DR - cOPYTECHNICAL REPORT TR-RD-SS--90-4

to THE RELEVANCE OF THE DE BROGLIE RELATION TO00 THE HUGONIOT ELASTIC LIMIT (HEL) OF SHOCKN LOADED SOLID MATERIALS

LN

James P. Billingsley~James M. Oliver

Aeroballistics Analysis BranchSystems Simulation and Development Directorate~Research, Development, and Engineering Center

MARCH 1990

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TECHNICAL REPORT TR-RD-SS-90-4

THE RELEVANCE OF THE DE BROGLIE RELATION TOTHE HUGONIOT ELASTIC LIMIT (HEL) OF SHOCKLOADED SOLID MATERIALS

James P. BillingsleyJames M. OliverAeroballistics Analysis BranchSystems Simulation and Development DirectorateResearch, Development, and Engineering Center

MARCH 1990 7 -

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11. TITLE (include Security Classification)The Relevance of the De Broglie Relation to the Hugoniot Elastic Limit(HEL) of Shock Loaded Solid Materials

12. PERSONAL AUTHOR(S)

Billingsley, James P. and Oliver, James M.13a. TYPE OF REPORT 13b. TIME COVERED 14. DATE OF REPORT (YearMonth, Day) S. PAGE COUNT

Final FROMJUl_89 TOQa§ 1 1990 March 2716. SUPPLEMENTARY NOTATION

17. COSATI CODES 18. SUBJECT TERMS (Continue on reverse if necessary and identify by block number)FIELD GROUP SUB-GROUP De Broglie Equation, Particle Velocity,Wave Velocity,

Hugoniot Elastic Limit, Iron, TNT, Alluminum Alloys,Elastic-Plastic Shock Waves, 6061, 2024

19. ABSTRACT (Continue on reverse if necessary and identify by block number)

Under moderate shock loading (a few kilobars) where Elastic-Plastic Shock Phenomena

(EPSP) occur in solid materials, two stress waves are created. The first one is es-

sentially an elastic wave and has become known as the Hugoniot Elastic Limit (HEL).

The HEL particle velocity, UPHEL , can be closely related to the De Broglie particlemomentum wave velocity, V, = h/(2md, ), where, h is Planck's constant. m is the mass ofan atom, and di is the closest distance between atoms in the structure. Classic EPSP

test data for iron, aluminum alloys (2024, 6061), and pressed TNT are compared to com-

puted values of VI and 2V1 . Qualitative remarks with respect to particle stabilityare made which are based primarily on Fitzgerald's analysis ([17]), Chapter 3.)

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•~~~ ~ *i -!- -I" -iiI I III I_-I

TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS ................................ iv

I. INTRODUCTION ....................................... 1

II. THE De BROGLIE PARTICLE VELOCITY, V1 = H/(2MDI) ... 2

III.FITZGERALD'S PARTICLE VELOCITY STABILITY CRITERIA 5

IV. EXPERIMENTAL HEL UPHEL DATA AND COMPARISON WITHV1 MAGNITUDES .................................... 6

A . Iron ................................................ 6B. Aluminum Alloys ...................................... 7

1. 2024 Alloy ......................................... 72. 6061-T6 Alloy ...................................... 7

C. Pressed TNT ......................................... 13

V. CONCLUSIONS ........................................ 15

IV. RECOMMENDATIONS .................................. 16

REFERENCES .......................................... 17

,..!1

LIST OF ILLUSTRATIONS

Figure B1te

1 Elastic and Plastic Wave Front Velocity Versus Particle Velocityfor Shocked Irorn .. ......................................... 8

2 Elastic Wave Particle Velocity Versus Specimen Thickness forShocked Iron ................................................ 9

3 Elastic and Plastic Wave Front Velocity Versus Particle Velocityfor Shocked 2024Aluminum Alloy ............................. 10

4 Elastic and Plastic Wave Front Velocity Versus Particle Velocityfor Shocked 6061-T6 Aluminum Alloy ......................... 11

5 Elastic Wave Particle Velocity Versus Specimen Thickness forShocked 6061-T6 Aluminum Alloy ............................ 12

6 Elastic and Plastic Wave Front Velocity Versus Particle Velocityfor Shocked TNT (o = 1.648g/cc) ............................. 14

iv

I. INTRODUCTION

Many solid materials under moderate shock loading exhibit a two-wave structure where thefirst one is essentially an elastic wave followed by a slower plastic deformation wave front. Ref-erences [1] through [10] contain experimental data which illustrate this elastic-plastic-shockphenomena (EPSP). Selected wave and particle velocity results from this classic experimentalinformation for iron, aluminum alloys, and pressed TNT are shown in Figures 1 through 6. Vari-ous theoretical aspects of EPSP are described in References [7] through [14].

The elastic wave generated under EPSP conditions is called the Hugoniot Elastic Limit(HEL) to distinguish it from a dynamic elastic limit which could occur from less severe transientload conditions [13]. It is noted in reference [12], page 191, that the HEL condition can producethe same effect as a phase transition. That is, an additional wave is formed.

The main purpose of the przsen, repcrt is to provide evidence that the HEL particle velocity,UPHEL, is fundamentally determined (or essentially limited) by the De Broglie momentum velo-city-wave length relation which applies for any type of particle motion (Reference [15], page479; and Reference [16], page 433). Consequently, new insight with respect to the EPSP is pro-vided by the fact that experimentally observed steady state, or stable, values of UpHEL comparefavorably with the De Broglie velocity, V1. Vi is computed for an unperturbed atomic lattice (seeSection II). These comparisons are shown herein for shock loaded iron, aluminum alloys, andTNT, which exhibit EPSP and HEL stress waves.

II. THE De BROGLIE PARTICLE VELOCITY, Vi = H/(2MDI)

Fitzgerald ([17], Chapters 1 and 3) delineated the importance of the De Broglie momentumwave-wave length particle velocity, Vi, with regard to impacted solid material behavior. The DeBroglie velocity, V1, is:

h h2md, (1)

Where:

Vi = Limiting particle velocity which can occur without permanent lattice distortion(plastic flow); or the limit particle propagation velocity in a stationary lattice.Units are cm/sec or km/sec

h = Planck's Constant

= 6.6262 • 10-27 (gram)(cm 2)/sec

m = Mass of one atom, grams

di = Closest distance between the atoms in a crystal lattice, or the atomic spacing in aslip direction, units are cm or angstroms, A0, (1 AO - 10- cm)

k, = 2 di = wave length associated with the momentum, mV1. It is the shortest wavelength possible in an undistorted lattice or stationary lattice, cm or AO.

Table 1 lists Vi information for iron, aluminum, and TNT (Po = 1.648 g/cc). The values foriron and aluminum are from Reference [17]. Vi for aluminum was employed for the aluminumalloys (2024 and 6061) since they contain a very high percentage of aluminum. For TNT, an av-erage value of the mass (may) of an atom and the average distance between atoms (dlav) wascomputed in the manner outlined in Reference [18]. These average values (mav and dlav) wereemployed in Eq. (1) to compute Vi for TNT.

Table 2 lists longitudinal elastic wave velocity, CL, information for the materials consideredin this report. Also shown in Table 2 is the elastic wave pressure, Pv, corresponding to the wavevelocity, CL, and the particle velocity, Vi. This is given by:

Pvi a p0CLV1 (2)

where po is the material density (grams/cc).

t ~ T-4 00

0 0 --

r--

00

0 0

c#~ 0 0000

C) C 0

00 0to0

60

00 C

-o C

TABLE 2. Tabulation of CL and Pv1 .

TABULATION OF CL AND Pvj

Material po CL Vi Pv1

g/cc km/sec km/sec kbars

Iron 7.84 6.04 0.0144 6.82[21

2024 -T4 2.70 6.41 0.0258 4.46[7]

6.236061 -T6 2.70 [8] 0.02584.34

TNT 1.648 2.798 0.0832 3.84[101

=( eo CL V1 (G) cc = MBARS-c u -sec u - sec

P(KBARS) = 10 3 P(MBARS)

I KBAR = 14,504.0 PSI

4

III. FITZGERALD'S PARTICLE VELOCITY STABILITY CRITERIA

Fitzgerald, utilizing his concept of reversed lattice motion ([ 17], Chapter 3) shows thatparticle velocities from approximately 0.50 Vi to 0.75 Vi are in an unstable region and can jumpto values higher than Vi. Fitzgerald cites experimental results from Professor Bell and co-work-ers at Johns Hopkins University which support the analytical prediction.

Particle velocities in the immediate region around Vi appear to be stable.

In addition, Fitzgerald also showed 2Vi was an important unstable velocity where exces-sive distortion would occur. This analytical result also compared favorably with experimentalresults obtained by Bell and co-workers for aluminum and copper. These experimental datawere acquired from end-to--end cylindrical rod impact via diffraction grating instrumentation.

Professor Bell summarizes his results in Reference [21] and lists three critical or transi-tion particle velocities which were experimentally discovered in pure aluminum. They are listedas follows and compared to Vi for aluminum (2580 cm/sec from Reference [17])

Vrl = 1478 cm/sec = 0.573 V, - 0.6 V

Vcr2 = 3350 cm/sec = 1.298 V1 - 1.3 Vi

Vcr3 = 5080 cm/sec = 1.969 V1 - 2.0 V1

The comparison is remarkably good, particularly for VCRI and VCR3.

Thus there are three important velocity regions:

1. 0.50 V1 to 0.75 Vi; unstable, may jump to a magnitude greater than V1 .

2. V, vicinity; apparently stable.

3. Vi to 2Vi; unstable, particle velocities in this region will approach Vi under "longterm" operating conditions. Ample time is necessary to allow particle momentum sharing witha sufficient number of lattice masses, (see [17], pages 72 to 74). This, by definition, is a relax-ation time.

IV. EXPERIMENTAL HEL UpHEL DATA AND COMPARISON WITH ViMAGNITUDES

A. Iron

References [1] through [6] document experimental investigations of shocked iron. Fig-ure 1 depicts experimental data points for HEL wave velocities as a function of the particle ve-locity. No actual data points for the plastic wave data are shown. Instead, a recommended [6)linear fit to the plastic wave ve!ocity is illustrated. Only a few data points (with considerablescatter) were available for particle velocities less than 0.06 km/sec and the primary focus of thepresent study was the elastic wave behavior.

Various investigators found that the HEL for iron depended on the test specimen thick-ness. Figure 2 illustrates experimental UpHEL data as a function of test specimen thickness. Avery corroborative data point in Figure 2 was found in Figure 14 of Reference [19). Reference[19] is an historical review article concerning shock physics investigations performed at LosAlamos National Laboratory. The initial particle velocity (free surface velocity divided by 2)was practically identical to Vi for this specimen thickness (24.7 mm).

V1 and 2Vi indicators for iron are shown in Figures 1 and 2. The experimental values forUpHEL vary from approximately 0.5 Vi to 2.4 Vi. Most of the UpHEL magnitudes lie between V1and 2V1.

As shown in Figure 2, UpHEL - 2Vi for the thin specimen and UpHEL approaches V1 forthicker specimen. Thus, the initially overdriven "elastic" wave UpHEL is not stable near 2V1. Itdecays or relaxes to a stable value (Vi) provided that the sufficient time (travel distance or speci-men thickness) is available. This experimentally observed behavior corroborates remarks madein Section III with respect to the unstable region 3 where Vi < Up < 2Vi.

See also the initial remarks in Section II of Reference [201 with respect to the asymptoticparticle velocity decay exhibited by the data in Figure 2. This reference employs dislocationtheory to model the observed phenomena.

An earlier phenomenological model for the relaxation time, T, is developcd in Reference14]. Reference [4] equations for - and specimen thickness, L, can be written in terms of V1 and2Vi. This is because 0.0140 and 0.0280 mm/4L-sec appear as constants in the equations and V1and 2VI are equal to 0.0144 and 0.0288 mm/li-sec respectively. The significance of the number0.0140 mm/L-sec is that it was the observed steady state particle velocity, which is shown inFigure 2 to be V1.

6

B. Aluminum Alloys

1. 2024 Alloy

Reference [7] documents Fowles' classic experimental and theoretical study ofEPSP for both hard and soft 2024 aluminum alloy specimens. Initial and final values of the elas-tic wave free surface velocities (2 - UpHEL) were tabulated in Reference [7]. These UpHEL dataare plotted in Figure 3 which apparently indicates that over-driven transient behavior was exhib-ited by the "elastic" wave particle velocity data for both hard and soft specimens. However, un-like iron, none of the UpHEL magnitudes exceed 2Vi. The general relevance of the De Broglievelocity, V1, to UpHEL behavior is substantiated by these data for 2024 aluminum alloy. Certain-ly, the UpHEL magnitudes compare reasonably well with Vi.

2. 6061-T6 Alloy

Experimental EPSP results for 6061-T6 aluminum alloy are documented in Refer-ences [8] and [9]. Elastic and plastic wave velocities and corresponding particle velocity infor-mation from reference [8] is graphically illustrated in Figure 4. UpHEL data from Reference [9]are plotted versus specimen thickness in Figure 5. This UpHEL information was obtained fromthe initial step rise of the transient free surface velocity (Ufs) data in Figures 4, 5, 6, and 7 ofReference [9]. The initial Ufs was carefully scaled from these figures and then UpREL was com-puted via the well known relation:

Up = Ufs/2 (3)

As illustrated in Figure 4 (Reference [8] data), the particle velocities for elasticwaves ( with no following plastic wave) do not exceed V1 by more than approximately twentypercent. This indicates that the limiting particle velocity for a true elastic wave was essentiallyVi. When the loading increased to until EPSP began, the "elastic" wave velocity and particlevelocity behaved in an erratic manner. This may be indicative of the instability that could be ex-pected where V1 < UpHEL < 2V1.

Compared to iron (Figure 2), UpHEL for 6061-T6 (Figure 5) exhibits very weak de-pendence on the specimen thickness (or time). The trend is that shown by the data collectedfrom Figure 4 of Reference 9. UpHEL magnitudes vary from about 0.80 Vi to 1.30 Vi. Thus for arather wide range of specimen thicknesses and input shock pressures (Pmax), UpHEL was withinthirty percent of Vi. The asymptotic stable value of UpHEt. was approximately 0.85 Vi, or essen-tially Vi.

These HEL data for 6061-T6 confirm the importance of Vi and substantiate the re-marks in Section III with respect to particle velocity stability when Vi < UpEL < 2Vi.

V1 2V,

7I

I IhI 2md

A ,-++ ++ I + 0. km

6------- ----- 4------ = -0.---4secAI

CL = 6.04 I AlU, Al I

5

PLASTIC WAVE

REPRESENTATION

U, =4.63 + 1.33 x U,

FROM [51, [6]

3

km

sec

2II

SYM SOURCE

x [2]I [3]+ [4]* [5]

0 0.01 0.02 0.03 0.04 0.05 0.06kmsec

Figure 1. Elastic and Plastic Wave Front Velocity Versus Particle Velocity for Shocked Iron.

8

%00

000

p-i p- Q~i >

~tn

C6 Cqz

k'I.

,P U

40

U >

u0

00

z4.)

-Z z-

000

Ui V

4f4.

100

uu t uu uu -°P PCi00

U0

'"1 ,,U

En 4 - 0. i

000

°

00

C"4 .r '

x, .. .

CD >

00

X + +m

S I-+ + IV ++r

Uc

- -- -- -- -- -- @--0(

X x

I, x

cu

~- "?f-r-0 C;

+ C d, -

>.~ X+*4 -

I Z

CsCD C)

120

C. Pressed TNT

Very interesting experimental EPSP results for pressed TNT (po = 1.648 g/cc) are re-ported in Reference [10]. Elastic and plastic wave velocities are plotted as a function of theirassociated particle velocities in Figure 6. This figure is similar to Figure 6 of Reference [10] andFigure 11-13 of Reference [11]. These EPSP data for this polycrystalline organic material exhib-it metallic like behavior (see Figures 1, 3, and 4).

Only the Vi indicator line is drawn in Figure 6 which depicts the pressed TNT EPSPwave and particle velocity experimental results from Reference [10]. Even though there is aspecimen thickness size effect on both the HEL wave and particle velocities, none of the HELparticle velocities exceeded Vi. The highest value of UpHEL was 0.0739 km/sec or 0.89 V1.

13

e rz,

u

OEI-11-

P4 - 4~ -

+ 4

C) 0 0 CW)6

>d

40

ci 4~ C4-

14d

V. CONCLUSIONS

Although this is not an exhaustive investigation, enough information has been presented toindicate that:

A. A basic reason for EPSP to occur is because the De Broglie relation Eq. (1) specifiesthe limiting velocity (Vi) at which an atom of mass, m, can travel in an undistorted or stationarylattice structure where di is the smallest distance between atoms. The plastic wave particle ve-locity is Up = h/(2mdl I' ) so that the lattice spacing must change from di to d,' to accommodatethe irresistibility driven particle. This is a rather violent and catastrophic event whose conse-quences are the same as a polymorphic phase change. A phase change causes a two-wave struc-ture since the De Broglie relation must be satisfied by adjustments made to the atomic spacingin the new lattice structure. Thus there is one wave fron the old lattice structure plus an addi-tional wave from the new structure.

B. UpHEL for the stronger metals (iron 2024-T4 and 6061-T6) was generally between Viand 2W. Apparently strong metallic atomic lattice bonding requires that UpHEL must be "over-driven "greater than V1 to displace or distort the lattice plastically (reduce di to di' ) and thusallow a plastic wave particle velocity which is Up = h/(2mdi').

C. The "weak" materials (annealed 2024 alloy and pressed TNT) did not require "overdri-ven" HEL particle velocities for formation of plastic waves. Essentially V1 was an upper boundon HEL particle velocities for these materials.

D. Vi is the steady state stable UpHEL for iron and 6061-T6, even when UpHEL had beenoverdriven as much as 2.4 V1. If sufficient time (or specimen thickness) was available, UpHELdecayed and asymptotically approached V1 as a lower bound. This is consistent with Fitzger-ald's [17] stability analysis for Vi < Up < 2V which indicates that this is an unstable region.

E. Although UpHEL magnitudes may reach 2.4 Vi as shown for iron, most UpHEL values donot exceed 2V. This is consistent with Fitzgerald's analysis showing the importance and criti-cality of 2V.

15

IV. RECOMMENDATIONS

Additional EPSP data for other materials should be investigated for the De Broglie velocity,Vi, influence at the HEL conditions. Certain anomalous behavior observed at low level shockloadings could possibly be explained as a Vi effect. If this is suspected, then Vi s'lould be com-puted via Eq. (1) and compared with experimental particle velocities (Up) in the region of inter-est.

16

REFERENCES

1. Minshall, Stanley, "Properties of Elastic and Plastic Waves Determined by Pin Contractorsand Crystals." Journal of Applied Physics, Vol. 26, No. 4, pp. 463-469, April 1955.

2. Bancroft, Denison, Eric L. Peterson, and Stanley Minshall, "Polymorphism of Iron at HighPressures." Journal of Applied Physics, Vol. 27, No. 3, pp. 291-298, March 1956.

3. Hughes, D. S., L. E. Gourley, and Mary F. Gourley, "Shock Wave Compression of Ironand Bismuth." Journal of Applied Physics, Vol. 32, No. 4, pp. 624-629 April1961.

4. Taylor, John W. and Melvin H. Rice, "Elastic Plastic Properties of Iron." Journal ofApplied Physics, Vol. 34, No. 2, pp. 364-371, February 1963.

5. Barker, L. M., and R. E. Hollenbach, "Shock Study of the cc *> c Phase Transition inIron." Journal of Applied Physics, Vol. 45, No. 11, pp. 4872-4887, November 1974.

6. Barker, L. M., "cc - Phase Hugoniot of Iron." Journal of Applied Physics, Vol. 46,No. 6, pp. 2544-2547, June 1975.

7. Fowles, G. R., "Shock Wave Compression of Hardened and Annealed 2024 Aluminum."Journal of Applied Physics, Vol. 32, No. 8, pp. 1475-1487, August 1961.

8. Lundergan, C. D. and Walter Herrman, "Equation of State of 6061-T6 Aluminum atLow Pressures." Journal of Applied Physics, Vol. 34, No. 7, pp.2046-2052, July1963.

9. Johnson, J. N. and L. M. Barker, "Dislocation Dynamics and Steady Plastic WaveProfiles in 6061-T6 Aluminum." Journal of Applied Physics, Vol. 40, No. 11, pp.4321-4334, October 1969.

10. Wasley, R. J. and F. E. Walker, "Dynamic Compressive Behavior of a Strain - RateSensitive, Polycrystalline, Organic Solid." Journal of Applied Physics, Vol. 40, No. 6,pp. 2639-2648, May 1969.

11. Wasley, Richard J., Stress Wave Propagation in Solids. an Introduction, MarcelDekker, Inc. New York, 1973.

12. Duvall, George E., "Some Properties and Applications of Shock Waves." Responseof Metals to High Velocity Deformation., Proceedings of a Technical Conference,July 11-12, 1960, Editors P. G. Shewmon and V. F. Zackay, Interscience Publishers, Inc,.pp. 165-203, New York, 1961.

17

13. Minshall, Stanley F, "The Dynamic Response of Iron and Iron Alloys to Shock Waves."Response of Metals to High Velocity Deformation, Proceedings of a TechnicalConference, July 11-12, 1960, Editors P. G. Shewmon and V. F. Zackay, IntersciencePublishers, Inc., New York, pp. 249-274, 1961.

14. McQueen, R. G., Marsh, S. P., Taylor, J. W., Fritz, J. N., and Carter, W. J., "The Equation ofState of Solids From Shock Wave Studies." Hypervelocity Impact Phenomena, edited byRay Kinslow, Academic Press, New York, pp. 293-417, 1970.

15. Moore, Walter J., Physical Chemistry, Prentice - Hall Inc., Englewood Cliffs, N. J., 1962.

16. Daniels, Farrington, and Robert A. Alberty, Physical Chemistry, John Wiley andSons, Inc., New York, 1966.

17. Fitzgerald, E. R., Particle Wave and Deformation of Crystalline Solids, IntersciencePublishers, Inc., New York, 1966.

18. Billingsley, J. P. and C. L. Adams, Applications of Crystal Lattice DisintergrationCriteria to Compute Minimum Shock Induced Reactive Conditions in Solid ExplosiveMaterials, RD-SS-10, Aeroballistics Analysis Branch, Systems Simulation andDevelopment Directorate, Research, Development, and Engineering Center, U. S.Army Missile Command, Redstone Arsenal, AL, March 1989.

19. Taylor, J. W., "Thunder in the Mountains.", Shock Waves in Condensed Matter 1983,Elsevier Publishing Co., Inc., New York, 1984.

20. Taylor, J. W., "Dislocation Dynamics and Dynamic Yielding." Journal of AppliedPhysics, Vol. 36, No. 10, pp. 3146-3150, October 1965.

21. Bell, J. F., The Physics of Large Deformation of Crystalline Solids, Springer-Verlag,New York, Inc., 1968.

18

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