Thermo-viscoplastic behavior of 304 austenitic stainless steel at various strain rates and temperatures: Testing, modeling and validation

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Bin JIA, Alexis RUSINEK, Raphaël PESCI, Slim BAHI, Richard BERNIER - Thermo-viscoplastic

behavior of 304 austenitic stainless steel at various strain rates and temperatures: Testing,

modeling and validation International Journal of Mechanical Sciences Vol. 170, p.105356

-2020

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Thermo-viscoplastic

behavior

of

304

austenitic

stainless

steel

at

various

strain

rates

and

temperatures:

Testing,

modeling

and

validation

B.

Jia

_,

_A.

_Rusinek

_,

_R.

_Pesci

_,

_S.

_Bahi

_,

_R.

_Bernier

a ENSAM-Arts et Métiers ParisTech, Laboratory of Design, Manufacturing and Control (LCFC), Metz 57070, France

b ENSAM-Arts et Métiers ParisTech, Laboratory of Microstructure Studies and Mechanics of Materials (LEM3), UMR CNRS 7239, Metz 57078, France c Lorraine University, UFR MIM, Laboratory of Microstructure Studies and Mechanics of Materials (LEM3), UMR CNRS 7239, Metz 57078, France d Institute of Fundamental Technological Research, Ul. Pawinskiego 5B, Warsaw 02-106, Poland

e Chair of Excellence, Departamento de Ingeniería Mecánica, UC3M (Carlos III University, Madrid) Avda. de la Universidad 30, Leganés, Madrid 28911, Spain

a b s t r a c t

Thispaperpresentsasystematicstudyofthethermo-viscoplasticbehaviorofa304austeniticstainlesssteel(ASS). Theexperimentswereconductedovera widerangeofstrainrates(10−3 _s−1 _to3270s−1 _{)andtemperatures}

(-163°Cto172°C),forwhichthedeformationbehaviorof304ASSbecomesmorecomplexduetothe strain-inducedmartensitictransformation(SIMT)effect.Dynamictestsatlow/elevatedtemperatureswereconducted usingtheHopkinsontechniquecoupledwithacoolingdevice/heatingfurnace,andtemperaturedistribution withinthespecimenwasverifiedtobeuniform.Experimentalresultsshowedthatthestrainhardeningrateof 304ASSwasstronglyaffectedbySIMTeffect.Forquasi-statictests(10− 3 s− 1 to1s− 1 )atlowtemperatures (-163°Cto-20°C),thestress-strainrelationsexhibitedanS-shapeandasecondstrainhardeningphenomenon. Thestrainratesensitivityandtemperaturesensitivityof304ASSwerealsodifferentfrommetallicmaterials deformedbydislocationglide.Severalunexpectedphenomenaincludingthenegativestrainratesensitivityand thechangingtemperaturesensitivityfromquasi-statictodynamictestswereobserved.Basedonexperimental results,anextensionoftheRusinek-Klepaczko(RK)modelconsideringSIMTeffectwasusedtosimulatethe deformationbehaviorof304ASS:itpredictedflowstresscurvesof304ASSabove-60°Ccorrectly.Inaddition,to validatetheextendedRKmodelandtheidentifiedmodelparameters,numericalsimulationsofballisticimpact testsof304ASSplatesatvarioustemperatureswerecarriedout,showingagoodagreementwithexperiments.

1. Introduction

As a representative of transformation induced plasticity (TRIP) steels[1],304ASShasauniquecombinationofhighstrengthandhigh ductility[2].ItsbeneficialmechanicalpropertiescomefromtheSIMT effect,which meansthatuponplasticdeformationprocesstheinitial austenitephase (𝛾)transforms intothe stablemartensite phase (𝛼′); thus,bothimprovedworkhardeningrateandsignificantlyenhanced ductilitycanbeachieved.304ASSiswidelyusedinmanyengineering areasrangingfromliquefiednaturalgasstorageatcryogenic tempera-tures[3,4]tocrash-resistantstructuresinautomotiveindustryatroom temperature[5,6] andnuclearfacilitiesatelevatedtemperatures[7]. Therefore,much workhasbeendonetoinvestigate thedeformation behaviorof304ASSundervariousstrainratesandtemperatures.

The effect of temperature on the deformation behavior of 304 ASSisdifferentfromthecommonlyusedalloyswiththedeformation mechanismof dislocationglide.Asshown in Fig.1, attemperatures

lowerthanM_d ,thetemperaturebelowwhichmartensitic transforma-tion occursautomaticallyor can betriggered byplasticdeformation [5,6],stress-strainrelationsof304ASSexhibitanS-shapeandasecond hardeningphenomenon.Thisiscausedbythemartensitic transforma-tion effect. At temperaturesabove M_d , the deformation mechanism of 304 ASSchanges intodislocationglide [5]. Withincreasing tem-perature, dislocationannihilation due tothe crossslip andclimbof dislocationsisaccelerated[8];thusthestrengthof304ASSdecreases without the second hardeningphenomenon. Byun etal. [9] studied thequasi-statictensionbehaviorof304ASSattemperaturesbetween −150°Cand450°C.Itwasobservedthatboththeyieldstressandthe ultimate tensile strength decreased with increasing temperature T0, andtheoptimalductilitypeakedat20°C.Attemperatureslowerthan 20°C, a second hardeningphenomenon accompanied by martensitic transformation was observed. Hamada et al. [10] investigated the quasi-staticdeformationbehaviorof201and201Lausteniticstainless steels at temperatures ranging from −80°C to 200°C and obtained similarresults:theflowstressincreaseswithdecreasingtemperature. Zheng andYu [11] studiedthe quasi-statictension behaviorof 304 ASSat temperaturesbetween −253°Cand20°Candfoundthatboth theflowstressandthestrainhardeningrateincreasedwithdecreasing

Fig.1. SchematicillustrationofthedeformationmechanismofTRIPsteelsatdifferenttemperatures:stressassistedplasticity(𝑀𝑠−𝑀𝑠𝜎),strain-inducedplasticity

(𝑀𝜎

𝑠−𝑀𝑑),anddislocationglideplasticity(>Md)[12].

temperature. In addition, due to the thermally induced martensitic transformationbeforetests,theyield stressesat −253°Cand−196°C weresignificantlyhigherthanfortheothertemperatures.

Theeffectofstrainrateonthedeformationbehaviorof304ASScan beexplainedintwostrainregimes:attheinitialstageofdeformation, theconstitutivebehaviorof304ASSisdominatedbydislocationslip. Theflowstressincreaseswithincreasingstrainrateasthetimeavailable foradislocationtowaitinfrontofanobstaclefortheadditional ther-malenergyisreduced[8,13].However,whenitcomestolargestrains, thestrengtheningeffectoriginatedfromSIMTisinhibitedbyadiabatic heating,andthecorrespondingstrainhardeningratedecreases. Consid-eringtheeffectsofstrainrateonbothdislocationslipandSIMT,either positive[14,15]ornegative[3,16]strainratesensitivityatlargestrains wasobserved.Lichtenfeldetal.[17] investigatedthetensionbehavior of304Lstainlesssteelundervariousstrainrates(10−4_s−1_to4×102 s−1_{)atroomtemperature.Theyieldstressincreasedcontinuouslywith} increasingstrainrate.However,theultimatestrengthdeclinedfirstin quasi-staticstrainrates(10−4_s−1_to10−2_s−1_{)andthenbegantoincrease} indynamicstrainrates(10−1_s−1_to4×102_s−1_{).Asimilarphenomenon} wasobservedbyIshikawaandTanimura[18] in304Nstainlesssteel.

Fromtheliteraturereview, thedeformationbehaviorof 304ASS hasbeen investigatedmassively.Previous studiesmainlyfocused on quasi-staticbehavioratvarioustemperaturesordynamicbehaviorat roomtemperature.Althoughthecombinedeffectsoftemperatureand dynamicloadingonmechanicalproperties of304ASSarefrequently encounteredsuchasliquefiednaturalgasstorage(−163°C)andsheet metalforming(20°Cto300°C),thecorrespondingdeformationbehavior isnotclearlyunderstood.Inaddition,attemperaturesaboveM_d ,the deformationbehaviorof 304ASSissimilartothatof thecommonly usedsteels.However,attemperatureslowerthanM_d ,thedeformation behaviorof304ASSbecomescomplexduetotheSIMTeffect.Hence,a deformationbehaviorstudyof304ASSespeciallyfocusingon temper-aturesbelowM_d (140°Cforthestudiedmaterial)canbeinteresting.

In this paper, the compression behavior of 304 ASS has been studiedoverawiderangeofstrainratesandtemperatures.Inindustrial applications,strainrates can reach2000s−1_{witha commonregime} lyingaround10−3_–10s−1_{[19].Consideringthestrainratelimitations}

oftheadoptedSHPBtechniqueandtheflowstresslevelofthestudied 304 ASS,fourquasi-staticstrain rates(10−3 _s−1_{, 10}−2 _s−1_{, 10}−1 _s−1 and1s−1_{)andthreedynamicstrainrates(1550s}−1_,2370s−1_,3270 s−1_{)wereselected.Forthetestingtemperature,muchworkconcerning} deformation behavior of 304 ASSat room or hightemperature can be found, butstudiesat lowertemperatureis rare.Inthiswork, six temperaturescoveringlow temperature(−163°C,−60°Cand−20°C), room temperature (20°C) and temperatures respectively below M_d (88°C)andslightlyhigherthanM_d (172°C)wereconsidered.

First,self-designedheatingfurnaceandcoolingdevicearecoupled to theconventional SHPB device fordynamic compression tests not only at low butalsoat elevatedtemperatures. Theset-up reliability is verified by both experiments and numerical simulations. Then, compressiontestsof304ASShavebeenconductedandtheeffectsof temperatureandstrainrateonthedeformationbehaviorof 304ASS arediscussedin detail. Anextension of theRusinek–Klepaczko(RK) constitutive model [20,21] considering martensitic transformationis chosen to describe the temperature and strain rate-dependent con-stitutive behavior. Finally, toverify theprediction capability of the extendedRKmodel,numericalsimulationsofballisticimpacttestsof 304ASShavebeencarriedoutandcomparedtoexperiments.

2. Materialbehaviorandset-updescription

2.1. Materialandspecimens

Thematerialconsideredhereinisacommercial304stainlesssteel produced by Thyssenkrupp Materials, available in annealed plates (dimensions: 1000×100×12 mm3_{). According to the manufacturer’s} specification,thechemicalcompositionofthesteelisgiveninTable1.

Table1

Chemicalcompositionofthe304ASSgivenbythesupplier(wt%).

Material C Cr Mn N Ni Co Cu Fe

Fig.2. ThenormaldirectionIPFof304ASS.

Fig.3.SchematicdiagramoftheconventionalSHPBset-up.

Themicrostructureofthe304 ASSwas characterizedbyelectron backscatterdiffraction (EBSD) technique. Byapplying a step sizeof 1𝜇mandconsideringamisorientationangleof5°,thenormaldirection inversepolefigure(IPF)isshowninFig.2.Itisseenthatthe304ASS consistsof100%austenitephasewithanaveragegrainsizeof28𝜇m.

Thespecimensusedforcompressiontestsarecylinders3mmhigh and6mmindiameter.Asreportedin[22],aheighttodiameterratioof s0=0.5helpstoreducethefrictionandtheinertiaeffects.Thespecimens weremachinedfromtheas-receivedplatealongtherollingdirection usingwireelectricaldischargemachining(WEDM)technique.Toavoid bucklingand toensure a uniform stress state inside thespecimens, the specimen end faces were coated with lubricant. To ensure the reliabilityof theexperimental results,three testswereperformedfor eachtemperatureandstrainratecombination.Theaveragecurvefor eachconditionisdepictedasthereferenceinthiswork.

2.2. Quasi-staticanddynamiccompressiontests

Quasi-static tests at room temperature were conducted using a Zwick/Roell200 kNuniversal testing machine. Fortests atlow and hightemperatures,acoolingdeviceoraheatingfurnacewasused.

Tostudythedynamicbehaviorof304ASS,compressiontestswith averagestrainrates around103 _s−1 _{andinitialtemperaturesvarying} between −163°Cand172°Cwere carriedout using an SHPBset-up. TheSHPBdevice, Fig.3,consistsof twolongelasticbars(length of L_B = 1500mmanddiameterofD₀ = 20mm),a compression

speci-mensandwichedbetween themandaprojectile.Whentheprojectile impacts the inputbar withan initial velocity V0, an incident wave

𝜀I = 𝜌C0V0/2Eis induced with a celerity 𝐶0= √

𝐸∕𝜌 (E and 𝜌 are YoungmodulusanddensityoftheSHPBbars,respectively).Duetothe cross-section differenceand the mechanical impedance between the barsandthespecimen,partoftheincidentwaveisreflectedbackas 𝜀R (t)andtheresttransfersintothesecondbaras𝜀T (t).Usingthethree

waves measurement, the average stress-strainrelationsof the tested materialmaybedetermined.Acompletedescriptionofthestresswave analysistodefinematerialbehaviormaybefoundin[23].

For dynamic tests at elevated temperatures, a heating furnace coupledtotheconventionalSHPBset-upisadopted.Anillustrationof thefurnaceisshowninFig.4.Duringtheheatingprocess,thespecimen togetherwithpartoftheincidentandtransmitterbarsareheated.Two thermocouplesareused tomonitorthetemperatureof thespecimen and the furnace environment, respectively; the latter can provide feedbacktothetemperaturecontrollertoadjusttheheatingrate.The maximumtestingtemperaturebytheheatingfurnaceiscloseto250°C. Concerning dynamic tests at low temperatures, a cooling device basedonpulsedcryogenicgasmethodhasbeendeveloped,Fig.5(a). The coldnitrogengas flows froma liquidnitrogentank through an aluminumpipeintothecoolingbox.Thetemperatureinsidethe cool-ingboxis monitoredbyathermocoupleconnectedtoa temperature controller.Oncetheenvironmentaltemperaturereachesthesetvalue, thetemperaturecontrollercutsoff thepowerofthepumptostopthe

Fig.4. Heatingfurnaceforhightemperature tests:(a)aschematicillustrationand(b)the completedevice.

Fig.5.Coolingdeviceforlowtemperaturetests:(a)between−90°Cand−20°C and(b)−163°C.1)liquidnitrogentank,2)coolingbox,3)pumpfornitrogen gasflow,4)temperaturecontroller,5)pipeforthenitrogengasflow,6) ther-mocouple.

nitrogengas flow. Bythis method, thetemperature varies precisely from −90°C to −20°C. In addition, to decrease strongly the initial temperature,thecoolingboxisfilledwithliquidnitrogendirectly.By thismethod,aminimumtemperatureof−163°Ccanbereached.

Since the heatingfurnace and cooling device for high/low tem-peraturetestsarehome-developed,theirreliabilityshouldbechecked before testing.Hence, the temperatureevolution of theheating fur-nace/coolingdevicehasbeenmeasuredexperimentallyandispresented inthenextsection.Inaddition,thetemperaturedistributioninsidethe specimenwasfurtherestimatedbyfiniteelementmethod(FEM)based onthermalheattransferapproach.

3. Calibrationandheattransfermodelingoftheheating furnace/coolingdevice

Thetemperatureevolutionofthespecimenandtheenvironmentis measuredexperimentally,andtheresultisshowninFig.6.Bysetting theenvironmentaltemperaturesoftheheatingfurnace/coolingdevice to be 200°C, 100°C, −25°C, −68°C and −163°C, the corresponding temperatures of the specimen are 172°C, 88°C, −20°C, −60°C and −163°C.ThewaitingtimeT_w forthespecimentemperaturetobestable changes depending on the initial temperature T0. It increases from

T_w =270sat−163°CcontinuouslytoT_w =1820sat172°C.Bychanging theenvironmentaltemperatureoftheheatingfurnace/coolingdevice between −163°C and 200°C, the corresponding temperature in the specimenvariesfrom−163°Cto172°C.

To analyze the temperature distribution in the heating fur-nace/coolingboxmoreprecisely,numericalsimulationsusingCOMSOL

Fig.6. Temperatureevolutioninthespecimenwiththeenvironmental temper-aturevariesbetween−163°Cand200°C.

Multiphysics havebeenconducted.Thethermaltransferis described bythegeneralizedtransientheatequation,Eq.1.

𝜌.𝐶𝑝 (𝑇).𝜕𝑇_𝜕𝑡 −∇.(𝑘(𝑇).∇𝑇)=0 (1)

Thethermal conductivityk(T)andthespecificheatC_p (T) of304 ASSmay be foundin [24]. Theboundaryconditions aredefined in Eq.(2)andFig.7.

- Naturalconvectiveheatfluxq_c onthefreesurfacesoftheheating furnace/coolingdeviceandtheSHPBbars.

- Forced convective heat flux q_f through the free surfaces of the specimenandtheSHPBbarsbyhotair,coldnitrogengasorliquid nitrogen.

- Thermalcontactheatfluxq_int betweeninterfacesofthespecimen andtheSHPBbars.

- Hotairinflowandoutflowwithcertaintemperaturesandvelocities. ⎧ ⎪ ⎨ ⎪ ⎩ 𝑞𝑐 =−ℎ_𝑐 (𝑇−𝑇0 ) 𝑖𝑛𝜕Ω_𝑐 𝑞𝑓 =−ℎ_𝑓 (𝑇−𝑇0 ) 𝑖𝑛𝜕Ω_𝑓 𝑞𝑖𝑛𝑡 =ℎ_𝑖𝑛𝑡 (𝑇−𝑇0 ) 𝑖𝑛𝜕Ω_𝑖𝑛𝑡 (2) whereh_c =10W/(m2_•K)andh

int =105_W/(m2_{•K)arethenaturalheat} convectionandthelayerconductancecoefficient,respectively.Valuesof h_c andh_int maybefoundin[25].Theforcedheatconvectioncoefficient h_f isequalto818W/(m2_{⋅K)whencoolingdownthespecimenbyliquid} nitrogenand109W/(m2_{⋅K)whennitrogengasorhotairisused.Values} ofh_f wereobtainedbyfittingnumericalresultstotheexperimentaldata. A comparisonof temperatureevolution betweenexperiments and numericalsimulationsisshowninFig.6.Itisclearthatagood agree-mentisobtainedforallthefivetemperatures.Therefore,thenumerical

Fig.7. BoundaryconditionsfortemperaturedistributionanalysisusingCOMSOLMultiphysics:(a)lowtemperatureand(b)elevatedtemperature.

simulationscanpredictthetemperatureevolutionanddistributionof theheatingfurnace/coolingdevicecorrectly.

Basedonthenumericalresults,thetemperatureevolutionoffour positionsshowninFig.8(a)wasrecorded;theresultsarethenplotted inFig.8(b).ItisseenthattemperaturesinpointsAandBarealways the same, indicating a uniform temperature distribution within the specimen.Incomparison,temperaturesin points C andD arelower duetotheheatexchangebetweentheSHPBbarsandtheenvironment. Thetemperaturedifferencebetweenthefourpointsincreaseswhenthe testingtemperaturedeviatesobviouslyfromroomtemperature,anda strongtemperaturegradientalongtheSHPBbarsisobservedforT₀= −163°Cand200°C.

AccordingtoFig.8(b),thetemperaturesinthefourpositionsare stableaftera waitingtime of 2000s.At T_w =2000s, thetemperature distributionof thespecimenandtheenvironmentisshowninFig.9. Fortestsatlowtemperatures,asshowninFig.9(a),(b)and(c),astrong temperaturegradientformsalongtheSHPBbars,butthetemperature distributionwithinthespecimenisuniformwithamaximum fluctua-tionof0.05°C.Fortestsatelevatedtemperatures,asshowninFig.9(d) and(e), atemperature gradientis observed from thebottom tothe topof theheatingfurnace,butthetemperaturewithin thespecimen remainsuniformwithafluctuationof1.2°C.

4. Experimentalresultsanddiscussion

4.1. Forceequilibriumstatewithinthespecimen

Thetypicalwavesignalsfromonetestat1564s−1_and172°Care showninFig.10(a).Inordertoobtainanaccuratedeformationbehavior descriptionusingtheSHPBtechnique,forceequilibriumstatewithinthe specimenisnecessary.Hence,aparameterproposedbyRavichandran andSubhash,R(t),isusedtoevaluatetheforceequilibriumcondition. 𝑅(𝑡)=|||| | Δ𝐹(𝑡) 𝐹𝑎𝑣𝑔 (𝑡) || || |=2 || ||𝐹𝐹11((𝑡𝑡))+−𝐹𝐹22((𝑡𝑡)) || || (3)

whereF1(t)andF2(t)arethetwoforcesactingontheendfacesofthe specimen,respectively.TheyarecalculatedasF1(t)=EA(𝜀I (t)+𝜀_R (t)) and F2(t) = EA𝜀T (t), whereE and A are Young’smodulus and the

cross-sectionalareaoftheSHPBbars,respectively.ΔF(t)andF_avg (t)are respectivelythedifferenceandtheaverageofthetwoforces.Aforce equilibriumstateisachievedwhenR(t)isclosetozero.

BasedontheexperimentalresultsinFig.10(a),thevariationoftrue stress,strainrateandforceequilibriumcoefficientwithtruestrainis cal-culatedandshowninFig.10(b).ItcanbeseenthatR(t)isalwayslower than0.1,exceptafluctuationatatruestrainof0.02.Therefore,force

Fig.8. (a)Positionsoftemperaturemeasurementand(b)temperatureevolutionofthefourpositions.

equilibriumstateisachievedfromtheverybeginningofthetestand maintainsuptothemaximumstrain.Theverifieddynamicforce equi-libriumconditionensuresthattheexperimentallymeasuredstress-strain relationsgiveanaccuratedeformationbehaviordescriptionof304ASS. 4.2. Truestress-truestrainrelationsof304ASS

The true stress-true strain curves of 304 ASS at temperatures between−163°Cand172°Candstrainratesfrom10−3_s−1_to3270s−1 areshowninFig.11.Itisseenthatforallthetestedstrainratesand temperatures, theflow stress increases continuously with increasing strainbutthestrainhardeningratesdiffer.Inquasi-statictests(10−3_s−1 to1s−1_{)conductedat172°C,thestrainhardeningratedecreasesslowly} withincreasingstrain;asthetestingtemperaturedecreasesto88°C,the flowstressincreasesalmostlinearly withaconstantstrainhardening rate; with further decrease of testing temperature, the stress-strain curves exhibitan S-shape and a second hardeningphenomenon. As discussedintheintroductionsection,thisphenomenonisalsoobserved byseveralotherauthors[14,17,26,27]inTRIPsteelsandisattributed tothe SIMTeffect. Fortemperatures belowM_d , which is 140°Cfor the304 ASS, martensitic transformation can be triggered by plastic deformation. A mixture of martensite and austenitephases is much harder than pure austenite phase. Therefore, the strain hardening

rate is enhanced. At 172°C, a temperature above M_d , the deforma-tion mechanism changesinto dislocationglide: thestrain hardening rate is controlled by a competition between dislocation generation, accumulation andannihilation [8]. As is pointedout by theunified constitutivemodelproposedbyLinetal.[8],thedislocationsgenerate andaccumulatequicklyatsmallstrains,andthentheincreasingrateof dislocationdensitydecreasesgraduallyatlargestrains.Therefore,small strainhardeningrateatlargestrainsisobserved.Concerningdynamic tests(1550s−1_to3270s−1_{)atvarioustestingtemperatures,theresults} areshowninFig.11(e),(f)and(g).Itisseenthatthesecondhardening phenomenonisnotobviousanymoreasmartensitictransformationis stronglyinhibitedbytheadiabaticheatingeffect.

4.3. Strainratesensitivityof304ASS

Thevariationsofflowstresswithstrainrateasafunctionoftesting temperaturefortruestrainsof0.05and0.2areplottedinFig.12(a)and (b),respectively.Theyrepresentrespectivelythedeformationbehavior of pure austenite phase and a mixture of austenite andmartensite phases.

AsshowninFig.12(a),thevariationofflowstresswithstrainrate atatruestrainof0.05isconsistentwithmetallicmaterialsdeformed bydislocationglide:theflowstressfirstremainsconstantinquasi-static

Fig.9. Temperaturedistributioninthe heat-ingfurnace/coolingdevicewiththespecimen temperaturesstableat:(a)−163°C,(b)−60°C, (c)−20°C,(d)88°Cand(e)172°C.

tests(10−3 _s−1 _to1s−1_{) andthenincreasescontinuouslyin dynamic} tests (1550 s−1 _{to 3270 s}−1_{).A phenomenonshould be noticed: in} dynamicstrainrateregime,thestrainratesensitivityiscoupledwith temperature.Theflowstressincreasesmoreslowlyatlowtemperatures than that at high temperatures. For thermo-viscoplastic behavior modeling,anitemdescribingthecouplingrelationshipbetweenstrain rateandtemperaturesuchastheArrheniusequationisneeded.

Theevolutionofflowstresswithstrainrateatatruestrainof0.2is showninFig.12(b).Inquasi-staticstrainrates,thestrainratesensitivity isdifferentfromthatatthetruestrainof0.05.Fortemperaturesbelow −20°C,theflowstressdecreaseswithincreasingstrainrate,indicating a negative strain rate sensitivity. Inparticular, at −163°C,the flow stressinquasi-statictestsisevenhigherthanthatunderdynamictests. Fortemperaturesabove20°C,theflowstressdoesnotchangeobviously

Fig.10. Atypicalcompressiontestat1564s− 1 and172°C:(a)thewavesignalsand(b)thetruestress,strainrateandforceequilibriumcoefficientvstruestrain curves.

withincreasingstrain rate.Indynamic strainrates,apositive strain ratesensitivitysimilar tothat atthetrue strainof 0.05isobserved. Butthestrainratesensitivityismorepronouncedatlowtemperatures insteadofthepreviouslyobservedhightemperatures.

4.4. Temperaturesensitivityof304ASS

Fig.13showstheevolutionofflowstresswithtemperatureat differ-entstrainratesandtwostrains.AsshowninFig.13(a),theflowstress decreaseswithincreasingtemperature.Thedecreasingtendenciesare around1.40MPa/°Cand1.74MPa/°Cforquasi-staticanddynamic strainrates,respectively.However,from−60°Cto−163°C,thedynamic flowstressincreasesbymerely20MPa:thestrongtemperature sensi-tivitydisappears.AsimilarphenomenonhasbeenobservedinHSLA-65 steelbyNemat-NasserandGuo[28] whenthecompressionbehavior wasstudiedover awiderangeof strainratesandtemperatures. Ac-cordingtothethermallyactivateddislocationmotiontheory[29],the totalflowstresscanbedividedintotwoitemscalledtheinternalstress 𝜎𝜇andtheeffectivestress𝜎∗.Thetwopartsdescriberespectivelythe

strainhardeningeffectandthethermalactivationprocess,Fig.14.𝜎𝜇is

independentofstrainrateandkeepsalmostconstantatdifferent temper-atures,while𝜎∗_{isstronglyaffectedbytheeffectsofstrainrateand} tem-perature.Duringquasi-static(10−3_s−1_to1s−1_{)testsatvarious} temper-atures,𝜎∗_{firstremainsconstantandthenincreasessignificantlyat} tem-peratureslowerthan0.1T_m (T_m referstothemeltingtemperatureof al-loysandis1300°Cfor304ASS).Hence,from−60°C(0.13T_m )to−163°C (0.06T_m ),theflowstressincreaseswithdecreasingtemperature.When thestrainratechangesfromquasi-statictodynamic(1550s−1_to3270 s−1_),𝜎∗_{increasescontinuouslybutthestressdifferencebetweenvarious} temperaturesbecomessmaller.Therefore,thepreviousobservedstrong temperaturesensitivityalmostvanishesindynamictests.

Fig. 13(b) shows the evolution of flow stress with temperature at a true strain of 0.2. Undertheeffects of strain rateon both the transformation process and the dislocation motion, the flow stress decreaseswithincreasingtemperaturebutthedecreasingtendenciesin quasi-staticanddynamictests aredifferent:theflowstressdecreases fasterintheformerconditionthaninthelatter.

5. Thermo-viscoplasticbehaviormodelingof304ASS

Motivatedbyscientific andengineeringdemands, alargenumber ofconstitutivemodelshavebeenproposedtodescribethedeformation behaviorof materials. Basically, they can be dividedinto two cate-gories:thephenomenologicalandthephysicalbasedmodels.Thefirst

categorydescribesflowstressofmaterialsaccordingtoempirical obser-vationsanddoesnothaveanyphysicalbackground,e.g.,Johnson-Cook model [31], Khan–Huang model [32], Fields–Backofen model [33], Molinari–Ravichandranmodel[34],Voce–Kocksmodel[35],Arrhenius equation[36]andtheirvariants[37–40].Anothergroupofconstitutive models are builtaccording to deformation mechanism of materials, suchasZerilliandArmstrongmodel[41], Rusinek–Klepaczkomodel [20],Voyiadjis–Almasri model[42] andBodner–Partommodel[43]. Inaddition,artificialneuralnetworkmodelsareincreasinglyusedin areasofconstitutivebehaviorpredictionastheyprovideacompletely differentapproachtomaterialsmodelingthanthetraditionalstatistical or numericalmethods[44]. Adetaileddescriptionofdifferent kinds ofconstitutivemodelswiththeiradvantagesanddisadvantagescanbe foundin[45].

According to the deformation behavior analysis of 304 ASS in Section 4, a coupling effect of strain rate and temperature exists. Moreover,thedeformationbehaviorisstronglyaffectedbymartensitic transformation.Therefore,aconstitutivemodeltakingthetwoeffects intoconsiderationisneededforaccuratedeformationbehavior model-ing.Inthissection,anextensionoftheoriginalRKmodelconsidering martensitic transformation[21] is used todescribe thedeformation behaviorof304ASS.

Atypicaltruestress-truestraincurveof304ASSat−20°Cand10−3 s−1_{isshowninFig.15.Thecurveisdividedintotwoparts:withinatrue} strainof0.08,nomartensitictransformationoccursduringthe deforma-tionprocess;forlargerstrains,thedeformationmechanismchangesinto acompetitionbetweendislocationslipandmartensitictransformation. Thetruestress-truestraincurvewithoutphasetransformationcanbe extendedbythesamestrainhardeningrate.Theextendedcurveis as-sumedastheflowstressof304ASSwithoutmartensitictransformation andcanbedescribedbytheoriginalRKmodel.Thedifferencebetween therealandtheextendedstress-straincurveissupposedtobecaused by martensitic transformation.The corresponding flowstress can be describedbyanextendeditemcoupledtotheoriginalRKmodel.

Theitemdefiningmartensitictransformationisbasedonthe prob-ability thattheaustenitephase will transforminto martensitephase undercertainconditions.No accuratephase fractionmeasurementis needed todefine the modelparameters. Various martensite fraction measurement techniquescan beused buteach ofthemhasthe limi-tationsanditisprettydifficulttoobtainaccurateproportionsofeach phase:

- Metallographicobservationisconvenientbuttheresultsarestrongly affectedbysamplesurfacepreparationtechniques.

Fig.11. Truestress-truestrainrelationsof304ASSasafunctionoftemperatureatstrainratesof:(a)10− 3 s− 1 ,(b)10− 2 s− 1 ,(c)10− 1 s− 1 ,(d)1s− 1 ,(e)1550s− 1 ,(f) 2370s− 1 and(g)3270s− 1 .

Fig.12. Variationofflowstresswithstrainrateasafunctionoftemperaturefortruestrainsof:(a)0.05and(b)0.2.

Fig.13. Variationofflowstresswithtemperatureasafunctionofstrainratefortruestrainsof:(a)0.05and(b)0.2.

Fig.14. Decompositionofthetotalflowstressintotheinternalstress𝜎𝜇and

theeffectivestress𝜎∗ usingthethermallyactivateddislocationmotiontheory

Fig.16. ComparisonofflowstresscurvesbetweenexperimentsandRKmodelfor(a)10− 3 s− 1 ,(b)10− 2 s− 1 ,(c)10− 1 s− 1 ,(d)1s− 1 ,(e)1550s− 1 ,(f)2370s− 1 and (g)3270s− 1 .

- ForEBSD,thesevereplastic deformationmakestheindexingrate decreasealot,sothemeasurementresultsarerepresentativeonly undersmallstrains.

- ThedetectionareaofXRDislimitedtothesamplesurface. - Magnetic permeability measurement is also used to calculate

martensitefractionbuttheinversemagnetostriction phenomenon cannotbeavoided.

Withoutmartensitefractionmeasurement,theerrorcausedby inac-curatemeasurementmethodscanbelargelyavoided.

5.1. TheextendedR-Kconstitutiveequation

Inspiredbythethermally-activateddislocationmotiontheory[29], theoriginalformoftheRKmodelisgivenasasumoftwocomponents: theinternalstress𝜎𝜇(𝜀𝑝 ,̇𝜀𝑝,𝑇)andtheeffectivestress𝜎∗₍̇𝜀𝑝_{,𝑇),Fig.14.} Theformercomponentisinducedbylong-rangebarrierstodislocation motionandisindependentofdeformationconditions.Incontrast,the lattercomponentiscausedbyshort-rangebarrierstodislocationmotion such asthe interactions of dislocationsand crystaldefects, andcan beovercomebythermalactivationprocess[13].Thetwocomponents aremultipliedby a parameterE(T)/E0 torepresent thetemperature dependenceofYoung’smodulus.

̄𝜎(𝜀𝑝 ,̇𝜀𝑝 ,𝑇)=𝐸(𝑇) 𝐸0 [( 𝜀𝑝 ,̇𝜀𝑝 ,𝑇)+𝜎∗ ( ̇𝜀𝑝 ,𝑇)] (4)

AstrainhardeningequationsimilartotheSwiftlawisusedto de-scribetheinternalstress𝜎_𝜇(𝜀𝑝 ,̇𝜀𝑝 ,𝑇).

𝜎𝜇

(

𝜀𝑝 ,̇𝜀𝑝,𝑇)=𝐵(̇𝜀𝑝,𝑇)(𝜀0+𝜀𝑝 )𝑛 (̇𝜀 𝑝_,𝑇 )

(5) where𝜀0 referstothevaluecorresponding totheyieldpointduring quasi-statictests.

Theeffectivestress𝜎∗₍̇𝜀𝑝 _{,𝑇)definestheflowstressinducedby} ther-malactivationprocessusinganArrheniusequation:

𝜎∗(̇𝜀𝑝_,𝑇)_=𝜎∗ 0 ⟨ 1−𝐷1 ( 𝑇 𝑇𝑚 ) loglog (_̇𝜀 max ̇𝜀𝑝 )⟩𝑚 ∗ (6) where𝜎∗

0istheeffectivestressat0K,D1andm

∗_{arematerialconstants.} Todefinethestresscomponentcausedbymartensitic transforma-tion,athirditem𝜎_{𝑇 𝑟} (𝜀𝑝 ,̇𝜀𝑝 ,𝑇)iscoupledtotheoriginalR-Kmodel.

̄𝜎(𝜀𝑝 ,̇𝜀𝑝 ,𝑇)=𝐸(𝑇) 𝐸0 [ 𝜎𝜇 ( 𝜀𝑝 ,̇𝜀𝑝 ,𝑇)+𝜎∗ ( ̇𝜀𝑝 ,𝑇)]+𝜎_{𝑇 𝑟} ( 𝜀𝑝 ,̇𝜀𝑝 ,𝑇) (7) 𝜎𝑇 𝑟 ( 𝜀𝑝 ,̇𝜀𝑝,𝑇)=𝜎₀𝛼𝑓 ( 𝜀𝑝 ,̇𝜀𝑝 )𝑔(𝑇) (8) where𝜎𝛼

0 referstothemaximumstressincreasecausedbymartensitic transformation.Thevalueshouldbe obtainedbymechanicaltests at thelowesttemperatureofinterest.

𝑓(𝜀𝑝 ,̇𝜀𝑝 )isaphenomenologicalfunctiontodescribetheeffectsof strainrateandstrainontheprobabilityofmartensitictransformation. Itisgivenas

𝑓(𝜀𝑝 , ̇𝜀𝑝)=[1−𝑒𝑥𝑝(−ℎ(̇𝜀𝑝 )𝜀𝑝 )]𝜉 (9) ℎ(̇𝜀𝑝 )=𝜆0exp

(

−𝜆 ̇𝜀𝑝) (10)

where𝜉 isaconstantthatdefinesthestrainvalueforwhichthe austen-itephasestartstotransformintomartensitephase.Thevaluecanbe determinedaccordingtotheinterruptincreaseinstrainhardeningrate ofstress-straincurves.ℎ(̇𝜀𝑝 )isafunctionthatdescribesthestrainrate dependentmartensitictransformation.𝜆0and𝜆 aretwoshapefitting pa-rameters.Adetaileddescriptionconcerningthefittingresultsofphase transformationusingℎ(̇𝜀𝑝 )canbefoundin[21].

Todefinetheeffectoftemperatureonthetransformationprocess,a temperaturefunctiong(T)isproposed:

𝑔(𝑇)=1− ( _𝑇−𝑀 𝑆 𝑀𝐷 −𝑀𝑆 )𝑛 (11)

Fig.17. ThedescriptionerroroftheextendedR-Kmodelinpredictionofthe experimentaldata.

whereM_S refers tothetemperaturebelowwhich theGibbsfree en-ergybetweenaustenitephaseandmartensitephaseishighenoughfor martensitictransformationtooccurspontaneously.M_D isthe temper-atureabovewhichmartensitictransformationdoesnotoccuranymore andthedeformationmechanismchangesintotwinningordislocation slip.nrepresentsthestrainratesensitivityofthetransformation pro-cess.

ThedeterminationofextendedRKmodelparametersisdividedinto twosteps:firstofall,theextendedcurvesof304ASSunderdifferent strainratesandtemperaturesareregardedasthestress-strainrelations withoutmartensitictransformationandarethendefinedbytheoriginal RK model. A detailed description of the fitting procedures can be foundin[30].Afterthat,thestresscomponentcausedbymartensitic transformation is defined by the third item𝜎Tr of theextended RK

model. Inbothsteps, aleastsquaremethod isusedtominimize the errorbetweenthecalculateddataandtheexperimentalresults.

ThetotalnumberofmaterialconstantsoftheextendedRKmodel is12including4parameterstodefinemartensitictransformation.The fittedmaterialparametersoftheoriginalR-Kmodelandtheextended itemareshowninTables2and3,respectively.

5.2. Comparisonbetweenexperimentalandpredictedflowstressof304 ASS

A comparison between experiments and predicted stress-strain curvesaswellasthecorrespondingpredictionerrorsareshowninFigs. 16 and17, respectively.Thepredictionerror Δis usedtoassessthe fittingresultsandisdefinedas

Δ = 1 𝑁 ∑𝑁 𝑖 =1 || || | 𝜎_𝑖 𝑒𝑥𝑝 −𝜎_𝑖 𝑝𝑟𝑒 𝜎_𝑖 𝑒𝑥𝑝 × 100%||||_| (12)

ItisseenfromFigs.16 and17 thatagoodagreementisachieved for testing temperatures between −60°C and 172°C. The prediction errors are respectively 7.6% and4.9% for quasi-static anddynamic tests. According to J.A.Rodríguez-Martínez et al. [46], an obvious temperatureincreasestillexistsinquasi-statictestsofTRIPsteels,and it affects martensitic transformation significantly. However, in this study,thequasi-statictestsareconsideredasisothermal.Therefore,the predictedresultsforquasi-statictestsarenotasgoodasthedynamic ones.Inaddition,duetotheunexpectedlowstrainratesensitivityof 304 ASSat−163°C,Fig.13, thepredictedflowstressis significantly higher than theexperimental ones with anaverage error of 21.2%.

Table2

ThefittedparametersoftheoriginalRKmodelfor304ASSwithoutmartensitictransformation.

E0 (GPa) Tm (K) 𝜃∗ (-) 𝜎∗0 (MPa) D1 (-) m∗ (-) B 0 (MPa) v (-) n0 (-) D2 (-) 𝜀 0 (-) 210 1700 0.9 658 0.58 1.84 1693 0.41 0.39 − 0.19 0.023

Fig.18. ComparisonofstrainratesensitivitybetweenexperimentsandtheextendedRKmodelattruestrainsof(a)0.05and(b)0.2.

Table3

Thefittedparametersoftheextendeditemdescribingthe marten-sitictransformation.

𝜎0 (MPa) MD (K) MS (K) 𝜆 (-) 𝜆0 (-) 𝜉 (-) 𝜂 (-) 1026 413 20 0.60 21.17 10.76 0.82

ThisismainlybecauseintheextendedRKmodel,aphenomenological insteadofphysicalfunctionisusedtodescribethemartensitic transfor-mationprocess.Thefunctionworkswellwithinalimitedtemperature regime. Compared to sophisticated physical models [47–49], the phenomenological approach simplifies the finite element (FE) code implementationprocessandhelpstoreducethecomputationaltime. However,forabettermartensitictransformationbehaviordescription, afurtherimprovementoftheextendedRKmodelisneeded.

Tocomparetheexperimentaldatawiththepredictedonesindetail, theevolutionofflowstresswithstrainratefortwostrains0.05and 0.2areshowninFig.18.ItisclearinFig.18(a)thattheextendedRK modelgivesasatisfactorypredictionofstrainratesensitivityatvarious temperatures, except −163°C. In dynamic strain rates, the coupling relationshipbetweenstrainrateandtemperatureisalsocaptured:the materialshowshigherstrainratesensitivityatelevatedtemperatures.In Fig.18(b),thenegativestrainratesensitivityinquasi-staticstrainrates isnotpredictedcorrectly.Asexplainedbefore,thisisbecausein quasi-statictests,theexperimentallyobservedtemperatureriseisnottaken intoconsiderationwhenperformingconstitutivebehaviormodeling.

Concerning predictions of temperature sensitivity, a comparison betweenexperimentsandtheextendedRKmodelisshowninFig.19. Itisclearthatthemodeldefinesthetemperaturesensitivityaccurately between −60°Cand172°C.InFig.19(b),thedecreased temperature sensitivitywithincreasingstrainrateisalsocaptured.Onthewhole, theextendedR-Kmodelpredictsthedeformationbehaviorof304ASS correctlyfortestingtemperaturesT0≧−60°C.

6. ValidationoftheextendedRKmodel

Theutilityofaconstitutivemodelliesinnot onlyitscapacity of fitting experimentally obtained results but alsoits abilityto predict

deformationbehaviorbeyondthetestingconditions. Toevaluatethe extendedRKmodelandthepreviouslydefinedmodelparameters, nu-mericalsimulationsof304ASSplatesimpactedbyaconicalprojectile atdifferenttemperatureshavebeencarriedout.

6.1. Numericalmodeldescription

The numericalsimulationswereperformed basedon theballistic impactresults of304ASSreportedbyJiaetal.[24].During experi-ments,304ASSthinplateswereimpactedbyaconicalprojectileunder sub-ordnance velocitiesrangingfrom80 m/sto180 m/sandinitial temperaturesbetween−60°Cand200°C.Eachplateisasquarewitha sidelengthof130mmandathicknessof1.5mm.Theconicalprojectile isacylinderwithadiameterof12.8mmandaheightof25mm.Atthe topoftheprojectile,aconicalnosewithanangleof72° ismachined.A detaileddescriptionoftheballisticimpactset-up,theconicalprojectile andthecoolingdevice/heatingfurnaceforlow/elevatedtemperature testscanbefoundin[24].

A3Dfull-sizefiniteelement(FE)modelconsistingoftheprojectile andthetargethasbeenbuiltusingsoftwareABAQUS/Explicit,Fig.20. Thegeometryanddimensionsofthetargetareexactlythesameasthe experimentalones.Thetargetissetasadeformablebody,andthe con-stitutivebehaviorischaracterizedbythepreviouslydefinedextended RKmodel.Toreducecalculationtime,theprojectileisregardedasa rigidbodywithaconstantmassof29g.Thefouredgesofthetargetare fixedandnodisplacementisallowed.Theconicalprojectileisplaced perpendicular to the target with a predefined velocity. For contact betweentheprojectileandthetarget,penaltymethodwithafriction coefficientof0.1isadopted,avaluefrequentlyusedfordrysteel-steel contact [50]. The perforation process is assumed as adiabatic, no heattransferbetweenthetargetandtheprojectileorthesurrounding environmentistakenintoconsideration.

Accordingtoameshconvergencestudy,theoptimalmeshdensity distributioninthetargetisshowninFig.21.Thecentralpartofthe tar-getisbuiltwith107380C3D8Relements(8-nodelinearbrick,reduced integrationelement)withaninitialelementsizeof0.2mm.ALE adap-tivemeshingisusedtomaintainahighmeshqualitythroughout the analysis.Intheexteriorarea,wherenoprojectile/targetimpactoccurs,

Fig.19. ComparisonoftemperaturesensitivitybetweenexperimentsandtheextendedRKmodelattruestrainsof:(a)0.05and(b)0.2.

Fig.20. Thefiniteelementmodelofballisticimpacttests.

Table4

Failurestrainsof304ASSplatefordifferenttesting tem-peratures.

Initial temperature ( °C) − 60 − 20 20 200 Failure strain, 𝜀 f 0.49 0.55 0.6 0.67

C3D8Relementswithaninitialelementsizeof1.5mmareadopted.In thewholetarget,fiveelementsacrossthethicknessdirectionareused. Thismeshdensityisrecommendedbyseveralauthorswhenmodeling ballisticimpactbehaviorofthinmetallicstructures[50,51].

Tobeabletocapturethefractureprocess,afailurecriterionwith element deletion is necessary. According to the work of Kpenyigba etal.[51],aconstantfailurestrainforeachprojectileshapeisableto producenumericalresultsinagoodagreementwithexperiments.Inthis work,aconstantfailurestrainisassumedforeachtestingtemperature. Accordingtoanoptimizationofballisticcurves,thefailurestrainsfor differenttestingtemperatureswereestimated,Table4.

Fig.22. Comparisonofballisticcurvesbetweenexperimentsandnumericalsimulations:(a)−60°C,(b)−20°C,(c)20°Cand(d)200°C.

6.2. Numericalresultsintermsofballisticcurvesandfracturepatterns TheexperimentalresultsintermsofballisticcurvesV_R -V₀are pre-sentedinFig.22.Thecurvesarethenfittedtotheequationproposed byRechtandIpson[52],Eq.(13).InEq.(13),theresidualvelocityof theprojectileV_R iscalculatedasafunctionoftheinitialvelocityV₀, theballisticlimitvelocityV_bl andafittingparameter𝛼.

𝑉𝑅 =(𝑉0𝛼−𝑉𝑏𝑙 𝛼

)1∕𝛼 ₍₁₃₎

Acomparisonbetweentheexperimentalandthenumericalballistic curves is shown in Fig. 22. A good agreement is observed for all thefourtemperatures. Withtheincreasingtesting temperature from −60°Cto−20°C,20°Cand200°C,thepredictedballisticlimitvelocities decreasegraduallyfrom108m/sto106m/s,100m/sand87m/s.The predictionerrorsofV_bl forthefourtemperaturesarerespectively4.9%, 2.9%,4.2%and6.5%.Inaddition,theevolutiontendencyofV_R with V₀ isalsocapturedcorrectlybynumericalsimulations.Underallthe fourtemperatures,the V_R -V0 curves show aparabolic shape.Fitting thenumericalV_R -V0curvesintoEq.16,thefittedvaluesofparameter

𝛼 fordifferent temperaturesareshownin Table5. Itis seenthat in bothexperimentsandnumericalsimulations,valuesof𝛼 increasewith increasingtesting temperature,indicatinga deterioratedballistic im-pactresistanceandalowerballisticlimitvelocityathightemperatures. Therefore,theevolutionsofV_bl and𝛼 areconsistentwitheachother.

Table5

Parameter𝛼 underdifferenttestingtemperaturesinbothexperimentsand numericalsimulations.

Initial temperature ( °C) − 60 − 20 20 200 Values of 𝛼 using experimental V R - V 0 2.285 2.239 2.713 2.766 Values of 𝛼 using numerical V R - V 0 2.028 2.005 2.712 2.660

ForacompletevalidationoftheextendedRKmodel,thepredicted failure mode is compared to the experimental ones, Fig. 23. It is seen thatpetalling resulting fromradialnecking duringthepiercing process isobserved, thesameasexperiments. Plasticdeformationis onlyobservedinpetalsoftheperforatedspecimens,especiallyonthe fracturesurfacewherethematerialexperiencedthelargestdeformation untilfailure;plastic deformationin theotherparts ofthespecimens isprettylimited,withamaximumvalueof1.1%fordifferenttesting temperatures.

Thenumberofpetalsvariesunderdifferenttestingtemperatures.A comparisonofthenumberofpetalsbetweenexperimentsandnumerical results isshown inTable 6. Duringnumerical simulations,thevalue decreasescontinuouslyfrom6at−60°Cto4at200°C.Inexperiments, asimilartendencyisobservedbutthecorrespondingvaluesare com-parativelysmaller.Inanycase,tohaveabetterunderstandingofthe

Fig.23. Comparisonoffracturepatternbetweenexperimentsandnumericalsimulations:(a)−60°C,(b)−20°C,(c)20°Cand(d)200°C.

Table6

Experimentalandnumericalnumberofpetalsasafunctionoftesting tem-peratureatV₀ =110m/s.

Initial temperature ( °C) − 60 − 20 20 200 Number of petals in experiments 5 4 3 3 Number of petals in numerical simulations 6 5 4 4

failuremode,afurtherstudyonthefailurebehaviorof304ASSasa functionofstrainrate,temperatureandstressstateisnecessary.

7. Conclusionsandremarks

Thethermo-viscoplasticbehaviorof304ASShasbeensystematically studiedoverawiderangeofstrainrates(10−3 _s−1 _to3270s−1_)and

temperatures(−163°Cto172°C).Dynamictestsatlow/elevated tem-peratureswereconductedusingtheHopkinsontechniquecoupledwith specificallydesignedcoolingdevice/heatingfurnace,andreliabilityof thetechnique wasverified bythermal simulations.The deformation behaviorof 304ASSwasanalyzedin termsofstrainhardeningrate, strainratesensitivityandtemperaturesensitivity.Basedon experimen-talresults,anextensionoftheRKmodelconsideringSIMTeffectwas usedtodescribetheconstitutivebehaviorof304ASS.Thecorrectness of the extended RK model was further verified through numerical simulationsof ballistic impact tests at various testing temperatures. Severalnoteworthyconclusionsaredrawn:

1 Withthedevelopedcoolingdevice/heatingfurnace,dynamictests attemperaturesrangingfrom−163°Cto172°Ccan beconducted usingtheHopkinsontechnique.Accordingtothenumericalresults, thetemperature distributionwithin the compressionspecimen is uniformwithamaximumfluctuationof1.2°C.

2 Thedeformationbehaviorof 304ASSis dominatedbya compe-tition between dislocation glide and martensitic transformation. Bothstrainrateandtemperaturehavesignificanteffectsonthetwo mechanisms.Hence,thedeformationbehaviorof304ASS,mainly representedbythestrainhardeningrate,strainratesensitivityand temperaturesensitivity,isdifferentfrommetallicalloyscommonly deformedbydislocationslip.Severalunexpectedphenomenasuch astheS-shapedstress-straincurves,thenegativestrainrate sensi-tivityandthechangingtemperaturesensitivityfromquasi-staticto dynamicstrainrates,areobserved.

3 From−60°Cto−163°Candunderdynamic strainrates,the tem-peraturesensitivityof304ASSwaslowerthanthatfortheother temperatures. According to the thermally activated dislocation motiontheory,thisismainlycausedbythecomparativelysmaller effectivestress𝜎∗_{increasefrom−60°Cto−163°C.}

4 The extended RK model was used to describe the constitutive behaviorof304ASSwithanadditionalitemlinkedtomartensitic transformation.Themodelpredictedtheflowstresscurvesof304 ASSabove−60°Ccorrectlywiththeseveralunexpectedphenomena beingcaptured.Thepredictionerrorsforquasi-staticanddynamic testswere7.6%and4.9%,respectively.

5 The utility of the extended RK model was further evaluated by numericalsimulationsofballisticimpacttestsatvarious tempera-turesbetween−60°Cand200°C.Thenumericalresultsintermsof ballisticcurves werecompared toexperiments:both theballistic limitvelocitiesV_bl andtheballistic curves V_R -V0 werepredicted accurately. In addition, the failure process was also captured: boththefailurepatternandthenumberof petalsunderdifferent temperatureswerepredictedcorrectly.

Authorstatement

TheideaandstructureofthearticlewereproposedbyAlexisRusinek andRaphaëlPesci.Theset-updesignandexperimentswereconducted byBinJia,RichardBernierandSlimBahi.BinJia,AlexisRusinekand Slim Bahi performed constitutive behavior modeling and numerical simulationsofballisticimpactbehavior.Thepaperwritingwasmainly donebyBinJiawithsuggestionsandmodificationsfromAlexisRusinek andRaphaëlPesci.Allauthorsreadandapprovedthefinalmanuscript.

DeclarationofCompetingInterest

Theauthorsdeclaredthattheyhavenoconflictsofinterestinthis work.Wedeclarethatwedonothaveanycommercialorassociative interestthat represents a conflictof interestin connectionwith the worksubmitted.

Acknowledgements

PartofthisworkwasfinanciallysupportedbytheChinaScholarship CouncilunderGrant201606220056.

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