t. s. duffy- strength of materials under static loading in the diamond anvil cell

8/3/2019 T. S. Duffy- Strength of Materials Under Static Loading in the Diamond Anvil Cell

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CP955, Shock Compression of Condensed Matter - 2007,edited by M. Elert, M. D. Furnish, R. Cliau, N. Holmes, and J. NguyenO 2007 American Institute of Pliysics 978-0-7354-0469 -4/07/$23.00

STRENG TH OF MA TERIALS UNDER STATIC LOADINGIN THE DIAMOND ANVIL CELLT. S. Duffy^

Departmet ofGeosciences, Princeton University, Princeton, NJ 08544

Abstract. Strength properties of materials under static and dynamic loading are essential quantitiesfor characterizing mechanical behavior. Quantitative measurements of static strengths achieved in thediamond anvil cell can be made using x-ray diffraction in a radial geometry for samples under non-hydrostatic c omp ression. Resu lts for several metals up to 68 GPa show that the ratio of yield streng thto shear mod ulus ranges from 0.01-0.03 and increases weakly with compression. Trend s at lowerpressure for Re, W , and 8-Fe are consistent with available data above 1 Mb ar. Strong ceramics such asBgO, c-Si3N4, and TiB2 achieve yield strengths as large at 10% of the shear modulus at pressures up to~70 GPa. Strengths of mate rials used as pressure media are quantitatively com pared and evaluated.Keywords: yield strength, x-ray diffraction, high pressure, metals.PACS: 62.50.+P, 62.20.Fe, 61.10.-i.

INTRODUCTIONMechanical properties and strength at ultrahighpressures have been of interest since the advent ofhigh-pressure techniques [1]. Applications includeoptimizing the design and performance of statichigh-pressure apparatus and improving ballisticsperformance under dynamic loading. Basicquestions regarding strength include how pressureaffects yield strength and how the strengths ofdifferent classes of materials compare at ultrahighpressures [2]. Development of new methods for

dynamic loading under shock and quasi-isentropicconditions has also spurred renewed interest inmaterial strength [3,4].In the diamond anvil cell, qualitative methodsto identify non-hydrostatic stresses were developedfrom observation of pressure gradients andlinewidths of ruby fluorescence peaks.Quantitative measurements of shear strengthfollowed from considering the balance between theradial pressure gradient and the shear stress at the

anvil-sample interface which yields the relation [5](Fig. 1):h dPT = , (1)2 dr

where r is the shear stress, dP/dr is the radialpressure gradient, usually obtained from rubyfluorescence measurements, and h is the samplethickness. A second method for stressdetermination uses measurements of the ellipticityof diffraction rings formed by x-rays which passperpendicularly to the axis of the diamond cell [6].Renewed development of this radial diffractionmethod [7] in recent years has led to rapid growthin characterization of strength effects at highpressures as well as opportunities for studyingelasticity, equations of state, and deformationmechanism s at extreme conditions.

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THEORY d^ (hkl) = d (hkl)[l + (1 - 3cos t//)Q(hkl)], (3 )In radial x-ray diffraction experiments, asample is compressed under intentionally non-hydrostatic conditions and the elastic lattice strainis measured as a function of the angle from the

loading axis using synchrotron x-rays (e.g. [7,8])(Fig. 1). The stress state in a polycryslaUine sampleunder uniaxial compression in the diamond anvilcell is described by a maximum stress along thecell loading axis, Oj, and a minimum stress in theradial direction, aj. The difference between Oj andoi is the differential stress, t:f = (T3 - (T j < 2 r = 7 , (2)

whe re 7 the yield strength of the sample. Theequality in Eqn. (2) holds for a Von Mises yieldcondition and t could be less than the yieldstrength.

Figure 1. Schematic illustration of diffraction geometryin a radial x-ray experiment in the diamond cell with EDdetector, r, z are the radial and axial coordinates,respectively, x is the cell rotation angle, P(r) is the radialpressure distribution, and 6* is the diffraction angle.

The observed J-spacings (d) for a set of planes(hkl) measured by x-ray diffraction are a functionof the angle t// between the diamond cell axis andnormal to the diffraction planes (Figs. 1 and 2):

where dpQikl) is the d spacing resulting from thehydrostatic component of stress, and Q(hkl) isgiven by:a (1 - a )QQlkl) : 3 [iG^Qikl) 2G (4)

GgQikl) is the aggregate shear modulus of grainscontributing to the diffraction intensity under thecondition of constant stress across grain boundaries(Reuss limit). Gy is the shear modulus underisostrain conditions (Voigt bo und).According to Eqn. (3), djhkl) should varyl inear ly with l -3 co sV (F ig . 2) . At ^ = 54 .7 (1-3cos^ y/ = 0), the position of the observed x-raydiffraction line reflects the d spacing due to thehydrostatic component of stress only.

l-3cosVFigu re 2. Variation in rf-sp acin g with angle from loadaxis for the (110) diffraction peak of tungsten [9].Pressures in GPa are indicated on the right.

The aggregate polycrystalline sample in thediamond anvil cell is generally assumed to beunder isostress conditions. In this case, a is equalto 1 in Eqn. (4) and the differential stress can beexpressed as:6G^ < Qihkl) >, (5)

where represents the average QQikl)value over all observed reflections, and G;; is theReuss bound on the aggregate shear modulus of thepolycrystalline sample. If the differential stress, t,has reached its limiting value of the yield strength

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at high pressures, 6 = t/G, will reflect theratio of yield strength to shear modulus.Orientation dependence of strength may beimportant in detail, but here we focus solely onmean strength.Microscopic deviatoric stress also exists in apolycrystalline sample under compression as aresult of grain-to-grain contact and/or strength ofthe pressure medium. This stress field leads tobroadening of x-ray diffraction peaks.Experimental studies in both large-volumeapparatus and the diamond cell have also usedmeasurements of broadening to constrain strengthfrom the microscopic stress distribution [9-11].

EXPERIMENTAL METHODSUpon loading in the diamond anvil cell, anequilibrium pressure gradient is established (Fig.1). The stress distribution is assumed to be radiallysymmetric about the load axis and the axial stressgradient is negligible [12]. In the earliest stages ofcompression, the sample undergoes compaction,followed by elastic and then plastic deformation.The sample can undergo considerable plasticdeformation even at low pressures. Strain rates arenot directly measured but are low (


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At 37 GPa, the strength of gold is just 0.4-0.6GPa, significantly below other metals examined.The predicted strength at 1 Mb ar from Eqn. (6) isabout 1.2 GPa. Platinum is also widely used as apressure calibrant and laser absorber in thediam ond anvil cell. Desp ite a relatively low shearmodulus (G=61 GPa), Pt appears to supportrelatively large differential stresses [14] (Fig. 3).

ro 0.03 -SZ!0 / ^ 0.02 - ,, ',"7; Mo . .2 0 01 - i 4 * i * Al l

oooc 1 1 1 1 1 10 10 20 30 40 50 60 70Pressure (GPa)

Fig ure 4. Ratio of differential stress to shear modulus asa function of pressure for selected metals.Figure 4 shows the ratio of differential stress toshear mo dulu s for the meta ls studied here. This

quantity is determined directly from the mean slopeof the d sp ac in g- ^ relation (Eqns. 3 and 5). Theincompressible metals (Re, W, Mo) exhibitdifferential stresses that are about 2- 3% of theshear modulus under these loading conditions.Furthermore, t/G tends to increase weakly withpressu re. That is, the differential stress increases ata faster rate than the shear mo dulus . This likelyreflects the contributions of both pressure andstrain hardening to the measured yield strength.In the case of Au, the t/G values are less than

0.01 up to 37 GPa. Pt, on the other hand , exhibitst values that exceed 3% of the shear modu lus. Inview of its importance, further studies on Pt arewarranted to confirm these observations and extendthem to higher pressures.Measurements of static strength at pressuresabove 1 Mbar are limited at present (Fig. 5).Meaurements of diffraction of 8-Fe and W at ^ =0 and 90 were carried out to pressures as high as

300 GPa [17]. In another study, the yield strengthof Re at 100 GPa was estimated from comparisonof finite element models with experimental gasketthicknesses [18]. Differential stresses have beenmeasured in (Mg,Fe)Si03 and MgGeOs post-perovskites (CalrOs-type) using radial diffractiontechniques [15,19] at 104-157 GPa, although thesevalues are probably lower bounds to the yieldstrength because the samples were laser heated andthis would be expected to partially releasedifferential stresses.

Pressure (GPa)

Figure 5. Summary of differential stress measurementsat pressures above 1 Mbar. Lines are fits that connectlow-pressure and high-pressure measurements. pPv -germanates and silicates in the post-perovskite structure.Experimental results are from [8,9,11,15,17-21].

In Fig. 5, measurements [17,18] at P>1 Mbarfor W, 8-Fe, and Re are compared with lowerpressure studies [8,9,11,21]. The high-pre ssuredata are broadly consistent with trends observed atlower pressures (e.g., tRe>tw>te-Fe)- Yield strengthcontinues to increase to Mbar pressures such thatfor 8-Fe and W differential stresses at ultrahighpressures are about 2-5 times larger than observedfor these metals near 50 GPa. For W, t values at300 GPa are -3-5% of the expected shear modulusat this pressure, suggesting that t/G continues toincrease with compression to ultrahigh pressures.Moreover, the t values measured at these extremeconditions may be a lower bound to the yieldstrength as anvil deformation may limitdevelopment of differential stresses [17].

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Radial x-ray diffraction experiments have alsobeen carried out up to 68 GPa on a variety ofstrong ceramics including BgO [22], Si3N4 (spinelstructure) [23], TiB2 [24] and silicates such asringwoodite [25], stishovite [26], and perovskite[27] (Fig. 6). Differential stresses in theseceramics greatly exceed those observed in metals.The strongest materials measured to date includeBgO and TiB2 which sustain differential stressesnear 28-30 GPa at mean pressures of 60-65 GPa.For co mpa rison, the differential stress in TiB2 wasdetermined to be 42 GPa at 61 GPa along theHugoniot [28]. The values of t/G for thesematerials increase under static loading anddifferential stresses reach 8-10% of the shearmodulus at the highest pressures for BgO and cubicSi3N4, considerably greater than values observedfor metals.

Pressure (GPa)

Figure 6. Differential stresses in selected strongceramics including oxides, borides, nitrides, andsilicates. / values for Re metal are shown forcomparison. Rwringwoodite; Pv- perovskite; c-Si3N4-cubic (spinel-type) silicon nitride. Data are from [22-25,27].Hydrostatic pressure conditions are optimum

for many types of high-pressure experiments butare difficult to achieve at room temperature . Avariety of soft pressure transmitting media havebeen employed to achieve quasi-hydrostaticcondition s in diamon d cell experim ents. In laserheating experiments, an insulating medium isneeded to reduce axial temperature gradients [29].While qualitative constraints on relative strengthsof various pressure and insulating media have long

been known, the radial x-ray diffraction methodnow allows quantitative limits to be placed on themaximum differential stresses that may besustained. Resu lts for several mate rials used aspressure media are summarized in Fig. 7. Mg O isrelatively strong but there are large differences instrengths reported in diamond anvil [30] and large-volum e press experim ents [31]. These stressdifferences may be related to higher levels ofplastic strain achieved in the DAC experiments[31]. In addition, the evolution of grain size withpressure may play a role [32].Limited data on NaCl suggests that it supportsrelatively small differential stresses (0.5-0.8 GPa at19 GPa confining pressure) [33]. At modestpressures, solid argon has the lowest observed tvalue (0.4 GPa at 28 GPa pressure), although thereis evidence that the strength of Ar begins toincrease rapidly above 30 GPa [34]. While CaO isnot commonly considered as a pressure medium, itis notable that at 40 GPa and above, differentialstresses in CaO are actually lower than those in Ar[35]. This shows how quantitative measurementsof differential stresses are valuable forcharacterizing material behavior and identifyingcandidates for insulating and pressure transmittingmedia over different ranges of compression.

XPressure Media

Pressure (GPa)Figure 7. Differential stresses in materials commonlyused as pressure transmitting or insulation media indiamond anvil cell experiments. Refs: MgO [30,31],NaCl [33], Ar [34], CaO [35].

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CONCLUSIONSIn general, strength increases with compressionat a rate greater than the shear modulus, implyingsignificant strain hardening under diamond cellloading conditions. At pressures of 20-80 GPa,

metals typically exhibit strengths of 1-3% of theshear modulus, G. Strong covalent oxides,silicates, and nitrides such as ringwoodite, boronsuboxide, and spinel-structured silicon nitridepossess yield strengths that can approach 10% ofthe shear modulus. The maxim um strength valuewe have measured is 28 GPa for BeO at 65 GPaconfining pressure. The general increase instrength with pressure for all material classesimplies that most materials are "strong" at Mbarpressures if one recognizes that the strength ofhigh-grade steel at ambient pressure is ~2 GPa.Furthermore, the consistency of general strengthtrends across material classes suggests thatreasonable empirical predictions of strength in themulti-Mbar pressure regime can now be made.

ACKNOWL E DGE ME NT SFunding was provided by the NSF and theCamegie-DOE Alliance Center. S. Dorfmanprovided assistance in figure preparation.

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t. s. duffy- strength of materials under static loading in the diamond anvil cell

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