PHYSICAL SCIENCES
118
Pene'IQ.ion of an Initially Radial Shock Waft
Through an Aluminum-Glass Interfacel
B. A. SOBEK, JR. aDd F. O. TODD, Beputment 01
Oklahoma State UDlvenlt7, Stillwater
.1IJaI.,a
IN'J'BODUCTlON AND STATEMENT OF THE PRoBLEM
ThIs paper presents an analytical study and computer solution for the
propagation of an tnttlaUy radial shock wave in aluminum through a
plane interface into glass. The problem originates from the investigation
ot micrometeoroid impacts on photomultiplier tubes flown on space vehlclea.
Mtcrometeoroids are arbitrarily defined as particles with a mass of leu
than 10·· grams. Theoretical considerations indicate that these parUcles
may impact on the space vehicle with velocities In the range from 30,000
to 240,000 teet per second (Colllns, 1960). There Is a lower limit on the
mass of these particles of about 10 0M grams because of the Poytlng-Robertson effect (Beard, 1961). ThIs effect predicts that the radiation from the
sun will push very small particles outside the orbit ot the earth. A typical
micrometeoroid may be assumed to have a mass ot 10·t grams and a velocity of 118,000 feet per second. According to Whipple, some mlcrometeorolds
with a relative density of 0.05 are presumably aggregates of material from
the tails of comets. The densities ot micrometeoroids appear to vary
rather continuously from this low density up to that of iron-nickel With
a relative density of 8.0. A stone meteoroid would have a density of
about 4. The following calculation Is not applicable for the very low
density mlcrometeoroids.
Experiments have been reported in which simulated micrometeoroide
are made to impinge at different velocities on a target. The penetration
of high velocity projecWes produces craters which are very nearly hemispherical in shape and which show no apparent evidence of shear. Interpretation of avallable data suggests basic hydrodynamic theory as the
first approximation to represent the cratering mechanlsm (Todd, 1960).
It is useful because it permits an analytical study to be made and the
results to be compared with experiment.
A typical crater from the impact of a glass sphere on an alumlnum
plate Is shown in crou-section in Fig. 1. The initial velocity ot the glau
sphere was 28,500 feet per second. This sUde was prepared in order to
show the relatively unitonn thickness of the plastically deformed crystat.
around the crater. The surface of the cavity under the projectile material
Is quite smooth. This Is evidence of plastlc flow of the target materlaI
during impact (Summel'8, 1959).
The basic assumption for the present calculation follows trom IUCh
experimental evidence. The impacting micrometeoroid la auumed to iDi·
tlate a rad1al shock wave in the aluminum which propagatea away from
the point ot impact as a hydrodynamic abock wave in a non-vl8coua Ould.
The nature ot the phyalcal processes during the lnlUal phase8 of impact
is not well understood: and consequenUy, the true condiUona at the time
ot contact between mlcrometeoroid and target cannot be specified for an
accurate analytical solution. The proposed problem 18 IIOlved, however,
with approximate initial condttiona in order to obtain information that
will asa1st in evaluating the relaUve importance of other poulble mechanlmna and to provide a buI8 tor a more exact solution when tbe true
174
PROC. OF THE OKLA. ACAD. OF SCI. FOR 1962
Fig. 1.
Crater Fonned by a Glass Missile Impacting at 28,000 Feet Per
Second.
deUneatlng conditions are known. As the shock moves from aluminum
into glass, the prime interest for this calculation is the reflection and
penetration of this wave at a plane interface. The interface moves during
the impact, which differentiates the problem from the common seismic
problem of reflection at an interface between two different media.
HYDBooYNAMIC EQUATIONS AND CONDITIONS
The preasure drop across a shock front is very much greater than the
elastic Umtt8 of either aluminum or glass. Under these conditions, Bjork
(1958) and others have assumed that the shock front is propagated as a
hydrodynamic wave through a non-viscous fluid which may be described
by the equations of fluid flow. These differential equations may be expressed in Eulerian or Lagrangean form. The Eulerian equations are
written with respect to fixed space coordinates. The Lagrangean equatlOM describe the motion in tenns ot the paths of motion of individual
particles. In problema with two spatial coordinates and particularly for
a multilayer target, the Eulerian form of the equations is perferred in
order to 8lmpllty the calculations on a digital computer. The hydrodynamic dltterentlal equations. in Eulerian coordinates, are presented in Fig.
2. The ortatn of each equation is indicated.
_the
(1H2) has ahowD that the propagation of a shock wave through
any medium can be aolved provided the equation of state is known for each
dlatlnct material CODJIldered. In addition to the application of these equationa. Hugonlot showed that the entropy must iDcreaae &Cross the shock
front. TheIle two equations are presented 1D FIg. a. TIle formal repre-
PHYSICAL SCIENCES
1Ta
Conservation of Mass
-M...
at
D
~
-
rot
M.e!!l.
roe
_
(1)
Conservation of Momentum
_
_
o(reuu)
ror
~
or
~
roO
(2)
o(reuw)
ror
(3)
Conservation of Energy
-~
ror
~
roe
o(rpuy.)
~
roe
r(~r
P density
u
radial velocity
p
pressure
w
tangential velocity
internal energy
t
time
.E
r
ra~ial
coordinate
e
allgulilr coordinate
Fig. 2. Hydrodynamic Flow equations in Polar Coordinate..
8entaUon is the same for either aluminum or gJ&u. The Jut relation in
this figure may be derived from the conservation equatiOD8 and Sa the
Rankine-Hugonlot energy equation. Th18 expreulon relata the lDtemal
energy, pressure, and specific volume at the crest of a .bock front to the
corresponding values tor the material which" yet unct1Aurbed. The Ihock
pressure-volume relationship of a given material .. unique and .. called
the Hugoniot equation ot state. A comprehenllve equation of .tate Sa not
Immediately avallable tor either material.
The Mle-GrQne18en (GrOne1sen, 1928) equation of state .. ODe of the
most general relations possible. It relate. prasure, volume and eDefJY at
points in P, V, and B space to the value. of p., V. &lid E. OIl the HugcmIot
curve. For both materials, experimental lDtormation .. avatJabJe tor the
low, less than one megabar, preuure region (Wackerle, 1H2; WaJU, 19A.)
Theoretical predlctfons, available from the \&Ie of the Tbomu-Ferm1 lItaU8t1cal method, claim vaUdlty wIleD preMUl'M exceed twenty .mep.ben.
176
PROC. OF THE OKLA. ACAD. OF SCI. FOR 1962
Iquatloll of State In Cruaeu.1l Yom
p-~ =yp (
E-EJ
(5)
CoDdltlon aero•• Shock Front
S(entropy)~ 0
(6)
IaDkllle-Bul01l1ot Relation (Derived)
(Eli ~)= -i(~- ~)('{_\O
(7)
'tV and I are pre.sure, volume, and internal energy
y
sublcrlpt 0 i. in front of Ihock
lublcript H i. peak, or .ini~ value. in the
shock front.
il the Crunei••n ratio and changes .lowly with
the denltty, only.
Fig. 3.
Additional Equations for Solution of Shock Propagation.
Adequate representation between these two extremes is made possible by
interpolation. Equation of state curves for aluminum and glass appear
in Figs. 4 and G. The heavy Une in each figure is the Hugoniot curve along
which the material is compressed by a shock wave. The lighter curves
are typical adiabata along which the material expands after it has been
compreued by the ahock front.
NUMERICAL SOLUTION OF FLow EQUATIONS
To obtain a numerical solution on a digital computer, von Neumann
and Richtmyer (19GO) suggest that the dltterentlal equations be modified
by the introduction of a pseudo-vIscosity term. As a consequence, the
shock front changes from a true discontinuity to a narrow zone of large
aradtenta. The explicit form of the dlaalpative term in this problem is
baaed on a suggestion by Landshoff (19M). This term is taken as q, and
is given by Equation 8 in Fig. 6. This pseUdo-viscosity term is added to
the pressure. 'P, in the preceding ditferentlal equations e¥J)ressing conse.....
vation of momentum and energy. TOe modlfted equations also appear in
Ftc. 6.
For conveDlence in acallng numerical magnitudes for a computer calculation and to pruerve computer storage, the relevant equations are
converted to a nondimenalonal form. A dimensionless solution also permits aeaunc to problema involving different impacting velocities and
dttfereDt m- of particles. The resulting equatiODB are then transformed
to f1nlte difference equations by methods almllar to tho8e used by Longley
(1868) tor cylindrical coordinates and prepared for solution by a digital
comPuter.
PHYSICAL SCIENCES
177
r
~
"-
~
~
~ ~
~
~
~
~~
y.
~~
I
~~ .
,~,.~
I
1'~c~~
-1 'q:
!~~~
~!U ~
1/
1/ JI
~~~~
~~~~
~ "
~~,:"
, ~ ~~~,
~
\1
II
t()'~
// ~
1/ JV ~
/ 1/ /A'l
V
V
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~~
I"-
/
:::.-
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---
.-"i
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--
VV
rII
~~ /
7/
/'
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~~
./
1/ . / V
'/
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./
,,/
~ ....
...... ....-: V
\t
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~7t11fi1b'~W' ,.,nS;S-i"~
BoUNDARY CONDmON8 AND SOLUTlON8
The finite difference equations yield a solution by the u.ual iterative
method employed with digital computers. To apply thue equaUOn8. conditions at time, t, equal zero must be completelyapeclfted. The auumption
that a micrometeoroid produces a radlal shock 1Illlustrated in J'lg. 7. Tbe
lower part of the figure shows the init1al pressure diatribuUon and the
modJ.tted profile produced by the q term which permits a numerical metbocS
of solution. This pressure 18 14.1 megabanr, or a UtUe over 1. miWoD
atmospheres, developed by the typical m1crometeorold of 10-' gl'lUU fm·
pactlng with an 1n1tial velocity of 118,000 feet per 1eCOnd. TIle radb... of
the shock front 11 13 per cent greater than the radiU8 of the micrometeoroid (Todd, 1961).
At later times, the shock wave propagates in alumtnwn and decreueI
in amplitude 88 the energy of the abock .. 8pread over greater .,.."
After tile shock radius bu become about one-tltth greater tbaD ft. iDWaI
value, the shock front encounters the alumlnum-gJau interface. '!'be
pressure and the partlcle veloclty are contlDuous &Cro.I tile materta1
PROO. OF THE OKLA. ACAD. OF SCI. FOR 1962
178
A'
...:
----
II-
- - -
~
-
I(i
~
~
-
~
-
~ t-
N
~ 0
t-.~
~
Q~~Qcri
~ Z
a:
9
(!)Cf)f&i~~
~
I
I
I
~j::!-~
LlJti W
(!)
~
t-
Z
~~-b~
(/)UJZZV
~
:::::)
~~~OO
a:::I:i15~Z
Ut<t
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~
"'.
o_c ~~
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~
0r-O- 0 -
.o~cI V
vV
"'- ~ /'"
./
./
../
.-""
to-:
I
I ~i
-
.«1
[let: r, I
,.. p
l~
r""..............
~
V
V
,,/
.-.!!I
~ ~, ""'/
Q)
""
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.=
>
:IE
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t= ~
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a:
III
i
3 to)e
..J
l&J
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-;J'
"- ::s
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/ }
:::I ...
;;
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....0
<
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ad
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boundary; but the denalty and the intenal energy are discontinuous. The
Introduction of material of a different density destroys the radial symmetry of the flow. Fig. 8 lUustrates density-radial distance curves at
three different times. The ratio of the denslty to the nonna} aluminum
denalty of 2.'laG gr&mII per cubic centimeter 18 indicated on the vertical
axis. For example, at the start of the calculation, the density of the
shocked alumlnum is ".32 grams per cubic centimeter which 18 approximately two and one-balf times nonnal aluminum density. The top third
of the ftIure shows the lnltial profile and a protUe after "8 iteration cycles.
The shock wave has not yet encountered the interface and the protlle i8
identical along any radius. In the center graph, the leading point of the
shock wave on the 90 0 line hal reached the Interface. The shock front i8
penetrating the interface and the building of a shock wave in the glass
can be noted. A protUe alODg the 6 0 line Is al80 shown. Here, the disturb&Dee Is entlreJy In alumlDum. The bottom portion of the figure again
allow protllee on the 6 0 . and too radial linea after 96 cycles. The strength
of the 8hock in aJaas baa lDcreued while the strength of the shock in
PHYSIC AL SCIENC ES
Dissipa tive term
I'll
,~hil:h is signifi.c ant only fol' laue "eloc1ty lradhn u
(8)
when
Vis
the materin l velocity vector, A1 and A2 are conetant e detor.i.n ed
by trial and error.
Conserv ation of Momentum
~
_
~r
(} (pw)
~
~_ ~(Pwu)
ror
r~e
'" _ £ll!+gl
(9)
(10)
r~e
Conserv ation of Energy
q[r(p+g)u}
o(PE) •
ref
of (P+q)w)
roe
(11)
Fig. 6.
o(rPuE)
r~r
o(PwE)
rae
Introdu ction of Viscous Dampin g Term for Integrat ion Acrou
Shock.
ately behincS
aluminu m has decrease d. The density of the materia l lmmed.l
probablU ty
the
s
indicate
Thl8
line.
5°
the
on
than
lower
is
e
the interfac
m u a
aluminu
the
into
back
ted
propaga
being
wave
ion
of a rarefact
outmoved
bu
e
interfac
The
e.
interfac
the
at
n
reflectio
shock
result of
dlvl~
the
of
u
poalt1o
new
and
old
The
passes.
front
shock
ward as the
Une between aluminu m and glau have been marked in the figure.
of the deJWty.
The behavio r of the internal energy 1.1 limilar to that
materia l bounda ry
The velocity and pressure are continuo us acrou the cbaract
er u u.o.e
and profiles ot these variabl u have the I&me general propea on later
of the density front in aluminu m. calculaU oll8 are in
phases of this problem .
a mscroThe assume d radial .hock front reaulUn g from the impact ofthe
alumithrou&'b
ted
propap
been
baa
target
er
multilay
a
on
id
meteoro
e. The
intertac
um-glau
alwn1n
the
on
impact
to
made
and
layer
num
lJI the I'~ u4
inciden t shock tront produce s a tl'lUllJlli tted 8JIock wave
m.
aJumtDu
the
into
back
g
travelin
wave
ion
rarefact
d
a reflecte
PROC. OF THE OKLA. ACAD. OF SCI. FOR 1962
180
r-Compressed Zone
of Target
Aluminum
-------------A.
Assumed
Material
Glass
Initial Boundary Condition
.
•..
I
d:•
:I
Radial Direction
R-U3r
B. Assumed Step Shock Wave
Ftc. '1.
AuumpUQIUI for Impact of a Sphere
PHYSICAL SCIENCES
181
cYCLE-o----'
I
I
I
I
CYCLE 4&
I
--A-tO Alumlnwn L
- - A •.79 Glass
>-
_
-Aluminum
CYCLE 78
to-
(/)
~
- - p. -
1.0 Aluminum
- -P. • .79 GIG••
___J\
2.4
,.-8· 5°
,
\
2.0
\
\
\,
- -p. .1.0 Aluminum
- - A • .79 GIG..
OLD
RADIAL DISTANCE
Fig. 8.
I.
'---
HNE'-W-.g08
PropagaUon of Density J'ront.
~=.:::-_-
182
PROC. OF THE OKLA. ACAD. OF SCI. FOR 1962
Ul'F.&\TVU CITED
Beard, D. 1961 (Jan.).
SCR-262.
loUd interplanetary matter. Sandia Corporation,
Sethe, H. A. 1942. The theory of shock waves for an arbitrary equation
.tate. OSRD No. M6 (Division B of National Defense Research
Committee) •
of
Bjork, R. L. 1968. Effects of a meteoroid impact on steel and aluminum
in 8P&CC!. Engineering Dlv18Ion, Rand Corporation, P-1662.
CoUIn8, R. D., Jr. and W. Kinard. 1960 (May). NASA TN 0.230.
GrOnei8en, E. 1926. ZUBtand der Festen Karpers, in "Handbuch der
Phyatk", vol. 10, Julius Springer, BerliD. p. 1·52 (Translation by S.
Re1u, NASA republication RE 2-18-59w, Feb., 1959).
Landshotf, R. 1956 (Jan.). A numerical method for treating fluid flow
In the presence of shocks. Los Alamos Scienti!ic Laboratory, LA·
1930.
Longley, H. J. 1959 (Apr.). Methods of differencing in Eulerian hydrodynamics. Los Alamos Scientific Laboratory, LAMS·2379.
Smnmers, J. L.
1959 (Oct.). NASA TN 0.94.
Todd, F. C., H. R. Lake and B. A. Sodek, Jr. 1960 (Dec.). Progress
report No.1, contract NASr-7, Oklahoma State University Research
Foundation.
Todd, F. C., Shl-yu Wu, H. R. Lake and B. A. Sodek, Jr. 1961 (Mar.).
Progreu report no. 2, Contract NASr·7, Oklahoma state University
Research Foundation.
von Neumann, J., and R. D. Rlchtmyer. 1950. A method for the numer·
lcal calculations of hydrodynamlcal shocks. J. Appl. Phys. 21: 232-237.
Wackerle, J, 1962.
83: 922-87.
Shock wave compression of quartz.
J. Appl. Phys.
Walsh, J. M., N. H. Rice and R. G. McQueen. 1958. CompressloD of solids
by strong shock waves, Solid State Physics, Advances in Research
and Appllcations. 6: 1-68, Academic Press, New York, N.Y.