CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
2002 American Institute of Physics 0-7354-0068-7
THE HEL UPPER LIMIT
J.P. Billingsley
U.S. ArmyAMCOM, AMSAM-RD-SS-AA,
Redstone Arsenal, Alabama 35898, U.S.A
ABSTRACT. A threshold particle velocity, Vf, derived by Professor E.R. Fitzgerald for the onset
of atomic lattice Disintegration Phenomena (LDP) is shown to exceed and/or compare rather well with
the maximum experimental Hugoniot Elastic Limit (HEL) particle (mass) velocities (UPHEL) for
selected hard strong mineral/ceramic materials.
INTRODUCTION
velocity. (2) The wavelength (X) of a propagating
disturbance along a row of particles CANNOT be
less then 2di where di is the repetitive spacing
between particles, because there is nothing to
vibrate in the space between the particles [8, 9,
10].
This work can be considered as an extension
to the results reported in References [1] and [2]
which delineated the relevance of the Debroglie
Monument-Wavelength Equation to the HEL
decay phenomenon. Earlier, Reference [3] in
1989, Reference [4] in 1990, Reference [5] in
1994, and Reference [6] in 1995 had all suggested
a tie in between the Debroglie relation and HEL
phenomena. In particular, Reference [5] contains
comparative information that provided the genesis
and some of the data for Reference [7] in 2001
and the present document. In [5], it was suggested
that Vf could be an upper limit for HEL particle
velocities, UPHEL-
When A, = 2db then V{ = h/(2mdO = the
largest possible particle velocity without violating
Rule (2). The importance and influence of this
conflict cannot be over emphasized. It led Dr.
E.R. Fitzgerald [8, 9] to derive a relation for the
maximum particle velocity, Vf, that the lattice
could sustain before chaotic conditions (LDP)
occur. This relation is:
= V,
Cs is an elastic wave related velocity that is a
measure of the lattice cohesive energy. It is given
ANALYSIS
by
There are two important particle (mass)
velocities (V{ and Vf) connected with the
Debroglie-Fitzgerald Particle Momentum Wave
(PMW) theory of stress-strain wave motion. They
are associated with atomic lattice instability and
final breakdown (LDP) caused by a conflict
between TWO physical laws that are: (1) the
Debroglie momentum (mV)-wavelength (A)
relation that is X = h/(mV), where h = Planck's
constant, m = particle mass, and V = particle
* Ct
Cs where CL = the longitudinal elastic wave velocity
and Q = transverse elastic shear wave velocity.
APPLICATION
So Vf, for materials known to have very large
experimental UpHEL magnitudes, was computed
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and compared to these UpHEL data. These hard
strong mineral/ceramic materials were: diamond,
single crystal and fused quartz, aluminum oxide
(sapphire and alumina), silicon carbide, boron
carbide, and three different types of Partially
Stabilized Zirconia (PSZ). These comparisons are
graphically summarized in Figures 1 and 2.
Reference [7] contains more detailed comparative
information and lists the sources of the
experimental data.
Letter, Vol. 15, No. 11, November 1989, Pages
878-880.
4
Billingsley, J.P. and Oliver, J.M., "The Relevance
of the Debroglie Relation to the Hugoniot Elastic
Limit (HEL) of Shock Loaded Solid Material,"
U.S. Army AMCOM, Technical Report RD-SS90-4, March 1990.
5
Billingsley, J.P., "Possible V! and Vf Effects in
Shock Loaded Materials from Polymers to
Diamond," U.S. Army AMCOM, Technical
Report RD-SS-94-4, February 1994.
Cherepanov, G.P., Balankin, A.S., and Ivanova,
V.S., "Fracture Mechanics - A Review,"
Engineering Fracture Mechanics, Vol. 51, No. 6,
August 1995, Pages 997-1033.
DISCUSSION
Figure 1 shows the comparisons when Vf is
greater than UpHEL for eight examples. Figure 2
depicts the comparisons for three examples where
Vf is exceeded by the experimental UpHEL
magnitudes.
7.
Fitzgerald, E.R., Particle Waves and Deformation
of Crystalline Solids, Interscience Publishers, a
Division of John Wiley and Sons, Inc., New York,
1966.
When these cases, in Figure 2, are carefully
considered, along with those where Vf is greater
than UPHEL, it is apparent that Vf is a good
measure of these maximum UpHEL values [7].
Thus even in these most extreme cases,
judiciously computed Vf values provide some
rationale for the anomalously large UpHEL
magnitudes. Consequently, Vf could, at least, be
employed to estimate an upper bound on UpH£L
for untested materials. Finally, it is worth noting
that "Vf effects," or LDP, may be responsible for
"failure waves" experimentally observed [11] in
glasses and ceramics.
9
Billingsley, J.P., "The Hugoniot Elastic Decay
Limit," Proceedings of the Shock Compression of
Condensed Matter - 1997 Conference, AIP
Conference Proceedings 429, New York, 1998,
Pages 199-202, Ed. by Schmidt, Dandekar, and
Forbes.
2.
Billingsley, J.P., "The Decay Limit of the
Hugoniot Elastic Limit," International Journal of
Impact Engineering, Vol. 21, No. 4, April 1998,
Pages 267-281.
3.
Balankin, A.S., "Synergetics and the Mechanics of
a Deformable Body," Soviet Technical Physics
-
Fitzgerald, E.R., "Particle Waves and Phonon
Fission in Crystals," Physics Letters, Vol. 10, No.
1, May 15, 1964, Pages 42-43.
10. Girifalco, L. A., Statistical Mechanics of Solids,
Oxford University Press, New York, 2000, Pages
98-99.
jj
REFERENCES
1.
Billingsley, J.P., "The Upper limit of the Hugoniot
Elastic Limit," U.S. Army AMCOM, Special
Report RD-SS-01-01, May 2001.
736
Brar, N. S., "Failure Waves in Glass and Ceramics
Under Shock Compression," Proceedings of the
Shock Compression Condensed Matter - 1999
Conference, AIP Conference Proceedings 505,
2000, Pages 601-606, Ed. by Furnish, Chabildas,
and Hixon.
v
KM/SEC
f
1.10
I
DIAMOND
"PHB.
|
0.90
Q8H
BORON
CARBIDE
U
I?
PHEL
J
I
I
0.500
0.574
SILICON CARBIDE
P0 » 3.177 G/CC
I
UPHEL
___„,.,
|
-'og^,
0.542
QUARTZ
(X-CUT)
I
UPHEL
I
0.500
0.55
SAPPHIRE
(Z-CUT)
f
,T^PHEL
i I
|
0.45
0.54
ALUMINA
= 3.92 G/CC
UPHEL
|
a
I
6".4>3
0.459
PSZ.9.6MOL
%Y2O3,<100>
V,
"PHB.
I
|
0.313
0.435
PSZ.3.0MOL
%Y 2 0 3
I
u
Pna
6.362
Vf
UpHEL
KM/SEC
FIGURE 1. Vf and UPHEL Comparison Where Vf > UPHEL
737
Up HEL
FUZED
QUARTZ
vf
U
KM/SEC
Vfi
0.383
Vf,
0.746
1
1
"HEL
.
1
0.82
SUPRA - PIEZO VITREOUS
TRANSITION
0.57
QUARTZ
(Z-CUT)
1
"PHEL
1
0.700
0.436
PSZ, 9.6 MOL
%Y203,<110>
1
"PHEL
1
0.516
Vf
UpHEL
KM/SEC
FIGURE 2. Vf and UPHEL Comparison Where Vf < UPHEL
738