MPa.Due to highconfining pressure and lower shear stressinthe centr al area, the iceis less damaged{the critica l stress requiredforcrack nucleatio nincreases with confiningpressure(Kalifaetal.,19S9)). Nearthe edge of theint erface. thereisless confining pressure and higher shearstress,so theicehas becomemoredamaged and thecrushedlayer is thicker.Recrystallizatio nduetopossib le pressuremelting hasalso beenrepo rt edduringindentationtests(G agnonand Sinha,19!)I).which means thatfriction betweenthe indenterand theice may bewrysmall.

(aj

as

(b)

Figllrt·;U:Phctogrephofire samples :(b)befurt·test;(b)itft~'rn-st.,

Figure3,2:'It-sfsetupformeasuring theaxial str ain.

39

FiguH':l.:~:'Ii,,.1setup for measuring bothaxie!andleterel strei n.

lJ

1.1

~

•

i

^IJ

~

OJ

' . ^{. 01}

^0.015

^00'

S. . .

Figur~3.4:StrC!l!· strain curvefor uciaxiel testNo.1.

1.1

~

IJ

1

,.,

' . ^..1

^•O!

. ..

s....

Figure3.5:StresNtraincurve(oruniu:iaJ.te!ltNo,2.

'0

9

""'"

'000

b

~

¹⁵⁰⁰Time (sc",)

, LL~(

^{2000 2S00 3000}

J

³⁵⁰⁰

·'1

Figure 3.6:Applied str esshistoryandstrai n responseofcree ptest No. 3.

:,["'"

j

° 0 sos

^{1500 2000 1500 3000 "00 4000 .500 5000}

Time ("'''-)

Figure 3.7:Appliedstresshist oryandst ra in responseofcreeptest No...

g

^{1.5 I.S MPa 2.0MP.}

i

^{I •}

^{r. 7S}

^{MPa tOMPajl}

:::0$ .pMPa I

.., -n25MPal (U.5MPa 0.25MPal

°0 500 1000 1$00 2000 2500 3000 3.500 4OO(l 4500 5000 Time(sec.)

:==J

2000 2500 3000 )SOl) 4000 4500 .5000 Timc (Jcc.)

Figure3.8:Applied stresshistoryandstrain responseofcreeptest No.5, '2

' t

~ I.'

j

^I

¥ , a·.525 MP• 00

ijo.s M r.

SOIl 1000

f..

SIlO

::I ... I

LJ

^{1.500 2000 2500 3000}

.

Time (sec.)

~

TllDc(sc c.)^{1.500 2000 2500 :JOOO}

I

Figure3.9: Appliedstresshistoryandstrainresponse ofcreeptestNo.6.

1:1

eJ"

^{1500 '---.2000}=----.."'-~lOOO2S00

I

Time (sec.)

' [

~ ^I.S

j ':2SMP. fSM Po

00 SOO 1000

I-

'00

:F

TlIlle(sec.)^{IS00 zeoo )0l1O}

^I

Figure3.10:Applied stresshistory andstra inresponseofcreep testNo.7.

us

Strain

Figure 3.11: Stress-strain curve for uniaxialtestNo.8.

fr~sh...aierice

/

Figure3.12: Stress-straincurvesofconstantstrainrate testsonbothfreshwater andmultiyear ice at strainrate of 10-^{4,5- 1 .}

40 60 80 100

Time (sec.)

Figure 3.13: Str ain responses ofint act andpredamaged ice forstressat0.75 MPa..

i"la~

z,

I.'

0.'

,r' ~

£j ~~:::~~~::

^tee

^j

Inlact lce

Hi

ElaJlic Slrain (a)

dO"' CreepS trBin

(b)

Figure3.14: Creeptests:(a) stressversuselast ic strainforintact and prcdamagcd ice;(b)stress versustotalcreepstrainat20 seconds afterthe applicationofthe load forintactand preda magedice.

/

lACK"A~~

//

/

/ /

/ ~~~fU

Figure3.15:Schem at icoftheactua to r inden te r system(Frederk ingetal.,1990a ).

,.

DIMENSIONSIN...

M"IE~IA\;CASUtuMINUlol

Figure :).16: Schematic of the sphericalindenter and the locationsoflocal pressure cells(Frederking et al.,1990a).

NRC TEST NO.I

[I]

! [

] _]

00 50 '50 is 20 2.S 30 3$

Time(st c.) Time(sec.)

(al (b)

Figure3.17:The total load versus time recordsof test: (a) NRC1;(b) NRC2 (Frederking etal.•1990a).

Chapter 4

Constitutive Modelling

4.1 Ice Mod e l

As discussedinthepreviouschapters.the deformationof iccisa complexpro- cess, especially whencracking activ ityoccurs.Theproper ties of iceare strongly influencedbythe presenceof cracks and damage.Theidealizedmechanicalmodel, calledBurgers'model(see Fig.2.1).consistsof combination of a Maxwellanda Kelvinunits,with anonlineardashpotineach unit (see alsoJonlaan andMcKenna, 1988b).

Muchwork hasbeen done to model theprimary and secondarycreepillicc and othermaterials.AKelvin unitwith a power-lawstress-dependent creep com pliance, as proposedby Jordaanand McKen na (198gb, 1989),Jordaan etal.,(1990a, bj,has beenshownto beappropriate(ordescribing theinitial primary creep underrapid loading.Thisalsoprovidesan expedient ccrnputaticnalsolution forthe primary creepstra in.With this model. at thebeginning ofeach time increment, theprogram only needs to readthe stresses,strains and other modelpar ameters which arestored as statevariablesfromthe previousstate,insteadofrequ iringaccess to thewhole storageof past history.Allofthe statevariables willbe updated at theend ofeach

'8

Inerement,

IntheCa5C'ofuniaxialstress,to tal axialstrain isgiven in terms of threecom- pcnents.Le.

('.1) where the elastic componentis givenby

('.2) whereUlis the axielstressandEistheelastic modulu s.Fromthe laborato ry resul ts ofcreep tests,Young'smodulusofmulti-yearice forthestaticcase is approximately 8000 MPa at·20·C.Sincethe primar y creep properti esofice wereestima tedbased ontheseresults.forconsisten cy, thisvalue is used in themodelling.

The delayedelastic andseconda ry creepstrain re tesarc definedas

t.~⁼lTl/~u .And (4.3)

(4.4)

whereIltlandiJ...1Metheviscositycoefficientsof the Kelvin unitandMaxwell unit.respectively.Assumingthatthestrains ofthe dashpots inbothunitsfollow thepower-lawrelationwith stress,as giveninEq. (2.5).thedelayed elas tic strain rate isgiven as

(1.5)

where

i ;

^{is a}creep reference rate;0'0

=

^{1MPa,isa constantwith unitstress;and}

u t

^{isthest ress}in the dashpotin Kelvinunit, whichis calculated by