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I PICATIMINY ARSENAL TECMW4ICAL REPORT 62M

THE GRUNEISEN CONSTANT

POROUS MATERIALS

II1111!.. .I

'I'ENERGY DEPOSITION MATERIALS

I BY

I PAUL HARRIS

AUGUST 1971

APPROVED FOR PUBLIC RELEASE: DISTRIPUTION UNUMITED.

PICATINNY ARSENALDOVER. NEW JERSEY c

NATIONAL TECHNICAL'NFOPMAT;ON SFRV/ICE

UNClASSIFIEDSWcuty (1assdiea.

Ii , @GMATMsu ACT"RV I CM1. M ) MMMW RE'@UT *tCU.-01 C LASSSFICATSO{U.S. Army, Picatiumy- Ar 1u, Dower, New Jersey M791 Jnafm

mTE GRME N VI"NST OF PROUS MATERiLS N ENERGY DEPQSrMlN

*mATRILS f U)4ni~ 4W M omr* MO tye so FOA se din..) Delailei proposal &it usual enerdy depositioad tOe urn! ao tntmrok dop nt ]id a true Gmweiven aM~eter of porous matter.

a, iviwoewsi Mes, ANN. amt sn. na

Harris, PaulC Man o CATE &.TOTAL me- es P*%6S a a oCr*

August 1971 1- 28 21$a. CetTRACT C a K fAV o N/A go, @touhTewftAgPOOT uIMmeat4)

& ~i~juc mcN/APicatiy Arsna Technical Report 4255

r10 AV^ IL AUTL;iWTATM

WTCE

Approved for public release; distribution unlimited.I. suPPi gugumm' WO SPOING MUTANT ACTIVITY

J'lcainny Arsenal, Dover, N. J. 07801

MS APSTtACT

It is propoeed that the usual energy &Mpsition, e.g., electron beam, experiments, aswell as the usual shock Introduction, e.g., flyer plate, experiments dD not yield a trueGruneisen parameter of porous materials. It is proposed tha"t In a porous material stress

relief occtrs wit:&'.n the time that energy to being deposited, or before the final state isreached in a flyer plate experiment. Both types of experiments are analyzed and correlatedI and agree with the ideas put forwarV here. Energy deposition experiments of Shea,Mazella. and Avrainl (porous PETN), and flyer plate experimetits of Boade (sinteredI porous copper), are oonsidered.

Wave propagation in porous materials is considered in some detail, and it is concludedthat in such materials macrosropic measurements can not always be used to determinemicroscopic parameters. It is proposed that a porous material acts as a diffractiongrating with respect to a shock wave, and that unidirectional one-dimensional strain doesnot hold,

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Sec-r C'.tsiona io

PICATINNY ARSENAL TECHNICAL REPORT M25

THE GRUNEISEN CONSTANT OF POROUS MATERIALS IN ENERGYDEPOSITION MATERIALS

BY

PAUL HARIS

AUGUST 1971

Approved for public release- di!Ftribu!ion unlimited.

CONCEPTS AND EFFECTIVENESS DIVISION, NEDPICATINNY ARSENALDOVER, NEW JERSEY

ACKNOWLEDGMENTIi

The author wishes to thank Mr. Louis Avrami of Picatmy Arsenal'sExplosive Laboratory for suggesting the problem to him, and for manyhelpful discussions.

TABLE OF CONTENTS

PAGE NO.

List of mustrtoms 4

List of Tables 5

Abstract 6

Summary 7

Section L Introduction 8

IL The Grianelsen Constant 10

IL Microscopic Effects and the Grh'nelsenConstant 13

IV. Shock Propagation in Porous Materials . 19

V. Microscopic Effects on Shock Propagation 20

References 23

Distribution List 25

I3I

LIST OF ILLUSTRATIONS

FIGURE NO. PAGE NO.

I An Averaging Model for Effective Pressure Dueto Energy Deposition in a Porous Solid 1

2 Shock Velocity versus Particle Velocity forSintered Porous Copper 17

3 Model for a Porous Solid 19

4

LIST OF TABLESTABLE NO. PAGE NO.

I Electron Mean Free Paths in Some Materialsas a Function of Energy 11

I Energy Deposition Properties of PETN 12

if GrUneisen parameters for Sintered PorousCopper 18

5

I

ABSTRACT

It is proposed tlXat the usual energy deposition, e.g., electron beam, experi-ments, as well as the usual shock introduction, e.g., flyer plate, experiments do notyield a true Grilneisen parameter of porous materials. It is proposed that in a porousmaterial stress relief occurs within the time that ehergy is being deposited, or be-fore the final state is reached in a flyer plate experiment. Both types of experimentsare analyzed and correlated and agref -vth the ideas put forward here. Energy deposi-tion experiments of Shea, Mazella, and Avrami (porous PETN), and flyer plate experi-ments of Boade (sintered porous copper), are considered.

Wave propagation in porous materials is considered in some dotail, and it isconcluded that in such materials macroscopic measurements can not always be used todetermine microscopic parameters. It is proposed that a porous material acts as adiffraction grating with respect to a shock wave, and that unidirectional one-dimensionalstrain does not hold.

6

In this report, we have looked at some of the details associated with shockpropagation in porous materials. We have inquired into effects resulting fromshock generation by 'instantaneous" energy deposition or by shock introduction,e.g., via a flyer plate, at a iree surface of the sample.

By considering the physics of energy deposition and shock propagation in aporous medium, we have shown that conventional techniques do not yield valid measure-ments of the true solid-state Grilneisen parameter, i. e., the relation between pres-sure and specific rnergy at constant volume, because on a microscopic scale de-termined by particle size, stress relief occurs before the final state is reached.

This is true for the case of shock introduction and energy deposition. In the case of*energy deposition, simple models which attempt to consider the result of stress

relief within deposition time are presented. While the models are inadequate, theyserve as a starting point for understanding the physics and for correlating the energydeposition and shock introduction cases.

Energy deposition experiments of Shea, Mazalle, and Avrami (14), and shockintroduction experiments of Boade (18) are analyzed. In addition, we have appliedsome elementary considerations of a porous material as a diffractive medium tocomment upon the difficulty of utilizing macroscopic experiments for investigatingmicroscopic parameters. The difficulty being that local unidirectional one-dimensionalstrain cannot be assumed to hold. In this regard, we considered some flyer platework in polyurethane as reported by Butcher (21).

In the Introduction, we have considered the importance of porous materialsfrom an academic and f -om an engineering point of view. Porous materials can beused for shock nditatjuzzan ed/or as thermal standoff materials. Consequently, the

U. S. Army and the shock wave community would like to understand as much as pos-sible about porous solids. Considerable more work is needed.

7

L INTR ODUCTION

Porous materials are any materials that happen to have a mass density lessthan the maximum possible equilibrium mass density at the pressure and tempera-ture in question. Examples of such materials are foams, e. g., polyurethane. lessthan nornial density explosives (1), and porous metals, e. g., porous tungsten (2).

Porous materials are of both engineering and academic kiterest. Polyurethanefoams are utilized as shock mitigators by having the shock energy absorbed in theprocess of compacting the foam to its normal solid density, Porous metals haveengineering applications but are also of extreme academic importance. If a solidof normal density is shock-loaded, its final state lies in a single curve (the Hugoniotcurve) in the PV plane. On the other hand, when a solid such as porous tungstenis shock-loaded, the final state reached will reside on one of a continuum of Hugoniotcurves, depending upon the initial porosity, the Irreversible processes which occurduring compacting yield PVT states for the compacted solid different from those oc-curring in the shock-loaded material having initial normal density. Thus, by shock-loading an initially porous material to bevond complete compaction, it is possibleto investigate broad regions In PVT sp _: This academic property of porous ma-terials was first pointed out by Zeldovich (3).

Another important application of porous materials is in the area of explosives.It is well kno(n that the initiation sensitivity of explosives to mechanical shock dependsupon the final density to which a granular explosive may be pressed (4). This areaalso has both academic and engineering applications.

Soviet and Western interest in inert porous materials, judging from we openliterature, has been in different pressure regimes. Western interest+ (5, 6, 7, 8, 9) hasbeen strong "n thie area of shock propagation in the pressure range at which compaction

occurs. O the other band, the Russians appear to have their main effort (2,10,11) inthe pressure range beyond which compaction has already occurred. Among the interestingirgredients of high-pressure physics which the Hussians have studied via porous ma-terials is the electronic contribution to the Grineisen constant (2).

There are two general ways by which porous materials may be examined experi-mentally. In one method, energy is dumped into the material in a very small time,more or less uniformly as a function of position and the shock effects accompanying thert-sulting thermal expansion may then be studied. A typical procedure for dumping inthe energy utilizes an electron beam machine (12) which produces (typically), Mev

'The references in this report, for the most part, are meant to be illustrative rathertfhan exhausTive.

8

electromw in large numbers within a pulse width from 20 naoseconds to 70 nanoseconds

The electrons are scattered by the solid parts of the porous material, resulting in an

increase in the internal energy of the material. The second general method involves

the impacting of the porous material with a fyer plate (13) of known mechanical prop-

erties. Instead of the shock originatig within the material as in the energy dumping

method, the shock now originates at ihe Interface between the flyer and the sample

target.

In this report, we wish to specifically consider an aspect of some electron beam

experiments( 4 ) in porous samples of the explosive PETN (Pentaerytbritol Tetra-

nitrate). Some other experimental work will also he considered.

1 9

IL THE GRUNEISEN CONSTANT

There are two possible definitions for the GrUneisen parameter, one micro-scopic, and the other macroscopic. The microscopic Involves a model for inter--actions on an atomic level, while the macroscopic definition involves measurableparameters.

In the macroscopic definition, the GrUneisen constant, r, is defined by:

p ,

where p is pressure, p is mass density, and E is internal energy +. Eq. (1) saysthat if one deposits .hermal energy in a time small compared to the time necessary(defined beiow) for significant mass motion, then the pressure in the solid changes.The quantity r determines the change in pressure, and is a property of the structureof the material.

The time criteria mentioned above is usuIly taken as the time necessary foran acoustic wave to traverse an electron scattering mean free path in the solid. Letsuch a time be ti . If the pul&e duration is T, then the constant volume definition ofEq. (1) requires-

T > t1. (2)Physically, this means if Eq. (2) holds, then any energy inhomogeneities introducedby the electron beam are not relieved, by acoustic signals before the energy is de-posited. In other words, the energy can be considered as deposited instantaneously'if Eq. (2) holds.

+ Actually, the Griineisen parameter is a tensor quantity, and the stress tensor should

be substituted for the pressure in Eq. (1). Such an anisotropy effect could be im-portant in this report where -e are interested in a pressure range comjiarible tothe yield strength of the solid particles of PETN. Perhaps more important, the pres-sure range of interest is comparable to the pressure used in pressing the PETN toan initial density. Such pressing undoubtedly introduces anisotropy into the materialwhich could conceivably show up in the experiment. In the Shea, Mazella, and Avramiexperiment (141, peak pressures generated were in the one-kilobar range, while pres-sing pressures were in the one to ten-kilobar range. Because of a lack of sufficientexperimental evidence to decide this question at the present time, we will assumehere that Eq. (1) is correct in its present isotropic form.

§For those computer codes which can handle energy deposition and stress propaga-tion simultaneously, T should be replaced in Eq. (2) by the time cycle. If the energydeposition is by other than electrons, the appropriate mean free path must be used.Since this report makes a. point of physics, rather than technique, the possibility ofhandling hydrodynamics and energy deposition simultaneously is delayed until thediscussion section of the report.

10

One of the ma points of this report is that the time criteria of Eq. (2)is, in general, not valid for a poro.s material All it takes to invalidate that cri-teria is that the average dimension of the solid particles making up the material besmall compared to the appropriate radiation mean free path. For such small par-

ticles, the eaergy deposition-nduced thermal stresses are relieved by therma! ex-pension prior to completion of the energy deposition process. Table I below liststsome electron energy absorption mean free paths for a variety of materials andeneries.

TABLE I

ELECTRON MEAN FREE PATH AS A FUNCTION OF ENERGY

Material Mean Free Path Electron Energy

Al 100&I 0. 4 MevPETN 500p 0. 4 MevBe I, 000 & 0. 6 MevQuartz Crystal 4, 000#1 2 MevPETN 1,000J 4 MevLead Styphnate I, 000 p 4 Mev

As contrast to the typical dimension of 103A, i. e., 0. 1 cm, of Table I, anacoustic wave propagating at 3 x 105 cm/sec can relieve the stress to the center ofa 180A particle in 30 nanoseconds. Consequently, an electron beam machine havinga pulse width of 30 nanoseconds (FWHM) would not be able to utilize the criteria ofEq. (2) for experiments with porous solids having particle sizes of 180#, or less. Itis interesting to note that the 30-nanosecond figure is typical of the faster machinesin use today. In the work by Shea, Mazella, and Avrami (hereafter called SMA), theaverage grain size of the PETN was observed to be 2 50U. The electron beam was at0. 4 Mev with a pulse width of 25 nanoseconds (FWHM). Thus, if the PETN in questionwas porous enough so that a significant numb6r of grains were not in intimate contact-along a grain boundary, then the time criteria of Eq. (2, would be invalid.

We hasten to say that the work of SMA is still interesting in that it yieldsthe m'gnitude of the pressure pulse resulting from a thermal energy deposition. In-deed, the experimenters were aware of the point with respect to time criteria madeabove at the time of their experiment, and that is why they speak in terms of an '"e-fective" Griineisen constant in their paper. The object now, however, is to see whetherthe measurements made can yield any useful information concerning microscopicprocesses. Table II lists the pertinent results of the SMA experiments.

II

TABLE Ha

ENERGY DEPOSITION PROPERTIES OF PETNb iI

POO/PO Density Sound Speeds Effectivegm/cm 3 cm/sec x 10-5 r

0.95 1.67 2.8 1.20.90 1.59 2.4 ± 0.1 0.510.87 1.54 1.8± 0.3 0.150.84 1.48 1. 7 * 0. 3 (0. 07 to 0. 23)

aThis table was taken, with minor changes, from reference 14.

b Po denotes the normal density. poo initial porous material density.

Although SMA report an experimental error in the measured r of 20 percent

or more, the trend as a function of density is clearly discernible.

12

IU. MICROSCOPIC EFFECTS AND THE GRUNELJEN CONSTANT

In this section, we discuss some of the physics which might be importantin interpreting the measured or effective Griineisen constant. Integrating Eq. (1)gives+

P = Pi r (E-E) (3)

where pi denotes the initial density at which energy is deposited in the constant volumeapproximation, r is assumed independent of energy density, and Ei is the specificenergy density prior t energy deposition. That r is indeed independent of E hasbeen borne out experimentally by the success of the linear P vs E curve in fittingthe data (14).

There are two relevant microscopic effects associated with the energy deposi-tion aspect of porous solids. First, there is the possibility that surface energy ef-fects, because of the large internal surfa'e area expected in a porous solid, can playa role in determining the pressure, P. Second, is the expected and mentioned stressrelief in the solid particles of the porous material.

Let us write (15) for the surface energy, Y.:

aoY7 s = 20 (4)

where ao is the lattice parameter, and Y is Young's modulus for the solid particles.Using values for a typical solid:

Y = 1012 dynes/cm 2 , ao = 2 x 10- 8 cm,we find that:

T= 10 ergs/cm 2 "

The heat of formation, A Hf = 5 x 104 cal/gram formula weight. For a solid of po -5 gm/cm3 and a gram formula weight of 50, we have an Eo of:

E0 = 5 x 103 cal- 2 x 1011 ergs

+In Eq. (3), we assume that the thermal pressures generated are considerably larger

than the pressure before deposition, I. e., atmospheric pressure, and also largerthan anyinternal stresses generated in the process of pressing the porous materialto its initial density. There is a possibility that this last assumption is invalidsince typical measured pressures (SMA) were in the fraction of a kilobar range,while pressing pressures were in the one to ten-kilobar range.

13

for one cm 3 of a typiral solid. If the same cm 3 of material is now considered tohave 103 cm 2 of internal surface area, S, (a factor of 103 more chan external surfacearea):

YsS = 106 ergs.

Comparison of Eo and ysS would seem to indicate that surface energy effects canbe neglected. This view is further supported by the idea that surface energy is notexpected to change during energy deposition; thermal expansion should allow stressesbuilt up by deposition to be more or less instantly relieved so that neglecting tem-perature, the surface remains in a constant state.

Let P1 denote the pressure in the normal density solid, and P2 the pressurein the corresponding porous solid of initial density Poo :

P1 Po ri (E - Eo) (5a)

P2 Poo r2 (E - Eo) (5b)

Since it has been shown that surface energy considerations are not important, thesame specific internal energy has been used in Eqs. (5a) and (5b). Now let V. bethe volume occupied by the solid particles, and V that occupied by the voids. Wewish to investigate the relation between I and r2 as a function of various modb!qfor averaging the pressure over the solid particles of the porous material. Theaveraging takes place with the mass motion frozen at the time at which energy deposi-tion ends. One of the simplest averaging models is given by:

2(n) p Vs , (6)2V s + Vp

whece P2 (n ) simply denotes the pressure with this particular averaging model. Eq.(6) says that the pressure is reduced from that of the normal density solid by beingsmeared out over the entire volume to the solid particle. If m is the mass of the solidparticle:

m = P (V + V) (7a)

m = P0 Vs (7b)

r (n) = r /Poo n-i (8)2 1(Po

Although the model given by Eq. (6) is unphysical, it is still interesting to note thatit takes an exronent n greater than ten to fit the data of Table II, while n = 1 yields aGrUnoisen parameter independent of porosity.

14

The next averaging model we consider is more physical and is shown in fig-

ure 1 following:

Pressure

I I I . .. TI ... -

VOID VOID VOID

I I

I D SOLID

0 A _-Distance

FIGURE 1: An Averaging Model for Effective Pressure Due to

Energy Deposition in a Porous Solid

Averaging the pressure over the solid particles:

1s PdV, (9)

P 1 ar(10)

where we are considerig the parti'cls tj be spherical in Eq. (10). Using V

4 # a. 3 = m/po, we thus find:3

a CP1 8 4 2 r"()

P2 = (V+V) o 4r (1 - -2

+ V 0 a a8s ps 2foo

2 P 4 s s =a53F o,) P (12)12 - 1 Vp([ -2 1(V +V) 3 5 \0I 1

- 2

r2 5 r l1 (13)

~15

The model leading to Eq. (13) is equivalent to assuming that a relief wave propagatestoward the center of the particle, and that the wave decreases in amplitude as it propa-gates toward the particle center. It would also appear that a further assumption, thatthe wave just reaches the center of the particle at the end of the energy deposition, hasbeen made. Once again, the averaging model is unphysical but interesting. We findan effective Grtineisen parameter which is dependent upon the spherical geometry, butindependent of porosity. If anything, we have shown that the naive approaches willnot work.

We feel that this type of work should be carried further. A more powerful andmore physical approach than that presented above would use a computer program forstress wave analysis. The program would have to handle energy deposition and hydro-dynamics at the same time (17). Having such a code, the proposal is then to look atthe pressure time distribution in a slab of solid material, i. e., po = p , where theslab is so thin that the acoustic transit time through it is small compared to the timeassociated with energy deposition.

There is a definite lack of data specifically relating to Grilneisen constant of -porous materials before compaction occurs. It is possible, however, to analyze ex-isting shock velocity versus particle velocity data to obtain sub information. Figure2 following, is based upon data taken from Boade (18) on sintered porous copper. Ifwe let Us be shock velocity, and up be particle velocity, then the extrapolations (dashedlines) in figure 2 can be represented by:

Us = A + B up (14)

Let r o be GrUrneisen parameter corresponding to the extrapolated zero pressurestate. It is easily shown (19) that:

2 (5ro =2B 3 (15)

It is then a simple matter to apply Eq. (15) to figure 2.

For the data shown in figure 2, experimental points only exist for the 67. 7percent, 82. 8 percent, and solid density sample. The other curves are theoretical.Applying Eq. (15) to the mentioned data gives the results shown in Table IIL

16

Solid Copperp 0 = 8. 93 g/cm 3

6 = 4.02+1.46up

I99% ,

~95%

2 82.9% .

0 // 6 7.7 to,

O .5 J. .5Particle Velocity, mm//g sec

FIGURE 2-, Shock velocity vs particle velocity for sintered porouscopper (from Boadel18). Percentage porosity is indicatedby poo0/po , and dashed lines are meant to extrapolate tozero pressure data.

17

TABLE III'Criieisen Parameter for Slntered Porous Copper

Io P00 P 1"o

0.677 2.790.829 4.671.00 2.25

The data thus show that the Grilneisen parameter decreases as the degree ofporosiiy increases, This is similar to the averaging model of Eq. (8). On the otherhand, the r o value corresponding to the solid density copper is lower than that ofany of the experimentally observed porous density Io values. This last point is bothunexpected and interesting. It correlates, however, with an observation made bySMA (14) that the calculated Gfiveisen parameter for normal density, i. e., solid,PETN is 0. 8 (with perhaps 20 percent error), which is smaller than the r associatedwith Poo/Po = 0. 95, as shown in Table IL For the moment, this phenomenon is notunderstood.

18t

IV. SHOCK PRGPAGATION IN POROUS MATERIAIS

In this section we would like to discuss the relationship between the Grineisendata obtained from the impact experiments of Boade, and the energy deposition dataobtained by SMA. We will then further discuss an aspect of the physics associatedwith shock propagation in porous materials.

Consider a porous material modeled as shown in figure 3:

Shock

Propagation

Direction

FIGURE 3: Model for a Porous Solid. (The shaded areas representsolid particles, and the unshaded areas are voids.)

If a shock of amplitude too small for compaction propagates into the material shownin figure 3- we first expect something akin to the Hugoniot state of a normal densitysolid to be transmitted across the contacting surfaces of the solid particles. Thepressure corresponding to such a Hugoniot state would then be relieved in time byrarefactions entering the solid particles through their free surfaces. For solid par-ticles with diameter small compared to the width of the shock front, the final statewould thus be a function of the Hugoniot state in the normal state plus a relief factor.

Since the Us versus up relation is measured for the final state, and thus sois the Griineisen relation, the degree of porosity must play a role in determining r.Greater porosity means greater relief of the normal density pressure state, a re-duced a Us value, and thus a reduced r. The same physical phenomenon appears

I Up

to be occurring in the shock propagation determination of r' as occurs in the energydeposition case.

19

V. MICROSCOPIC EFFECTS ON SHOCK PROPAGATION

An observer riding on a shock front in a porous material such as shown infgure 3 would see a succession of diffraction-grating planes. The effect of diffractionin general is to change 'he geometry of the wave front. In our case, the change isfrom plane wave geometry to aonplane wave geometry. Further specializing, we cansay that on a microscopic particle size level, the mass disturbance is not ,midirectionone-dimensional strain* even though it may be unidirection one-dimensional strain ons macro:co pic level. One-dimensional strain occurs when the dimensions of thesample pe!pendicular to the direction of shock propagation are large compared to thesample dimensions parallel to the direction of shock propagation. If the relative sizeof the dimensions are reversed, then the propagation is clarsified as one-dimensionalstress. On a microscopic level, for spherical particles with some void separationbetween them as shown in figure 3, we thus expect a change in the geometry of the wavetjort.

If one is only interested in characte :.-ing the macroscopic shock properties ofa porous solid, then the unidirection one-dmentional strain view of the problem (withappropriate perpendicular and parallel dimensions) is valid. If, however, one wishesto use the data from the macroscopic experiment to determine microscopic parameters(such as a relaxation time), large errors could result. Physically, the reason forsuch errors is that diffraction modifies the way in which the prikeiple of the conserva-tion of momentum is applied.

In figure 4 following, we illustrate the result of microscopic diffraction. Or-iginaliy, a plane wave is propagating in the z direction. After diffraction has cccurred,smlll segment of the wave front, with area (&)4 moves in the direction (normal tothe t axis).

By one-dimnnotal strain we imply plane wave propagation with particle motion only inth- di rection of the wave propagation. The one-dimensional strain case is the typical'onfigt,.i ion i,. shock wave theory and experiment.

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iX

FIGURE 4: Diagram for Calculating the Results of Diffraction byParticle Structure In a Porous Solid

For brevity, assume that the local particle motion is also in the 4 direction.This simplification uegle.ts the possibility of local transverse waves being generateddue to the porous structure of the medium. In other words, we have local one-dimensional strain in other than the z direction. Ifrg represents the momentum flux (20)

in the g direction, then (20) :

fft (AS) p +Pu u e (A S)g (16)

where p is pressure, p is mass density, and ut the particle velocity in the di-rection.

(& s) s)Z cos (17)

Let Pz be the momentum transferred per time across (A S) z .

;Z M P r A S)z = Pt cos (18)

z (&S) 4 oo 8(19)

.'Wz =(p + p ut ut) 0os 2 e ,1 (20)

or

rz = p c0s2 + p Uz Uz . (21)

Two points now need to be made. Fi rst, when employing the conservationof momentum in shock wave problems, people usual.y equate the right hand side ofEq. (21) on each side of the shock front (with 0=0). Second, even though no net de-viation from macroscopic one-dimensional strain in the z direction is observed, anonzero average value for cos 2 0 is expected to exist.

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Bucht r 42 ' Uses Eq. (20) with 0 = 0 tc investigate shock prapqtlon In0. 96 gmicc poiyurethane ioam. he was interested in the relaxation Urns v", for voidcollapse (closure) and generated an equation of the form:

A (G "eq)I oUz = (22a)

(B+ Ca)

where A, B. and C may be regarded here as constants. Our point is that an equationmore in the form of:

uz = A(creq!coo2 9/,ro (22b)

(B + Co cos 2 O)

should have been used. In Eqs. (22), a denotes stress, oae an equilibrium stressvalue, and cos 2 e an average value of ;---S. Such a correction, at leaet in the nu-merator of Eq. (22b), would result in a larger value of io than previously estimated.A larger v;due of To would b-- in keeping with some other ideas (based upon a "sim--plified gap model') put forward by Butcher (21) . At the moment, we do not have agood model for calculating eos 2 0.

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REFERENCES

1. V. A. Vasil'ev et al, Combustion, Explosion, and Shock Waves 3, 371 (1967).

2. K. K. Krupikova, Soviet Physics - JETP 15, 470 (1962).

3. Ya. B. Zel'dovich, Soviet Physics - JETP 5, 1103 (1957).

4. M. W. Evans et al, Fourth Symposium on Detonation, 12-15 October 1965,White Oak, Silver Sprinm, Marysland, p 359.

5. J. Thouvenin, FGurth Symposium on Detonation, 12-15 October 1965, White Oak,Silver Spring, Maryland p 258.

6. H. P. Lavender, Thaoretical-Experimental Correlation of Large Explosivel,-Induced Transient Deformations of Polyurethane Foam Metal Slabs, Air ForceDynamics Laboratory, Wright-Patterson Air Force Base, Ohio. AFFDL-TR-67-106, AD822911, dated September 1967.

7. B. M. Butcher and C. H. Kannes, Dynamic Compaction of Porous Iron, SandiaLaboratories, Albuquerque, New Mexico, SC-R" 37-3040, dated April li68.

8. W. Hermann, Journal of Applied Physics 40, 24S (1969).

9. L. Seaman and R. K. Linde, Distended Material Model Development, Vol. 1,Stanford Research Institute, Menlo Park, California, Contract No. F29601-67-C-0073, Report No. AFWL-TR-68-143, Vol. 1, dated May 1969.

10. S. B. Kormer et al, Soviet Physics JETP 15, 447 (1962).

11. V. N. Zharkcv and V. A. Kalinin, Equations of State for Solids at High Pressuresand Temperatures (Consultants Bureau, New York, 1971).

12. R. A. Graham and R. E. Hutchison, Applied Physics Letters 11, 69 (1967).

13. For a general review of flyer plate technology see, for example, H. M. Berkowitzand L. J. Cohen, A Study of Plate Slap Technology, Part I, McDonnell-DouglasAstronautics Company, Huntington Beach, California, Report No. AFML-TR-69-106, Part I, dated June 1969.

14. J. H. Shea, A. Mazella, and L. Avrami, Fifth Symposium on Detonation, Pasadena,California, 18-21 August 1970.

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REFERENCES (Continuecd

15. A. S. Tetelman and A. J. McEvily, Jr., Fracture of Structaral Materials,(John Wiley and Sons, Incorporated, New York, 1967).

16. D. R. Stull and G. C. Sinke, Thermodynamic Properties of the Elements(American Chemical Society, Washington, D. C. 1956).

17. The PUFF V-EP Code of Kaman Nuclear Corporation does handle such problems.

18. R. R. Boade, Experimental Shock Loading Properties of Porous Materials andAnalytic Methods to Describe These Properties, Sandia Laboratories, Albuquerque,New Mexico, Report SC-DC-70-5052, dated 1970.

19. Y. K. Huang, Journal of Chemical Physics 51, 2573 (1969).

20. L. D. Landeu and E. M. Lifshitz, Fluid Mechanics (Addison-Wesley, New York,1969), Section 7.

21. B. M. Butcher, The Description of Strain Rate Effects in Shocked Porous Ma-terials, Seventh Sagamore Army Materials Research Conference, Raquette Lake,New York, dated September 1970.

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