Compressional Sound Velocity, Equation of State, and Constitutive

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 100, NO. B1, PAGES 529-542, JANUARY 10, 1995

Compressional sound velocity equation of state and constitutive response of shock-compressed magnesium oxide ThomasS. Duffy 1 and ThomasJ. Ahrens SeismologicalLaboratory, California Institute of Technology,Pasadena

Abstract. Wave profileand equationof state (EOS) data are reportedfor low- porositypolycrystalline magnesium oxide under shockcompression. The Hugoniot equationof statebetween 14 and laa GPa is Us - 6.87(10)+ 1.24(4)up,where the numbers in parenthesesare one standard deviation uncertainties in the last digit(s). Reverse-impactw•ve profilesconstrain the compressionalsound velocity, Vp, at 10-27 GP• to 4-2%. Measured Vp valuesare consistentwith ultrasonic data extrapolatedfrom 3 GPa. By combiningthe Hugoniotresults with ultrasonicdata, the adiabatic bulk modulus and its first and secondpressure derivatives at constant entropyare 162.5(2)GPa, 4.09(9),and-0.019(4) GPa -•. The shearmodulus and its firstand second pressure derivatives are 130.8(2) GPa, 2.5(1),-0.026(45) GPa -•. PolycrystallineMgO has • compressiveyield strengthof 1-1.5 GPa at the elastic limit whichincreases to 2.7(8) GP• alongthe Hugoniotand is similarat unloading. Wave profilesfor MgO at 10-39 GPa are describedusing a modified elastic-plastic model. There are significantdifferences in the dynamicresponse of single-crystal and polycrystallineMgO.

Introduction wave velocity is studied in shock-compressedMgO to 27 GPa. The results provide a test of finite strain ex- The most directapproach for understandingthe com- trapolation and place constraintson higher-orderelastic position and structure of the Earth's deep interior is constants. We report direct measurementsof the com- through measurementof elastic wave velocities in min- pressionalwave speed in a potentially significantlower erals and metals at high pressuresand temperatures. mantle material at lower mantle pressures. Laterally averagedseismic wave velocitiesare known to MgO has been the subject of extensive theoretical betterthan 1%throughout the Earth [Layet al., 1990]. and experimental study as a result of its wide pres- Small lateral deviations of wave velocities have also now sure and temperature stability range and simple struc- beenextensively documented [Romanowicz, 1991]. Un- ture. While single-crystalMgO has been studied pre- der static pressure,there are few measurementsof elas- viously under shock compression,we report the first tic wavevelocities above i GPa in mineralsof planetary equationof state (EOS) and waveprofile measurements interest. on low-porositypolycrystalline MgO. Significantdiffer- Soundvelocities in shock-compressedmaterials were ences in the shock responseof polycrystals and sin- first measuredby Al'tshuleret al. [1960]. Subsequent gle crystals have been identified in several materials work of particular geophysicalsignificance included the [Mashimo,1993]. There has alsobeen much recent in- studyof Fe by Brownand McQueen[1986]. Sound ve- terest in the dynamic properties of ceramicswhose re- locitiesmeasured under shockcompression have been sponseto dynamic compressionappears to be quite vari- usedto placeexperimental constraints on the tempera- ablebut is largelyunknown [Rosenberg, 1992; Mashimo, ture coefficientof compressionalvelocity at ~100 GPa 1993]. An abbreviatedreport of this work is given by [Duffy and Ahrens,1992, 1994a]. This is an impor- Duffy and Ahrens[1994b]. tant quantity for interpretationof seismictomography. In this study, the effect of pressureon compressional Experimental Method

Samples INowat GeophysicalLaboratory, Carnegie Institution of Washington, PolycrystallineMgO sampleswere obtained (Cercom D.C. Incorporated,Vista, California)as 1.25-inch-diameter Copyfight1995 by the AmericanGeophysical Union. hot-presseddisks. The manufacturer'sspecifications in- dicated that the density was within 1% of crystal den- Papernumber 94JB02065. sity and chemicalpurity was greater than 99.5%. The 0148-0227/95/94JB-02065505.00 latter was confirmed by electron microprobe analysis

529 530 DUFFY AND AHRENS: SHOCK COMPRESSION OF MgO which revealeda small amount of CaO (~0.3%) and Ar + ion laseris focusedonto a diffuselyreflecting sur- no other impurities at detectable levels. Bulk densities face. Target motion inducesa Doppler shift in the re- were measuredby weighingthe ~10 g sampleswith a flectedlaser light which generatesinterference fringes in microbalancesensitive to 4- 10-4 g. The averagecrys- a wide-angleMichelson interferometer. The fringesare tal density was found by Archimedean immersion to recordedusing photomultiplier tubes and digital oscil- be 3.571(4)g/cm a, and the averagebulk densitywas loscopes.The relationship between surfacevelocity and 3.562(6)g/cm a, whichare within 0.4% and 0.6%,re- numberof fringesis [Barker and Hollenbach,1972] spectively,of the X ray densityof 3.584g/cm a. The numbers in parenthesesare one standard deviation un- u(t- r12) = kF(t), (1) certaintiesin the last digit. Optical examinationof thin sectionsrevealed a structureof colorless,roughly equant where u is the particle velocity, t is the time, r is the grains with approximate dimensionsof 5 ttm. Sample lag time (~1 ns) of the interferometer,k is the velocity- flatness variations were less than 0.01 mm. per-fringeconstant, and F(t) is the numberof fringes Ultrasonic sound velocity measurementswere per- recordedup to time t. The fringe constant k can range formed along the cylindricalaxis (in the directionof from ~100 m/s/fringe to over1 km/s/fringe and is con- shock wave propagation)by S. M. Rigden at Aus- trolled by inserting fused silica etalons in one arm of tralian National University. Measurements between the interferometer. The VISAR we constructed is simi- 30 and 70 MHz revealed little dispersionin velocity lar to that describedby Barker and Hollenbach[1972]. (4- 0.005 km/s) and a compressionalwave velocity of We also incorporated the push-pull modification and 9.81(2) km/s. This is 1.1% greater than the Voigt- data reductionalgorithm of Hemsing[1979] and the Reuss-Hill averagefor a single crystal of this material polarization-randomizationscheme of Asay and Barker [Jacksonand Niesler, 1982]. This differenceexceeds [1974]. the Voigt and Reussbounds on the soundvelocity and Two types of impact geometries,forward and reverse, probably results from internal strains, small amounts of were used. The reverse-impactexperimental setup is il- impurities, or preferred orientation. These effectsall oc- lustrated in Figure la. In this arrangement,the sample cur in polycrystallineMgO samples[Tagai e! al., 1967; is inserted in the projectile and used to impact a thin Schreiber and Anderson, 1968; $petzler and Anderson, aluminum buffer with an LiF window epoxiedto it. The 1971]. VISAR monitors the interface between the window and buffer. An array of electrical shorting pins is used as Equation of State Experiments part of a capacitor-dischargecircuit to trigger recording instrumentation and to measure impact planarity. Equation of state experimentswere conductedusing The sampleis backedby low-densityfoam which serves both a propellant and a light gasgun. The 40-mm bore to introducean unloadingwave into the specimen. propellantgun couldlaunch ~100-g projectilesto veloc- A Lagrangianwave propagationdiagram for the re- ities up to 2.5 km/s, while the two-stagelight gasgun verse-impactexperiments is shownin Figure lb. At im- can accelerate~20-g projectilesto 6.5 km/s. Projec- pact (t=0), an elasticprecursor and a shockwave prop- tile velocity was measuredby recordingX ray shadow- agate throughthe sample,and a shocktravels through graphs immediately prior to impact. The MgO targets the buffer. It is assumedhere that the impact stress were mounted on tantalum or aluminum 2024 driver is sufficientto overdrive the precursorin aluminum but platesand placedin an evacuated(~100-mm Hg) sam- not sufficientto overdrivethe precursorin MgO. At time ple chamber. Two pairs of fiat mirrors were mounted tl, the shockreaches the buffer-windowinterface and is on the rear surfaceof the target and driver. An inclined recordedby the VISAR. The slight impedancecontrast wedgewas mounted in the center of the target. Light between LiF and aluminum causes a weak rarefaction to from a 10-kV Xenon flash lamp was directed onto the propagatethrough the aluminum which returns to the rear of the target, and reflected light was returned to reflectingsurface at t2. The elastic precursortraveling a continuous writing streak camera. Shock wave ve- through the samplereaches the sample-foaminterface, locities were measured by recording the reduction in reflectsfrom it as a rarefaction wave, and interacts with reflected light intensity due to the destruction of the the oncomingshock. The shockwave alsoreflects from mirrors by the shockfront and by the changein extinc- the foam interface and producesa rarefaction which fur- tion angle of the inclined wedge. Further details are givenby Ahrens[1987]. Wave Profile Experiments Projectile A seriesof wave profile measurementswas undertaken using shock wave velocimetry. These measurements yield a continuousrecord of particle velocityat a sample interfaceduring both the loading and unloadingcycles of the experiment. The method for recordingparticle / velocity histories was the velocity interferometersys- Buffer Window Light to VISAR Foam Sample tem for any reflector(VISAR) [Barkerand Hollenbach, 1972]. In this technique,200-300 mW of light from an •'igure la. Reverse-impactexperimental geometry. DUFFY AND AHRENS- SHOCK COMPRESSION OF MõO $31

Sample I Buffer Window Flyer Sample Buffer/ Window

USb

VPs •-• Vpb

Uss U• t• Distance Distance Figure lb. Lagrangian distance-time diagram for Figure 2b. Lagrangiandistance-time diagram for for- reverse-impactexperiments. The sample-foambound- ward impact experiments.The symbolsare identifiedin ary is at left edge of figure. Vp0 is elastic precursorve- Figure lb; h2 is thicknessof buffer-sampleinteraction locity, Us is shock velocity, Vp is compressionalsound region,and V* is the averagevelocity in the interaction velocity, tl-t3 are wave arrival times. hi is the point region. The subscriptfrefers to flyer properties. where reflectedprecursor and direct shockinteract. The subscript b and s refer to the buffer and to the sampleø the shockfront arrives and further reducesthe particle velocity. The forward impacts have the disadvantagethat the ther unloadsthe material. The unloading of the sample unloading portion of the waveform is significantlyaf- is recordedas a decreasein particle velocity beginning fected by wave interactions generated at the sample- at t3. buffer interface. These interactions are eliminated in The forward impact experimental setup is shown in the reversegeometry allowing determination of the un- Figure 2a. In this case,the flyer plate impacts a sam- loading wave speed. The peak stressattainable in the ple to which a thin aluminum buffer has been epox- reverse-impactarrangement using an A1 buffer is lim- led. Sample-bufferand buffer-windowepoxy layers were ited to ~27 GPa with the present 40-mm gun. The ~10 ym thick in all VISAR experiments.A Lagrangian initial conditionsfor the VISAR experiments are listed distance-timediagram for this geometryis shownin Fig- in Tables i and 2. ure 2b. Symmetric impact resultsin an elastic precursor followed by a plastic shock propagating through both the MgO flyer and the MgO sample. The impedance Results contrast between the sample and buffer causesa partial unloadingwave (dashedlines) to propagateback For the EOS experiments,the shockand flyer veloci- through the sample. The precursor and shock travel ties were combinedwith impedancematching [Ahrens, through the buffer and reach the buffer-window inter- 1987]and the Rankine-Hugoniotequations to constrain face at t l and t2. The impedance contrast between the the particle velocity, stress, and density of the shock- buffer and window has been neglectedin this analysis. compressedstate (Table 3). The EOS propertiesof the The elastic precursorin the flyer eventually reachesthe flyer and driver plates are listed in Table 4. The elastic back surface and reflects from it. It interacts with the precursorwas resolvedin one experimentfrom which a oncomingshock wave at position hi and later interacts precursorvelocity of 9.77(12)km/s wasobtained, which with the buffer-sample reflection at h2. It reachesthe agreeswith the ultrasonicallymeasured sound velocity. buffer-window surface at t3 and producesa decreasein The equationof state resultsare shownin Figures3 and particle velocity. Subsequently,the unloadingfan from 4. The buffer-window particle velocity histories mea-

Pins sured in the reverse impacts are shown in Figure 5a. Projectile The sharp jump in particle velocity is the arrival of the shockfront (tl). This is followedby the velocityplateau where some variations in particle velocity are evident. This may reflect differential motion of grains, material reorganization,or heterogeneousfaulting and has been observedpreviously in velocity profile measurementson ß . Window ceramics[Kipp and Grady, 1989]. Aluminum buffers I % Sample Buffer Lightto VISAR Foam Flyer were usedto smoothout suchirregularities. The arrival of the unloading wavesproduces a decreasein particle Figure 2a. Forward-impactexperimental geometry. velocitystarting at t3 which displaysan S-shapedstruc- 532 DUFFY AND AHRENS' SHOCK COMPRESSION OF MgO

Table 1. Summary of Reverse-Impact Experiments

MgO A1 6061 Buffer LiF Window Impact Sample Shot p0, Thickness, p0, Thickness, p0, Thickness, Velocity, Peak Stress, g/cm3 mm g/cm3 mm g/cm3 mm km/s GPa 848 3.561(2) 4.025(2) 2.681(4) 1.955(2) 2.631(1) 12.133(3) 1.432(10) 15.2(2) 851 3.569(8) 3.032(6) 2.681(3) 1.962(2) 2.630(1) 12.138(2) 2.366(35) 27.4(6) 853 3.558(7) 3.030(2) 2.684(3) 1.887(2) 2.629(4) 8.128(3) 1.612(19) 17.4(3) 854 3.568(3) 4.018(2) 2.687(5) 1.880(3) 2.626(1) 12.142(2) 1.020(9) 10.4(2) 856 3.570(4) 4.025(4) 2.685(6) 1.953(4) 2.628(1) 12.131(2) 2.165(20) 24.7(4)

Numbersin parenthesesare onestandard deviation uncertainties in the last digit(s). ture. The final particle velocity level is dependent on referenceto Figure lb. The time differenceAt between the properties of the foam backing of the flyer. the unloadingwave arrival t3 and the shockarrival t l The forward impact velocity histories are shown in is determined from the wave profile. The Lagrangian Figure 5b. The first arrival is the elasticprecursor (tl unloadingvelocity is given by in Figure2b). The amplitudeof this waveis a measure xs-hl of the largest elastic stress that can be supported by the buffer under compressiveloading. This is followed •r; - zxt+•/vs• - (• +a•)/•ro - •/•r;•' (•) by the plasticshock front (t2) and the plateauregion. where Vl•l; and V•,l;bare the Lagrangiansound veloci- Velocity variations in the plateau region are generally ties in the sampleand buffer, a:• and a:• are the sample smaller for these experiments, probably becauseof the and buffer thicknesses,Usb is the buffer shock veloc- longer propagation distance. The shape of the unload- ity, and Vl•0 is the elasticprecursor velocity. h• is de- ing wave profile (rs) is similar to the reverseexperi- fined by the point of intersectionof the elastic wave ments, but the initial releaseis not as sharply defined reflected from the rear surface of the flyer and the on- in these experiments. coming shock wave Hugoniot states were not obtained directly from the VISAR results as the impact times were not recorded. hi= zs(Vpo- Uss) (3) (•,o + vs.) ' However, peak particle velocitiesfrom VISAR shots are consistent within experimental uncertainties with val- where Us• is the sampleshock velocity. The ultrasoni- ues from the MgO equation of state and impedance cally determinedcompressional wave velocity was used matching(see discussionsection). The agreementbe- for the precursorvelocity. Shock states in the buffer tween measured and calculated interface velocities can and samplewere determinedfrom impedancematching also be seen in the wavecode simulations discussed be- using the data of Table 4. low. Calculated MgO longitudinal stressesachieved in Compressionalvelocities in the buffer were deter- the VISAR experiments are listed in Tables 1 and 2. mined from the time difference between the shock ar- The stressrange coveredby these experimentsis 10-39 rival, tl, and the arrival of the reverberationthrough GPa. the buffer,t2 (Figurelb) using Unloading wave velocitieswere determinedfrom the 2Zb reverse-impactexperiments. For an elastic-plasticma- V_pZ:b---- •. (4) terial, the initial unloadingvelocity is the compressional t2 -- tl sound velocity Vio and for a fluid it is the bulk veloc- Only experimentsin which the reverberationarrival was ity Vls [Zel'dovichand Raizer, 1967]. The methodfor clearly evident were used here. Lagrangian velocities determining the unloading velocitiescan be seen with were converted to Eulerian velocities Vl• by multiply-

Table 2. Summary of Forward-Impact Experiments

Flyer MgO Sample A1 6061 Buffer Impact Sample Shot po, Thickness, p0, Thickness, po, Thickness, Velocity, Peak Stress, g/cm• mm g/cm• mm g/cm• mm km/s GPa 857 3.563(4) 3.024(2) 3.561(4) 4.030(4) 2.683(3) 1.944(2) 1.520(20) 21.4(3) 859 3.563(4) 3.026(3) 3.564(2) 4.023(2) 2.679(3) 1.942(2) 2.432(30) 37.6(8) 861 3.558(4) 3.030(2) 3.551(6) 4.025(2) 2.684(3) 1.980(2) 1.015(8) 13.5(2) 864 3.836(12) 3.217(10) 3.566(5) 3.780(4) 2.687(4) 1.946(3) 2.471(32) 38.9(1.0) Flyerswere MgO exceptfor shot864, which used A1203. LiF windowshad nominal density of 2.63g/cm 3 and thicknessof 8.15 mm. Numbersin parenthesesare onestandard deviation uncertainties in the last digit(s). DUFFY AND AHRENS: SHOCK COMPRESSION OF MgO 533

Table 3. Results of Equation of State Experiments Shot Flyer/Driver uyp, p0, up, Us, a, p, VI-, km/s g/cms km/s km/s GPa g/cms km/s 233 Ta 4.876(3) 3.566(12) 3.367(12) 11.042(73) 132.6(8) 5.130(27) 840 Ta 1.291(11) 3.552(10) 0.893(8) 7.96(11) 25.3(3) 4.001(15) 841 Ta 2.521(30) 3.561(6) 1.742(23) 9.01(13) 55.9(9) 4.415(25) 843 Mg/A12024 1.713(17) 3.566(12) 0.513(10) 7.53(12) 13.8(2) 3.827(16) 9.77(12) Hereuyp is the impactvelocity, up is the particlevelocity, Us is the shockvelocity, Vpo is the elasticprecursor velocity,and the subscriptzero refersto ambientpressure conditions. Numbers in parenthesesare one standard deviationuncertainties in the last digit(s). ing by the ratio of densitiespo/p. The Eulerian com- pressurephase change. For MgO, the bulk velocityfrom pressionalvelocities determined for the buffer material, ultrasonicdata [Jacksonand Niesler, 1982]is 6.73(1) A1 6061, are shownin Figure 6. Also shown are some km/s, whichis 1.9%above the Us-upintercept. A1 6061 soundvelocities determined in separateexperi- There is considerableevidence that single-crystalMgO ments(T. S. Duffy,unpublished data) aswell as data of has little or no strength when shockedabove its elastic Asay and Chhabildas[1981] for this material. The vari- limit of 2.5 GPa. The Hugoniot stress-densitystates ation of Vp with Hugoniotstress in A1 6061 (Figure6) are nearly coincident with a 300 K hydrostatic com- can be describedby pressioncurve at low stresses(Figure 4). The static isotherm was computedusing ultrasonic elasticity data lnVp- 1.8796+ 0.03031na+ 0.01771n 2a, (5) [Jacksonand Niesler, 1982] corrected to isothermalcon- where Vp is in km/s and •r is in GPa. The Eulerian ditions and extrapolated using the Birch-Murnaghan unloadingvelocities in MgO determinedfrom (2) are equation. Collapse to the hydrostat is also supported shownin Figure 7 where the axial stresshas been con- by featuresobserved in waveprofiles recorded for single- verted to mean pressure. Uncertainties in the velocities crystal MgO at stressesof 5-11 GPa [Grady, 1977]. are typically 4-2%. These include stress relaxation behind the precursor, shock velocitiesbelow the local bulk sound speed, and Discussion inferred stress-densitystates coincidentwith the hydro- star. These features are characteristicof strength col- lapsein brittle singlecrystals [Mashimo, 1993]. Equation of state A least squaresfit to the Hugoniot data for polycrys- While single-crystaland porouspolycrystalline MgO talline MgO yields(Figure 3) have been studied extensivelyusing shocktechniques, Us = 6.87(10)+ 1.24(4)up, (7) the present •.tudy represents the first EOS determination for low-porosity(<1%) polycrystallineMgO. A which is significantlydifferent from the fit to the single- least squaresfit to the single-crystalMgO data listed crystal data. The intercept here lies above the ultra; by Marsh[1980] yields (Figure 3) sonic bulk sound speed, which is commonly observed Us = 6.61(5)+ 1.36(2)up. (6) in materials with significantshear strength along the Hugoniot[McQueen el al., 1970]. Comparisonwith the The interceptof the Us-uprelation corresponds to the 300 K hydrostat showsthat Hugoniot states for poly- bulk soundvelocity for a material undergoingno high- crystallineMgO lie abovethe hydrostat at low stresses.

Table 4. Equation of State Standards

Material p0, co, s 70 Y0 Y, Refer- g/cma km/s GPa ences

. MgO 3.562(6) 6.87(10) 1.24(4) 1.52 0.18 1.25 1 A16061 2.683(3) 5.349(56) 1.338(20) 2.10 0.34 0.3 2 Ta 16.65(3) 3.293(5) 1.307(25) 1.60 0.34 0.75 3 LiF 2.64(2) 5.15(3) 1.35(1) 1.63 0.22 0.2 2 Mg AZ31B 1.775(1) 4.52(5) 1.26(2) 1.43 0.30 - 2 A12Oa 3.83(1) 6.91(8) 1.44(4) 1.27 0.24 5.8 2 Here co and s are Hugoniotequation of state constants,y0 is the Griineisenparameter, v0 is Poisson'sratio, and Y is the yield strength. The referencesfor Hugoniotproperties are 1, this study; 2, Marsh [1980];and 3, Mitchell and Nellis [1981]. The numbersin parenthesesare one standarddeviation uncertainties in the last digit(s). 534 DUFFY AND AHRENS: SHOCK COMPRESSION OF MgO

2.0 11- ©Polycrystal (ThisStudy) o SingleCrystal (Marsh,

851

856

ß-• 9 853 ts ts 848 > • 8 854

0.0

6 i I i I t i 0.5 1.0 1.5 2.0 0.0 0.5 1.0 !.5 2.0 2.5 3.0 3.5 Time (•sec)

ParticleVelocity (kin/s) 2.0 , , , , , Figure 3. Shock velocity-particle velocity data for MgO. Solid symbols are data of this study. The tri- angle is bulk sound velocity from ultrasonic data. Solid 864 curve is a least squaresfit to polycrystalline data, and dashed curve is a fit to single-crystaldata.

An estimate of the shearstrength can be madefrom the stressdifference between the Hugoniot and isotherm after thermal and porosity effects have been subtracted using the Mie-Griineisen equation. For the three lowest stress data, the average corrected stressdifference between the Hugoniotand isothermis 1.8(5) GPa. The yield strength, Y, can then be determinedusing 1.25 1.50 1.75 2.00 2.25 2.50 Time (p,sec) ¾- •(•-3 •) , (8) Figure 5. Interfaceparticle velocityhistories for (a) where er is the axial stress and P is the pressure. reverseand (b) forwardimpact experiments. Shot num- For polycrystallineMgO, this yieldsY - 2.7(8) GPa bers are listed next to each profile. Times t l-t3 corre- at stresses of 14-56 GPa. Thus the EOS data reveal spond to those in the x-t diagrams of Figures I and 2. significantdifferences in the behavior of single-crystM Starting times are arbitrary. The elastic precursor is labeled HEL.

140 and polycrystalline MgO. Similar differenceshave been I I I I I I I observedin the shock responseof other oxides such as 120- ßPolycrystal (ThisStudy) A1203 and ZrO2 [Grady,1977; Mashimo, 1993]. o SingleCrystal (Marsh, 1980) It is of interest to comparethe dynamic yield strength lOO of MgO with recent static measurementsin the diamond anvil cell and multi-anvil press. Meade and Jean- • •o loz [1988]measured static shear stressesin the range • Hugoniot •,'*' of 1-4 GPa at pressuresof 10-40 GPa in diamond anvil cell experimentson MgO. More recently, measurements of the static strength of MgO to 227 GPa have shown 40 that the strength is at least 11 GPa at that pressure [Duffy el al., submittedmanuscript]. Microscopic yield 20 strengths of 2-4 GPa were obtained in multi-anvil experimentson MgO to 500øCat 8 GPa [Weidnerel al., 0 1994]. The dynamicyield strengthsobtained here are •.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 similar to static values despitelarge differencesin strain Density(g/cm 3) rates between static and dynamic experiments. Differ- Figure 4. Stress-densitystates in shock compressed encesin the strength of single and polycrystalsof MgO MgO. The ambient pressuredensity is shownby the tri- were alsoobserved under static conditions[Meade and angle. The solid curveis the polycrystallineHugoniot, Jeanloz,1988]. and the dashedcurve is the 300 K isothermcomputed Reduction of the Hugoniot data to an adiabat was from extrapolated ultrasonic data. carriedout usingMie-Grfineisen theory [Heinz and Jean- DUFFY AND AHRENS: SHOCK COMPRESSION OF MgO 535

I I I I Table 5. Elastic and Thermodynamic Properties of Aluminum 6061 MgO

Property Value Reference p0,g/cm 3 3.584(1) 1 Kos, GPa 162.5(2) 1 (OKos/OP)s 4.09(9) 1 8 - (02Kos/OP2)s,GPa -1 -0.019(4) 3 Go, GPa 130.8(2) 1 OGo/OP 2.5(1) 1 O•Go/OP•, GPa-1 -0.026(45) 3

ß This Study 7o 1.52(2) 2 o Asayand Chhabildas, 1981 q; 7 = 70(Po/P) q 1.0(5) 4

I I I I Referencesare 1, Jacksonand Niesler[1982]: 2, Sum- 10 20 30 40 ino et al. [1983];3, this study; and 4, assumed.Num- Stress(GPa) bers in parenthesesare one standard deviation uncer- taintiesin the last digit(s). Figure 6. Compressionalwave velocitiesin 6061 aluminum. The solid squareis the ambient pressureultrasonic velocity. in the 41 and 42 terms, respectively. f is the Eulerian strain given by loz, 1984; Ahrens and Jeanloz,1987]. The Griineisen parameter 7 and the exponent specifyingits volume de- i p pendenceq are listed in Table 5. The reducedHugoniot f- • •00 - i . (10) data are displayedin a normalized pressure-strainrep- resentation[Birch, 1978] in Figure8. In coordinatesof The normalized pressure F is Figure 8, the Eulerian finite strain equation of state is P

F - ao+ alf + a2f• + ..., (9) F= 3f(1+ 2f)5/•' (11) Truncationof (9) after the linear strain term yields where a0 is the ambient pressureadiabatic bulk modu- the third-order Birch-Murnaghan equation, while inclu- lus, Kos, and its first pressurederivative at constant en- sion of the quadratic term givesthe fourth-order form. tropy,K•) s - (OKos/OP)s,and second pressure deriva- tiv6at constantentropy, K•)' s - (02Kos/OP2)s,appear

I I I I I ' 'Ks"= -0.019 GPa '•' 1'

- t - 180

160

• 8 140 Ks'=--4'0; / ...... '..... I 120 ß This Study o Grady, 1977 - _ I I Chop½las,1992I ...... 0.00 0.05 0.10 0.15 0.20 I• I I I I 0 5 10 15 20 25 30 Eulerian Strain, f Pressure(GPa) Figure 8. Hugoniot EOS data reducedto a principal isentropeand plotted in normalized pressure-strainco- Figure 7. Compressionalwave velocities in MgO. Solid ordinates. Solid symbols are polycrystalline data and symbolsare polycrystallineMgO, and open symoblsare open symbols are single-crystal data. Long dashed single-crystalMgO shockedalong [100]. Solid curves curve shows3-GPa ultrasonic data extrapolated using are third-order finite strain extrapolations of ultrasonic a third-order Birch-Murnaghanequation. Solid curveis data at 300 K for Vp and VB, usingparameters of Table a fourth-order Birch-Murnaghan fit to shock data with 5. Long dashed curve showsmaximum expected effect Kos and K•s constrainedat ultrasonicvalues. Short- of Hugoniot temperature on Vp. Short dashedcurve is dashed curve showsfourth-order Birch-Murnaghan fit extrapolatedcompressional velocity along [100]. usingparameters of Chopelas[1992]. 536 DUFFY AND AHRENS' SHOCK COMPRESSION OF MgO

Analogousfinite strain expressionscan also be devel- -0.023GPa -•. Bukowinski[1985], using the augmented oped for the variation of sound velocity with strain planewave (APW) method,obtained (02KoT/OP2)T [Sammiset al., 1970]as discussedbelow. = -0.030 GPa-• for MgO. By includinga correction Ultrasonic sound velocity measurements have been term which accountsfor the energy differencebetween made on MgO singlecrystals to 3 GPa [Jacksonand the adiabat and the isotherm, isothermal moduli and Niesler, 1982], which constrainthe aggregateelastic derivatives can be extracted from reduced Hugoniot moduli to 0.1% and their pressurederivatives to ~4% data [Heinz and Jeanloz,1984]. For the single-crystal (Table5). A third-order(a2-0) finitestrain extrapola- shockdata, this gives (02KoT/OP2)T ---0.022(4)GPa -•, tion of these data is compared with the reduced Hugo- which is similar to values from first-principle calcula- niot data in Figure 8. A small correctionhas beenmade tions. to convert the pressure derivative of the bulk modu- The valueof K•'s whichresults when the finitestrain lus to constant entropy conditions. Both the reduced expressionistruncated at thirdorder is-0.025(1) GPa -•, single-crystaland polycrystal Hugoniot data are con- which differsonly slightlyfrom the valuefound here us- sistent with extrapolated ultrasonic data. The study of ing shockcompression data (-0.019(4)GPa-•). This Jacksonand Niesler[1982] possesses significant advan- indicates that the fourth-order term is at best only tages over earlier high-pressureultrasonic experiments marginallyrequired for describingMgO, in agreement on MgO. In particular, Jacksonand Niesler[1982] per- with the resultsof Jacksonand Niesler[1982]o formed their measurementsto significantlyhigher pres- The value of the second pressure derivative result- sure (3 GPa) than other workersand made correc- ing from the reducedshock data is sensitiveto the tions for transducer bond shifts. The secondpressure choiceof the logarithmic volume dependenceof the derivatives of the bulk and shear moduli required for Griineisenparameter q [Jeanloz,1989]. The rangeof q the fourth-order finite strain expansion are not well- valuesconsidered here (0.5-1.5) is consistentwith ambi- resolved in their data. ent pressurethermoelastic data [Isaaket al., 1989],the- The ambient pressure adiabatic bulk modulus re- ory [Isaaket al., 1990;Anderson et al., 1993],Hugoniot ported by Jacksonand Niesler [1982](162.5 GPa)is data for porousMgO [Carter et al., 1971],and shock consistent with other ultrasonic and resonance studies temperaturedata [Svendsenand Ahrens, 1987]. In or- of MgO [Suminoet al., 1983]. However,static compres- der for the reduced shock data to be consistent with sion studies have consistentlyreported a significantly the secondpressure derivative of Chopells[1992], it is higherbulk modulusfor MgO (KT ~ 180 GPa) [e.g., necessaryto have q values of-1 to-3. Such values are Perez-Albuerne and Drickamer, 1965; Weaver et al., inconsistent with the above constraints and are physi- 1971]. A recentstatic compressionstudy to pressuresin cally implausiblebecause they lead to negative sound excessof 200 GPa has shownthat the high bulk modu- velocitieson the Hugoniot[McQueen et al., 1967;Jean- lus is due to the static shearstrength of MgO [Duffy et loz, 1992]. Furthermore,a room temperatureequation al., submittedmanuscript]. This studyalso showed that of state constructedby correctingChopelas's [1992] re- when this strength is accountedfor, a singleequation of sults to the isothermal moduli deviates strongly from state can successfullydescribe the results of shock, ul- measuredpressure-volume data above50 GPa at 300 K trasonic, and static compressionexperiments for MgO. [Duffyet al., submittedmanuscript]. Since the single-crystal Hugoniot equation of state data do not show evidencefor material strength, it Compressional Wave Velocity is possible to combine the Hugoniot and ultrasonic data to constrain the fourth-order term in the Birch- The compressionalwave velocities measured under shock compressionare compared to third-order finite Murnaghanequation [Jeanloz et al., 1991].The results strain extrapolationof ultrasonicdata in Figure 7. The of the constrainedfourth-order fit are shown in Fig- Hugoniot measurementsagree with the ultrasonicex- ure 8 and do not differ greatly from the third-order trapolationswithin their experimentalprecision. This fit. The second pressure derivative of the bulk mod- supportsthe use of third-order finite strain theory for ulusK•' s is determinedto be-0.019(4) GPa-•. In extrapolating compressionalwave velocitiesin MgO contrast,Chopells [1992] obtained (02Kos/OP2)T -- over pressuresnearly a factor of 10 greater than the -0.054(16)GPa -• fromfluorescence sideband measure- experimentaldata and into the pressurerange of the ments on MgO to 20 GPa at room temperature. The lower mantle. The dependenceof compressionalsound secondderivative of Chopells[1992] is not consistent velocity on Hugoniot pressurein MgO to 27 GPa is with high-pressureHugoniot data (Figure8). The pres- given by surederivatives of ChopellS[1992] have been corrected to constantentropy conditions (K•s - 4.04, K•s = -0.052GPa -•) in makingthis comparison. In Vi• - 2.2790- 0.0624In P q-0.03441n 2P, (12) The elastic properties and equation of state of MgO have been calculated to 150 GPa using an ab initio whereVp is in km/s and P is in GPa. potential-inducedbreathing (PIB) model [Isaaket al., Hugoniot sound velocity measurementscan be in- 1990]. The secondpressure derivative of the isothermal verted to constrain elastic moduli and their pressure bulkmodulus at constanttemperature, (02KOT/0P2)T, derivativesalong the Hugoniot. In the region where foundin that studyis-0.026 GPa-• and the derivative thermal effects are small, these are comparable to ul- of the adiabaticmodulus is, (02Kos/OP2)T---- trasonicdeterminations. By analogyto (9), the fourth- DUFFY AND AHRENS: SHOCK COMPRESSION OF MgO 537

order finite strain expressionfor compressionalvelocity associatedwith shock fronts are treated by the method can be written [Duffy and Ahrens,1992] of artificial viscosity. In general, the stressconsists of a pressure(volumetric) term and a deviatoricterm a• (14- 2f) 5/2 -- aœo 4.aœ1f 4.aœ2f •, (13) a = P + •', (17) where both stressesand pressuresare taken to be pos- where itive in compression. The volumetric portion is de- aLo - Kos + 4/3G0, (14) scribed by the Mie-Grfineisen equation all -- 3I•'os(I•s 4. 4/3G[) - 5(1•0s4. 4/3Go), (15) P - PH+V( E - EH, (18) 9 2 9 where V is the volume, E is the energy, and the sub- aL2-- •Kos(K•s4-4/3G•0 + •Kos(K• s- 4) script H refers to the reference Hugoniot state. The Grfineisenparameter is modeledassuming q=l (Table x(K•s+ 4/3G•)+ (Kos+ •Go), (16) 5)s ? 4 P7 = p070. (19) where G is the shearmodulus and primes representfirst The bulk sound velocity is given by and second pressure derivatives, respectively. As with EOS data, the Hugoniot soundvelocities were combined with ultrasonic moduli and first pressurederivatives to s constrain the second pressure derivatives of the mod- and the compressionalsound velocity is uli. The fit yieldsK•s + 4/3G• --0.053(60) GPa-1. Togetherwith the value of K•s --0.19(4) GPa-1 deter- (21) minedabove, this implies that G•' - - 0.026(45)GPa -1. Vf- 3(1- 14- •) V• ' The elasticproperties of MgO are listed in Table 5. The where •, is Poisson's ratio. PIB modelcalculations yield G•• --0.020 GPa-1. The The stress deviators are obtained from a relation of fluorescencesideband measurements of Chopelas[1992] the form yieldG• • --0.034(10) GPa-1. Comparison of Hugoniot data with 300 K extrapola- Oa'lOt = 2G(OelOt- g), (22) tions of ultrasonic data can be biased because of thermal where e is the engineeringstrain, t is time, and g is a effects on the Hugoniot. Between 10 and 27 GPa, the relaxation function. continuum Hugoniot temperatures in MgO are calcu- A simple descriptionfor solidsis the elastic-perfectly lated to lie between 320 and 408 K using the method plastic model. Such a material behaveselastically until of McQueenet al. [1970]. From the ambientpressure its yield point, after which it deformsplastically, main- value of the temperature coefficientof compressionalve- taining a constantoffset of 2Y/3 from the hydrostat. locity for MgO [Isaak et al., 1989],an upper boundto Upon unloading,the material again behaveselastically temperaturecorrection can be determined(Figure 7). until it reachesa state of stressof 2Y/3 below the hy- The thermal effect on the velocitiesis significantly less drostat, after which it decompressesirreversibly to zero than experimental uncertainties. stress. It is also of interest to compare sound velocities in The yieldstrength at the Hugoniotelastic limit (HEL) single-crystaland polycrystallineMgO. Unloading wave (assumingavon Misesyield condition) can be obtained velocitiesin single-crystalMgO shockedalong [100]to from 5-11 GPa are ~5% below expected values based on y = (1- 2•,) extrapolatedultrasonic data [Grady,1977] (Figure 7). (1-•) a•/.r. (23) This suggeststhat the strength of the single-crystalma- The HEL amplitude of polycrystallineMgO was not di- terial has substantially but not completely recoveredin rectly measuredin these experiments,but the compres- the ~500-ns time interval between shock and release. sive wave structure was recorded at the buffer-window The recovery may be related to the relatively large interface in the forward impact experiments. thermal diffusivity in single-crystal MgO which leads The HEL amplitudes were modeled using WONDY to rapid thermal equilibration[Grady, 1980]. Polycrys- simulations and an elastic-perfectly plastic model for talline MgO, on the other hand, initially unloads in a both MgO and A1 6061. The model parameters are purely elastic fashion. listed in Table 4. The results of simulations for two for-

Constitutive Behavior ward impacts are shownin Figure 9. The yield strength of A16061 was fixed at 0.3 GPa [Asayand Lipkin, 1978]. Numerical simulations of the particle velocity his- The yield strength of polycrystallineMgO was varied tories were carried out using the one-dimensionalfi- between 0.5 and 2.5 GPa, and the predicted compres- nite differencewave code WONDY [Kipp and Lawrence, slye waveformsare shownin Figure 9. For yield stresses 1982]. This programsimulates plate impact and other above about 1.5 GPa, significant secondary structure experimental geometries by solving the equations of is evident due to the onset of plastic deformation in conservationof mass, momentum, and energy together the A1 buffer. No such secondary structure is observed with an appropriate constitutive law. Discontinuities in the measured wave profiles. For yield stressesbe- 538 DUFFY AND AHRENS: SHOCK COMPRESSION OF MgO

0.4 in Figure10. The yield strengthof MgO wasassumed to be 1.25 GPa in the first simulation(Figure 10a) and 2.5 GPa in the second(Figure 10b). The EPP model El 0.3 with Y - 1.25 GPa doesa poorjob of matchingthe ob- servedwave profile. While the initial elasticrelease is well modeled,the bulk of the unloadinghistory arrives too late in the simulation. This indicates that the un- o ,_, 0.2 loadingbehavior of MgO is not significantlydispersive. Much better agreementbetween calculation and ex- perimentis obtainedwhen the initial yield strengthis o.1 2.5 GPa (Figure10b). Thisfigure shows that an elastic- plasticmodel is generallyappropriate for describingthe dynamic responseof MgO. A detailed comparisonof o.o measured and calculated wave profiles indicates that o.6 0.8 1.o 1.2 these are some important deviationsfrom the model, Time (•s) however.The initial yield strengthis a factor of 2 larger than that inferred from the HEL amplitude. This im- Figure 9. Comparisonof compressivewave structure pliesthat MgO undergoessignificant strain hardening. transmitted through aluminum buffer with wave code The EPP modelpredicts several distinct wave arrivals simulationsfor two forward impacts. Dotted curvesare data (shotnumbers shown next to eachrecord). Solid which are not observed.In order to improve agreement, curvesare simulationsusing different values for the yield we incorporatedmodifications to the elastic-perfectly strengthof polycrystallineMgO. Computedprofiles for yield strengthsof 0.5, 1.25, and 2.5 GPa are shown. 1.0 lowwell 1.0below GPa, thethe experimentally predicted amplitudeobserved ofvalue.the HEL Yieldlies '•0.e• stressesin the rangeof 1.0-1.5 GPa yield HEL ampli- v tudesthat are reasonably consistent (:k10 m/s) with • 0.e experimentalvalues. We thereforeadopt Y - 1.25(25) •o

GPa as representingthe possiblerange of compressive :> 0.4 yield strengths of polycrystalline MgO ' This implies a ßO Hugoniotelastic limit of 1.6(3) GPa. The HEL of single- • crystalMgO shockedalong [100] was found to be 2.5 2 0.2 GPa for 3.3-mm-thicksamples at peak stressesbetween • 4.8 and 11.2GPa [Grady,1977]. Ahrens[1966] observed . . i i I i I elastic wave amplitudesof 3.5-8.9 GPa in single-crystal 0.0 0.0 0.5 1.0 1.5 2.0 MgO shockedto peak stressesbetween 16.6 and 42.3 Time (•zs) GPa. The precursor amplitude in the polycrystalline material is significantly less than single-crystalvalues. •.0 This has been observed before in other materials, no- b

tablyA1203 [Graham and Brooks, 1971' • Mashimo et al.• •'• 0.8 1988] and is a reflectionof differentyielding processes • in singleand polycrystals. • Waveprofile and equation of statedata for ceramics • 0.• (e.g'• SiC• B4C, TiO2 • ZrO2• and A120•) suggestquite o0 variedmaterial response. Some materials (SIC, A12Oa) •. are consistentwith relativelysimple material models ß 0.4 suchas a modifiedelastic plastic model [Mashimo et al., •

1988;Kipp and Grady,1989]. Other materials exhibit •• 0.2 anomalousdispersion that impliessignificant deviation • from elastic-plasticbehavior [Kipp and Grady,1989]. In order to assesswhether an elastic-plastic model 0.o . . ,•, i I i is appropriate for magnesiumoxide, numerical simula- 0.0 0.5 1.0 1.5 2.0 tions of the experiments were undertaken utilizing an Time (•zs) elastic-perfectlyplastic (EPP) modelfor MgO. A1 6061 Figure 10. Calculatedand measuredparticle velocity and LiF were also modeled as EPP with the parameters histories at the aluminum-LiF interface for experiment listed in Table 4. The foam model of Grady and Furnish 848. Dotted curve is experimental profile, and solid [1988]was used to describethe flyer-backingmaterial. curve is numerical simulation. MgO, A1, and LiF are The predicted particle velocity histories are compared treated as elastic-perfectlyplastic. Yield strength of to measured values for a representative reverse impact MgO is (a) 1.25GPa and (b) 2.5 GPa. DUFFY AND AHRENS: SHOCK COMPRESSION OF MgO 539

plasticmodel that have been successfullyapplied to Table 6. BauschingerModels for MgO and A1 6061 metals. In particular,we includeda Bauschingeref- fect and strain rate dependentstress relaxation. In the MgO A1 6061 Bauschingereffect, the yield stressis reducedwhen the Model 1 Model 2 direction of plastic deformation is reversed. This behavior is a consequenceof the micromechanicsof defor- ai Y/, GPa ai Y/, GPa ai Y/, GPa mation,such as dislocationinteraction, slip bands, and twinning. Flow stressanisotropy is implementedinto 0.7 1.5 0.10 2.95 0.581 0.169 the wavecodeusing a multielementkinematical model 0.12 4.0 0.20 3.0 0.190 0.350 [Herrmann,1974] in whichthe equilibriumstress devi- 0.10 8.0 0.20 3.1 0.078 0.472 0.08 12.0 0.35 3.2 0.052 0.977 ator, ae,• is givenby - - 0.09 3.5 0.063 1.649 - - 0.06 4.1 0.036 2.589 ere' - • a i •ri ' , (24) wherethe ai are normalizedweighting factors. Each elementalstress deviator a i• is subject to avon Mises yield condition t2 reproducesthe strong elastic releaseobserved in these • _<(2/3Y•) •. (25) experiments. The primary differencebetween the model Strainrate dependenceis alsotreated through the and the experimentalrecords is that the modelpredicts deviatoricstress. The stressdeviator is obtainedby a secondelastic releaseoriginating from the reflection relaxationfrom some instantaneous value to an equilib- of the shockwave at the foam-flyerinterface. In fact, riumvalue, using the relaxationfunction in (22). The the releaseat this point is highly dispersive. relaxation function has the form Model 2 fits the forward impacts and the lowest- amplitudereverse experiment. The Bauschingerparam- 0.1 -- O'eI etersfor this modelare listedin Table 6, and the model g-- Gt,' (26) predictionsare comparedto representativeexperimen- tal data in Figure 11b. The agreementbetween data where tr is a characteristic material relaxation time. and calculationis goodfor the unloadingportions of Beginningwith the elastic-perfectlyplastic model, the waveform.The risetime of the shockfront is longer the waveprofiles were fit iterativelyby adjustingthe pa- in the modelcalculations than in the experimentaldata. rameters of the Bauschingermodel and the relaxation Figure 12 showsrepresentative stress-strain histories time constant.For A1 6061,the Bauschingermodel of at the center of the MgO samplesfor the forward and Lawrenceand Asay [1979]was used. Very little rate reverse impacts computed from the wave code simula- dependenceis requiredin fitting the profiles. A time tions. Here the strainis definedas r/ - 1 - po/p. The constantof 5 ns was found to improvethe fit slightly Bauschingereffect is responsiblefor smoothingthe tran- for approximatelythe final 20% of the unloadinghis- sitionfrom elasticto plasticstrain upon both loading tory. The Bauschingereffect, on the otherhand, is sig- and unloading. This effectis much more pronounced nificant in MgO. in the forwardimpact model (model 2). The higher- We were unableto fit all of the data with a single stressreverse-impact experiments (848 and 851) retain model. The reason for this can be seen with reference a cleartransition from elastic to plasticunloading. The to Figure 5. The reverseimpacts exhibit significantly stress-strain states inferred from the wavecode simula- stronger and sharper initial release behavior than the tions lie abovethe hydrostaticcompression curve, as forward experiments. The forward impacts involved inferredfrom the equationof state experiments. a propagationdistance through MgO of about 7 mm, whilein the reverseexperiments the MgO propagation Summary distancewas 3-4 mm. This may indicatethat the wave The Hugoniotof low-porosity(0.5%) polycrystalline profilesare not steady.A secondpossibility is that the MgO between 14 and 133 GPa is partial releasegenerated at the sample-bufferinterface in the forwardexperiments produces damage or defects Us = 6.87(10)+ 1.24(4)up, whichaffect subsequent unloading waves. The lowest- amplitudereverse impact (854) is similarto the forward which differssignificantly from that of single-crystal impacts in that it has a weak and diffuse elastic release. MgO. Compressionalsound velocities were determined In the forwardimpact experiments, the two highest- from unloading wave profiles at 10-27 GPa. Within stressexperiments (859 and 864) show nearly featureless their +2% uncertainties,these data are consistentwith unloading,particularly shot 864. Thesepoints illustrate ultrasonicdata extrapolated from 3 GPa. Thissupports that the shapeof the profilein MgO is dependentboth the useof third-orderfinite strain theory for comparing on peak stressand impact geometry. low-pressurelaboratory measurements with seismolog- The modelingprocedure converged on two modelsfor ical data to pressuresof at least ,-,25 GPa for close- MgO.The reverseimpacts were fit usingmodel I (Ta- packedsolids such as MgO. By combiningHugoniot ble 6) and are displayedin Figure 11a. This model equationof state, Hugoniotsound velocity, and ultra- 540 DUFFY AND AHRENS: SHOCK COMPRESSION OF MõO

1.6 i , , 1.6 i

859 851 , a

848 o 0.8 •0 0.8

854 -

0.4

0.0 , . lJ.. i i i 0.0 0.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 Time (/•s) Time (/•s)

Figure 11. Numericalsimulations of (a) reverseand (b) forwardimpacts. Symbols are experimental data, and solid curve is computedvelocity history. Profile of shot 859 was scaledby 1.05 to bring its amplitude into agreementwith calculated profile.

sonic data, the adiabatic bulk modulus, and its first ferences are also characteristic of the behavior of rela- and secondpressure derivatives at constantentropy are tively low-strengthceramics like MgO. 162.5(2)GPa, 4.09(9), and-0.019(4) GPa -1. The cor- From wave profile and equation of state data, we respondingvalues for the aggregateshear modulusare infer that for polycrystalline MgO, the yield strength 130.8(2)GPa, 2.5(1),and-0.026(45) GPa -1. at the HEL is 1-1.5 GPa, only ~0.5 the single-crystal The dynamic responseof MgO is stronglydependent value. The strength at the Hugoniot state is larger on crystal state. PolycrystallineMgO is characterized (2.8(7) GPa) and this strengthis largely maintained by a low Hugoniot elastic limit, significantHugoniot uponrelease. For single-crystals,the largestrength (2.5 shearstrength, work hardening,and fully elasticrelease. GPa) at the HEL is reducedto near zero alongthe By comparison,single-crystal MgO has a large elastic Hugoniot, but the strength is partially recovereddur- limit, no strength along the Hugoniot, and quasi-elastic ing release(with a reducedshear modulus). Thus the release[Grady, 1977]. However,it shouldbe notedthat shock-compressionprocess is complex for both forms wave profile data for the single crystal cover a lower of this material and each exhibits its own time depen- stress range than the polycrystal data. Nevertheless, dent strengthbehavior. While there are differencesdue the differences seen here are similar of those observed to experimentalconfiguration, the polycrystallineMgO in single-crystals.andpolycrystals of A1203 and ZrO2 wave profilesare generally consistentwith the behavior [Mashimo,1993]. We havedemonstrated that suchdif- of an elastic-plastic material.

30 , ! ! • / • / 40 b ' ' ' 85 / 25

• •o

• to

5 5

0 ' o 0.00 0.05 0.10 O.15 0.00 0.05 O.10 O.15 Strain Strain

Figure 12. Stress-strainhistories from numericalsimulations of (a) reverseand (b) forward impactparticle velocity histories of Figure11. Dashedcurve is the 300 K (hydrostatic)isotherm. DUFFY AND AItRENS: SHOCK COMPRESSION OF MgO 54!

Acknowledgments. We thank S. M. Rigden for per- Science and Technology- 1993, edited by S.C. Schmidt, forming the ultrasonic sound velocity measurements. M. J. W. Shaner, G. A. Samara, and M. Ross, pp. 1107-1110, Long, E. Gelle, and A. Devora provided experimental as- American Institute of Physics, New York, 1994b. sisrance. The comments of R. Jeanloz and two reviewers Grady, D. E., Processesoccurring in shockwave compression of rocks and minerals, in High-Pressure Research: Appli- are appreciated. We also acknowledgeL. Barker and O. B. cations to Geophysics,edited by M. H. Manghnani and S. Crump for helpful discussions.This research was supported Akimoto, pp. 389-438, Academic, San Diego, Calif., 1977. by the NSF. Division of Geological and Planetary Sciences, Grady, D. E., Shock deformation of brittle solids, J. Geo- California Institute of Technology contribution 5397. phys. Res., 85, 913-924, 1980. Grady, D. 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