LA-12641

1

1

II1

I

UC-700andUC-704Issued:November1993

DeviatoricStressesand PlasticStrainRatesin StrongShockWavesfor SixMetals

DavisL Tonks

LosAlamosNATIONAL LABORATORY

10sAlsmos,NewMexico87545

UIS?I?WUTION!W?YISIIOINMFNTN UBWWTE$

II

Deviatoric Stresses and Plastic Strain Rates in Strong ShockWavesforSixMetals

Iby

DavisL. Tonks

ABSTRACT

The strongshocktheoryof D. C. Wallace[Phys.Rev.B24,5597(1981)andPhys.Rev.B24,5607(1981)]is used to calculatethe shockstructurefor 1100Al, 2024Al, CU.Fe, Ta, and U.Emphasisis givento the behaviorof plasticity,i.e.,averagedeviatoricstresses,plasticand totalstrains,andstrainrates,whicharegivenin figuresfora numberof shockstrengths.Thisinformat-ion willbe usefulformodelingplasticityin metalsunderextremeconditions.It wasusedforpartof the PTWmodelformechanicalbehavior.

I. INTRODUCTION

In recentyearsWallacehaspresenteda theoryforstrongshockwavesin metals.192Thisthe-ory incorporatesbothheat flowandplasticflowas dissipativemechanisms.Heatflowis neces-saryat the beginningof theshockwavetodrivethepressureuptovaluesunattainablebyelasticdeformationalone.Plasticflowis necessarylateron in the shockwaveto producethe enormousamountsof heatto do this.

Weakshocks,314on the otherhand,requireonlyplasticflowto producethe necessaryshocksteepness.since this is less than that achievableby elasticstrainsalone.The transitionbetweenweak and strong shockwavesoccurswhenthe elasticprecursorin the weakshockbecomesabsorbedintothemainshockriseas thissteepenswithshockstrength.Thisconditionis calledan“overdrive” shockwave.

When the strongshock is steady,as is assumedfor mostexperimentallymeasuredstrongshockwaves,’ the shockvelocitycan be usedto integratetheequationsof motionto obtainthenormalstressas a functionof the volumetricstrain.Whhthis behaviorknown,the plasticflowbehaviorcanbededucedfromthatof theheatflowwhichis knownunderstrongshockconditions1,2SincetheShmk-ve]mit}’‘ Pficle-fromfairlyreliableextrapolationsusingsolidstatephysics.velocitybehavior is knownexperimentally,this meansthat the plastic flow behaviorcan beinferredfromthe knownheat flowbehavior,the shock-velocity/ particle-velocltybehavior,andthe knownthermoplasticbehaviorof the material.The latter is not knownas well as one wouldlikeunderMbarshockconditions,but isknownwellenoughto makeestimates.

Usingthis theory,the deviatoricstresses,(deviatoric)plasticstrains,and plasticstrainratescan,in principle,be found.ThethermoplasticbehaviorunderMbarshockconditionsis notknownwellenoughtocalculatethedeviatoricstresscontinuouslythroughtheshockpath,butanestimateof its averagethroughtheshockpathcanbe made.1$2Thisestimateis basedon theheatingeffectof plasticwork.Theaverageddeviatoricstresscanthenbe usedtocalculateapproximatetimeandspaceincrementsthroughtheshockpathfromwhichapproximationsto theplasticandtotalstrainratescanbe obtained.

The purposeof thisreportis to carryout theabovecalculationfor six metals:1100Al, 2024Al,Cu,Fe,Ta,andU.Thefocushereison theplasticbehaviorin termsof theaveragedeviatoricstress,plasticand total strains,andstrainrates.Thesewillbe reportedin figures.The heat flowand entropy are not so reportedbut can be reproduced,if needed,using the equationsgivenherein.

The emphasishere is on the calculationaldetailsrather than the generaltheoxywhich isdescribedadequatelyin Refs.1and2.Theequationspresentedherewillprovidethereaderwithareasonableideaof thegeneraltheory,however.

SectionII containsan enumerationof theequationsthatwereintegratedto obtainthe infor-mationin the figures. Someexplanationof theirphysicalbasisis given. SectionIII containstheresultsforeachmetal,togetherwitha listingof theparametersusedforeachin thecalculation.

L Equations Used and the UnderlyingPhysica!Basis

The equationsgivenhereweresuppliedto theauthorLj Wallace,sMostappearin thepub-lishedpapers,1,2but thoseinvolvingthe !ine~ approximationfor tk shock ve!ociw-pficlevelocitydo not.

Theintegrationthroughtheshockpathdependson anapproximationto the truepathin a plotof thetemperature,T,versusthecompression,E.TheapproximatepathispicturedinFig. 1,‘whichis reproducedfromFig. 1 of Ref. 2.2This approximationwacfoundby investigatingrigorousboundson thispathof temperatureversuse fromthebehaviorof variousidealmaterials,e. ., an

Finviscidfluid.1The approximatepatb is boundedfairly closely by these considerations andshouldgiverealisticresultsforthepresentcalculations.

Thefirstpartof theapproximatepathofT versuse followsthatof J nonplasticsolid,i.e.,onein whichonlyelasticstrainsandheatconductionoccur. Thisnonplastic-solidpathis followedtoa certainpoint,calledpointc, at whichplasticflowis assumedto becomeimportant.The pathofT versuss frompointc to theendof theshockwzveis takento be a straight-linefrompointc tothe end point on the Hugoniot,which is knownfromthe experimentalshockvclociiy-particlevelocityrelotion.Thisstraight-lineportioniscloselyhoundedusdescribedbyWallacein Ret’.2.2Pointc and the straight-lineportionare determinedby requiringthat the hnc be tungentto thenonplitsticsolidcurveat pointc.

In Fig. 1theapproximateshockpathusedforPt inRef.2 isshown.Thispathfollowsthelinelabelednonplasticsolidin Fig. 1untilthepointhbcled bys,. is reached.Theretifter,thepath fol-lowsthestraight-lineindicatedby thedashed1ineinFig. 1. Fig. 1idsoshowsthepathof an invis-cid fluid,which is an upperbound,and me line whoseordinateis the Hugoniottemperature,whichis alsoan upperbound.

Thus,the integrationup theshockpathconsistsof twostages. The firstfollowsthenonphus-tic solidandthe secondfollowsthestraight-lineportionwhereplasticflowis occurring.The firstpart iscalledthe (heat)conductionfrontandthesecondiscalledthe flowregion.

Beforelistingtheequationsof the shockpath,the modelingof the thermoelmticdescriptionwillbcdescribed.Theproductof py, wherep is thedensityandy is theGruneisenparameter,isawumedmbeconstant.Hence,ptiy(,,thevalueof thisproductforthe initialcondition,labeledbya. givesy as a functionof p. The ratioof GLB,whereG is theshearmodulus,andB is the bulkmodulus,is assumedto beconstant.ThisgivesGas a functionofB.Bon theRayleighlinecanbeexpressedin the followingwaywhenpy = paya is assumedandwhenthe shockvelocity,D, isgivenby

D = C +sV, (1)

whereV is theparticlevelocity,andc ands areconstants:

B = payaD*&(1 -e) +pac~(1 -e) [1 + (S-ya)s] (1 -s&)-S, (2)

wheree, thecompression,is 1- pa/p.

3

xmg

i=

Pf 2.5 Mbar #-/0’ -\

/ \/ \

,Z INVISCID \ .

/

/

/

FLUID \

\

TH —

NONPLASTIC

.

I I 1 I I0.1 0.2

e

Fig. 1. UpperA lowerboundson theshockpath:temperatureversuscompression.Theshockpathusedhereis thatof thenonplasticsolid10IIowwxIby thedashedIincto theHugoniotpoint.

The assumptionthat G/Bis a constantis madefromlackof information.However,it doesseemto give retiscmableresultsmostof thetime. It givesresultsmorereasonablethanthoseofanothermodel(notgivenhere)thatincludedtemperatureandderwitydependence,butproducednegati~oe(unphysical)strainrates.Thecalct.tkltionsare rathersensitivein thismannerto G.

Wal!ace’s~oments onG as a functionofT andP,pressure,are~ follows.5“G(R7’)1sdiffi-cult to estimate. TheT-dependenceof G is smallandeasy to estimate.but the p-dependenceis]mgeand cannotbe estimatedfor 8p/p 20.1. The onlywaywe willget G(P) is byelectronicstructurecalculations,in thedistantfuture. In themeantime,our bestestimateforG is stillGfl =constant,on theRayleighlineforoverdrivenshocks.”Supportfor this1sRev.B24.5607(1981),AppendixB.z

Theheatcapacityat constant\“olume,C,.is givenby

c,, = 3Nkb+ rT,

where3Nk~equals5.9858cal/moleK. rT is theelectroniccontribution.

r(p) s r (poJ (P(/P)~$

in D. C. Wallace,Phys.

r is givenby:

(3)

(4)

where~ isa constantand p. is thedensityat zerotemperatureandpressure.

r~p(,)= ,.,t#/.3) ,y@z(~F], where11(e}.) is thedensityofelectronicstatesat theFe~i

energyfor p ~=po. gequals2/3.5/3.and7/3fornearly--free-electronmetals,d-bandmetals,and

f-bandmeta!’,.b 3NkLis thelatticecontributionto C,.andis quiteaccurate(to* 5%).5FurthercommciitsbyWallaceon themodelingfor C,,areas follows:“Theelectroniccontributionis acrudeapproximation,butis thebestwecandowithoutextensilreelectronicstructurecalculationsforeachmetal. 1estimateit to beaccurateto~ 40%on theRayleighline,forshocksto melting.Notefor mostcases,thetotal C,.willstillbequiteaccurate.

““ForAl and Cu. I take n (&F)from ‘“CalculatedElectronicPropertiesof Metals,”by V.L.IMoruzzi.J. F..lartak.andA. R.Williams(Pergamon.hT.Y..1978).7

‘“Feundergoesu transitiontx+ c [beeto hcp)at 300Kand 130kbar. Foroverdrwenshocks,Fe mdergoes two more transitionson the Hugoniot:&+ y (hcp to fee) at 2.0 M&w and~+ liquid at2.43 Mbar. Hence,Fe is close-packedin all the overdrive shockregime. Sincen (~~) is aboutthesamefor e andy phases,I ignorethedifference.I taken (SF) at P. fromDOA. Boness,J. hf. Brown.and A. K. McMahan,Phys. Earth and PlanetaryInteriors42, 227(.1986).8

‘“Tiand U presumablyundergotraditions on the Hugoniotto bccbeforemelting. I am notawareof any informationon this. Taken (EF) fromthe recentworkof OlofErikssongandalsoforTa.

The figuresto come containvaluesfor melt temperaturesunder variousconditions.Themodelusedtocalculatethemisthefollowing:l” ~

2~m-3Tin(p) = Tm(P$m)(p/p~m) , (5)

whereTn,is themelttemperature,ympisa specialGruneisenparameterfor themeltlaw,and p~mis thedensityat meltof thesolidat zeropressure.Themelttemperaturedoesnotfigureinthecal-culationsalongtheshockpath(Rayleighline).

The thermalconductivityK canbe takento be a constantfor theconditionshere.*Wallace’scommentson the valueschosenare as follows:s“K entersonlyin the time-dependenceof theshockprocess,whichit determinescompletely.I expectT-dependenceof Kto be negligible,andK to increasewith p as p to pz.l’zSeealsomy shocktheorynotes,SectionN, for moredetail.Since p nearlydoublesat meltingon theHugoniot,the p-dependenceof Kis worthincludingintheestimate.Dothisby taking,for K= consfant,twicethehigh-temperaturevalueat normalden-sity; then K shouldbe accurateto withina factorof twofor all the shockcalculations.Notethat,explicitly,all the thermodynamiccoefficientsare designedto apply in the regionbetweentheHugoniot,andtheRayleighlinefcr theshockwhichreachesmeltingon theHugoniot.”

Theshockpathcalculationsdependon knowingtheendpointon the Hugoniot.Thispointisdeterminedin theliquidapproximation,i.e.,onlypressureis involved,nodeviatoricstresses.Forstrongshockwaves,wherethedeviatoricstressis smallcomparedto the averagestressor pres-sure,thisshouldbean adequateapproximationforthepresentpurposes.

Given PH, the pressureon the Hugoniot,onecan calculatethe quantitesD,&wSH,-Sa, .andTH,whereS is theentropy,bymakinguseof the followingqlliltiOllS:

D = c/(1 - S&H) (6)

PH = pac%H/ ( 1- SEH)2

c%;dc~dSH=

TH[ 1_ S&H) 3

dTH= yaTHd~H+ THdSH/c,..

(7)

(8)

(9)

The H subscriptaboverefersto the Hugoniot.The last two equationsare to be integratedsimultaneouslyup to thepointon theHugoniot.Theseequationsrelyon Eq.(1),the linearshock-velocityJ particle-velocityrelationgivenearlier,and on thejump conditionsthroughthe shockfront.Sa is theinitialentropy.

The followingthreeequationscan be integratedsimultaneouslyto producethe pathof thenonplasticsolid. Thisportionof theshockpathis theconductionfront. J is theheatcurrent.

6

dJ~D= {D~[l - ( 1+ ;:)yat]} -C2 ( I + ~;) [1+ (s-ya)&] (l– S@-3 }ds/ya (10)

dS = ~~TpaD1 dJ

dT = yuTd&+——C\,paD“

(11)

(12)

Eq. (10)arisesfromthe effectsof strainsandheaton the averagestress(pressure). Eq. (11)describesthe productionof entropydue to the heat flux,the onlydissipativemechanismof thenonplasticsolid.Eq. (12)describesthechangein temperaturedueto changesin entropyandtotaIstrain.Allof theseequationsarespecializedto theRayleighline.

Thepointcon theapproximateshockpath,wherethenonplasticsolidbehaviorendsandthestraight-lineportionof theplasticflowregionbegins,is determinedby the solutionto the follow-ingequation:

dT1TH_ TC

z ,. = EH-ec(13)

wherethec subscriptdenotesquantitiesevaluatedat pointc.

In the flowregion,thetemperatureversuss is assumedto havethe followingbehavior:

T(e) = T, +X8, (14)

whereX is (TH- Tc) ‘(&H-&c)andintegratedalongthestraight-linepath:

T, is TC-XEC.The followingthreeequationsare to be

TdS = C,.(X-yaT)d& (15)

dS = TdS/T (16)

dw = {paydTdS+[(1 -s)-1 (B+ ;G) -paDz]ds } /(2G) , (17)

whereB(e) is givenbyEq. (2)and~ is theplasticstrain.

Eq. (17)isbasedon theeffectofentropyandstrainon thepressurealongtheRayleighline.

Whentheabovecalculationis finished,(%),theaveragevalueof thedeviatoricstress(calcu-latedaccordingto theheatingeffectof plasticwork)canbeevaluatedas follows:2

(JC/pa)D+ ;TdS

<T> =( Vc + V/f) ;H

(18)

7

where VC+ VHequals(2 -Cc- CH)/Pa.

Using (?), approximatetimeandspaceincrement..canbecalculatedthroughthe plasticflowregion. Thesecan thenbe usedto obtainapproximateplasticand total strainrates. Thisproce-durecanbe accomplishedusiiigthefollowingequation:

dZ = (-K)dT/[ (1 -e)~] , (19)

whichis Eq. (14)fromRef.2. Here,dZ is theLagrangianspatialincrement.FromEq. (14)(thisdocument),dT canbe re-expressedas Xds.ThetimeincrementcanbeobtainedfromdZbydividingit by theshockvelocity,D. Toevaluatetheaboveexpression(19),J is needed.J canbeapproximatelyobtainedintheflowregionbyintegratingthefollowingmodifiedversionofEq.(5)fromRef.2, in which<%>is substitutedfor ~:

dJ/ (paD)- TdS- (Vc+ V/./)<My, (20)

whereVCUd VHarethespecificvolumesatpointc andon thetheHugoniot,respectively.Theapproximateplasticstrainratethatcanbeobtainedfromtheaboveequationcanbe averagednumericallyto obtainan averagestrainrate.

This numericalaveragingwas done with both plasticstrainand compressionused for theweightof averagingto producetwoversionsof theaverageplasticstrainrate throughthe plasticflowregion: (~)V. and (~)c(In the approximateshockpathusedhere,no plasticflowoccursintheconductionfrontregion.)

A roughvalueof theaverageplasticstrainrateis givenby

[

~@JC/paDl ( 1‘&H) WHW() = - y’,— AW ] (~”- &C)~’ (21)

whereAWstandsfor theatomicweightingtimole. Thisrelationcanbeobtainedfromthe aver-ageof ~ w“ithrespectto &,shownbelow,by applyingcertainapproximations.

(w) = ~wdc’’%-ec) . (22)c

Equation(21)canbe obtainedfromthe aboveby the followingsteps.First,replace~ in theintegrandby d~ Idt.Thensubstitutefor dt the form- (KXcfe)/ (D ( 1– S)J) describedabove.Then,in thisexpression,replaceJ byJ~2, itsapproximateaveragethroughthepathof integration.Next, replace l-& by its approximateavenge, (1-eH), ~d fin~y, ~Pl= ~~~V/~~ byv~l (EH- Q .

8

—-----.———

Theunitsiritendedforuseinallof theaboveequationsare thefollowing.ctnips forD; Mbarfor stress:cat!hole K forentropy;Kelvinfor temperature;cal/molefor dJ/ (paD) ;andcaL/s.cmfor ~X.Whentheseunitsareused,~ comesout in unitsof 1012/s, whichwillbe usedin the fig-ures,

11. Results

Fig. z for R is reproducedherefrGmRef.2 to showtypicalresultsthroughtheshockfrontforthetemperature,entropy,T.CC, and v a functionof&fora shockstrengthof 0.5 Mbar.The ver-ticallinelabeled&cdividesthe (heat)conductionfrontfromtheplasticflowregion.Tcan be cal-culatedin the formerbut onlyits averagecanbe obtainedin the latter,as mentionedpreviously.Plasticflowis assumedto occuronlyin theflowregion.

Theremainingfiguresarearrangedforthedisplayandinterpretationoftheaveragedeviatoricstresscr>, obtainedbytheheatofplasticwork.Thisquantityis, in effect,averagedwithrespectto plasticstrainthroughthe plasticregion. Theotherquantitiesaregivento helpunderstandtheconditionsunderwhich<- arose.Forconsistency,theyshouldalsobeaveragesthroughtheplas-tic flowregion. However,in preliminarycalculations,whenthequantitiesG, P,T. and Tmwereaveragc~withrespectto v ands, theresultsdifferedbyonlya fewpercentfromthevaluestakenat the pointalongtheshockpathwhereone-halfof the finalplasticstrainis achieved. Basedonthis similarity.for simplicity.all quantitiesgiven to help interpretc= are taken at this point,labeledb}’d. They can be consideredto be verygoodapproximationsto averagesthroughtheplasticflowicgion.Hugoniotpressuresandtotalstrainsarealsogivenin figuresto showwhatshockstrengthsare involved.

As describedearlier,theplasticstrainratewasaveragedthreedifferentw-ays.The firsttwo,yieldingc@c and CQ>V,involveaveragingapproximatestrainrate valuesthroughthe shockfront using total and plastic strainsas weightingfactors. These two versionsare seen in theremainingfiguresto be similar.ThethirdwayproducedC*>O ofEq.(21),whichis an approxi-mateversionof <@t. Inthefigures,thisversionisseento departfromtheothertwoforweakershockwaves. All threeof theseversionsdependuponusing<e in placeof ~ itselfin Eq. (20)describingdissipation,sincet itselfcannotbecalculatedwithsufficientaccuracy.2

In the figures,thelabelZog<dV/dt>on they-axisis usedtodenotethesethreeaverages.

Table1givesthevaluesof physicalparametersusedto generatetheremainingfigures.

The ya - valueusedforFe (2.5)in thestrongshockconditionis theupperboundsuggestedbyBrown and McQueen.13This y-value is C]OWto that of the liquid State,whosey-value isexpectedto be approximatelythatof theclose-packedphasesof Fe shockedabovethe transitionto the E -phaseat 130kbar.Whenthe value 1.7,the lowerboundof Brownand McQueen,wasusedin the calculation,negativestrainratesresu!teddue to the temperatureat pointc (at theendof theconduction-frontportion)beingalreadyhigherthanthaton the Hugoniot.Theslopeof thetemperatureversus&plotwasnegativefor theplasticflowregime. This indicatesthat 1.7is toolowfor strongshockwaves.

ThevaluesusedforGZBarethoseunderambientconditions.At firstsight,valuespertainingto highcompressionsandtemperaturesmightseemmoreappropriate.However,it is veryimpor-tant in thecalculationto accuratelycalculatethe(initial)thermalconductionportionof the shock

9

.

500

1-

300

200

0.02

c

. .

Pt 005 Mbar

1---- -- <r>—-——-— ———-

9

I0.02 0.04 0.06 0.08 0.10

).0

.0

.0

)

).08

).04 *

).

Fig.2. Tcmperaturc.T;entropy,S; pltisticstrain,V: anddcviatoricstress,~ plottedversuscompression.&;cidculatcdfora 0.5 Mharshockin Pt. Thedottedlinegivestheaver-age ~ in the flowregion.

pathtogetinaccuratetemperatureatpointc,wheretheplasticflowportionbegins.Thistemper-aturevalue,togetherwiththetempe.mtureon theHugoniot,determinetheplasticflow.It is the G/Bbehaviorcloserto ambientconditionsthatis importantfor theconductionfrontandfor thetem-peratureat pointc. WhenvaluesforGIBappropriateforstrongcompressions,i.e.,lowervalues,weretriedin the calculation,negativeplasticstminratesresulteddueto TCbeingalreadyhigherthanthetemperatureon theHugoniot. .

IQuantity [ 11OOA1I 2024AI

I AW(gm/inoZe) I 27.0 I 27.0

c(cm@) 0.5386]] 0.5328]2

s 1.3391’ 1.33812

pa (gticm3j 2.71311 2.785

Ya 2.0 2.0

m

R)(WCM3) 2.73410 2.73410

I 23I 2.3 .,

(10A caZ/..oleK2) I I

g~36 336

pSm(g~Cm3) 2.54210 2.54210

~.(psm)(m 933.01n 933.010

%n 2.210 2.210

I K(CahOn~) I 1.1 .I 10

Table1:MaterialParametersUsedinCalculi

Cu Fe Ta u

63.54 55.85 180.948 238.04

0.3933’1 0.395513 0.32931’ 0.248714

1.501] 1.58013 1.30711 2.2014

8.93310 7.87~0 16.75’0 19.07

2.0 2.513 1.7 2.1

0.35 0.5 0.36 0.75

9.0210 7.9210 16.810 19.2

1.6 I 1103 6.8 12.4

~36 I5/36 I5/36 I7/36 I8444)10 7.25410 15.3 17.5

1357.0]0 1809.0]0 3287.010 1407.()2.010 ~,710 1.610 2.1

1.8 0.2 0.3 0.2

ionThroughShockFront.p. is atzerotemperatureandpressure.Theinitialtemperaturefor theshockfrontcalculationwas~en to be 2!?3KAllunreferencedvaluesareprivatecommunicationsfromD. C.Wallace.

To assessthe sensitivityof the resultsto the valuesof GLBand y, thesevalueswere variedaboutthe standardvaluesin Table1. ChangingGA?by 10%producedthevariationin<-/Gdof5 to 10%andpr~ducedthevariationin log <~>t, of a fewpercent.AnexceptionoccuredforCufor the strongestshockcalculated,for whichthe 10%changein G/Bproduceda 4% variationin<~>c. Changingy by 10%producedthevariationin CT>/Gd of2 through490,-q forFeandTaforwhichthe 10%changein y producedthevariationin <-/Gal of 5 through10%.The10%changein y producedfor mostmaterialsthe variationin log <~>c ofonlya fewpercen~exceptforCufor thestrongestshockcalculated,forwhichthisquantityvariedby 7%.

11

Table2 givesvaluesof SOand PO,thecompressionandpressureat theoverdrive thresholdwherethestrongshocktheorybreaksdown.Fromthistable,thereadercanjudgehowcloselytheoverdriventhresholdhas been approachedin the figures.Except for Cu and T~ it was notapproachedveryclosely.

9

Quantity 1100Al 2024Al Cu Fe Ta TJ

&. 0.127 0.127 0.116 0.143 0.136 0.133

POMbar) 0.146 0.146 0.235 0.293 0.366 0.314

Table2: SOand PO,thecompressionandpressureattheoverdrivethshold fortie sixmew.

Figures3 through7 showcalculatedresultsfor 11OOA1.All but Figure7 involve (t) and<d~/dt>, whichare averages throughthe plasticflowregion,and involvequantitieslabeledwithad-subscript,whicharegoodapproximationsto averagesthroughtheplasticflowregion,asdiscussedearlier.<r>/Gd is presentedsince~ is commonlyrecognizedto scalewith G. Figure7 givespart of the Hugoniotto showthe finalstrengthof the shockwavesfromwhichthe otherquantitiesweretaken. Theshockstrengthsof Fig.7 are thesameonesusedin Figs.3 through6.Thesameis truefor thecorrespondingfiguresfor theothermaterials.

The temperatureplotsshowthatthecalculationswereextendedupto thep~intwheremeltingoccurredat the pointof half-finalplasticstrain.Meltingoccurredsoonerthanthis on the Hugo-niot,of course.

Figures8 through12showcorrespondingresultsfor 2024Al. Theystronglyresemblethosefor 1100Al, sincethephysicalpropertiesof thetwoaresimilar.

Figures 13through18showresultsfor copper. Notein Fig. 13that the estimatefor log ~takenfrom Eq. (21)departs,for the weakershockwavesnear the overdriventhreshold,signifi-cantlyfromthetwonumericalaverages.ThisshowsthattheestimateofEq. (21)departs fromtheotherestimatesneartheoverdriventhreshold,wherethegeneraltheorybreaksdown.

Figure 18has logscaleson bothaxesto showthat <d~/dt> and <%>/Gdroughiyobeya15frompressure-shearpowerlaw. The fivesquaresymbolslowerdownaredataof Cliftonet al.

plate-impactexperiments. Note that thesedataand the calculatedstrongshockpoints seemtoapproximatelyfollowa powerlaw. Thisfeatureoffersa wayof modelingbothsetsof datapoints.

Figures 19through23 showresultsfor iron. Note that the estimate(21) for log ~ departsradicallyfrom the two numericalaveragesfor the lowestshockpressure,which is somewhatclose,but not righton topof, theoverdriventhreshold. Thevaluefor ya for ironunderambientconditionsproducedunphysicalresultsin thecalculation.Thevalueappropriateto close-packedphaseswas used,whichis closeto that of the liquid. Sinceiron undergoesphasetransitionstoclose-pack4 phasesat theshockstrengthscalculatedhere,theclose-packedvalueis theappropri-ateone. The erroranalysisdid not showlargesensitivitywhenvtiations aboutthisvalueweremade.

12

—

. . ..

Shock Resultsfor 1100 Aluminum

o 0“0a

1100Al

a

e

@o

I 8 I

0.04 0.0!5 0.06 0.07 0.08 0.09<T>/Gd

Fig.3. Average(throughshockpath)plasticstrainrateversusaveragedeviatoricstressdividedbyGdforvariousshockstrengthsfor 1100Al.Thesquares,triangles,andcirclescorrespondto strainratesaveragedusingEq. (21),averagednumericallyusingthevolumetricstrain,andusingtheplasticstrain,remq

o

m

0.

Qo0

ectively.

1100Alu

o0

0

0

0

0

0

0

0

0 0

00

00

00

00

3 0.5 0.7 0.9 1.1 1.3P4(Mbar)

Fig.4. PlasticStdIIS(CirCkS)andVOhUIWtriCStrains(~UtiI’42S)atthepointOfhdf-totd plastics&in alongthe shockpathplottedversusthepressureat thispointfor 1100Al.

1100Alo

00

00

0

0

0

0

c!

o 0

0 0

0 0

0 0

0 0

0.3 0.5 o.? 0.9 1.1P,(Mbar)

Fig.5.Theshearmodulus(circles)andthebulkmodulus(squares)at thepointofhalf-totalplasticstrainplottedversusthepressureat thispointfor 1100Al.

Bo

0 c

o

0 0

0

0 0

0

00

0 n

3

0

0

0

, I #

,3 0.5 0.7 0.9 1.1 1.3P#fbarj

Fig.6. Themelttempemtum(circles)andtemperature(squares)at thepointof half-finalplasticstrainplottedversusthepressureat thispointfor 1100Al.

14

b?o

n

n

1100Al o0

0

00

a

o

0

0

i 1

0.20 0.25 0.30 0.35 0.40&h

Fig.7. Hugoniotpressureplottedversusvolumetricstrainforthesameshockwavesusedtogen-eratetheearlierfiguresfor 1100Al.

Shock Results for 2024 Aluminum

0.04 0.05 0.06 0.07 0.08 0.09<r>/Gd

Fig.8.Averageplasticstrainrateversusaveragedeviatoricstressdivit!~.1h-tGdforvariousshockstrengthsfor 2024Al. Averagesarethroughtheshockpath. Thesquare~,triangles.andcirclescorrespondto strainratesaveragedusingEq.(21),usingthevolumetricstrain,andusingtheplas-tic strain,respectively.

3

2024 Al o0

qo-

GG

a0

z

c1

c

+“og ~

cc= 3

0 c 0

a 0

3 0

6- G3

G)

00 4 I I

0.3 0.5 0.7 0.9 i.i 13P,(Mbar)

Fig.9. Plasticstrains(circles)andvolumetricstrains(squares)at thepointof half-totalplasticstrainalongtheshockpathplottedversusthepressureat thispointfor 2024Al.

16

2024 Al ao

0c

o

0

n

c

o0 c

G o~

o 0

Gc

o

ii

0.3 0.5 0.7 0.9 1.1 1.3P,(Mbar)

Fig. 10.Theshemmodulus(circles)andthebulkmodulus(squares)at thepointof half-finalplas-tic-strainplottedversusthepressureat thispointfor2024Al.

[.c

2024 Al c o0

0

0

!3o

0 0

0

2

c

c!

o

0.3 0.5 0.7 0.9 1.1Pd(Mbar)

3

Fig. 11.Themelttemperature(circles)andtemperature(squares)at thepointof half-finalplasticstrainplottedversusthepressureat thispointfor2024AL

2024 Alo

0

00

c1

c

o

c

o

0

0

0.20 0.25 0.30 0.35 0.40&h

Fig. 12.Hugoniotpressureplottedversusvolumetricstrainfor thesameshockwavesusedtogen-eratetheearlierfiguresfor2024Al.

Shock Results for Copper

Ileu0

m

C!u

8

a

a

0.00 0.02 0.04 0.06 0.08 0.10<T>/Gd

Fig. 13.Averageplasticstrainrateversusaveragedeviatoricstressdividedby GdforvariousshockstrengthsforCu.Averagesarethroughtheshockpath. Thesquares,triangles,andcirclescorrespondto strainratesaveragedusingEq. (21),averagednumericallyusingthevolumetricstrain,andusingtheplasticstrain,respectively.

2

=.c

cc

m

L1

o

a

o

oG

c

c o

o

o

Fig. 14.Plasticstrains(circles)andvolumetricstrains(squares)at thepointof half-totalplasticstrainalongthe shockpathplottedversusthepressureat thispointfor Cu.

19

o

(SIo

0

0.0 0.5 1.5 2.0P,(&ar)

Fig. 15.Theshearmodulus(circles)andthebulkmodulus(squares)at thepointof half-fial pl=-ticstrainplottedversusthepressureat thispointforCu.Thesequantitiesareve.ygoodapproxi-mationsto averagesthroughtheshockpathwithrespectto plasticor volumetricstrain.

I Cuc

=

o

0

0

0 0 !Jcs oT o 0

g o

F; 3 0

e. o’ o

8-

230

0

0

a00 , , I

0.0 0.5 1.5 20P&a~

Fig. 16.Thecalculatedmelttemperature(circles)andtemperature(squares)at thepointof half-finalplasticstrainplottedversusthepressureath pointforCu.

20

Cu

c

0.10 0.15 0.20 0.25 0.30 0.35 0.40

&h

Fig. 17.HugoniotpressurepiottedversusvolumetricstrainforCuforthesameshockwavesusedtogeneratehe earlierfiguresforCu.

a

cl

r-1= ‘1’

.-

t . . .

+10-3 “ ‘ “ ‘ “10-2 lb-’<e/Gd

Fig. 18.Averageplasticstrainratesplottedversusaveragedeviatoricstressesdividedby GdforCu.Averagesare throughtheshockpath.Thefiveplusesarepressure-sheardata Theotherpointsme thesameas thoseof Fig. 13.

21

Shock Results for Iron

o

Q

o

)7

Fig. 19. AverageplasticstrainrateversusaveragedeviatoricstressdividedbyGdforvariousshockstrengthsforFe. Averagesarethroughtheshockpath. Thesquares,triangles,andcirclescorrespondto strainratesaveragedusingEq.(21),usingthevolumetricstrain,andusingtheplas-tic strain,respectively.

on.a o

❑o

m

Fe o

G

Cl

o

c

c

cc)=c

0=o

c

0.0 0.5 1.0 i.s 2.0 i.5

P#fbar)

Fig.20. Plasticstrains(circles)andvolumetricstrains(squares)at thepointofhalf-totalplasticstrainalongtheshockpathplottedversusthepressureat thispointforFe.

22

0

0

0

0

0

0

00

0

u

I%o

0

0

00

00

0

0

0

00

f I ,

0.0 0.5 1.0 1.5 2.0P~(Mbafi

Fig.21.The shearmodulus(circles)andthebulkmodulus(squares)at thepointof half-finalplas-tic-strainplottedversusthepressureat thispointforFe.

I

o8

0

Fe

o

0

0

0

0

0

0

u

co

0

0 0

0

0

0

0

0

0.0 0.5 1.0 1.5 20 :

P~(Mba~5

I

Fig.22. Thecalculatedmelttemperature(circles)andtemperature(squares)at thepointofhalf-finalplasticstrainplottedversusthepressureat thispointforFe.

23

oA.

L90“

o0.,

ko

0

0

0

0

0

0

0

0

0

[

0.17 0.22 0.27 0.32 0.37 0.42

Fig.23.HugoniotpressureplottedversusvolumetricstrainforFe forthesameshockwavesusedto generatetheearlierfiguresforFe.

ShockResultsforTantalum

oa“-

0 0e

0.00 0:028 I

0.04 0.06 0:08 0.10 ( 12<T>/Gd

Fig.24.Averageplasticstrainrateversusaveragedeviatoricstressdividedby GdforvariousshockstrengthsforTa Averagesarethroughtheshockpath. Thesquares,trian#es, andcirclescorrespondto strainratesaveragedusingI@ (21),usingthevolumetricstrain,&d usingtheplas-tic strain,respectively.

o

0

00

Ga

oc

Go

c1o

G

0.0 1.0 20 3.0P4(Mbar)

~lg.~. Plastic-S (circles)andVOhUIEtIiCstrauis“ (Squares)atthepointofhalf-total plasticstrainalongtheshockpathplottedversusthepressureat thispointforTa.

25

... ... .. . .. . . -----

0c

o

‘Iho

0

0

0

0

0

0

0 0

0 00

00

00

00

0

o.@ 1.O 20 #.Pa(hfbafi

>0

Fig.26.Theshearmodulus(circles)andthebulkmodulus(squares)at thepointof half-finalplas-tic strainplottedversusthepressureat tbispointforTa.

Ta 0c

03

0

0 0

0

c 0

0

0

0

0

3

0

0c1

I

c

0

3

0

0.0 1.0 ZoP4(Mbar)

,0

Fig.27. Thecalculatedmelttemperature(circles)andtemperature(squares)at thepointofhalf-finalplasticstrainplottedversusthepressureat thispointforT&

26

Ta

c

c3

0

0

0

0

a

o

0

c

0.1 0.2 0.3 0.4

&b

,5

Fig.28. Hugoni~Jtpressureplottedversusvolumetricstrainfor thesameshockwavesusedtogeneratetheearlierfiguresforTa.

27

Shock Results for Uranium

u

f?

90

cl

0.020 0.025 0.030 0.<r>/Gd

135

Fig.29. AverageplasticstrainrateversusaveragedeviatoncstressdividedbyGdfor variousshockstrengthsfor U.Averagesare throughtheshockpath. Thesquares,triangles,andcirclescorrespondto strainratesaveragedusingEq.(21),usingthevolumetricstrain,andusingtheplasticstrain.respectively.

o

o

oa

c

0.3 0.4 0.5 0.6 0.7

Pd(Mbar)9

Fig.30.Plasticstrains(circles)andvolumetricstrains(squares)at thepointof half-totalplasticstrainalongtheshockpathplottedversusthepressureat thispointforU.

28

od

u o

0

00

0

0

c

o0

00

0

o

n

o0

0

0.3 0.4 0.5 0.6 0.7P~(Mbar)

Fig.31.Theshearmodulus(circles)andthebulkmodulus(squares)at thepointof half-finalplas-tic strainplottedversusthepressureat thispointforU.

4.——

3c

00

c0

CJ0

3

0 0 c

0

c!0

0

0.3 0.4 0.5 0.6 0.7P4(Mba~

Fig.32.Thecalculatedmelttemperature(circles)andtemperate (squares)at thepointof half-firdplasticstrainplottedversusthepressureat thispointforU.

u

o

0

0

0

c

o

0

El

3

3

0.16 0:17 0.18 0:19 0.20 Oj$l a&b

?2

Fig.33.Hugoniotpressureplottedversusvolumetricstrainfor thesameshockwavesusedtogeneratetheearlierfiguresforU.

c=- - - “ -

J? 3xx 3

Xx”cLxx

.

0.OO 0.05 0.10 0.15<T>/’d

Fig.34.Averageplasticstrainrate ((~)v) versusaveragedeviatoricstressdividedby Gdforvariousshockstrengthsfor thesixmetals.Averages are throughtheshockpathusingtheplasticstrain. Thesquares,circles,triangles,pluses,andX’scorrespondto 2024Al, Cu,Fe,T%andU,respectively.30

Figures24through28showresultsfortantalum.Theseweresomewhatsensitivetovaria-tions in y butnotenoughto warrantconcern.

Figures29 through33showresultsforuranium.Theseresultswerefairlyrobustagainstvari-ationsin physicalparametersin thecalculation.

Figure34showsCY>Vplottedversus<?>/Gd forall themetalstogether.Thisplotshowsthedegreeofuniversalityexhibitedbythesequantites.

111. Conclusions

CalculatedresuhsusingWallace’sstrongshocktheoryaregivenin thisdocumentfora varietyof metals.The resultsare not as firmas one wouldwishdue to lackof completeknowledgeofthermoelasticityat mcgabarpressures. However.the underlyingtheoryis soundand the erroranalysisshow”sacceptablesensitivityto thethcrmoelasticmodeling.These two features.coupledwiththeopportunityof obtainingforthe firsttimeiIglimpseof strain-rateplasticityeffectsundertheseextremeconditions.justifiesthiseffortto maketheseresultsmorewidelyavailable.

Acknowledgment

.1wouldliketo thankD.C.Wallaceforprovidingguidanceandaccessto researchnotebooks.Thisworkis verymuchindebtedto hisearliereffort.

31

I

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