Elastic Waves in Truncated Cones Experimental and theoretical investigation is undertaken by authors to study the propagation of waves produced by impact of projectiles on truncated cones V. H. Kenner and W. Goldsmith Nomenclature
ABSTRACT--An e x p e r i m e n t a l i n v e s t i g a t i o n of elastic w a v e s p r o d u c e d b y t h e axial collision of s t r i k e r s w i t h t r u n c a t e d 2024 a l u m i n u m cones w i t h a p e x angles of 0.48, 5.38, 20, a n d 30 deg was p e r f o r m e d . W a v e p r o p a g a t i o n was i n i t i a t e d a t t h e s m a l l e n d of all f o u r cones a n d a t t h e large e n d of t h e 0.48-deg a n d 5.38-deg cones. T h e s t r i k e r c o n s i s t e d of a ~/2-in.-diam steel ball or a soft p h e n o l - i m p r e g n a t e d fiber cylinder. I n m o s t cases, i m p a c t was caused b y firing t h e s t r i k e r f r o m a n air g u n a t a p p r o x i m a t e l y 1300 ips; in a n a d d i t i o n a l series of tests, a steel ball was d r o p p e d o n t h e cone. T h e m e t a m o r p h o s i s of t h e pulse a t t h e surface of t h e t a r g e t was r e c o r d e d using b o t h foil a n d s e m i c o n d u c t o r r e s i s t a n c e s t r a i n gages. D a t a were o b t a i n e d for p e r i o d s r a n g i n g f r o m 200 to 500 #sec; t h i s p e r m i t t e d t h e o b s e r v a t i o n o f several reflections f r o m t h e e n d s of t h e specimen. I n s e v e r a l instances, cylindrical a l u m i n u m r o d s were glued to t h e c o n e t o f o r m a c o m posite t a r g e t ; t h i s p e r m i t t e d o b s e r v a t i o n of t h e initial pulse i n c i d e n t on t h e conical s e c t i o n b o t h f r o m surface s t r a i n gage a n d s a n d w i c h e d c r y s t a l records. S t u d i e s were also c o n d u c t e d t o a s c e r t a i n t h e s t r e s s d i s t r i b u t i o n across t h e base of t h e 20-deg cone. I n i t i a l pulse r e c o r d s were e m p l o y e d t o p r e d i c t t h e surface r e s p o n s e in t h e t a r g e t u s i n g t h e o n e - d i m e n s i o n a l e q u a t i o n of elastic w a v e p r o p a g a t i o n in a cone of infinite length. Reasonable agreement between the data and the r e s u l t s of calculations b a s e d o n t h e a n a l y s i s was o b t a i n e d .
A c co c~ E f,F g r R t u v0 vy
= = = = = = = = = = = = = = e = h = = p = =
area of cross section phase velocity rod w a v e velocity s h e a r w a v e velocity Young's modulus function acceleration of g r a v i t y d i s t a n c e f r o m cone a p e x r a d i u s of cross s e c t i o n time displacement initial v e l o c i t y of s t r i k e r final v e l o c i t y of s t r i k e r t o t a l c o n e angle strain wavelength Poisson's ratio density stress
Introduction An experimental and theoretical investigation has been undertaken to study the propagation of waves produced by impact of projectiles on truncated cones. The various phases of this program consist of the surface measurement of strain and pressure in elastic materials, internal strain-gage measurement in cones exhibiting viscoelastic behavior, and photoelastic analysis of transients in pyramids.
V. H. Kenner is Research Assistant and W. Goldsmith is Professor of Applied Mechanics, Department of Mechanical Engineering, University of CaliforniaSBerkeley, Calif. Paper was presented at 1967 S E S A Annual Meeting held in Chicago, Ill., on October 31-November 3.
3T
o.s I-- cfl-cO~ . 1 ~0 n '
;"'K'\3~ .,i
0
TRANSVERSE
~"
g 1.5
-
,(a) Run 2A (light lines) ~(b) Run 5A (heavy lines)
~
,
.-~ ~ I.o -
AXIAL
0 -0.5 ~-I.( z w I--
442 I October 1968
I
\"\
=~
9 '~'~
/I !\
5~l--
/
3
~lI
(b)
YI/ ,, ,if!// Ill~ "', i/,iJJ
__l~2x I/; DIA. AL. PROTECTOR
Vo=,577 in.lsec.
\
I
- 11
l~
13
i_~ ,.
(;~ ~-0 ~--~,'---+~,----.~-. in./sec.
-I.! '2.(
II
GAGE
Z Q:: I-U')
[~6'~-9"
(b)
0.5
,~
i
0 - ~ ~ - 0 -t-t-'--* 3T--- ~ 2T---~ 1 ~ vf=328.2 l l I I in./sec.
,
~x.X ~'1
~ l / 2 x 3/4 DIA. AL. PROTECTOR Vo=1569 ~ _ in./sec ~ _3 _2 .I
u3
,. i'F~
.,~.
GAGE
i
~ -0.5 2.O
Eig..1--Typical strain histories f r o m -thec0.48-deg c o n e
.\ "k
~io) ~ j
0
~(0)" 50
I moo 150 t , TIME ,
200 ~sec
(~ I 250
300
0.2
~,%~ j -
T,A,,,SVERSE,OAOy J
0
~
-o.2
c~
0.~
"5
%o
Fig. 2--Strain history from the 5.38-deg c o n e - - R u n 2B
0
Z F.-co
-0.5 STEEL BALL'7 ,, ,. / , - 3 / 4 x I~'4 DIA. AL PROJECTOR
oo
z
Vo=,31o
//
,,,
i-
-I.O 12"-
l
-i.5 0
20
40 t
60 , TIME
80
I00
t20
#sec
T h e p r e s e n t p a p e r is c o n c e r n e d w i t h t h e first o f t h e s e t o p i c s . T h e d a t a o b t a i n e d c o n s i s t of s t r a i n h i s t o r i e s f r o m v a r i o u s p o s i t i o n s o n t h e l a t e r a l surface of t h e s p e c i m e n s a n d p r e s s u r e r e c o r d s f r o m p i e z o e l e c t r i c c r y s t a l s m o u n t e d on t h e e n d s o f t h e cones. A l t h o u g h a c o n s i d e r a b l e b o d y of l i t e r a t u r e h a s been published with regard to the propagation of e l a s t i c w a v e s in c i r c u l a r b a r s , o n l y a few c o m p a r a b l e s t u d i e s h a v e b e e n r e p o r t e d for cones. L a n d o n a n d Q u i n n e y 1 e x a m i n e d t h e w a v e p r o c e s s in c y l i n d r o conical steel sections of small apex angle and comp a r e d e x p e r i m e n t a l r e s u l t s to t h e p r e d i c t i o n s o f a o n e - d i m e n s i o n a l t h e o r y a p p r o x i m a t e l y v a l i d for small cone angles and large wavelengths. Donnell 2 d e v e l o p e d a s i m i l a r a n a l y s i s a n d also c a l c u l a t e d t h e wave transmission by approximating the conical s e c t i o n b y a series o f i n c r e m e n t a l c y l i n d e r s . F a v r e 3 derived various relations pertaining to reflection and t r a n s m i s s i o n o f w a v e s in e l a s t i c c o n e s a t d i s c o n t i n u i ties, w h i l e R e e d 4 e n g a g e d in c a l c u l a t i o n s p e r t i n e n t t o t h e p r o g r e s s i o n o f w a v e s in b a r s w i t h g r a d u a l c h a n g e s in s e c t i o n . W i t h t h e e x c e p t i o n o f R e f . 1, that utilized the original Hopkinson bar method of m e a s u r e m e n t w h i c h is r e s t r i c t e d t o t h e o b s e r v a t i o n o f t h e i m p u l s e , no e x p e r i m e n t a l r e s u l t s o f a n y t y p e a r e a v a i l a b l e for t h i s p h e n o m e n o n . T h e c u r r e n t inv e s t i g a t i o n is i n t e n d e d t o p r o v i d e f u r t h e r i n f o r m a t i o n in t h i s d o m a i n . M e a s u r e m e n t s were p e r f o r m e d o n t r u n c a t e d
2024-T351 a l u m i n u m c o n e s w i t h a p e x a n g l e s o f 0.48, 5.38, 20 a n d 30 deg, r e s p e c t i v e l y . W a v e p r o p a g a t i o n w a s i n i t i a t e d a t t h e s m a l l e n d of all f o u r c o n e s a n d t h e l a r g e e n d of t h e 0.48 a n d 5 . 3 8 - d e g c o n e s b y t h e a x i a l i m p a c t o f a p r o j e c t i l e e i t h e r fired f r o m a p n e u m a t i c g u n or d r o p p e d o n a p r e c e d i n g c y l i n d r i c a l s e c t i o n ; in a few i n s t a n c e s , t h e c o n e was struck directly. Surface strains were measured by means of resistance and semiconductor strain gages oriented along the direction of the generators o f t h e cone. E x c e p t for t h e 3 0 - d e g cone, a p a i r o f s t r a i n gages in t h e t r a n s v e r s e d i r e c t i o n w a s also u t i l i z e d . T h e p r o p a g a t i o n v e l o c i t y in t h e cones w a s also d e r i v e d f r o m t h e s t r a i n - g a g e r e c o r d s . T h e p u l s e i n c i d e n t o n t h e f r o n t of t h e c o n i c a l sect i o n w a s s t u d i e d in t h r e e w a y s . (1) I n s o m e cases, s t r a i n s w e r e d e t e r m i n e d on t h e s u r f a c e o f a c y l i n d r i cal s e c t i o n w h i c h p r e c e d e d t h e cone. (2) I n o r d e r t o c o m p l e t e l y define t h e s t r a i n i n p u t g e n e r a t e d b y the impact and eliminate the interference that reflections from the lateral surface of the cone would p r o d u c e i n t h i s q u a n t i t y if m e a s u r e d o n t h e s y s t e m p r o p e r , a c y l i n d r i c a l 2024-T351 a l u m i n u m b a r o f t h e s a m e d i a m e t e r a s t h e t r u n c a t e d e n d o f t h e 20deg cone was struck in a manner identical to that of the actual test. (3) A p i e z o e l e c t r i c c r y s t a l w a s sandwiched between the apex end of the 20-deg cone and a cylindrical frontal section of the same d i a m e t e r . P i e z o e l e c t r i c c r y s t a l s w e r e also m o u n t e d o n t h e d i s t a l l a r g e e n d o f t h e 2 0 - d e g c o n e to p r o v i d e
Experimental Mechanics I 443
I.C
.E_
~"~!,A,STEELBALL
x{}l(2}(3){4){5)(6)Tr
i
in/sec . . . . " ~(%,t) ~ --~j ~ =320 x 106si~(~g~ )
9- O.~- 'O
~
.~ ,
NOTE:
4
~
~ = = = ~
LINE dENOTES ARRIVAL OF REFLECTIONS FROM FREE END AT THAT STATION II
-C
20 t,
40 TIME,
60
80
/~sec
Fig. 3--Solid lines: strain history from the 5.38 deg c o n e - Run 4B. Dashed lines: corresponding plot of eqs (7) and (ZO)
i n f o r m a t i o n on t h e s h a p e of t h e wave f r o n t a n d on t h e v e l o c i t y of t h e d i s t u r b a n c e in the cone.
Experimental Apparatus and Procedure S c h e m a t i c d i a g r a m s of the cones used in this s t u d y a n d t h e l o c a t i o n of s t r a i n gages are p r e s e n t e d in Figs. 1, 2, 5 a n d 7. T h e m a t e r i a l utilized exhibits t h e following m e c h a n i c a l properties: 5, 8 Ultimate strength Yield s t r e n g t h M o d u l u s of elasticity, * E W e i g h t d e n s i t y , pg Poisson's ratio, # R o d wave v e l o c i t y
68,000 psi 47,000 psi 10.6 X 106 psi 0. 100 l b / i n . 8 0.33 202 > 10 a ips
T h e four cones were used r e p e a t e d l y ; w i t h t h e exc e p t i o n of s m a l l - e n d s h o t s on t h e 5.38-deg cone, no plastic d e f o r m a t i o n in t h e conical sections was ob* Average o f tension a n d compression moduli; the compression modulus is about 2 percent greater than the tension modulus.
served. I n general, B a l d w i n - L i m a - H a m i l t o n t y p e FAP-12-12 foil gages were used. However, m o s t s t r a i n s m e a s u r e d in the 20 a n d 30-deg cones were so small t h a t t h e use of s e m i c o n d u c t o r gages (Baldw i n - L i m a - H a m i l t o n t y p e SPB2-12-12) was required; these t r a n s d u c e r s were c a l i b r a t e d d y n a m i cally in t h e s t r a i n range of i n t e r e s t to insure t h e v a l i d i t y of results.~ T h e l e n g t h of all gages was 1/s-in., insuring an a c c u r a t e r e c o r d of s t r a i n s for t h e observed pulse l e n g t h s of 6 in. or greater. T T h e gages were b o n d e d to t h e specimens using either E a s t m a n 910 or E p o x y 150 adhesive. No difference in t h e d y n a m i c response of t h e gages m o u n t e d w i t h e i t h e r t y p e of glue was f o u n d for t h e s t r a i n levels a n d frequencies of i n t e r e s t in t h e present investigation. A t each s t a t i o n , two s t r a i n gages were placed on opposite ends of a d i a m e t e r of a cross section to e l i m i n a t e a n y possible a n t i s y m m e t r i c c o m p o n e n t s of t h e pulse. T h e e s t i m a t e d a c c u r a c y of gage p l a c e m e n t was • 1/32 in. along a g e n e r a t o r of t h e cone a n d i l deg along t h e circumferential direction. W i t h t h e e x c e p t i o n of three r u n s in which a plastic s t r i k e r was e m p l o y e d , t h e t a r g e t s consisted of cylindrical a n d conical p a r t s j o i n e d along a c o m m o n axis b y m e a n s of wax or glue, the cylindrical sections being of t h e same m a t e r i a l a n d d i a m e t e r as t h e a d j a c e n t e n d of the cone. I n these cases, a short p r o t e c t o r piece never longer t h a n 3/4 in. was placed a t the i m p a c t end a n d replaced for each r e p e t i t i v e s h o t in order to p r e v e n t plastic deformation to t h e cone or preceding i n s t r u m e n t e d sections. Piezoelectric X - c u t q u a r t z c r y s t a l s which were 1/2-in. d i a m e t e r b y 0.014-in. t h i c k with g o l d - p l a t e d l a t e r a l surfaces were e m p l o y e d a t the apex end of t h e 20-deg cone. F o u r s i l v e r - p l a t e d X - c u t q u a r t z f Although sensitive to temperature and nonlinear over the strain range generally acceptable with conventional strain gages, these gages proved to be quite adequate when used on a l u m i n u m over a low strain range.
25 rL r I [ z'~-J"~ vo- 68 n/see ., ~-,~'~ 2"---z'---2"-~ I
2.0
2
1
,.o
!
I
:
i
,
q.O
0
20
40
60 80 t , TIME , gsec
IOO
120
Fig, 4--Strain history from the 5.38-deg c o n e - - R u n 9B
444 1 October 1968
140
0
20
40
60
80 IO0 t . TIME ,
120 /~sec
140
160
Fig. 5--Strain history from the 20-deg c o n e - - R u n 1C
180
2C~)
EXPERIMENTALRESULTS, I/2 DIA STEEL BALL DROPPED I0" --RESULTSUSINGCYLINDERAPPROXIMATION SHOWNWITHRECTANGULARINPUTPULSEOF 40/zSEC DURATION, t/2" END PIECE (A) SAMEAPPROXIMATIONAS ABOVE, 26.9"ENDPIECE(B) NOTE: SYMBOLSA,B,s,c DEFINEDIN DIAGRAM - -
F STRAINI GAGE (S)~, J1 '
200 AT "~
_L
CRYSTAL ~-~~ I
d 150
o 100
. . . . . . . .
'
i
r-
2
cc 50 03
',
,
L_
0
0
20
40
60
80 100 t , TIME,
120 H-see.
140
160
180
200
Fig. 6--Solid lines: results of ball-drop tests for two end configurations (crystal pressure records have been divided by E, 10.6 X 10 B psi)
Dashed lines: Corresponding analysis using rectangular pulse, approximation to cone geometry shown, and one-dimensional transmission-reflection laws 1400
deg cone in which quartz crystals were employed, the projectile was fired from a p n e u m a t i c gun and the specimen was ballistically suspended; the equipment and procedure for these cases have been previously delineated. B-H E x c e p t for three shots, the projectile consisted of a 1/2-in.-diam steel sphere of hardness Rc 67. I n striking the 5.38-deg cone at the small end, a I/2-in. diameter b y 3/4-in.long phenol-impregnated fiber slug of mass 1.72 • 10 -G lb_sec2/in., slightly r o u n d e d at the contact surface, was employed to p r e v e n t plastic deformation. To avoid breaking crystals, tests requiring their use were performed b y dropping a 1/2-in.-diam steel ball from a height of 10 in. along the axis of the target. T h e response of the strain-gage recording circuits was checked b y using a pulse generator to put a representative k n o w n signal into the system. N o discernible distortion was observed at frequencies one order of magnitude larger t h a n the p r i m a r y components exhibited b y the pulse.
Theoretical Considerations The one-dimensional equation of wave propagation in an elastic cone neglecting the effects of lateral inertia, shear, and warping and assuming a spherical: wave front with a one-dimensional constitutive equation of the form
-
,, ,,, ~ ~_ 1200
= EE = E
1000
bp
(1)
br
is given by '~ ~ 800
b2u
2 5u
5 r 2 4- r 6OO
i 400,
i
,i
of>
!
.
.
.
.
10 ~u =
(2)
Co '2 ~ t 2
Here, G and e are the radial stress and strain components, u is the radial displacement in direction r, ro is the distance from the apex to the cross section where the load is applied, and E is Y o u n g ' s modulus, respectively. The general solution of eq (2) is
% z
5r
.
i
u =
1
[ F ( r + cot) + f ( r
-
(3)
cot)]
r
or, for p r o p a g a t i o n only in the positive r-direction, -200 0
' 20
' 40
60 t , TIME,
80 #sec
100
120
u =
140
Fig, 7--Strain history from the 30-deg cone--Run 1D
crystals, 3/8-in. diameter b y 0.021-in. thick, were m o u n t e d on the base of this cone and were backed by 31/2-in.-long aluminum cylinders of the same diameter. T h e length of these pieces was so chosen t h a t the double transit time in the cylinder would be approximately equal to the duration of the pulse observed at the first station on the cone. T h e configuration for the crystal experiments is shown in Figs. 6 and 9. W i t h the exception of a series of tests on the 20-
1
[[(r -
(4)
cot)]
r
As m a y be noted from Fig. 8, the strain input to the cone at the apex end for the present experimental conditions of initial quiescence is closely approximated b y the following b o u n d a r y conditions at the i m p a c t end: 0 =
~o sin 2 I ( r~ 0
forro -- c o t > 0 l f o r O > ro -- cot >__
-- c . t )
A
~
-a
forro -- cot <
-- A
(5) where eo is the strain amplitude and A is the wave-
Experimental Mechanics
I 445
length of the impulse. A combination of eqs (1), (4) and (5) leads to the relation '3 d(r
- = roeo sin ~ re
cot)
--
~
(6)
whose solution in t u r n gives bu br -
e -
(1/ro) sin ~/~ + (2~/A) sin/~ cos/~] + (2~r~ro/A2)(1- e ~A/~~ 1 47r 2
f
roeo r2
ro~ +
]
A2
(7)
F(2~/Aro) sin/~ cos/~ + (2~r2/A ~) X (c~
roe~ I
-
-- s-ln~-~) 4~ ~ - (2r~/A~)e~/~
(r + A~and
forr/co
\
co
/
27r2roeo [e ~A/~,~ r~A2[(1/ro 2) + (4~2/A2)]
-e
(€176
] X
(r - re)
(8)
for r+A t > --, -
-
co
where ~ ~
r--cot - A
F o r a wave traveling in the negative r-direction, the time origin is chosen at the instant the transient reaches the apex, leading to a b o u n d a r y condition at the cone base given b y
0
=
Results Representative records of strain vs. time at various stations on the four conical bars are portrayed in Figs. 1 to 6. T w o typical strain histories from the 0.48-deg cone are presented in Fig. 1 depicting the wave transit resulting from i m p a c t at either end. A 4-in.-long cylindrical section placed ahead of the apex end of this cone has no significant influence on the wave pattern. The average propagation velocity was found to be 211 X 10 ~ ips; this value appeared to be independent of the direction of transit, the axial location of the measuring stations, or whether the initial strain-record deviation or the pulse peak was utilized for this determination. The average of the observed values of the ratio of circumferential to longitudinal peak strain was found to be - 0 . 3 4 • 2 percent. Three tests involving the 5.38-deg cone are shown in Figs. 2, 3 and 4, respectively. A comparison of Figs. 2 and 3 indicates the effect of the length of a
forro + cot < 0
] 0
where a is the cone angle, co a n d cs are the velocity of propagation of rod and shear waves, respectively, and # is Poisson's ratio, and a particular assumption concerning the shear distribution across the section is employed.14 The second and third terms on the right-hand side of eq (11) represent the contributions of lateral inertia and shear, respectively. A t t e m p t s to program eq (11) in conjunction with the b o u n d a r y condition represented b y eq (5) at one terminal and a free end at the other terminal of a finite cone have proved to be unsuccessful thus far.
for O < ro + c o t <
A
2.C
forro + c o t > A z 0
(9) and a corresponding strain history given by eq (7) if /~ is replaced by/~* - ~ [ (r + c o t ) / A ] ~ which holds for - r / c o < t < [(A -- r ) / c o ] a n d 2~2roeo [e~*A/. . . . e [(8"- ~)A]/,~o] A2r2(l/ro 2) + (47r2/A 2)
e =
(r -- re)
(10)
for t > (A -- r)/co. E q u a t i o n s (7), (8) and (10) are valid only for one-directional propagation, i.e., up to the instant when the effects of initial reflection from the distal end become manifest at a given station. C o m p u t a t i o n s involving these relations will be presented subsequently. The equation of m o t i o n corresponding to eq (2), but including the effects of lateral inertia and radial shear, is given b y ~t 2
0 (J
-
-
-
%
I.O
Z C~ F.-
0.5
C~ L b r 2 + r Dr J + #2r tan2 2 X
I b3u r b4u ] a F~3u ~rbt 2 + 2 br2~)t2j -- p,2cs2r t a n 2 ~ LSr 3
rb4u]
+ ~ ~r4J (11)
446 I October 1968
0
JO
20
30
t , TIME , /zsec. Fig. 8 - - R e s u l t s of p u l s e - s h a p e d e t e r m i n a t i o n tests
40
300 200
I00 ::;===s::~(bl m
"
(a)
"N
qoo d -2oo 0
20
40 60 t, TIME , #sec
80
I00
Fig. 9~Stres.s distribution across base of 20 deg cone 1400
0.00<
1200 M .
CC~o,~V_,~ =3.75•
I 10001
,l
x1 8 o " ~
I
(e)
J
50 ~ ALUMINUMCONE
600 1
~5(X? 80OI 40( -
~,~ d e
600
I 4ool
30C-
"
20(
z <
Q~
~-
IOC -
200
Discussion
0-0 g-t00 ~
T w o r u n s i n v o l v i n g t h e 20-deg cone are d e p i c t e d in Figs. 5 a n d 6, r e s p e c t i v e l y . T h e first of these r e p r e s e n t s t h e r e s u l t s o b t a i n e d using t h e s t a n d a r d ballistic procedure, while t h e second, p e r f o r m e d b y m e a n s of a b a l l - d r o p test, i n d i c a t e s t h e n a t u r e of t h e reflection process in t h e cone giving rise to a reflected compressive pulse at a s t a t i o n in t h e s t r a i g h t section of t h e t a r g e t . T h e a v e r a g e of t h e p r o p a g a t i o n velocities b a s e d on pulse p e a k a n d initial rise were 214 X 103 a n d 235 X 103 ips, r e s p e c t i v e l y , while t h e r a t i o of c i r c u m f e r e n t i a l t o l o n g i t u d i n a l s t r a i n was found to be - 0 . 3 4 . A t y p i c a l set of d a t a for t h e 30-deg cone is p r e s e n t e d in Fig. 7; corresponding values for t h e p r o p a g a t i o n v e l o c i t y based on p e a k a n d initial rise of t h e pulse were 212.5 X 103 a n d 232 X 103 ips. N o d a t a for c i r c u m f e r e n t i a l s t r a i n were o b t a i n e d for this t a r g e t . As m e n t i o n e d previously, t h e s t r a i n pulse gene r a t e d b y t h e t y p e of collision e m p l o y e d in t h e cone e x p e r i m e n t s was a s c e r t a i n e d f r o m t e s t s on a c y l i n d r i c a l bar, w i t h t h e results shown in Fig. 8 for 4 runs. I n a d d i t i o n , p o i n t s on t h e e m p i r i c a l curve e= 1.73 X 10-3 sin s ( ~ / 3 5 . t ) r e p r e s e n t i n g an optim a l c o r r e l a t i o n w i t h t h e d a t a h a v e been superposed. A c o m p a r i s o n of c r y s t a l records at t h e t r u n c a t e d end of t h e 20-deg cone r e s u l t i n g f r o m ball d r o p t e s t s with a 1/~-in. a n d a 26.9-in.-long f r o n t a l c y l i n d r i c a l section, respectively, m a y be o b t a i n e d f r o m Fig. 6. T h e stress d i s t r i b u t i o n across t h e base of this cone, d e t e r m i n e d f r o m c r y s t a l s with b a c k e d - u p alumin u m cylinders, is p r e s e n t e d in Fig. 9. T h e two c r y s t a l s located on t h e axis of s y m m e t r y on the two end faces of t h e cone were also e m p l o y e d for t h e e v a l u a t i o n of t h e p r o p a g a t i o n v e l o c i t y t h r o u g h t h e specimen.
-200
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Fig. lO--Transients predicted by the one-dimensional theory, 20- and 30 deg cones
f r o n t a l c y l i n d r i c a l section; i n s p e c t i o n of these diagrams reveals a r e d u c t i o n in p e a k s t r a i n with an increase of c y l i n d r i c a l section l e n g t h of 4 in. of a b o u t 17 percent. F i g u r e 4 shows t h e m u l t i p l e t r a n s i t of a pulse p r o d u c e d b y t h e i m p a c t of a p h e n o l - i m p r e g n a t e d fiber slug on t h e small end of t h i s cone in the absence of a p r o t e c t o r . T h e a v e r a g e p r o p a g a t i o n v e l o c i t y in this specimen was 214.5 X 103 ips. b a s e d upon t h e pulse p e a k a n d 211.4 X 10 ~ips b a s e d u p o n the i n i t i a l rise, while t h e a v e r a g e of t h e r a t i o of circ u m f e r e n t i a l to l o n g i t u d i n a l p e a k s t r a i n was found to be - 0 . 3 1 :~: 2 percent.
N u m e r i c a l solutions of eqs (7), (8) a n d (10) h a v e been o b t a i n e d utilizing t h e o b s e r v e d s t r a i n i n p u t to the 5.38-deg cone a n d t h e force-time h i s t o r y d e r i v e d from Fig. 8 for t h e 20 a n d 30-deg cones. These are shown in Fig. 3 for t h e 5.38-deg cone in Fig. 10 for t h e 20 a n d 30-deg cones, t h e l a t t e r c o r r e s p o n d i n g to the e x p e r i m e n t a l r e s u l t s of Figs. 5 a n d 7, respectively. T h e j a g g e d lines on these d i a g r a m s i n d i c a t e t h e a r r i v a l of t h e first reflected pulse a t a given station, i n v a l i d a t i n g t h e s u b s e q u e n t a n a l y s i s for t h e cones e m p l o y e d in these experiments. T h e present t e s t s as well as previous i n v e s t i g a t i o n s s, 10 h a v e shown t h a t t h e d i a m e t e r of t h e t a r g e t for a p a r t i c u lar collision c o n d i t i o n does not, to a first a p p r o x i m a tion, affect t h e r e s u l t i n g i m p a c t force h i s t o r y prov i d e d t h a t t h e r a t i o of t a r g e t r a d i u s to w a v e l e n g t h is less t h a n 0.1 a n d t h a t the t r a n s d u c e r s are at least several b a r d i a m e t e r s a w a y from the c o n t a c t point. T h e c o m p a r i s o n between e x p e r i m e n t a n d t h e oned i m e n s i o n a l t h e o r y for the 5.38-deg cone is v e r y good; t h e differences between t h e e x p e r i m e n t a l a n d t h e p r e d i c t e d s t r a i n results are of t h e same m a g n i t u d e as are the differences in c o m p a r a b l e s t r a i n s between similar runs. A g r e e m e n t is n o t as
Experimental Mechanics ] 447
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Fig. l l - - P l o t s of dimensionless peak strain vs. distance from the first gage station for the four cones
Fig. 12--Common plot of dimensionless peak strain vs. distance from the first gage station for the 5.38, 20 and 30 deg cones (apex end struck)
good for the 20 and 30-deg cones, although the t h e o r y correctly predicts the tensile tail for wave propagation in these cones. Some variation in the strain level of runs performed under identical conditions was observed; this is a t t r i b u t e d to variations in the bond between protector pieces and specimen proper. T h e reason for this hypothesis is the absence of such variations in tests not utilizing composite sections as well as consistency in the magnitude of the peak strain between various stations of a particular run. The effect of variation in peak strain at the first gage station was eliminated b y plotting the theoretical and experimental results in dimensionless form using the peak strain at the first station as the normalizing factor; the results are shown in Fig. 11 for the four cones. As expected, correlation between the analytical results and experimental d a t a is satisfactory for the 0.48 and 5.38-deg cones, a m o u n t i n g to a m a x i m u m deviation of 10 percent at the farthest gage station, while the corresponding correlation for the two larger cone angles is m u c h poorer, with a difference approaching 100 percent. T o compare the data from several cones with a single theoretical curve, the results from the 5.38, 20 and 30-deg cones were plotted as follows: strains were normalized using a c o m m o n first observation station 2.94 in. from each cone apex, interpolating the actual data where necessary. These values were plotted against distance from this hypothetical initial station; the theoretical curve is based upon a t r u n c a t e d end 1.44 in. f r o m the apex, with results as shown in Fig. 12. I t should be noted that, in this plot, the pulse shape changes due to the dispersion caused b y the divergence of the cone are neglected, since in reality the distances from the apex to the t r u n c a t e d ends of the three cones are not identical; however, this effect is not likely to be significant. T h e P o c h a m m e r - C h r e e t h e o r y for straight bars leads to the surmise t h a t the one-dimensional t h e o r y for cones will be in serious error at the large ends of
the 20 and 30-deg cones. B y virtue of the trigonometric identity sin ~ 0 = 1/2(1 - cos 20), it is seen t h a t the wavelength A of the Fourier component of the sine-squared pulse is equal to the length of the pulse; thus, h ~ 7 in. For the two large-angle cones, the radius R of the section at the base of the cone is approximately 3 in. and thus R / A ~ 0.4. Plots of c/co from the P o c h h a m m e r - C h r e e theory 9 where c is the phase velocity, indicate t h a t the onedimensional straight-bar t h e o r y is inadequate for this value of R/A. Thus, it is not surprising t h a t a comparable t h e o r y for cones is also in error for this value of R/A. T h e wave-propagation velocities associated with the pulse peaks for the four specimens are in all instances in good agreement with the rod wave velocity for aluminum and t h a t measured in a straight calibration bar. However, for the 20 and 30-deg cones, the propagation velocity based upon initial rise is closer to the computed dilatational velocity of 246 • 10 ~ ips as is the value of 252 • 10 ~ips. based on crystal measurement along the cone axis. T h u s it is a p p a r e n t t h a t a measurable portion of the initial pulse travels t h r o u g h the specimen as though in an infinite medium, while the greater portion of the wave is subject to reflection at the lateral surface and propagates in a manner not unlike t h a t in a circular bar. As noted from Fig. 6, both pulse shape and peakstress input to the cone are affected by the length of the frontal cylindrical section of the composite specimen. T h e initiation of the reflected pulse in the straight bar corresponds to the double transit time from the station to the interface; the compressive nature of this pulse is the result of the divergent character of the conical section and not due to any acoustic m i s m a t c h at the junction. This can be demonstrated b y a model of the 20-deg cone involving a series of cylindrical sections as shown in this diagram. A rectangular pulse with length approximately t h a t of the observed incident wave was as-
448
I October 1968
s u m e d to strike: (a) a ~/~-in.-long s t r a i g h t section a t t a c h e d to t h e f r o n t of the cone, a n d (b) a cylindrical rod, m u c h longer t h a n the pulse, t h a t preceded t h e cone. T h e results for these cases utilizing t h e laws of reflection a t changes in cross-sectional area are also p r e s e n t e d in Fig. 6, a n d show t h e m e t a m o r phosis of t h e pulse resulting f r o m c o n t i n u e d reflection, a n d c l e a r l y explain the reason for t h e compressive c h a r a c t e r of t h e reflected wave in the s t r a i g h t section. F o r a s h o r t f r o n t a l section r e l a t i v e to t h e pulse length, continued wave reflections at t h e i m p a c t end, whose precise b o u n d a r y c o n d i t i o n d u r i n g c o n t a c t is u n c e r t a i n a n d which acts as a free surface a f t e r t e r m i n a t i o n of i m p a c t , will t r a n s m i t tensile-pulse c o m p o n e n t s into t h e cone. On t h e o t h e r hand, a f r o n t a l section c o n s i d e r a b l y longer t h a n t h e conical specimen will n o t t r a n s m i t such tensile c o m p o n e n t s to the section of i n t e r e s t d u r i n g t h e initial t r a n s i t a n d first reflection in the conical bar, as is e v i d e n t b y a c o m p a r i s o n of the r e s u l t s shown in Figs. 5 a n d 6. T h e impulse of t h e first t r a n s i e n t in t h e 26.9-in. s t r a i g h t section of 0.0058 lb-sec c o m p a r e s w i t h a m e a s u r e d change of m o m e n t u m of the s t r i k e r of 0.0062 lb-sec, a r e a s o n a b l e agreement. T h e impulse r e c o r d e d b y the c r y s t a l in Fig. 8 is 0.0098 lb-sec t h a t can be c o m p a r e d to t h e sum of t h e initial a n d reflected impulses o b t a i n e d f r o m s t r a i n - g a g e measurem e n t at the 6-in. s t a t i o n in t h e cylindrical f r o n t a l b a r of 0.0110 lb-sec. T h e d i s c r e p a n c y could be t h e r e s u l t of i n a d e q u a t e low-frequency response of t h e c r y s t a l t r a n s d u c e r t h a t would yield a lower indic a t e d p e a k strain. ~' T h e results of Fig. 11 indicate t h a t the stress dist r i b u t i o n is r e a s o n a b l y uniform across a section of t h e 20-deg cone, with the o u t e r m o s t c r y s t a l recording a p e a k value 15 percent lower t h a n at the center for t h e initial wave. This is a b o u t the same value as was o b t a i n e d in corresponding t e s t s on p y r a m i d s , ~6 a n d is less t h a n the value of 25-percent r e d u c t i o n of surface to axial s t r a i n p r e d i c t e d by t h e Pochh a m m e r - C h r e e t h e o r y for a value of R / ~ = 0.4.9 T h e small difference in arrival t i m e s of the pulse a t v a r i o u s positions across the t e r m i n a l section prev e n t e d an a c c u r a t e c o n s t r u c t i o n of t h e shape of the wave front. However, the d a t a of Fig. 9 i n d i c a t e t h a t the pulse p e a k t r a v e l s faster near t h e l a t e r a l surface t h a n it does at the center of the cone.
Conclusions T h e o n e - d i m e n s i o n a l t h e o r y was found to p r e s e n t an a c c u r a t e model of wave p r o p a g a t i o n in t h e 0.48 a n d 5.38-deg cones; t h e v a l i d i t y of this analysis decreased p r o g r e s s i v e l y for the 20- a n d 30-deg cones. T h e a s s u m p t i o n of a plane or s p h e r i c a l wave is considered to be a d e q u a t e , at least for the cones tested. T h e difference between the plane or spherical ass u m p t i o n is small for cones with an apex angle less t h a n 30 deg. F o r t h e cones tested, the a s s u m p t i o n of a oned i m e n s i o n a l c o n s t i t u t i v e e q u a t i o n is a d e q u a t e ; this
is i n d i c a t e d b y t h e facts t h a t (1) the stress is n e a r l y u n i f o r m across t h e base section of the cone a n d (2) t h e r a t i o of c i r c u m f e r e n t i a l to l o n g i t u d i n a l s t r a i n was v e r y close to t h e k n o w n value of P o i s s o n ' s ratio. I t is n o t e d t h a t use of t h e o n e - d i m e n s i o n a l c o n s t i t u t i r e e q u a t i o n w i t h o u t inclusion of l a t e r a l i n e r t i a or shear t e r m s leads to the conclusion t h a t t h e w a v e t r a v e l s a t r o d wave velocity, co. T h i s conclusion is n o t correct, since p o r t i o n s of t h e stress pulse were o b s e r v e d to t r a v e l a t velocities close to t h e d i l a t a t i o n a l w a v e velocity, CD. However, t h e pulse p e a k t r a v e l e d a t r o d wave velocity. T h e neglect of l a t e r a l i n e r t i a a p p e a r s to be j u s t i fled o n l y in cases where the d i a m e t e r of t h e cone cross section a n d c h a r a c t e r i s t i c w a v e l e n g t h of t h e stress pulse would j u s t i f y the neglect of l a t e r a l inert i a in c y l i n d r i c a l bars. T h i s criterion is n o t satisfled a t t h e large end of t h e 20- a n d 30-deg cones. I t is felt t h a t the g r e a t e s t i m p r o v e m e n t in t h e onedimensional t h e o r y would r e s u l t f r o m t h e inclusion of l a t e r a l inertia. All o t h e r i m p a c t c o n d i t i o n s being equal, t h e placing of a c y l i n d r i c a l section in f r o n t of t h e t r u n c a t e d end of t h e cone will increase t h e p e a k stress a t the t r u n c a t e d section. T h e increase will d e p e n d b o t h on t h e cone angle a n d the l e n g t h of t h e end section. A s i m i l a r section placed on t h e base of the cone will reduce t h e stress in t h e cone for i m p a c t in t h e o p p o s i t e direction, t h e change again d e p e n d i n g on cone angle a n d l e n g t h of the section.
Acknowledgment A p o r t i o n of this work was p e r f o r m e d u n d e r cont r a c t with the S a n d i a C o r p o r a t i o n , L i v e r m o r e , Calif.
References 1. London, J. W., and Quinney, H., "'Experiments with the Hopkinson Pressure Bar," Proc. Royal Soc. of Lo~don, Series A, 1{}3, 622 (1923). 2. DonnelL L. H., "'Longitudinal Wave Transmission and Impact," Trans. Am. Soc. Mech. Engrs., 52, 153 (1930). 3. Favre, H., "'Etude thdorique de l'influence d'une discontinuitd de la section droite d'une barre conique sur la propagation des vibrations elastiques longitudinales,'" Bull. Tech. de la Suisse Romande, 88 (24), 353 (1962). 4. Reed, R. P., "'Stress Pulse-Trains [rom Multiple Reflection at a Zone of M a n y Discontinuities. A Notation for Machine Solution," Sandia Corporation Research Report, 4462 (August 1962). 5. Alcoa Aluminum Handbook, Aluminum Company of Amerlca, Pittsburgh, Pa. (1959). 6. Alcoa Structural Handbook, Aluminum Company of America, Pittsburgh, Pa. (1930). 7. Dove, R. C., and Adams, P. H., "'Experimental Stress Analysis and Motion Measurement," Chas. E. Merrill Books, Inc., Columbus, Ohio (1964). 8. Cunningkam, D. M., and Goldsmith, W., "'Short-time Impulses Produced by Longitudinal Impact," Proc. Sac. for Exp. Stress Anal., 16, 153 (1959). 9. Davies, R. M., "'A Critical Study of the Hopkinson Pressure Bar," Royal Soc. London Philosophical Tzans., Series A, 240, 375 (1948). 10. Goldsmith, W., and Lyman, P. T., "The Penetration of Hard-steel Spheres into Plane Metal Surfaces," Jnl. Appl. Mech., 27~ 717 (1960). 11. Goldsmith, W., Polivka, M., and Yang, T. L., "'Dynamic Behavior o[ Concrete," ]~XP~RIMENTAL~IECHANICS, 6 (2), 65--79 (1966). 12. Kolsky, H., "Stress Waves in Solids," Dover Publications (1963). 13. Kenner, V. H., "'Wave Propagation in Conical Bars," M.S. Thesis, University of California, Berkeley (1967). 14. Bishop, R. E. D., "'Longitudinal Waves in Beams," Aero Quart., 3, part 4, 280 (1952). 15. Gurtin, M. E., "The Effects of Accelerometer Low-frequency Response on Transient Measurements," Proc. Soc. for Exp. Stress Anal., 18, 206 (1961). 16. Okada, A., Cunningham, D. M., and Goldsmith, W., "'Stress Waves in Pyramids by Photoelasticity,'" ]EXPER]~MENTAL MECHANICS, ~ (7), 289-299 (1968).
Experimental Mechanics ] 449
