t.j. ahrens- equation of state

8/3/2019 T.J. Ahrens- Equation of State

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"H ig h P re ss ur e Shock Compression of Solids'.'eds. J. R. Asay and M. Shahinpoor. Sprin~er-Verlag, NY. 75-114 [''lq 3)

CHAPTER 4Equation of S tateTJ. Ahrens

4 .1 . IntroductionT he use of p lan e sho ck w av es to determ ine the equ ation s o f state of con -d en se d m ate ria ls to v ery h ig h p re ss ure b eg an in 1 9 55w i th th e c la ss ic p ap erso fWa ls h a nd Ch ri sti an (1 955) and Banc ro ft e t a 1 .(1 956) .Wa ls h a nd Ch ri st ia nd es crib ed th e u se o f in -c on ta ct e xp lo siv es to d ete rm in e d yn am ic p re ss ure -volum e relatio ns for m etals an d com pare these to the then available sta-tic c omp re ss io n d ata . B an cro ft e t al , des cr ib ed t he f irs t p ol ymo rph ic phas ech an ge d isc ov ered in a so lid , v ia sh ock w av es-iro n. Two years la ter S ov ietwo rk er s (A l 't sh ul er e t a l. , 1 958) re po rt ed t he f ir st d at a f or i ro n t o p re ss ur es o fs ev era l m i lli on b ars (megabars ) a ct ua ll y e xc ee di ng t he p re ss ur e c ondi ti on sw ithin the center of the Earth. S ince that tim e the equations of state ofv ir tu a'l y hundr ed s o f c onden se d ma te ria ls h av e bee n s tud ie d, in clud ing e le -ment s, c ompounds , a llo ys , r oc ks a nd m ine ra ls . p o lymer s, f lu id s, a nd por ou smedi a. The se s tudi es h av e employ ed bot h c onve nt io na l a ud nucl ea r e xp lo si vesources. as w ell as im pactors launched w ith a range of guns to speeds ofapprox ima te ly 10 km/s, Rec en tly , Av ro rin e t a t ( 1 986 ) h av e re po rte d shock -comp ressio n d ata in lead to a reco rd p ressu re o f 5 50 M bar.

In th is c ha pte r w e d efin e wh at is m ea nt b y a s ho ck -w av e e qu atio n o f s ta te ,a nd h ow it is re la te d to o th er ty pe s o f e qu atio ns o f s ta te . We a lso d is cu ss th ep ro perties o f sh ock -comp ressed m atter o n a m icro sco pic scale. as w ell asd is cu ss h ow s ho ck -w av e p ro pe rtie s a re m ea su re d. S ho ck d ata fo r s ta nd ardma te ri al s a re p re se nt ed . The e ff ec ts o f pha se c ha nges a re d is cu ss ed . t he me a-s ur ement s o f s ho ck t emper at ur es , a nd sound vel oc it ie s o f s ho ck ma te ria ls a rea ls o d es cr ib ed . We a ls o d es cr ib e t he a pp lic at io n o f s ho ck -c omp re ss io n dat af or por ou s medi a.S ho ck -w av e d ata h av e s ee n mo st a pp lic atio ns in th e m ea su reme nt o f d e n-s ity a t h ig h p re ss ure . O th er p ro pe rtie s o f c omp re ss ed c on de ns ed m ate ria lswh os e m ea su reme nts a re d is cu ss ed in th is c ha pte r in clu de s ou nd sp ee d a ndtempe ra tu re . Rev iew a rt ic le s by Grady ( 1977).Yakushev ( 1978),Davi son andGr aham (1 979) ,Mu rr i e t a I. (1 97 4 ),A l 'ts hu le r (1 965) . and M i ll er a nd Ah re ns(1 99 1 ) summar iz e e xper iment al t ec hn ique s f or me asuri ng dynam i c y ie ld ing,1IjIIIII~ "_~ ~, _

75

-----'-..__ .................=..c"'-- ___, _, .. ,_.


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76 T.J. Ahrensabsorption spectra, index of refraction, electrical conductivity at radio andmicrowave frequencies, and viscosity. Most of these properties can be relatedto pressure-density behavior. These are among the many physical propertiesof solids and fluids which can and have been measured under shock-loadingconditions. Other properties studied under shock, which have less relation toan equation of state, include piezoelectricity, ferroelectricity, shock-inducedpolarization, and shock-induced demagnetization. Practical applications ofshock-wave phenomena in metal working, shock welding, and dynamic com-paction ofmaterials are given in Murr et a1.(1981).In addition to fundamental studies of the compression of condensed mat-ter, an important application of shock and related properties is found in

the accretion impact mechanics of terrestrial planets and the solid satellitesof the terrestrial and major planets from asteroid-sized protoplanetesimals(Gehrels, 1978).A related application of both shock compression and isen-tropic release data for rocks and minerals (Ahrens and O'Keefe, 1972;Ahrensand O'Keefe, 1977)is in the mechanics of both the continued bombardmentand, hence, cratering on planetary objects through geologic time (Roddy etal., 1977), as well as the effects of giant impacts on the Earth (Sharptonand Ward, 1990;Silver and Schultz, 1982).Finally, recovery and characteriza-tion of shock-compressed materials have provided important insights intothe nature of shock-deformation mechanisms and, in some cases, providedphysical data on the nature of either shock-induced phase changes or phasechanges which occur upon isentropic release from the high-pressure shockstate (e.g.,melting) (Steffler, 1972, 1974).Three pressure units are commonly in use in shock-wave research: kilobar(kbar), gigapascal (GPa), and megabar (Mbar). These are equal to 109, 1010,and 1012 dyne/ern", respectively. The shock pressure range ofprimary interestin this review article is ""100-....,4000 kbar.

4.2. Sho ck-Wave Equatio ns of StateThe propagation of a shock wave from a detonating explosive or the shockwave induced upon impact of a flyer plate accelerated, via explosives or witha gun, result in nearly steady waves in materials. For steady waves a shockvelocity U with respect to the laboratory frame can be defined. Conservationof mass, momentum, and energy across a shock front can then be expressed as

PI =Po(U - uo)/(U - U1),PI - Po =po(u1 - uo)(U - uo),E1 - Eo =(P 1 + Po)(l/po - 1 /pd/2 =H U t - UO)2,

where p, u, and E are density, particle velocity, shock pressure, and internalenergy per unit mass and. as indicated in Fig. 4.1, the subscripts 0 and 1 referto the state in front of and behind the shock front, respectively. Equations

-'--~.-----~.---...------.....----.-

(4 .1 )(4 .2 )(4 .3 )


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4. E qu atio n o f S ta te 77

TJA8415:1SFDF ig ure 4 .1 . P rofile o f a stead y sh ock w av e, risetim e 1'" im pa rtin g a p artic le v elo city ,e.g. ,Ill'pressure Pl' a nd in te rn al e ne rg y d en sity EI, p ro pa ga tin g w ith v el oc ity U. intom aterial th at is at re st at den sity Po a nd in te rn al e ne rg y d en sity Eo.

(4.1)-(4.3)are often called the Rankine-Hugoniot equations. It should beunderstood that in this section pressure is used in place of stress in theindicated wave propagation direction. Actually stress, in the wave propaga-tion direction, is what is specified by (4.2).A detailed derivation of (4.1).(4.2)and(4.3)isgiven inChapter 2 and in Duvall and Fowles (1963).Equation (4.3)also indicates that the material achieves an increase in internal energy (perunitmass)which is exactly equal to the kinetic energy per unit mass.In the simplest case when a single shock state is achieved via a shock front,

the Rankine-Hugoniot equations involve six variables (U , U1 Po, PI' E} -Eo . and PI); thus. measuring three, usually U, Ul' and Po. determines theshock state, Ph El - Eo, and Pl' The key assumption underpinning thevalidityof (4.1)-(4.3) is that the shock wave is steady so that the risetime t.,is short compared to the characteristic time for which the high pressure,density,etc., are constant (see Fig. 4.1). Upon driving a shock of pressure PIintoa material, a final shock state isachieved which isdescribed by (4.1)-(4.3).Thisshock state is shown in relation to other thermodynamic paths in Fig.4.2,in the pressure-volume plane. Here Vo = l /po and V I = l /pl ' In the caseof the isotherm and isentrope, it is possible to follow, as a thermodynamicpath. the actual isothermal or isentropic curve to achieve a state on theisothermor isentrope. A shock, or Hugoniot, state is different, however. TheHugoniot state, Plf Vi' is achieved via a thermodynamic path given by thestraightline called a Rayleigh line (Fig. 4.2). Thus successive states along theHugoniot curve cannot be achieved, one from another, by a shock process.TheHugoniot curve itself then just represents the locus of final shock statescorresponding to a given initial state.To demonstrate that the Rayleigh line actually represents the thermo-dynamic path to which material is subjected on being shocked from state

p = 0, V = Vo to P = PI' V = VI' wedemonstrate below that the shock wavesketchedin Fig. 4.1 must be steady. Moreover, the Rankine-Hugoniot equa-tions 4.1)-(4.3 not only describe the conservation of mass, momentum,


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--

78 T.J. A hrens

TJA84154SFDF ig ure 4 .2 . P re ssu re -v olume c om pre ssio n cu rv es. F or isen tro pe a nd iso th erm , th et he rmodynam ic pat h coi nc id e s w ith t he l ocus o f s ta te s, wher ea s f or s hock , t he th e rmo -d yn am ic p ath is a straigh t line to poin t PI' Vt > o n th e H ug on io t c urv e, w hich is th el ocus o f s hock s ta te s.

and energy from state 0 to state 1, but also describe conservation of thesequantities from state 0 to an arbitrary intermediate state such as state A (Figs.4 .1 and 4.2) (see e.g., Zel'dovich and Kompaneets (1960), Chap. I) , If thepressure and specific volume of state A, which can lie at any point along theRayleigh line, are given by P and V . the pressure at A can be written as

or (4.4a)where (4.4) is obtained by eliminating Ul between (4.1) and (4.2), and byassuming Uo = 0 and Po = O.We shall use up to denote particle velocity andV,to denote shock velocity. Equation (4.4)is recognized as the equation ofaline of slope - U.2jVl and its intercept is at a value of V =0 of V/ / V O . Sincestate A can represent any state between P = 0 and P = P I' (4.4)represents aseries of thermodynamic paths which the material follows on being shockedfrom state 0 to state 1. Thus, although the Hugoniot represents a seriesof thermodynamically defined states, these states are always achieved bya Rayleigh line thermodynamic path and not from another state on the curve.When centered at ambient conditions, this Hugoniot curve is called a prin-cipal Hugoniot. The isentropic centered at ambient conditions is similarlydesignated as a principal isentrope. Some principal Hugoniots for solid mate-rials are shown in Figs. 4.3 and 4.4 in the shock velocity-particle velocity andpressure specific volume planes.It has long been recognized that the kinematic parameters measured inshock-wave experiments U s and up can empirically be described in regionswhere a substantial phase change in the material does not occur as

( 4 .4)

(4.5)


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i,,ii!ji!i

)Iii!

IIIIIII

I

4 . E quation of State 79..-o~ 15. ! I I : 5co. c :VI

O~~~~~~~~~~~~~~~~~o 2 .68Particle Velocity (km/sec)F ig ure 4 .3 . S ho ck v elo city v ers us p artic le v elo city fo r se ve ra l sta nd ar d m ate rials .

As further discussed in several review articles on shock compres-sion (Al'tshuler, 1965; Davison and Graham, 1979; McQueen et al., 1970),Hugoniot data for many condensed media may be described over varyingranges of pressure and density in terms of a linear relation of shock andparticle velocity.Typical U.-uP data for a wide range of materials are given in Fig. 4.3 andTable 4.1.Here Cois the shock velocity at infinitesimally small particle veloc-ity,or the ambient pressure bulk sound velocity which is given byCo=~, (4.6)

Shock PressureversusSpecific VolumeCu

TJA93185SFD

F ig ur e 4 .4 . S ho ck p re ss ure v ers us s pe cific v olu me fors ev er al s ta nd ar d ma te ri al s.


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" '

!IIiIiIiII!IfIIIIII

jIIiIIIIiL.. ._

80 T . J. Ah ren sTable 4 . 1 . Equ atio n o f s ta te o f c on de ns ed s ub sta nc es .

Initial density CoMaterial (Mg/ml) (km/s) sAluminum,IAIAlloy 1100 2.707 5.386 1.339Tantulum, Tal 16.67 3.291 1 .308Platinum, pe 21.41 3.641 1.541Copper.Cu' 8.939 3.933 1.500Teflon4 (CF1) 2.152 1 .846 1.777Water,l H2O 0.9979 2.393 1 .333Fuzed quartz,' SiOz 2.204 1 .7054 0.7643Calcia, Ca06 3.345 5.766 1 .404

3.265 1.709IMitchell and Nellis (1981) .2 Holmes et a1.(1989).3 Mitchell and Nellis (1 98 2) 1 .5 < up < 7.1 km/s." Morris et al. (1 9 84 ) 0 .6 < up < 2 .5 km/ s.5 High-pressure stishovite phase data (Wackerle, 1962; Marsh, 1980;Jones et al., 1972and Trunin et aI., 1971).6 Low-pressure, BI, phase and high-pressure, B2, phase data (Jeanlozand Ahrens, 1980).

where K& is the isentropic bulk modulus, K$ =- V (d Pld V) . Upon substitut-ing (4.5) into (4.2),and denoting the shock pressure as P H ' this is given by

Thus, from the form of (4.7),shock pressure is given as the sum of a linearand quadratic term in particle velocity, based on the data of Table 4.1. Apressure-volume relation can be obtained by combining (4.6) with (4.1) toyieldwhereEquation (4.8) is often called the "shock-wave equation of state" since itdefines a curve in the pressure-volume plane (e.g.,Fig. 4.4).In order to relate the parameters of(4.5),the shock-wave equation of state,to the isentropic and isothermal finite strain equations of state (discussed inSection 4.3),it is useful to expand the shock velocity normalized by Co into aseries expansion (e.g , Ruoff, 1967;Jeanloz and Grover, 1988;Jeanloz, 1989).

V.IC o = 1 + su + s'Cou 2 + s"CJu 3 + ...,where U = uplCo . Using (4.10)and (4.2),we find that

p = u (1 + su + s'Cou2 + s"CJu 3 + +-).(4 .10)

Moreover, wecan define a reduced pressure as p =P H IPo C J . Upon substitut-(4.11)

---------

(4.7)

(4.8)( 4 . 9 )


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4. Equation of State 81ing UJfrom (4.10) into (4.9) and using the binomial theorem. we find that

'1 = u(1 - su + (S2 - s'C o)u 2 - (S3 - 2ss'Co + s"CJ)u3 +.,'). (4.12)The series of (4.12) may be inverted by the reversion method to yield a seriesof u('1) which upon substitution in (4.11) yieldsP = '1(1 + 2s'1 + (2s'Co + 3 s2),,2 + 2(s"CJ + 4ss'C o + 2s3 ),,3 + "') . (4.13)The isentropic pressure can be written by an expression analogous to (4.13)as

( 4 .14)which upon differentiation yields the isentropic bulk modulus

K. = PoCJ(A + (2 8 - A)" + (3C - 2B),,1 + (4D - 3C) '1 3+ "'). ( 4 .15 )The analogous bulk modulus along the Hugoniot

KH = - V(OP/OV)Hmay be obtained by differentiating (4.13) to yield

KH :=: PoCfi(1 + (4 s - 1)"+ (6s'C o + 9s 2 - 4S),,2+ (8s"CJ + 32ss'C o - 6s'Co + 16s 3 - 952),,3 + .n). ( 4 .16)The isentrope and the Hugoniot and isentropic bulk modulus are related via

K . =KH + (y /2) [PH - KH '1 /(l - '1 )] - (PH - P,)(y + 1 )- qo(1 - q ''1 + ...)].

Here we assume a volume dependence of the Gruneisen parametery = V(oP /oE) . =Y o exp {!~[~r-1]}. (4.18)

(4.17)

whereq' =d In rId In V.

Y o is the Gruneisen parameter under standard pressure and temperaturesand is given by ( 4 .19)where IX is the thermal expansion coefficient, KT is the isothermal bulk mod-ulus, and Cp and C. are the specific heat at constant pressure and volume. Wenote that the P , and PH can be related by assuming the Mie-Griineisenrelation YPH - P, =v(EH - E.),where EH is given by (4.3) and E. is given by

E. = - r y p. dV.J y o ( 4 .21)(4 .20 )

~~.~------ -----------_


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I1 _ , _ ~ . . _ , _ _ _ _ _ _

82 T.J. AhrensBecause the Griineisen ratio relates the isentropic pressure, P s, and bulkmodulus, K., to the Hugoniot pressure, P H ' and Hugoniot bulk modulus. KH it is a key equation of state parameter.Using (4.13),(4.15),(4.16), and (4.17) to solve for the coefficients of (4.14)yieldsA = 1, (4.21a)B = 2s, (4.22)C = 2s'Co + 3s2 - sYo/3. (4.23)D =2s"C5+(8s-Yo/2)soCo+(4s2-(9s+')'o)Yo/12)s+Yos(qo-l)/3. (4.24)

Ruoff (1967) first showed how fhe coefficients of the shock-wave equation ofstate are related to the zero pressure isentropic bulk modulus, KO$ and its firstand second pressure derivatives, Kos and K o s . via

Kos = POCl, (4.25)K o . = 4s -1, (4.26)(4.27)

The relationship of the above parameters to finite strain equations of stateis given in the next section.

4.3 . F in ite-S tr ain Equ atio ns o f S tateIn the following treatment we show how the usual Birch-Murnaghan finitestrain equation of state is derived and is related to the Hugoniot parameters.Using the Eulerian definition of finite strain.

f = [(V 0/V )2 /3 - 1)]/2. (4 .28)Birch (1978) showed that the increase of internal energy upon isentropiccompression is given by

E. = 9VoKo.[f2/2 + atf3/3 + alf4/4 + "'J, (4 .29)wherea 1 =3 (Kos - 4)(2, (4.30)a2 = 3[(KosK;;s + Ko .(K o . - 7) + 143(9)](2. (4.31)

Upon differentiation, at constant entropy, the isentropic pressure is givenby P , = 3Ko . f (2f + W/2[(1 + ad + a 2P + "']. (4.32)Similar relations along an isotherm can be developed.If the pressure-volume equations of state is given by the two parameterthird-order, Birch-Murnaghan, a2 = O.


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4. Equation of State 83The Birch-Murnaghan assumption, a2 = 0, thus places a constraint on

the values of Ko., K~., and Ko. which from (4.32) yieldsKoKos =K~.(7 - K~s) - 143/9. (4.33)

Moreover, upon comparing (4.32) with (4.14), it can be seen that (Jeanloz andGrover, 19S5) the Birch-Murnaghan equation (4.32) with a2 = 0 describesthe isentropic equation of state provided the linear shock-particle velocityrelation (4.5) describes the Hugoniot. In combination, these require that

a1 =6s - 15/2, (4.34)a2 =215/6 - 60s - 3 yos + 27s 2 + ISs'Co. (4.35)

Moreover, from (4.35), it follows thatY o =(162s2 - 3 60s + 215)/ISs (4.36)

and from (4.30) and (4.31) thatY o =(81K~ - 558K~. + IOSI)/(36(Ko. + 1 . (4.37)

4.4. Pressure-Particle Velocity CurvesHugoniot curves, such as those depicted in Figs. 4.3 and 4.4, may be trans-formed from the pressure-volume plane to the pressure-particle velocityplane (Fig. 4.5) using

Up = P1 - Po)(V o - Vd)1/2,V.= VOP1 - P o)/(V o - V d)1 /2 .

(4.38)(4.39)

Shock Pressureversus-. 800 Particle Velocity(0c ..

< . : ; I- 600

200

8

Figure 4 .5 . Shock pressure versus particle velocity for various standard materials.

TJA92186SFD


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" 84 T.J. AhrensWeassume that in (4.38)and (4.39),all velocities are measured with respect

to the same coordinate system (at rest in the laboratory) and the particlevelocity is normal to the shock front. When a plane shock wave propagatesfrom one material into another the pressure (stress) and particle velocityacross the interface are continuous. Therefore, the pressure-particle velocityplane representation proves a convenient framework from which to describethe plane impact of a gun- or explosive-accelerated flyer plate with a sampletarget. Also of importance (and discussed below) is the interaction of planeshock waves with a free surface or higher- or lower-impedance media.

4.4.1. Plane Impact of a Flyer PlateThe physical state of the sample before and after impact is sketched in Fig.4.6(a), Positive velocity, indicating mass motion to the right (in the labora-tory), is plotted toward the positive, u , axis. Hence, in the initial state 0,the target B is at up =0 and P = 0. whereas the initial state in the flyerplate 0' is up =Ufp and P = O.Upon interaction of flyer plate A with targetB, a shock wave propagates forward in the sample and rearward in the flyerplate. Because the pressure and particle velocity are continuous at the flyer-

S ho ck w av es

Pressure(~)lyer p la te T arg et

U 1 ufp Particle velocit y(~

u "

(a) (b)TJA84157SFDFigure 4.6.Diagrammatic sketch of impedance match method for obtaining Hugoniotstates. (a)Upon impact of flyer plate A onto sample B shock waves propagate forwardin the sample and rearward in the target. (b)Because the pressure and particle velocityare continuous at the flyer-target interface, the state achieved in the sample may beobtained from knowledge of the flyer plate velocity. ur". the initial sample density P O B .and the sample shock velocity UB Equation (4.2) states that the Hugoniot state liesalong the line through the origin of slope P O B U B , On constructing this line the shockstate Pi and u" liesat the intersection of the line P = POBUB with the known Hugoniotofthe flyer plate. The latter isoriented so that the effectof the shock is to decrease theparticle velocity from ur~to U I.


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4.EquationofState 85sample interface. the pressure and particle velocity behind the shock wavepropagating forward in the sample and rearward in the flyer plate. Becausethe pressure and particle velocity are continuous at the flyer-sample inter-face, the pressure and particle velocity behind the shock wave propagatingforward in the sample equal those behind the shock wave propagating rear-ward in the flyer plate. Note that the particle velocity of the flyer plate isslowed down from Ufp to U1 by the shock wave in the flyer plate. Since eachmaterial is shocked to pressure P 1 along its Hugoniot curve, the shock stateachieved is indicated in Fig. 4.6(b). The particle velocity of the target changesfrom up = 0 to up = Ut whereas the state in the flyer plate changes fromup = Ufp to up = U1 ' The state Pl' U1 is thus determined from the intersectionof the rightward-facing Hugoniot of the target material and the leftward-facing Hugoniot of the flyer-plate materials. Figure 4.6 may be used to dem-onstrate the usual situation in which the Hugoniot state achieved in the targetis unknown and the shock velocity in B is measured in addition to the initialdensity of the sample. In general, the Hugoniot of the flyer plate is known.The Hugoniot state is determined from the intersection of the Rayleigh lineof slope Po U with the Hugoniot of A as indicated in Fig. 4.6(b).If the Hugoniot of the flyer plate (A) and the target (B) are known and

expressed in the form of (4.7~ the particle velocity U1 and pressure Plof theshock state produced upon impact of a flyer plate at velocity, u(p' may becalculated from the solution of the equation equating the shock pressures inthe flyer and driver plate:

POA(Ufp - ud(CoA + S A(Ufp - ud) =PO BU1( COB + sBud (4.40)isUl =(-b - Jb2 - 4ac)/2a, (4 .41)

(4.42)(4 .43)(4.44)

wherea = SAPOA - POBSB .b = COAPOA - 2sAPOAUCp - POBCOB 'and

In the usual situation the Hugoniot of the flyer plate (A ) is known and theuncompressed densities of the flyer plate and target, POA, and POB ' and givenby an expression of the form of (4.5), but the Hugoniot parameters, COBand SB of the target are not. From the impedance match condition that thepressure and particle velocity at the flyer plate-target interface are equal. itfollows that upon measuring the density PO B and shock velocity U1 in thetarget, that the pressure equality in the flyer plate and target can be written as

P OA(Urp - ud(COA+ SA(Urp - u1 =PoUt V I' (4.45)which has a solution

u1 =Ufp + a' - (a'2 + (POB/POA)U1Ufp!SA,)1/2. (4.46)a' = (COA+ V 1POB/POA)/2sA, ' (4.47)

.--.-~.--.'''''''----.- ..~~--''_--'''''''''-.-_~'''.~~-'< ~ . ,_. ~


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86 TJ. Ahrens4.4.2. Reflection from a Free SurfaceOn reflection of a rightward-tra veling shock wave at a freesurface, a leftward-propagating rarefaction wave releases the pressure in the shock state along(assumedly) an isentrope to zero pressure. The mathematics describing themapping of the pressure-particle velocity unloading path to the pressure-volume plane is discussed in Chapter 2. If the material remains in the samephase and the shock process is nearly reversible (i.e., not in the porousmedium which undergoes permanent compaction), the resulting free-surfacevelocity is (4.48)Equation (4.48) would be an equality if, upon unloading, the entire shock-induced internal energy increase of the compressed material were convertedto kinetic energy. This is because the shocked material's internal energy perunit mass is, as indicated in (4.3), also equal to the shock-induced kineticenergy per unit mass. In the case of a truly elastic response, upon unloading,the internal energy density increase, in the shock state specified in (4.3),iscompletely converted to kinetic energy and the material velocity upon un-loading is exactly twice the shock particle velocity.In general, when a solid returns to its initial phase upon unloading fromhigh pressure, or when the postshock temperature is sufficiently low that

TJA84158SFD

(a)

(b) (c)Figure 4.7. Internal reflection of a shock wave from a free surface. (a) Reflection of ashock wave from a free surface causes a reflected rarefaction wave. As indicated in (b),this increases the velocity of the shocked material from "j to "ro. The path uponshocking is Rayleigh line 0-1, whereas unloading occurs along release isentrope curve1-0'. (c) Release isentrope path in P- V plane is indicated.

-~---~--~- ~----~--------------.---


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4. Equation of State 87A 8'

Aa) (b)Up TJA84159SFD

B

(c)F ig ur e 4 . 8. (a) Upon reflection of a shock wave bringing material A to state I againstan interface with lower shock impedance material (8), the reflected wave (rarefaction)propagates back into material A, bringing it to state 2 , and a shock, bringing material8 to state 2, propagates forward. (b) Upon reflection of a shock bringing material Ato state 1 against an interface with higher shock impedance material (Bt) , the reflectedshock propagates back into material A, bringing it to state 3, and a shock bringingmaterial Bt to state 3 propagates forward. (c) Pressure-particle velocity plane repre-sentation of relative Hugoniots of A, B , and B ' and reflected shock state (3 ) and releaseisentropic state (2) .

appreciable vaporization does not occur, Uf o exceeds 2u 1 by only a fewpercent(Walsh and Christian, 1955).A detailed treatment of the degree to which the"free-surface approximation" (4.48 is valid is given in Walsh and Christian(1955).An approximation which is often made in constructing Hugoniots andrelease isentropes in the pressure-particle velocity plane, such as those shownin Figs. 4.6-4.8, is that the release isentrope (and shock-compression Hugoniot)from a previous shock state can be approximated by the pressure-particlevelocity curve corresponding to the principal Hugoniot. This approximationisoften made for construction in the pressure-particle velocity plane as out-lined in the cases below.4.4.3. R eflec tion fr om a L ow er - or H igher -I mp ed anc e B ound ar yUnderstanding such interaction is important both in predicting the ampli-tudes of shock waves transmitted across interfaces (in the case where theequations of state of all materials are known), and in determining releaseisentropes or reflected Hugoniots (when measurement of the equation ofstateis needed). Consider first a shock wave in material A being transmitted to a

----_._----_._---


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TJA84150SFD

88 TJ.Ahrenslower-impedance medium, B. As indicated in Fig. 4.8,after interaction at theinterface, lower-amplitude shock is transmitted into B and a reflected wave(rarefaction) propagates backward in A. The pressure P 2 and particle velocityUz associated with both the shock wave transmitted to material B and thatassociated with the backward-propagating rarefaction wave are determinedin the pressure-particle velocity plane from the intersection of the releaseisentrope of material A, centered at state I, and the Hugoniot of material B,centered at state O.Here we use the continuity of pressure and particle veloc-ity at the interface ofmaterials A and B . In the case of shock-wave interactionwith a higher-impedance medium as indicated in Fig. 4.8(b),a shock is trans-mitted into B' as state 3 and a reflected higher-pressure shock propagatesbackward into material A. The pressure and particle velocity achieved atshock state 3 are determined from the intersection of the reflected Hugoniotof material A, centered at state 1,and the principal Hugoniot of materia) B',centered at state O.An important application of the impedance match method is demon-strated by the pressure-particle velocity curves of Fig. 4.9 for various ex-plosives. Using the above method, the pressure in shock waves in variousexplosives is inferred from the intersection of the explosive Hugoniot withthe explosive product .elease isentropes and reflected shock-compressionHugoniots (Zel'dovich and Kompaneets, 1960).The amplitudes of explosivelyinduced shock waves which can be propagated into nonreacting materials arecalculable using results such as those of Fig. 4.9.

1.5 2.0 2.5 3.0 3.5Port icle Ve lo c ity ( km / se c )

Figure 4.9. Shock pressure versus particle velocity for engineering materials, geologi-cal material, and explosive detonation products. Intersection of detonation productcurves with nonreactive media predicts shock pressure and particle velocity at anexplosive sample interface. (After Jones (1972).)

o


15/39

4. Equation of State 89

t-OZo(!)~%

o. .

)(

>

TJA84160SFD


16/39

90 TJ. Ahrens4.5. Shock-Induced Dynamic Yielding and Phase TransitionsBoth dynamic yielding and phase transitions give rise to multiple shock-waveprofiles as depicted in Figs. 4.10 and 4.11 (McQueen et al., 1970).Virtuallyall nonporus minerals and rocks in which dynamic compression has beenstudied demonstrate dynamic yielding, and most minerals and a large numberof compounds, elements, and organic materials demonstrate shock-inducedphase changes. In several interesting cases a separate phase transition waveshock front is formed, for example, in Fe (Bancroft et al., 1956).Si02 (in fused

~E la stic sh o c k , Unshocked

(a)

Unshocked

(b)

Unshocked

S ho c k (c )frontFigure 4.11. Diagrammatic sketches of atomic lattice rearrangements as a result ofdynamic compression, which give rise to (a) elastic shock, (b) deformational shock, and(c)shock-induced phase change. In the case of an elastic shock in an isotropic medium,the latera!"stress is a factor ,,/(1 - \I) less than the stress in the shock propagationdirection. Here \I is Poisson's ratio. In cases (b) and (c) stresses are assumed equal inall directions ifthe shock stress amplitude ismuch greater than the material strength.

TJA84095MFD


17/39


oQ). ,'10.0e~!''u 8.0o~. : . :r. >_ g 6.0[I)

FUSED /UARTZ__,~ lfI- e .,SP'-MIXED PHASE

STISHOVITETJA93187SFD

FU SED QU ARTZ,sro,

4.0 ~~'-'+'-'''''''''''.u.L/;'''''''''''''''''''''_'k''"''''''''''''''"'-'+'4 =>..........'-'-'-' ........... ';\-'.>.~'-'-'-I.Particle Velocity (km/sec)

Figure 4.12. Shock velocity versus particle velocity for fused quartz. Three regimes areindicated: low pressure. fused quartz regime, the mixed phase regime. and the high-pressure phase. stishovite regime.

quartz, not crystal quartz) (Wackerle. 1962) (Fig. 4.12), FeS (Ahrens, 1979),Mg2Si04 (Watt and Ahrens, 1983), and CaO (Jeanloz and Ahrens, 1980) (Fig.4.13).The general conditions required to produce multiple wave structureswhen phase transitions occur are discussed by McQueen et al. (1970).The condition that gives rise to multiple shock fronts (i,e., allows a shockwaveto bifurcate as indicated in Fig. 4.1O(bwilloccur when the second wavepropagation velocity (with respect to the laboratory) is given by (4.39).How-

"";,11.0 CALCIA.CaO< I . >.,S '~t- 9.0'0.Q)>. : . :g 7.0.c :(I)

TJA93188SFDParticle Velocity (km/sec)Figure 4.13. Shock velocity versus particle velocity for calcia, CaO, low-pressure Blregime, mixed phase regime, and high-pressure phase B2 regime, indicated.

-~--.------ ----- ..--~--.


18/39

92 TJ. Ahrensever, now we have three subscripts: 0, which as before indicates the initialstate; 1, which corresponds to an intermediate shock state; and 2, whichcorresponds to the final shock state. The second shock wave propagatesslower than the first shock wave into material already traveling at particlevelocity U 1 The condition for the formation of two shock waves, one fromstate P =Po , V = Yo, U =Uo to state P =PI ' V = VI' U = UI and the secondfrom state P =PI ' V = VI' U=U1 to state P = Pl. V = V1, U = Ul, is then

ul + V IP 2 - Pd/(VI - V ZW / 2 < VO((P 1 - P o)/ (V o - V I))I/2 , (4.49)or using (4.39)VIP l - Pd/(VI - V l1/2 < VIPI - P o)/(V o - VI1/1. (4.50)

Figure 4.10indicates that ifthe Rayleigh line (0-2) from the initial to the finalshock state intersects the Hugoniot at an intermediate shock state, two shockwaves will form, one from 0-1 and one from 1-2.Thus two shocks form forfinal shock states between 1 and 3. For states at stresses higher than state 3,only one shock forms. The deformations which accompany elastic shocks(discussed in Section 4.5, below), deformational shock and phase transitionshock. are sketched in Fig. 4.11.The onset of the phase change in fused quartz (zero pressure density, 2.204Mg/rn") to the high-pressure phase, stishovite (density, 4.35 Mg/m"), as canbe seen in Fig. 4.l2 gives rise to a two-wave structure. The slight decrease ofshock velocity at particle velocities in the range 1< up


19/39

50SiO.

TJA93190SFD


C a c n 1150 B 2 1 y .- - " 1\ TISHOVITES -


20/39

94 T.1.Ahrens1.0 r--~-----"--~--~--~--'

Molybdenum (1400oC)

II0.2! t:

21 CPe.

12 OPe .

TJA91132SFD 0.0 '-------'-_,_---'---...__------'0.& 1.0 2.0Time ~)Figure 4 .16 . Free-surface velocity profiles measured on 14000 C molybdenum. Thefree-surface velocity profile is characterized by an 0.05 km/s amplitude elastic precur-sor, a plastic wave front, and a spall signal (characteristic dip) upon unloading. Thedashed lines represent the expected free surface velocity based on impedance-matchcalculation [Duffy and Ahrens. unpublished].

is given byP,=vPH/(l - v),

where v is Poisson's ratio (0 < v


21/39


TJA84161SFDVo lume

Figure 4.17. Hugoniot curve relative to principal isentrope of solid and release isen-tropes. Above the Hugoniot elastic limit (HEL) the Hugoniot curve may be offset inpressure above the principal isentrope by an amount 41:/3where 1; is the maximumshear strength sustainable by the solid in the shocked state. On unloading, the releasecurve may be steep, indicating elasticlike behavior followed by deformational un-loading with decreasing pressure. Elastic followed by deformational unloading willoccur on unloading from shock pressures up to those at which the Hugoniot crossesthe fusion curve. Above this pressure unloading occurs in the liquid (melt) field.Thecritical isentrope crosses the Hugoniot at the point at which complete melt is retainedon release to zero pressure.

4.7. E quation of State of P or ous M ater ialsThe study of shock-wave equations of state of porous materials provides ameans to expand knowledge of the equation of state of condensed materialsto higher temperatures at a given volume than can be achieved along theprincipal Hugoniot. Materials may be prepared in porous form via pressing

,

\

- ~ - _ I


22/39

96 TJ. AhrensK)k ....... k-K K(k

'I< B .' c" \.c .. TJA93191SFDPOROUSMEDIUM

V oo 0" 0'"

VOQ VOLUME

Figure 4.18. Sketch of Hugoniots of material with different porosities. Isentrope formaterial is indicated by curve S, along OA. Principal Hugoniot, H, indicated by curveDB. Porous Hugoniots are indicated with initial volumes. V oo shown at o r , 0", and O"and curves DC', OC", and ~C'''. Normal Hugoniots are indicated for K > k whereK = = (2/yo + 1)and k = Voo !Y , whereas anomalous Hugoniots are shown for K < k.powder or fibers of the same composition as the solids into uniform sampleswith zero pressure specific volume, Voo , greater than the normal crystal spe-cific volume, Vo. If a porous material with an initial specific volume, Voo , iscompressed isentropically, weassume that the media displays no strength andcompresses to crystal volume, Vo, at zero pressure along 0' A. When com-pressed further, we assume states in the material lie along the isentrope, S ,along the curve OA.The Hugoniot is assumed to also lie along 0'0 at zeropressure, and then along the line OB.The Mie-Gruneisen equation of state (4.20) may be used to define theHugoniot curve from

P H =(K - l)P s - 2E . /V ,K - Voo /Vwhere K = (2/1 '0) + 1 ( Y o is assumed to be constant), and P, is the isentro-pic pressure and E. (4.21) is the energy change along the isentrope. For anyvalue of Voo the Hugoniot is explicitly specified by (4.53). In the case ofk 5: Voo /V :$ K , the Hugoniots have their normal behavior with V decreas-ing with increasing pressure. In the situation where K = Voo / V , (4.53) indi-cates that the P H -+ 00. Moreover, where k > K , Hugoniots of porous mediadisplay anomalous behvior. This is predicted by (4.53).In fact, as P H increasesthe specific volume increases.Important applications of (4.53) are the generation of porous Hugoniotsand their subsequent inversion via (4.53)to obtain Y o or r(V). Critical to thisapplication is the requirement that the length scale of the porosity or grain

(4.53)

------.------~----------------


23/39

TJA93192SFD 4. Equation of State 97

~.556 1 7 ~ ~i1,35 I~~ III~1,'5 III, . " I.I.~ I.~~ ,0 It. I40 " 0 $ ' 1 0 . 6 ,0 v ~ i "J 0 IJ 0,55 0.65 I r Jj, I " o f i. I " I II I 'I + +8 0,+ I> 1 r . . ~I +.c ~ .(~fI) I, _ -

dl_~~~~~-----3~'----------*~--------~5D ens ity - Mgfm3

Figure 4.19.Shock pressure versus density Hugoniot states for initially porous quartz.Density of starting material is indicated on various curves. Porous properties ofstishovite are represented by curves with 1.75, 2.13, and 2.65 MgfmJ, initial den-sity, whereas coesitelike properties are represented by 0.2-0.8 Mg/m3 curves (afterSimakov and Trunin (1990)).

(4 .54)

size o f po wders, 1 , b e sufficiently sm all that therm al equ ilibrium i s achievedbehind the shock state. The tim e for therm al equilibrium r, should be less thanthe shock duration tim e 1'D

where D is th e th erm al d iffu siv ity o f th e sh ock -c om pre sse d m ate rial.A n interesting application of the use of the H ugoniot data of porous m edia

to determ ine the G runeisen ratio is the data forporous quartz show n in Fig .4 .1 9. H ere the effective G runeisen ratio of the high-pressure phase is found tob e dependent on energy. The G runeisen ratio of the high-pressure phase isshown in Fig. 4 .20. N otably, the data corresponds to the initial densities of2 .65 (crystal quartz) and 2 .1 3 M g/m ! (20% porous) indicating a Gruneisenratio in the range 0.1 -0 .2 in the stishoviteregime. Data for more porous

IIIIIII


24/39

98 TJ. AhrensTJA93193SFDI~ _---u---------Ier-'l.-bi-gh-tc-~~-wre-fl:uid:. ~9-

~ 0 . 1 . r ;-==--r- coesite Q~----t~j __ = i ~ = = : : : : : = . . . .= ~.rL-:----+----+" stishoviteo D - - - ~ - - - - ~ ~ - - - - ~ - - - - - - ~

10 2 1 1 30 I/O 50 6DPressure-GPa

Figure 4.20. Gruneisen parameter versus pressure for different regimes are indicated.Pluses indicate properties of srishovite phase, half-filled circles and closed circlesindicate properties of high-density molten material, whereas open triangles and opencircles and upper branch indicate behavior of coesitehke phase (Simakov and Trunin,1990).

quartz, for example, 1.55Mg/m" down to densities of 0.8 Mg/mJ, appear torepresent the properties of superheated fluid which has a Gruneisen ratioin the range 0.2-0.3. Finally, Hugoniot data (or very porous quartz withdensities in the range of 0.2-0.8 Mg/m! corresponds to a moderate pres-sure, high-temperature, phase, coesite. This has a zero pressure density of 2.9Mg/m", The effectiveGruneisen ratio ofthecoesite phase appears to be -0.2.

4.8. Sound Speed Behind Shock FrontsAnother important method of determining the Gruneisen ratio in the shockstate is the measurement of sound speed behind the shock front. The techni-ques employing optical analyzers (McQueen et al., 1982)piezoresistive (Chap-

TJA84178SFD

vFigure 4.21. Pressure-volume paths used to relate the slope of Hugoniot (OP/iJV)H toisentropic sound speed C. in (4.57).


25/39


DRIVER LAGRANGIAN

TJA84179SFDDISTANCEFigure 4.22. Schematic distance-time diagram of the shock and rarefaction processfor ideal elastic-plastic flow.U, shock wave; C, and Cb Langrangian longitudinal andbulk wave velocities. The cross-hatched areas represent the region bounded by thelead and tail characteristics. The P versus t insert labeled "Lagrangian" representswhat an in-situ pressure gauge record might look like as it moves along in the flowand the corresponding pressure versus time at the shock front is indicated below. Asshown. the two in-situ gauges would record the arrival of the shock at 1 and 2 , theelastic unloading wave at I' and 2', and the deformational unloading wave at II


26/39

)00 TJ. Ahrensassociated with the intersection of the release isentrope and the Hugoniot(Fig. 4.21) may be related to the Gruneisen parameter at the Hugoniot statePh VI byi(Yo - Vd - ~[Pl +(!~tV } ( Y o - (VI + AV

= PlaV + V ;[_(OP ) AV _ (_ OP ) AVJ. (4.57)y oV H iJ V sHere the terms on the left-hand side represent the Rankine-Hugoniot energydifference between P~ and P l' the pressures along the Hugoniot at V ; and VI 'located infinitesimal specific volumes apart (by AV) , represented by path 4 inFig. 4.21. This energy is set equal to the energy difference between P s and P Ialong the isentropic path 3 (first term on the right-hand side) plus the energydifference associated with the pressure difference between P s and P~. This isrepresented by the second term on the right-hand side. The latter isthe energyequivalent of path 1minus path 2.Using (4.18)and (4.56)in (4.57),and takinginto account that AV is infinitesimal, it follows that

c; = VI {(:~t[V o - V d2~1 - I J + P12~J = C b, (4.58)where (dP/dV)H isthe slope of the Hugoniot curve at P l' Thus ifthe Hugoniotand bulk sound speed C. are measured, the Gruneisen parameter may bedetermined.Equation (4.58)is derived for the case of a bulk wave which is appropriateto a fluid. However, if unloading from the solid state occurs, and the solidretains rigidity, the "sound" velocity will be that of a longitudinal wave. Thus,the sound (elastic or plastic wave)speed in the high-pressure shock state is animportant property which can help determine whether a solid or a fluid phaseexists in the shocked state. Before the wave profile methods of Chapter 7where developed, simpler methods for measuring the sound speed in theshocked state were described by Al'tshuler et al, (1960)and Fowles (1960).Thesimple geometry which these papers describe is shown in Fig. 4.22. Here theflyer plate and sample are likematerials. Upon impact of the flyer plate. whichcan. in practice, be either explosively or gun launched with the sample, shockwaves propagate in both the flyer plate and sample. Upon reflection of theshock wave at the rear surface ofthe flyer plate, the resulting rarefaction wavepropagates forward and will catch up with the shock front propagating intothe sample. This catch-up occurs in the case of a plastic release wave as (4.58)can demonstrate that the rarefaction velocity C.. in the compressed state,exceeds the shock velocity. In the case of a longitudinal release wave the onsetof the release wave is even more rapid. The point at which the rarefactionwave originating at the rear of the flyer, or driver plate, overtakes the shockfront is measured. If C , is the release wave velocity, then j = 1or b, where1 indicates longitudinal and b indicates bulk sound velocity in the high-

~ ..-~.-.------~----------------------.--- .~---


27/39

4. Equation of State lOtpressure shock state

C,=J(K s + 4~/3)/(Jo (4.59)c, =JKs/(Jo ,an d (4.60)

where Ks and J t are isentropic bulk modulus and shear modulus, respectively.At this point it is useful to define the velocity of a sound (lor b) wavepropagating in a moving medium, which may also be compressed. The veloc-ity with respect to stations moving with the medium is termed Lagrangian,CL The position of a station (for the purpose of calculating the velocity) istaken to be specified by its initial position. Sound velocity with respect todistances, measured with respect to the laboratory, is termed Eulerian, C".As indicated in Fig. 4.22, C. is the Lagrangian wave speed of the head ofthe rarefaction wave propagating forward into the shock state. Thus, measur-ing the time intervals 1-1', 1'-1", 2-2', and 2'-2" at two different gaugepositions between the arrivals of the heads of the longitudinal rarefactionwaveand bulk rarefaction wave determines the rarefaction wave velocities C,and C h However, determining with high precision the onset ofthe rarefactionarrivals with particle velocity or shock pressure gauges is not simple becausethese devices have a response which is limited by their physical thickness.The thickness of the target at the point at which the rarefaction wavecatches up to the shock front and begins to attenuate rapidly the pressure ofthe shock front is readily definable in the time versus Lagrangian distancediagram of Fig. 4.22. In the Lagrangian (material) coordinates, the distance

d may be calculated from equal time paths for the direct shock dIU andthe initially backward-propagating shock and subsequent forward-reflectingrarefaction wave propagating at velocity c t from

d ridU = U + c t + c f (4.61)

or! ! _ = 1 (U - u) [ I + d JU U+ U Cj'

which corresponds to the Eulerian (laboratory) coordinates. Here, 1 is flyerplate thickness. Also,(4.62)

c r = (Po/p)Cf. (4.63)where Po and (J are the initial and high-pressure densities.McQueen et al. (1982) demonstrated that by placing a series of high-impedance transparent fluids (called optical analyzers) over the sample at aseries of thicknesses less than d in the target that the overtaking rarefaction(sound) velocity can be accurately obtained. Arrival of rarefaction wavesrapidly reduce the shock pressure. These wave arrivals could be very readilydetected by the change in light radiance caused by the onset of a decrease inshock amplitude when the rarefaction wave caught up to the shock front. The

Ii

\,I\III\


28/39

I,

._----_._--

102 TJ. Ahrens TJA84182SG

10

8

100 200PRESSURE (GPo)

Figure 4.23. Elastic wave velocities as a function of pressure along the Hugoniotof iron. The solid curve is the calculated bulk sound velocity. (From Brown andMcQueen (1982).)

sound velocity behind the shock front along the Hugoniot on crossing fromthe s-iron to y-iron to the liquid field was reported by Brown and McQueenet a l, ( 1 98 2) (Fig. 4.23).

4.9. Shock Temperatures4.9.1. TheoryFor many condensed media, the Mie-Gruneisen equation of state, based ona finite-difference formulation of the Gruneisen parameter (4.18),can be usedto describe shock and postshock temperatures. The temperature along theisentrope (Walsh and Christian, 1955)isgiven by

(4 .64)

where 1j is the initial temperature. For the principal isentrope centered atroom temperature, r . =To, v , . =Yo, initial volume, and ~ = V . compressedvolume. Forthe calculation ofposts hock temperatures r . = Tn, the Hugoniottemperature, v . . = Vn, the volume of the shock state, and v " =V~o,the post-

-------------_._---


29/39

TJA84183SG 4. Equation of State 103

if",::l' Eu3:. . . .

~ I CVI 1 1 ' ' \ - 4 ~w r ? t ~ 2 8 9 I M M - I C ".J I'~.. .~IC "/ V, Ito 1 til I" II~~J ~K~ 7 1 / 1 /i I ' , c p ~

I I I I V I'~I I ~ I I ~ " ~ ~J I ~ i'-:~'"

I I I y ' i"~. r ! o. . , I 1 7 I I',!~r 9~ ~ : / I r j p Z [ f f ~"., f / - y'-~ \-~~ Kr f l 1 - ' - " / f V I f : '~. . . . I t ' , I r. . . .~ 2 3 4 ,. I 2 3 4 ,. to 2 II 4 &

W A V E L E N G T H tum)Figure 4.24. The Planck distribution law spectral radiance of blackbody radiation asa function of temperature and wavelength. (AfterTouloukian and DeWitt (1972).Plenum Press.)

shock volume corresponding to the postshock temperature. For shockcompression to a volume V, p. is first obtained by using (4.55);then 4. theisentropic compression temperature at volume V, may be calculated byusing (4.64).Finally. using (4.18),the shock temperature TH is given byV I T "(PH - p.) = C v d T .Y T. (4.65)

It is useful to carry out both postshock and shock temperature measure-ments as they provide complementary information for the thermal equationof state, i.e.,y, aswell as C u ' Using (4.55),(4.64),and (4.65),shock temperaturesfor materials such as MgO have been measured and these measurementscompare favorably with the calculations (Fig. 4.27).

---;ii~I

!


30/39

ImpocfchQlYlber(vacuum)

Figure 4.25. Experiment!'-Iconfiguration for optical pyrometry of shock temperaturesinduced in transparent minerals. Upon impact ofprojectile with driver plate, a shockwave isdriven into the driver plate and then into the sample. Optical radiation fromthe sample is detected v ia s ix lens/interference filter channels and an array of sixphotodiodes. S ig n al s f rom photodiode circuits are recorded on oscilloscopes operat-in g in s in gle s we ep m od el. (After A hren s e t at (1982) .)

TJA84061SFDT40IO22Kp.84Gf>o

/10.150.10 T'2481t37K

P'4BGPo

4~ 600 7~ 900Wave le n g th ( om )F ig ur e 4 . 26 . S pe cia l radiance versus wavelength for CaAlzSilOs glass shocked to 48GPa and 84GPa. Best-f it ting Planck blackbody curves are shown in relation to theradiance data. (After Boslough and Ahrens (1984). the American GeophysicalUnion.)

104

- - - - - - - - - - - - - - - - - - - - - - . . . . . .- - - - - - - - - - ~ - ~


31/39

4. Equation of S tate 105" ! . : .000c o

gQ >k~ 0.. ,co 0k 0Q> '"~. . .f-o

TJA92037SFD

Pressure (GPa)Figure 4.27. Experimental and model pressure-temperature shock-compression re-sults for MgO. after Svendsen and Ahrens (1987).

In the case of molecular fluids such as water, a formulation based on thenear constancy of C,.at constant pressure is used (Bakanova et al., 1976;Riceand Walsh, 1957).Although there have been few data collected, postshock temperatures areverysensitive to the models which specify y and its volume dependence, in thecase of the Gruneisen equation of state (Boslough, 1988;Raikes and Ahrens,1979a;Raikes and Ahrens, 1979b). In contrast. the absolute values of shocktemperatures are sensitive to the phase transition energy Er R of Eq. (4.55),whereas the slope of the T H versus pressure curve is sensitive to the specificheat (see, e.g., Fig. 4.28).4.9.2. ExperimentsAlthough some measurements of shock temperatures in the metals havebeen carried out using thermistors and the thermoelectric effect(Bloomquistet at, 1979; Rosenberg and Partom, 1984), most determinations of shocktemperatures have been carried out by analysis of radiations from trans-parent media and more recently in metals. As a strong shock wavepropagates through a transparent material, the temperature of the materialin the shock-compressed state is calculated by measuring the radiativespectrum of the shock-compressed material as transmitted through thetransparent unshocked material assuming thermal radiation and a graybodyspectrum. The dependence of the radiative powder N ) . or wavelength A . for ablack, or grey, body is shown in Fig. 4.24 and is given by

N;.==eC1A . - S(expC2!}.T-l)-I, (4.66)where e is emissivity, C1 = 1.191 X 10-16 W m2jsteradian, and C2 = = 1.439 X10-2 m K. The figure demonstrates that a system which is sensitive to radia-

_.------~-~----~-- -_ ---------_ . . . . . ._._------------_._-_._------

IIIIiii

IIIIi\I\\\II

i\IIi!I\I


32/39

1 06 TJ. Ahrens

5000Mg2Si04ShockTemperolures

4000

wa:;: 30000 : : :wc,:iwt- 2000~uoJ:(/)

IIIO L I V I N E I.AHR ENS e1 a t. / I(1969)-""''YI 1I I: 1I

TJA93194SFD

500 1000 1500 2000SHOCK PRESSURE (kbor)

Figure 4.28. Measured and calculated shock temperatures versus pressure for for-sterite for low-pressure (olivine),mixed phase, and high-pressure phase regime (possi-bl y MgO periclase) + MgSi03 (perovskitej). Shock temperatures in the mixed phaseregime (Ahrens et al., 1969).

tion from the near infrared (-1 pm) into the visible (-0.5 um = = 500 nm)will record peak radiances for blackbodies (e = = 1) between '" 2000 K and5000 K. A number of radiative sensing systems utilizing interference filtersand photomultipliers or photodiodes has been used to record shock tempera-tures (Boslough and Ahrens, 1989;Kormer et at, 1965;Lyzenga and Ahrens,1979). A system of wideband beam splitters and interference filters in therange 450-900 nm is shown in Fig. 4.25. Sensitivity to radiation at relatively

Table 4.2.Properties of MgO.Property Unitsymbol

DensityIntercept of V.-V" relation!Slope of V,-Vp relationSpecific heat at constant pressureCoefficient of thermal expansionThermodynamic Griineisen's parameter

PoCob

3.5836.611.36937.42.71 .3

C,.G t:}'

I- - - ~ - ~ - ~ _ _ ~ . . 1


33/39

,' . 4. Equation of State 1 07low shock temperatures is achieved with this system because of the wideaperture (0.03 steradian), large bandwidth per channel (- 4 0 nm)), and largephotodiode detector area (-0.2 cml).Some typical shock temperature dataforCaAl2Si20a and MgO are shown in Figs. 4.26 and 4.27 and the sensitivityof the shock temperatures to transformation energy, in the case of Mg2Si04isdemonstrated in Fig. 4.28.Recently, several measurements have been reported describing shock tem-perature measurements in metals (e.g., Bass et al. (1990)) . Such experimentsare intrinsically difficult as metals are opaque and it is thus possible to collectradiation only from the surface of a metal and not from its interior. In orderto make a meaningful high-pressure measurements it is thus necessary toplace a transparent nonradiating window, or anvil. in close contact with themetal and drive a shock through the metal into the (transparent) window andobserve the radiation from the metal-anvil interface. Assumptions needs tobe made regarding the shock temperature in the anvil and the mismatch ofshock impedance and thermal diffusivities at the metal-anvil interface. Theconfiguration employed in these experiments is shown in Fig. 4.29. We shallconsider the situation where the shock impedance of the window is less thanthe metal sample, since both LiF and Al203 have been used as windowmaterials for the study of shock temperatures in iron and stainless steel.Upon

TJA90128SFDThin Metal

MaskThermalRadiation

IIl - L _Figure 4.29. Sample assembly for optical shock temperature measurements. The sam-pleconsists of a metal filmdeposited on a transparent substrate which serves as bothananvil and a transparent window through which thermal radiation isemitted. Rapidcompression ofgases and surface irregularities at the interface between the sample filmand the driver produce very high temperatures in this region. The bottom portion ofthe figure illustrates the thermal distribution across through the assembly. (After Basset a!.(1987).)

I _ ~ - - ~ - . - - - - -i - _ ._ _ . . . . ~ _


34/39

1 08 T J. A hrenswave interaction at the metal-window interface, the temperature in the metalis reduced from the Hugoniot state temperature, TH, isentropically accordingto

T R = T H exp{ - f : : t dV } ,where Vn and VR are the Hugoniot and partially released volumes of themetal, respectively. The pressure and temperature release described by (4.67)arises from the lower shock impedance of the window materials relative tometals such as iron and stainless .steel. The shock temperature and thermaldifTusivitymismatch of the interface is described by Grover and Urtiew (1974) .They showed that for thick meta) samples and anvils, the interface tempera-

(4.67)

TJA90133SFD9000

8 0 0 0

~ 7 0 0 0W 6 0 0 0a:: : : >I-5 0 0 0a:w 4 0 0 0n,:2 3000WI-

2000

1000

304 8TAINLESSSTEEL

o Interface T*: Melting T0- Hugoniot T

100 30000PRE SSURE (G Pa )

Figure 4 .30 . D ata from study of 304 stainless steel. From the raw interface tem peratures (circles), points on the m elting curve of the m etal (stars) are obtained using theanalysis o f T an and A hrens (1 990). A rrow s are calculated using the analysis of G roveland U rtiew (1 974 ). w hich ignores m elting effects, and provide bounds on the H ugonioitemperatures TH The arrows point in the direction of TH S ol id s ymb ol s representshots using A lz03 w indow s; open sym bols ind icate use o f a L iF w in dow . T heoreticav alu es o f T H (M cQ ueen et al., 1 970 ) a re given by the solid curve; the d ashed H ug onioiis i nf er re d s ho ck tempe ra tu re s.


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f:1l4. Equation of State 1 09

90 0 0

80 0 0

~ 7000u.j 6000a:: : >r- 5000a:w 4000a.~ 3000Wr-2000

1000

/TJA90134SFD

tlC B10 0 30 000PR ES SUR E (G Pa )

Figure 4 .31 . A comparison of the results from shock temperature measurements onFe. Hatchured area for Fe melting is defined by the results of Bass et al. (1981) ,Williamset at. (1981) , and the theoretical calculations ofMcQueen et at. (1970) predictthat the shock temperatures of solid stainless steel are lower than for pure, solid iron,as observed.ture, 1 1 , should be time independent and is given by

1 1 = = T R + ( T . . - T R )/(l + ( X ) ,where 'f a is the shock temperature in the anvil and( X = [(KRPRCR )/(K"p"C,,)]1/2,

(4.68)

(4.69)where K is the thermal conductivity, P is the density, and C the specific heat.Subscripts "R" and "a" refer to the partially released state in the metallicsample and the shock state in the anvil, respectively. In the case of AlzOl theshock-compression heating is considerably less than in iron or stainless steel.An important consistency check in these experiments is to demonstrate thatsimilar shock temperatures are inferred from experiments using different anvilmaterials. Melting of samples either in the shock state or upon partial releasecomplicates the above analyses (Tan and Ahrens, 1990).In the case of releaseto a partially or fully melted state, the interface temperature follows the fusioncurve in the temperature-pressure plane (Figs. 4.30 and 4.31).


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110 T.1. Ahrens'" 4.10. Acknowledgments

Supported under NSF and NASA. Contribution number 5084, Division ofGeological and Planetary Sciences. The helpful comments of an anonymousreviewer are appreciated.

4.11. Problems4.1. Using the parameters of Table 4.2, calculate the shock temperatures of MgO to

200 GPa and compare your results with those of Fig. 4.27.4.2. Calculate the final shock state pressure and density from the measured shockvelocity of 5.77 kmjs in a sample of glass (initial density 2.204 gJcml) which is

mounted onto a driver plate of pure Cu. The Cu driver plate is impacted at 4.5km/s by a Ta flyer plate. Use the impedance match methods.4.3. In 1963,McQueen, Fritz, and Marsh ( J. G eo phy s. R e s. 68 , p. 2319) suggested thatthe high-pressure shock-wave data for fused quartz (Table 1) and the data for

crystal quartz,po = = 2.65 g/cm", Co e= 1.74kmjs and s = 1.70,both described theshock-induced high-pressure phase of SiOz, stishovite ( P o = 4.35 g/crrr'), above50GPa. Assume ETR = 1. 5 kJ/g show that these shock data are consistent with aconstant value ofy = = 0.9 in the 50-100 GPa range.

4.12. ReferencesAhrens, TJ. (1979), Equations of State of Iron Sulfide and Constraints on the SulfurContent of the Earth, J. Ge op hy s. R e s. 84 , 985-998.Ahrens, TJ., Anderson, D.L., and Ringwood, A.E. (1969), Equation of State andCrystal Structures of High-Pressure Phases of Shocked Silicates and Oxides, Rev .

Geophys . 7,667-707.Ahrens, TJ., Lyzenga, G.A., and Mitchell, A.C. (1982), Temperatures Induced by

Shock Waves in Minerals, in H igh P re ssu re R e se arc h in G eo physic s (edited byAkimoto S. and M.H. Manghnani), Center for Academic Publications, Japan,pp.579-594.Ahrens, T.1., and O'Keefe, J.D. (1972), Shock Melting and Vaporization of LunarRocks and Minerals, T he M oo n 4 , 214-249.Ahrens, T.J., and O'Keefe, 1.0. (1977),Equation of State and Impact-Induced Shock-Wave Attenuation on the Moon, in Im pa cl a nd E xp lo sio n C ra te rin q (edited byRoddy D.J. et al.), Pergamon Press, New York, pp. 639-656.Al'tshuler, L.V. (1965), Use of Shock Waves in High-Pressure Physics, S ov ie t P hy s.

Uspekh i 85,52-91.Al'tshuler, L.V., Kormer, S.B., Brazhnik, M.I., Vladimirov, L.A., Speranskaya, M.P.,and Funtikov, A.I. (1960),The Isentropic Compressibility of Aluminum, Copper,Lead, and Iron at High Pressures, S ovie t P hys . J ET P 11,766-775.AI'tshuler, L.V., Krupnikov, K.K., Ledenev, B.N., Zhuchikhin, V.I., and Broznik, M.I.(1958),Dynamic Compressibility and Equation of State of Iron under High Pres-sure, S ovie t P hys . J ET P 34 (7),606-19.

-.


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4. Equation of State 1 1 1Avrorin, E.N., Vodolaga, B.K., Voloshin, N.P., Kuropatenko, V.F., Kovalenko, G.V.,Sirnonenko, V.A., and Chemovolyuk, B.T. (1986), Experimental Confirmation ofShell Effects on the Shock Adiabats of Aluminum and Lead, JETP Lett. 43,

308-311.Bakanova, A.A., Zubarev, V.N., Sutulov, Y.N., and Trunin, R.F. (1976), Thermo-dynamic Properties of Water at High Pressures and Temperatures, S ovie t P hy s.JETP 41,544-548.Bancroft, D., Peterson, E.L., and Minshall, S. (1956), Polymorphism of Iron at HighPressure, J. A pp l. P h ys . 27,291-298.Barker, L.M., and Hollenbach, R.E. (1972), Laser Interferometer for Measuring HighVelocities of Any Reflecting Surface, J. A pp l. P hy s. 43, 4669-4675.Bass, J.D., Svendsen, B., and Ahrens, T.1., (1987), The Temperatures of Shock-Compressed Iron, in H igh P re ssu re R ese ar ch in M in er al P hy sic s (edited by Mang-hnani M. and Y. Syono), Terra Scientific, Tokyo, pp. 393-402.

Birch, F. (1978), Finite Strain Isotherm and Velocities for Single-Crystal and Poly-crystalline NaCI at High Pressures and 300 K, J. G eo ph ys . R e s. 83, 1257-1268.Bloomquist, D.O., Duvall, G.E., and Dick, U. (1979), Electrical Response of a Bi-metallic Junction to Shock Compression, J. A p pf . P h ys .5 0 , 4838-4846.Boslough, M. (1988), Postshock Temperatures in Silica, J. G eo ph ys. R e s. 93, 6477-6484.

Boslough,M.B.,and Ahrens, T.1. (1984), Particle Velocity Experiments in Anorthositeand Gabbro, in S ho ck W a ve s in C o nd en se d M a tte r- 19 83 , (edited by Asay lR. etal.), Elsevier Science, New York, pp. 525-528.Boslough, M.B., and Ahrens, T.J. (1989), A Sensitive Time-Resolved Radiation Py-rometer for Shock-Temperature Measurements above 1500 K, R e v. S c i. lnstrum.60,3711-3716.Brown, lM., and McQueen, R.G. (1982), The Equation of State for Iron and theEarth's Core, in H ig h P re ss ur e R e se ar ch in G eo ph ysic s (edited by Akimoto S.andM.H. Manghnani), Academic Press, New York, pp. 611-622.Davison, L., and Graham, R.A. (1979), Shock Compression of Solids. P hys . R ep . 55 ,255-379.

Duvall, G.E., and Fowles, G.R. (1963), Shock Waves, in H igh P re ssu r e P hysic s a ndChemis t r y (edited by Bradley RS.), Academic Press, New York, pp. 209-292.Fowles, G.R. (1960), Attenuation oCthe Shock Wave Produced in a Solid by a FlyingPlate, J. A pp l. P hy s. 31, 655-661.

Gehrels, T. (1978), P ro to sta rs a nd P la ne ts , University of Arizona Press, Tucson,pp, 1-756.Grady, O.E. (1977), Processes Occurring on Shock Wave Compression of Rocksand Minerals, in H igh P re ssu re R esear ch: App lic at ions in G eop hys ic s (editedby Manghnani M.H. and S. Akimoto), Academic Press, New York, pp. 389-438.

Grover, R., and Urtiew, P.A. (1974), Thermal Relaxation at Interfaces FollowingShock Compression, J. A pp l. P h ys . 45, 146-152.Holmes, N.C., Moriarty, J.A., Gathers, G.R., and Nellis, W.J. (1989), The Equation ofState of Platinum to 660 GPa (6.6 Mbar), J. A pp l. P h ys . 66, 2962-2967.Jeanloz, R. (1989), Shock Wave Equation of State and Finite Strain Theory, J. Geo-

p hy s. R es 94, 5873-5886.Jeanloz, R., and Ahrens, TJ. (1979), Release Adiabat Measurements on Minerals: TheEffect of Viscosity, J. G eo ph ys. R e s. 84, 7545-7548.

I e . . _ ." e_ . ec .e ,, _ _ __ e __ . . - . . - "~ -- -- -- ~" - -


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1 12 T.J. AhrensJeanloz, R., and Ahrens, TJ. (1980), Equations of State of FeO and CaO, Geophys . J.R oy. A stro no m. S oc . 62, 505-528.Jeanloz, R, and Grover, R.(1988),Birch-Murnaghan and Us -Up Equations of State,

in P ro ce ed ing s o f the A me rican P hysica l S oc ie ty T op ical C onfe re nc e on S hockWaves in C ond ense d M atte r, M onte re y, C A, 1987 (edited by Schmidt S.C. andN.C. Holmes), Plenum, New York, pp. 69-72.Jones, O.E. (1972),Metal Response under Dynamic Loading, in B e ha vio r a nd U t il iz a-tio n o f E xp lo sive s in E ngine er ing D eS ign (edited by Henderson R.L.), UniversityofNew Mexico Press, Albuquerque, pp. 125-148.Kormer, S.B., Sinitsyn, M.V., Kirillov, G.A., and Urlin, V.D. (1965), ExperimentalDetermination of Temperature in Shock-Compressed NaCI and KCl and ofTheir Melting Curves at Pressures up to 700 kbar, S o vie t P hy s. U sp ekhi (Engl.transl.), 21, 689-700.Lyzenga, G.A., and Ahrens, T.J. (1979),Multiwavelength Optical Pyrometer for ShockCompression Experiments, Rev . Sci. Ins t rum. SO,1421-1424.Marsh, S.P. (1980),LASL S ho ck H ug on io t D ata , University ofCaJifornia Press, Berke-ley, pp, 1-327.McQueen, R.G., Hopson, lW., and Fritz, J.N. (1982), Optical Technique for Deter-mining Rarefaction Wave Velocities at Very High Pressures, R e v. S ci. lns tr um .53,245-250.

McQueen, R.O., Marsh, S.P., Taylor, J.W., Fritz, IN., and Crater, W.l (1970),TheEquation of State of Solids from Shock Wave Studies, in H ig h- V e l O ci ty Impa c tPhenomena (edited by Kinslow, R.),Academic Press, San Diego, pp, 249-419.Miller, G.H., and Ahrens, TJ. (1991),Shock-Wave Viscosity Measurement, R e v. M od -e rn P hy s. 63, 919-948.Mitchell; A.C.,and Nellis, WJ. (1981),Shock Compression ofAluminum, Copper, andTantalum, J. A p pl . P h ys . 52, 3363-3374.

Mitchell, A.C., and Nellis, WJ. (1982),Equation of State and Electrical Conductivityof Water and Ammonia Shocked to the 100 GPa (1 Mbar) Pressure Range, J.Chem. Phys . 76,6273-6281.

Morris, C.E., Fritz, IN., and McQueen, R.G. (1984),The Equation of State of Poly.tetrafluoroethylene to 80 OPa. J. Chem. Phys . 86, 5203-5218.Murr, L.E. (1981),S ho ck W ave s a nd H tq h-S tr am -R au : P he no me na in Meta l s , Plenum,New York, pp. 1-11Ot.Murri, W.J., Curran, D.R, Petersen, C.F., and Crewdson, RC. (1974), Response ofSolids to Shock Waves, in Advances in H ig h P r es su re R e se a rc h, Academic Press,New York, pp, 1-163.Raikes, SA, and Ahrens, T.J. (1979a), Measurements of Post-Shock Temperatures inAluminum and Stainless Steel, in H ig h P re ss ur e S cie nc e a nd T e chn olo gy (editedby Timmerhaus K.D. and M.S. Barber), Plenum, New York, pp. 889-894.Raikes, S.A., and Ahrens, T.J. (1979b), Post-Shock Temperatures of Minerals, Geo phys . J. R oy. A str ono m. S oc . 58, 717-748.Rice, M.H., and Walsh, J.M. (1957), Equation of State of Water to 250 Kilobars, J.Chem. Phys . 26, 824-830.Roddy, OJ., Pepin, R.O., and Merrill, RB. (1977), Im pa ct a nd E xp lo sio n C r ate ring ,Pergamon, Oxford, pp. 1-1301.

Rosenberg, Z., and Partom, Y. (1984),Direct Measurement of Temperature in ShockLoaded Polymethlmetacrylate with Very Thin Copper Thermisters, in Sho c k

..-----.- . . . .-..-....-.~-.---


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4. Equation of State 113Wa ve s in C o nd e ns ed M a tt er -1 9 83 (edited by Asay J.R. et at), Elsevier, Amster-dam, pp. 251.Ruoff,A.L. (1967), Linear Shock-Yeiocity-Particle-Velocity Relationship, J. App l .Phys . 38, 4976-4980.

Sharpton, V.L., and Ward, P.D. (Eds.) (1990), G lo ba l C ata str op he s in E ar th H is to ry ;An Inte rd isc ip lina ry C onfe re nc e o n Im pac ts, V olcanism , and M ass M or ta lity ,pp. 1-631, Special Paper 247, The Geological Society of America, Boulder, Colo-rado,1990.Silver,L.T., and Schultz, P. (Eds.) G eo lo gic al I mp lic atio ns o f Im pa cts o f La rg e A ste r-o id s and Come ts on the E ar th , pp. 1-528, Special Paper 190, The GeologicalSociety of America, Boulder, Colorado, 1982.Simakov.G.V.,and Trunin, RF. (1990),Compression of Super-Porous Silica in ShockW ave s,lzv. E ar th P hys . (Russian), 11,72-77.Stomer, D. (1972), Deformation and Transformation of Rock-Forming Minerals byNatural and Experimental Shock Processes, I, Fo rt sc h r. M in e r. 49, 50-113.Steimer,D. (1974), Deformation and Transformation of Rock-Forming Minerals byNatural and Experimental Shock Processes. II. Physical Properties of ShockedMinerals. Fo rt sc hr . M in e r. 51, 256-289.Svendsen,8., and Ahrens, TJ. (1987),Shock-Induced Temperatures ofMgO, Geophys .J. R oy. A stro no m. S oc . 91, 667-691.Tan,H., and Ahrens, TJ. (1990),Shock Temperature Measurements for Metals, HighP r es su re R e s. 2, 159-182.Touloukian, Y.S., and DeWitt, D.P. (1972), Thermal Radiative Properties of Non-metallic Solids, in T he rm op hysic al P ro pe rtie s o f M atte r, Plenum, New York,pp.3a-48a.Trunin, R.F., Simakov, G.V., and Podurets, M.A. (1971), Compression of PorousQuartz by Strong Shock Waves, Iz . E ar th P hy s. English Transl., #2,102-106.Wackerle,J. (1962),Shock-Wave CompressionofQuartz,J. A p pl . P h ys . 33, 922-937.Walsh,J.M., and Christian, R.H. (1955),Equation of State of Metals from Shock WaveMeasurements, P hy s. R e v. 97,1544-1556.

Watt, J.P., and Ahrens. TJ. (1983), Shock Compression of Single-Crystal Forsterite,J. Ge op hy s. R e s. 88, 9500-9512.

Williams,Q., Jeanloz, R., Bass, J., Svendsen, B., and Ahrens, T.J. (1987),The MeltingCurve of Iron to 250 Gigapascals; A Constraint on the Temperature at Earth'sCenter, Sc i enc e 236, 181-182.Yakushev, V.V. (1978), Electrical Measurements in a Dynamic Experiment, F iz. G o-r en iy a V zr yv a 14, 3-19.Zaitzev, V.M., Pokhil, P.F., and Shvedov, K.K. (1960), An Electromagnetic Methodfor Measurement of the Velocity of Explosion Products, D okl. A kad . N a uk. S S SR132, 1339-1340.Zel'dovich,YO., and Kompaneets, A.S.(1960), T he o ry o f D e to na tio n, Academic Press,New York.

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t.j. ahrens- equation of state

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