The dynamic plasticity of non-symmetrical free-flight collision impact

Int. J. mech. Sei. PergamonPress. 1969, Vol.11, pp. 633-657. Printedin Great Britain

THE DYNAMIC PLASTICITY OF NON-SYMMETRICAL FREE-FLIGHT COLLISION IMPACT JAMES F. BELL The Johns Hopkins University, Baltimore, Maryland (Received 25 M a r c h 1969)

S u m m a r y - - T h e major portion of the diffraction grating experimental studies of finite amplitude wave propagation in crystalline solids has measured the detail of non-linear wave fronts generated b y symmetrical free-flight axial collision of two identical specimens. I n the present paper a series of non-symmetrical free-flight impact experiments are described for a wide variety of materials and material conditions. I t is shown that the writer's generalized linearly temperature-dependent parabolic stress-strain function experimentally established from symmetrical measurements, m a y be introduced into the non-viscous finite amplitude wave theory to provide a detailed correlation between experiment and theory for the high strain rate non-symmetrical free-flight impact situation.

NOTATION B0 Co Cr Cr* E e L r R0 t Tc Tm ur v vI vH x fl

g ~s A A0 t~(0) p a a,~

0.0280, a dimensionless universal constant linear-elastic bar velocity finite strain plastic wave speeds plastic wave speed of one-half the m a x i m u m stress linear-elastic bar modulus coefficient of restitution specimen length 3.06, the dimensionless stress and strain ratio of the aggregate theory 1, 2, 3, 4 .... the finite deformation mode index initial specimen radius time time of specimen contact melting point in degrees Kelvin radial displacement normalized longitudinal particle velocity post-impact longitudinal particle velocity of struck specimen pre-impact constant velocity of the projectile specimen distance from the impact face angle of the specimen surface during wave propagation coefficient of the parabolic stress--strain function single crystal resolved shear strain nominal axial strain nominal axial strain rate nominal axial strain at one-haft the m a x i m u m energy of deformation nominal axial strain in a single crystal angle between the single crystal specimen axis and shear direction initial value of the angle between the single crystal specimen axis and shear direction linear-elastic isotropic shear modulus at absolute zero temperature mass density nominal axial stress nominal axial stress for single crystals 633

634

v ¢0

JAMES F. BELL

single crystal resolved shear stress angle between the specimen axis and the normal to the slip plane initialangle between the specimen axis and the normal to the slip plane INTRODUCTION

OVER a century ago, from 1829 to 1867, when Cauchy, Poisson and then St. Venant, were developing an infinitesimal linear-elastic one-dimensional theory of rod impact, the main problem considered was the symmetrical freeflight axial collision impact of two identical rods. Boltzmann, Voigt, Hausmaninger, Hamburger and Sears, between 1871 and 1906, were interested in the serious discrepancies Boltzman had found experimentally between observation and the then 4-year-old St. Venant theory; accordingly, their experimental studies also were symmetrical free-flight impact measurements. When one is questioning the applicability of a dynamic one-dimensional approximation, or the specific form of a dynamic governing stress-strain function, the symmetrical free-flight axial collision impact situation possesses enormous advantages over any other type of high strain-rate experiment. Not only do identical waves propagate away from the impact face in each specimen with a precise equipartition of deformation energies, but a constant stress and particle velocity are maintained behind these wave fronts. Bars can be made sufficiently long so that the non-linear complications of interacting reloading or unloading waves can safely be ignored. In addition, the impact face in the symmetrical measurement acts as an infinite barrier during impact contact, prohibiting the exchange of electrical, mechanical or thermal energies from one specimen to the other. At the impact face, the radial expansion of each specimen is identical; hence, no frictional difficulties are encountered. The absence of frictional complications is particularly important when the experimentalist is analyzing one-dimensional theories which do not assume lateral motion. In 1956, b y means of the then newly developed diffraction grating technique for accurately measuring finite dynamic strain, the present writer 1 was able to extend the symmetrical free-flight impact experiment to the domain of finite amplitude wave propagation. In each of the several crystalline solids studied, the finite amplitude wave theory independently developed 27 yrs ago by a number of different individuals 2-s was found to be applicable. The writer further discovered e-s that the governing stress-strain function for each metal studied is (~ =/~e½,

(1)

where =

(~)'/~ t~(o) Bo(1 - T/T.,).

Now that the finite deformation parabolic stress-strain function of a large number of crystalline solids has been firmly established through such direct finite amplitude wave propagation studies, H it becomes possible to examine a class of problems which require that the finite distortional deformation constitutive relations be known.

The dynamic plasticity of non-symmetrical free-flight collision impact

635

The non-symmetrical free-flight collision impact situation described in the present paper is one of a series of such problems which the writer has considered. In this experimental situation, of course, interface phenomena, and the difference in the manner and timing in the three-dimensional first-diameter wave initiation for the two specimens, were present. Whether the free-flight axial collision shown in Fig. 1 is symmetrical or non-symmetrical during impact contact, stress and velocity equivalence at the impact face furnishes 0`1 = 0'2' V H -- V 1 =

(2) (3)

V2,

VH

I

v, r o t - -

v2

BAR I

I

BAR 2 IMPACT FACE

FIO. I. Experimental configuration for the symmetrical or non-symmetrical free-flight

axial collision impact situation.

I f either the projectile specimen or the struck specimen has a sufficiently high elastic limit, then, instead of the parabolic stress-strain function of equation (1), we have the linear-elastic stress-strain function of 0` = E~

(4)

providing the linear stress-velocity relations, equation (5), and strain-velocity relations, equation (6), for the propagating linear-elastic shock front: 0` = p C o v ,

(5)

v = C0s.

(6)

For the finite deformation symmetrical situation, experiments of the writer have shown that C~(e) - d0`/de P

(7)

and

v:

f£

(8,

For the parabolic law of equation (1), equation (7) becomes

~2 C~ - 2~I - 2pa

(9)

and equation (8) becomes v2 = 8 ~

9p

= 8 o~

(t0)

9 pfl2"

Combining equations (9) and (10), one obtains a stress-velocity relation for the non-linear finite amplitude wave paralleling that of equation (5) for linear

636

JAMES F. BELL

elasticity: (11)

(7 = ~pCp(a)v.

I f the stress-strain function of b o t h bars in the n o n - s y m m e t r i c a l i m p a c t is parabolic, then, introducing equation (l l) into the equivalence relation of equations (2) and (3) provides Pl Cpl V2 :

(12)

piCpi.~p2Cp2VH

which, from equation (9), m a y be r e w r i t t e n

v, = g(pi)

~(P1) ~1

+ (p2)

v..

(13)

Given vg, v~ m a y be d e t e r m i n e d from equation (13) and v 1 from equation (3). W i t h these velocities known, the finite strain arrival times a n d t h e m a x i m u m strain m a y be calculated from equations (9) and (10), in order to m a k e a comparison with diffraction grating experiments. I f the projectile specimen material is linear-elastic a n d the struck specimen material is parabolic, equations (2) and (3) furnish e q u a t i o n (14),

pl Col V2 -----P1 C01 -~ ~P2 Cp2((7)

(14)

vH'

which, from equations (5) and (9), m a y be r e w r i t t e n as 9

t

(15)

VH = ~(~P2~) ( p l E 1 ) vt, +v~.

I f the struck specimen material is linear-elastic a n d the projectile specimen is parabolic, equation (15) becomes e q u a t i o n (16), VH :

~(P~E2)

(16)

~"

(~1~1)½ v. "t-V2"

I n the r e m a i n d e r of this paper, experimental d a t a in the struck specimen for a wide v a r i e t y of n o n - s y m m e t r i c a l i m p a c t situations are c o m p a r e d with prediction from e q u a t i o n (13) or (15) or (16), depending upon the n a t u r e of the non-symmetry. I t should be k e p t in mind t h a t for symmetrical free-flight i m p a c t collision, equations (2) and (3) provide v 1 = v2 = VH/2. SITUATION SPECIMEN

vs.

I. HARD POLYCRYSTALLINE SOFT POLYCRYSTALLINE

PROJECTILE STRUCK SPECIMEN

The first non-symmetrical free-flight impact situation considered is the collision of a linear-elastic projectile polycrystal with a parabolic finite deformation struck polyerystal. The maximum particle velocity of the struck specimen is given by equation (15) for the

projectile velocity VH shown in Fig. 2. In these experiments, the mass densities of the two specimens are identical. The numerical values of both E 1 and fl~ have been determined dynamically from prior symmetrical free-flight impact experiments using electric resistance gages for determining the linear-elastic strain, and 30,720 lines/in, diffraction gratings to determine the finite strain profiles and wave speeds. For the 70-30 a-brass experiment 9 of Fig. 2, the bar modulus of the projectile specimen is E = 10,900 kg/mm 2. The mode index r in equation (1) is r = 2, iz(0) = 4660 kg/mm 2, T,a---- 1188°K, providing a value of f~ at 300°K for the 10 in. long struck specimen of

The dynamic plasticity of non-symmetrical free-flight collision impact

637

64.2 kg/mm 2. For the two a l u m i n u m experiments in Fig. 2, the bar modulus of the 20 in. long a l u m i n u m hard projectile specimenis 7160 kg/mm 2. The modeindex r for the annealed 99.2 per cent purity struck polycrystal is r = 2, which for /~(0) = 3110 kg/mm 2 and T m = 932°K furnishes a 300°K value for fl of fl = 39.4 kg/mm ~. Substituting the numerical values obtained from symmetrical impact experiments into equation (15), and subsequently, into equations (9) and (10), provides the close correlation between experiment and theory shown in Fig. 2. Assuming linear-elastic unloading in the 7 0 - 3 0 (2- BRASS 3 0 0 ° K (Hartman)

ALUMINUM POLYCRYSTALS 9 9 . 2 % Purify , 3 0 0 ° K

E 0,040 -

theoretical ~) ~ e e

o o o_

~

, r=2 max

' 0 0 04 II

o

°° o

0'020

predicted

unloading

strain

ooooeo o

I

.

oo

.

.

.

.

.

.

.

t .

r I:'°' i

predicted time

.uno°.n

-

I O'OlO -

0 < 0

• I theorehcal ~max,r=2 J .... .

Test 1172 Hitter velocify :~2426 cm/sec Grating position l" I I I O0 200 T I M E , /J. sec

I

I

J 1"

J 7

/ I

~/ /

/ Y 0 ~oev 0

®Test,33, Hitter velocity= 333l cm/sec Grating position Z" o Test 1334 Hitter velocity = 3359 cm/sac Grating position 2 I I l I00 200 300 T I M E , /.L sec

FIG. 2. Diffraction grating strain-time profiles (circles) compared with prediction (solid lines), for a 30.5 em long, hard, a-brass projectile specimen striking a n annealed 25.4 cm long, s-brass polycrystalline specimen, and for two experiments in which a 50.8 cm long, hard a l u m i n u m projectile specimen is in axial collision with a completely annealed, 99.2 per cent purity, fine-grained a l u m i n u m polycrystalline struck specimen. a h u n i n u m experiments in Fig. 2, one sees that both the time of unloading from the free end of the projectile specimen and the magnitude of the unloading strains are in only approximate agreement with prediction. The solid lines in the strain-time profile and the m a x i m u m strain represent the predicted dynamic finite deformation behavior. The close agreement with the experimental data (circles) demonstrates that the influence of the nonsymmetrical unlubricated interface is minimal in this situation. As in the symmetrical impact situation, the strains above the mean energy g are delayed because of the threedimensional first-dia~neter wave initiation,s, 10,11 although traverse times beyond the first diameter are given by the wave speeds of equation (9). S I T U A T I O N 2. S O F T P O L Y C R Y S T A L L I N E PROJECTILE SPECIMEN vs. HARD POLYCRYSTALLINE STRUCK SPECIMEN The second non-symmetrical impact situation involves a reversal of the projectile and struck specimens, i.e. the parabolic annealed specimen now strikes the hard linear-elastic specimen. Using equation (16) to calculate the velocities, and electric resistance gages to determine the strain-time profiles, or, as plotted in Fig. 3, the stress-time experimental data, theory and experiment again m a y be compared. Many experiments have been reported upon by the writer s, 10-12 in which the soft projectile specimen was a 99.2 per cent pure a l u m i n u m annealed polycrystal colliding with a long hard a l u m i n u m bar. For variety, two different solids are considered in the non-

638

JAMES F. BELL

s y m m e t r i c a l h a r d b a r e x p e r i m e n t . T h e first, s h o w n i n Fig. 3, consists of a 99.99 p e r c e n t h i g h - p u r i t y a l u m i n u m c o l u m n a r m u l t i c r y s t a l s t r i k i n g a long, h a r d , linear-elastic a l u m i n u m p o l y c r y s t a l l i n e b a r ; t h e s e c o n d is a n a n n e a l e d 99.9 p e r c e n t p u r e , c o p p e r p o l y c r y s t a l l i n e p r o j e c t i l e s p e c i m e n s t r i k i n g a long, h a r d , linear-elastic c o p p e r s p e c i m e n . T h e finite comp r e s s i o n d e f o r m a t i o n of t h i s h i g h - p u r i t y p o l y c r y s t a l l i n e a l u m i n u m , as t h e w r i t e r h a s s h o w n i n a n e x t e n s i v e series of e x p e r i m e n t s , 8 h a s t h e m o d e i n d e x r = 6, w h i c h t o g e t h e r w i t h ALUMINUM COLUMNAR 99'99% Purity , 500°K

MULTICRYSTAL

(:7, psi

Io, o o o -

Test 7 2 4 ,

Gage. p o s i t i o n

I0"

Hitter velocity = 2407.6 8,000

-

.[

crn/sec theoretical Ti

6,000 -

,ooo-l

4,000

04~

theoretical O"ma ~

I00

0

COPPER Test 0", psi

200

r=6

I theoretical Tc

I

300 400 TIME,/./. sec

POLYCRYSTAL

, 99.99%

theoretical O'max I

-

PuPil) ' , 3 0 0 ° K

1241 , Gage p o s i t i o n I 0 " Hitter velocity = 1464,5 cm/sec

12,000 ---~ 8,000

I

700

r=4

. (

4,000

-

oi 0

I IO0

I 200

I

500 400 TIME, ~ sac

500

600

l

700

FIo. 3. S t r e s s - t i m e d a t a in t h e h a r d b a r are c o m p a r e d w i t h p r e d i c t i o n for e x p e r i m e n t s in w h i c h a h i g h - p u r i t y , a n n e a l e d a l u m i n u m p o l y c r y s t a l strikes an aluminum hard bar, and in which a hlgh-purity, annealed copper polycrystal strikes a hard copper bar. v a l u e s of ~(0) = 3110 k g / m m 2 a n d T m = 932°K, p r o v i d e s a p a r a b o l a coefficient fl in e q u a t i o n (1) of 17.5 k g / m m 2. T h e v a l u e of E 1 i n t h e s t r u c k h a r d l o w - p u r i t y a l u m i n u m b a r r e m a i n s a t E 1 = 7170 k g / r n m 2. (The m e a s u r e d s t r a i n s i n t h e h a r d , s t r u c k s p e c i m e n a r e m u l t i p l i e d b y E I in Fig. 3 to p r o v i d e a s t r e s s - t i m e history.) T h e p r e d i c t e d v a l u e of t h e stress is 3.52 k g / m m 2, w h i c h , as m a y b e seen f r o m Fig. 2, is in precise a g r e e m e n t w i t h t h e m e a s u r e d m a x i m u m stress for t h i s r = 6 p a r a b o l a coefficient for t h e m a x i m u m stress of t h e p r o p a g a t i n g w a v e in t h e soft b a r . Symmetrical free-flight i m p a c t e x p e r i m e n t s i n c o p p e r 8 h a v e r e v e a l e d t h a t t h e finite d e f o r m a t i o n m o d e i n d e x is r = 4, w h i c h , w i t h F(0) = 5160 k g / m m 2 a n d T,, = 1358°K,

The dynamic plasticity of non-symmetrical free-flight collision impact

639

provides a room-temperature parabola coefficient of 49.9 kg/mm% I n copper the measured dynamic value of the E-modulus in the hard specimen is 12,600 kg/mm=. Introducing these values into equation (16) provides the close correlation between prediction and electric resistance strain measurements in the hard bar, shown in Fig. 3. The infinitesimal strain measurement 25.4 cm from the impact face was multiplied b y the modulus E to provide a stress-time curve. One m a y conclude from Figs. 2 and 3 that the finite amplitude wave theory governed by the writer's parabolic law m a y be extended from the free-flight symmetrical impact situation to the non-symmetrical situation, irrespective of whether the hard or the soft specimen is the projectile specimen. The mass densities of both specimens, of course, are the same for all experiments thus far considered. S I T U A T I O N 3. R O O M - T E M P E R A T U R E SOFT POLYCRYSTALLINE PROJECTILE SPECIMEN vs. HIGH-TEMPERATURE SOFT STRUCK SPECIMEN The linear temperature dependence of the parabola coefficient of equation (1) was established experimentally from diffraction-grating measurements of finite dynamic strain at high strain rates during symmetrical free-flight collision impact experiments, v Two of a series of sixty-five such polycrystalline a l u m i n u m symmetrical experiments at numerous ambient temperatures up to within a few degrees of the melting point are shown in Fig, 4.

E 0'018

m

ta~et~

o9

%o ace see o o theoretical ~

1~

J

0"016

ALUMINUM POLYCRYSTALS 9 9 " 2 % Purity , 6 9 4 ° K Gage position 2"

0'014

E 0,014

0'012

0"012

0o010

0.010

0.008

0.008

theoretical Emax l

,ee=e e oe~ o

e~ • 00o

¢

0"006

0.006 Test 505 Hitter velocity=3208,9 cm/se¢

0"004

0,004

0"002

i

~o

theoretical E

----i e ~ ° '

Test 494 Hitter velocity= 1484.9 cra/u¢

/ 0-002 I s 7

I I 200 :500 TIME, ~M.MI¢ FIG. 4. Diffraction grating strain-time profiles (circles) for two 6Tin-

0

0 00-':' -

-

I

I00

I 200 TIME, ~ sec

I 300

0

0

I I00

metrical free-flight impact experiments at a high ambient temperature

are compared with prediction (solid lines). Both the projectile and struck specimens are at the same high ambient temperature. The twowave structure with the first m a x i m u m at the strain of the mean energy g is observable at both impact velocities.

43

400

640

J A ~ s F. B~.LL

The two experiments were at the same ambient temperature of 694°K ( T / T , , = 0.75) for both projectile and struck specimens, with diffraction grating measurements at the same position of 2 in. from the impact face in the struck specimen. The mode index at 694°K remains the same as at room temperature for this 99.2 per cent purity aluminum polycrystal, i.e. r = 2. Data for different impact velocities are shown to emphasize the point discovered earlier ~ that, at high ambient temperatures, the two-wave structure produced at al] 99.2 % Purity

ALUMINUM POLYCRYSTALS, High Temperature Tests

l Test 463 , 478 °K I H i t t e r velocity= 799cm/sec

Test 453 , 781°K Hitter v e l o c i t y = 1141 cm/sec E

[

theoretical

•004t-"

0.020 0

r=2

-]

r=z .

O/001)000

e'rnax-"/

!he0retical

~max

r- - ° -

0

o

0.015 o

o

o

.003 o

[oo

o o

0.010

O0

0 O0

.002

o

o

o00 I

0.005 o o o

0

-o o

I00

Grating I

200

Grating position = 3"

position = 2 . 5 "

I

300 TIME

o

, /J. sec

I

400

I

500

0

0

I

I

I00

200

TIME,/.L

I, 300

sec

FIG. 5. Measured strain-time profiles (circles) are compared with prediction (solid lines) for two annealed aluminum polycrystals at the temperature and impact velocities designated. The projectile specimen in each instance is an annealed polycrystaUine specimen of the same purity at room temperature. temperatures during the initial finite wave development in the first diameter remains distinct beyond the immediate vicinity of the impact face, where, at room temperature, coalescence would have occurred. The m a x i m u m of the initial wave is still t h a t of the strain of the mean energy g (g = 0"63~max). F o r a non-symmetrical impact when the projectile specimen is at room temperature, i.e. 81 = 39-4 kg/mm ~, and the struck specimen is heated to a high ambient temperature, the mode index for both specimens remains at r = 2, but the agreement between prediction and experiment for finite strain-time profiles and m a x i m u m strain (solid lines) is more approximate than for the symmetrical situation. The two experiments shown in Fig. 5, with different ambient temperatures, projectile velocities and positions of diffraction grating measurement, exhibit the two-wave structure, the predicted arrival times and the approximate finite strain predicted from equations (13), (1), (9) and (10). They also demonstrate the superiority of the symmetrical experiments in such high-temperature wave-propagation studies. The values of r, F(0) and T~ are the same for both specimens. Substituting the values of the ambient temperature of 781°K (T/T,~ = 0.84) for test 453, and 478°K ( T / T , , = 0.506) for test 463, in equation (1), provides the respective parabola coefficients of 9.2 k g / m m 2 and 28.5 k g / m m 2 for substitution

The dynamic plasticity of non-synunetrical free-flight collision impact

641

in equations (13), (9) and (10) to obtain the predicted behavior (solid lines). I t is interesting to note t h a t the test at 478°K has the same T / T m as the finite amplitude waves in lead at room temperature. T, s, 18 The diffraction gratings for all experiments of Figs. 4 and 5 were 0.005 in. long, with 30,720 lines]in, rulings. S I T U A T I O N 4. A L U M I N U M (99.99 P~.R CE~T P U R E ) S I N G L E CRYSTAL PROJECTILE SPECIMEN vs. HARD ALUMINUM POLYCRYSTAL A very important form of the non-symmetrical impact experiment is that in which a long, hard, linear-elastic bar is struck by a single crystal of the same solid with a known initial crystallographic orientation. To determine the axial parabola coefficient fl of the incident single-crystal specimen, one must first determine from the polycrystal stressstrain function, equation (1), the resolved-shear-stress, resolved-shear-strain function on the primary slip plane in the primary direction, and then de-resolve this function to the specimen axis. The writer has shown 8, 14, 18 that the first objective is met through the empirical use of the Taylor 18 and Bishop and HilP ~ aggregate ratios, a_ = ~

7,

=

(17)

where ~ = 3.06. Substituting equation (1) into equation (17) furnishes the resolved stress-strain function, r = ~),

•

(18)

The writer has shown, further, 8, 14, 15 that the single crystal axial stress as and strain es arc given b y the Boas and Schmid relations: 18 ~" =

as

c o s ¢0 c o s ~, c o s ~0 cos4 o,

c o s ;~

y = cos4 l+

s =

sin Ao

=

(19)

cos 40

The specimen axis de-resolved parabola coefficient of the single crystal projectile specimen for substitution in equation (16) thus is given by equation {20): = ~s

(20)

~S t"

For the hard a l u m i n u m struck specimen in the two room-temperature experiments shown in Fig. 6, the bar modulus is E = 7160 kg/mm ~. Substituting the X-ray determined initial Sehmid factor, cos 40 cos A0, and the measured projectile velocity v~ in the above equation provides the remarkable correlation between theoretical and experimental m a x i m u m stress shown in Fig. 6. These data obtained b y Gillich19 in the writer's laboratory several years ago also exhibit the complexity of the first-diameter wave initiation in terms of the initial peak stress collapse. The mode index of these single crystals is r = 6, corresponding to dynamic and quasi-static compression data in polycrystals of the same 99.99 per cent purity. 8 (See Fig. 3 above.) The a l u m i n u m values of/~(0) and T~ are the same, of course, for both single crystal and polycrystal. SITUATION 5. LOW-PURITY vs. HIGH-PURITY ALUMINUM

ALUMINUM SINGLE

POLYCRYSTAL CRYSTAL

The extension of the diffraction grating measurement of finite strain to the study of axial large deformation waves in single crystals, 19 while offering no new optical difficulties, was complicated b y the difficulties of growing the necessary 1 in. diameter, 10 in. long single

642

JAMES F. BELL

crystals. To achieve a symmetrical impact, it would have been necessary to obtain large projectile and struck specimens of identical orientations. Furthermore, as the writer has shown,S, 14, 15 from the study of resolved quasi-static data, variability in the deformation mode index r could be expected from one measurement to another. Since the quasistatic resolved data had been found to be predictable through the aggregate ratios [equation (17)] and the axial single-crystal deformation in terms of macroscopic single slip

( Gillich )

ALUMINUM SINGLE CRYSTALS 99"99% Purity , 5 0 0 ° K at, psi Gage position :5" 9,000

Test 5.79 Hitter velocity= 1570 cm/sec

8'000 I I c r, psi

7,000 Cos ~o Cos XO= "4639

Test 581 Hitter velocity = 1656 cm/sec

6,000

Cos ~o Cos Xo= -4 406 4,000

theoretical O-max

4,000

5,000

5,000

2,000

2,000

1,000

1,000

0

0

I00

.200

:300 400

500

TIME, ~ sec

600

700

0

theoretical O'ma x r=6 ~

J

0

200

"'1

400

1

600

I

800

I

I

TIME, /.z sec

F~G. 6. A comparison of measured stress-time histories in a 6 ft long, hard polycrystalline struck specimen. The projectile specimen in each instance is a 10 in. long, high-purity single crystal with the designated initial orientation and projectile velocity.

[equation (18)], Gillich's correlation of experiment and finite amplitude wave theory was based upon comparing strain arrival times with prediction for strain below g, assuming an infinite step at the impact face. Because of the orientation non-symmetry, and, hence, the u n k n o w n division of m a x i m u m particle velocity between the specimens, the correlation of m a x i m u m strains was not possible in Gillich's dynamic single crystal study, la His assump. tion of an infinite step at the impact face, based upon the present writer's polycrystalline observations of such a behavior below g, required the additional assumption that for the single crystal the effects of linear-elastic single-crystal anisotropy in the first diameter could be ignored. I n view of the high degree of linear-elastic isotropy in aluminum, this would seem to be a reasonable assumption. Gillich's experiments did demonstrate the applicability of the non-viscous finite-amplitude wave theory, the non-sensitivity of stage I I I finite deformation to increases of strain rate of eight orders of magnitude, and the fact that macroscopic single slip governed the dynamic wave speeds. The two non-symmetrical experiments described in this section were not included in the previous description of dynamic single-crystal data. s, lg They were obtained in the very early phases of the single-crystal work to demonstrate that diffraction grating

The dynamic plasticity of non-symmetrical flee-flight collision impact

643

determination of dynamic finite strain in single crystals was possible. The projectile specimens for both of the measurements of Fig. 7 were 99.2 per cent purity a l u m i n u m polycrystals, for which r = 2, F(0) = 3110kg/mm 2 and T m = 932°K, providing at 300°K, f~ = 39.4 kg/mm 2. The parabola coefficients for the single crystal were obtained b y de-resolving, b y means of equation (19), the resolved parabolic stress-strain function, equation (18), obtained from the aggregate ratios of equation (17) from the polycrystalline stress-strain function of equation (1). The most important new aspect of the experiments of Fig. 7 is that now it is possible to demonstrate the agreement with prediction of m a x i m u m strains, a n agreement which is even more remarkable considering the wide difference in initial orientation for the two tests shown. ALUMINUM

SINGLE

CRYSTALS ,

99.99%

Purity

(Gillich)

E 0.060

0.050

oOOo°

~m~x'-"/'/

theoretical

Test 437 , Hitter velocity = 3116 cm/sec Grating position= 2,We"

/ / / --/

u o

o 0 o

oo

0.040 °°

0,050 -

, / ~

/

0"020- COS~oC°s~'o~"0"419

/ °°

/

0.010

E

o

0"030

0'020

°°

~

Test 454 Hitter velocity = 1153 cm/sec Grating oosition --2"

theoretical Emaxl r=5 1

0.010 ~ - C ~

O(~nu on nnu o ~

50

I I00 -

I 150 TIME, ~ sec

I 200

I 250

0

0 ~

-.

-°50

0 0

0

1013

150

TIME, ~ sec

FIG. 7. Annealed a l u m i n u m polyerystalline projectile striking aluminum single-crystalline specimen with the initial orientations and projectile velocities shown, providing the diffraction grating strain-time profiles (circles) which are compared with prediction (solid lines). Substituting in equation (13) these single crystal parabola coefficients, and the measured projectile velocities vR for each test, provided the predicted arrival time (solid lines) and m a x i m u m strain shown in Fig. 7. Also shown are the two very different initial crystallographic orientations and Schmid factors for these measurements. Since it is now possible to predict m a x i m u m strains, the differences in integral mode indices, i.e. r = 6 for test 437 and r = 5 for test 454, could be cross-checked between arrival times or wave speeds and m a x i m u m strain, despite the fact that, in test 437, firstdiameter difficulties similar to those shown in the initial stress detail of Fig. 6 limited the arrival time correlation to the very low strains. This correlation of both conditions of the finite amplitude wave theory (shown in Fig. 7) in terms of the aggregate ratios and macroscopic single slip for stage I I I deformation at high strain-rates removes any question which might have been raised regarding the

644

JAMES F. BELL

limiting of t h e earlier comparisons to arrival t i m e vs. strain d a t a a n d load bar m a x i m u m stress data. Despite t h e observed correlations for b o t h arrival times and m a x i m u m strains in t h e present n o n - s y m m e t r i c a l i m p a c t experiments, t h e e x p e r i m e n t s p r o v i d e f u r t h e r evidence of additional difficulties, in t e r m s of t h e distortion of t h e u p p e r p o r t i o n of t h e s t r a i n - t i m e profiles in first-diameter w a v e initiation, w h e n c o m p a r e d w i t h the s y m m e t r i c a l free-flight i m p a c t situation. SITUATION

6. A N N E A L E D ALUMINUM POLYCRYSTALS ANNEALED COPPER POLYCRYSTALS

vs.

I n all of the n o n - s y m m e t r i c a l i m p a c t situations considered above, t h e mass densities of b o t h t h e projectile and struck specimens were t h e s a m e in e v e r y instance, w h e t h e r t h e specimens were g o v e r n e d b y a linear-elastic stress-strain function, w h e t h e r a t high t e m p e r a t u r e or r o o m t e m p e r a t u r e , and w h e t h e r t h e specimens were single- or polyerystais. F o r t h e n e x t series of t h i r t y - o n e n o n - s y m m e t r i c a l free-flight axial i m p a c t e x p e r i m e n t s considered in this paper, however, t h e ratio of mass densities has t h e v e r y large v a l u e of 3.3. I n fourteen of these experiments, the projectile specimen was an annealed copper polycrystal, a n d t h e initially s t a t i o n a r y s t r u c k specimen was an annealed a l u m i n u m polycrystal. I n the r e m a i n d e r of the e x p e r i m e n t s t h e situation was reversed, w i t h an a l u m i n u m projectile specimen and a copper s t r u c k specimen. T h e projectile v e l o c i t y VH, w h e t h e r a l u m i n u m or copper, was m a i n t a i n e d a t a p p r o x i m a t e l y t h e s a m e value of VH -----2700 c m / sec. A 0-005 in. long, 30,720 lines-per-in, diffraction grating was located at 5 cm (two diameters) f r o m the i m p a c t face in n e a r l y e v e r y e x p e r i m e n t to p r o v i d e a strain vs. t i m e and surface angle vs. t i m e m e a s u r e m e n t of the finite-amplitude w a v e front b e y o n d the first d i a m e t e r in the s t r u c k specimen. S y m m e t r i c a l free-flight i m p a c t e x p e r i m e n t s in t h e 99.2 per cent pure c o m p l e t e l y annealed polycrystaUine a l u m i n u m considered h a v e revealed a t h a t t h e large d e f o r m a t i o n m o d e index for this solid is r = 2. F r o m the e x p e r i m e n t a l fact t h a t tz(0) -- 3110 k g / m m ~, t h e a m b i e n t t e m p e r a t u r e , T = 300°K, and t h e m e l t i n g point, Tm -- 932°K, one obtains t h e p a r a b o l a coefficient of fl -- 5.60 × 104 psi or fl = 39.4 k g / m m ~ for t h e s i t u a t i o n in e q u a t i o n (13). S y m m e t r i c a l free-flight i m p a c t e x p e r i m e n t s in 99.9 per cent pure annealed polycrystalline copper 8 h a v e p r o v i d e d a d y n a m i c m o d e index, r = 4, which, w i t h the e x p e r i m e n t a l values of/~(0) -- 5160, T = 300°K and Tm = 1358°K, p r o v i d e a p a r a b o l a coefficient [equation (1)] of f~ = 7.1 × 104 psi or fl = 49-9 k g / m m 2. The mass density of a l u m i n u m , which was used in e q u a t i o n (13), was 0.000253 lbsec2/in 4, a n d for copper, 0.000833 lb sec~/inL Substituting these values in e q u a t i o n (13) provides a ratio VH/V 2 of 1.433 w h e n the projectile specimen is copper a n d t h e struck specimen is a l u m i n u m , and VH/V 2 = 3"31 w h e n t h e projectile specimen is a l u m i n u m a n d t h e struck specimen is copper. I t is i m p o r t a n t to note, p a r t i c u l a r l y in v i e w of the instability complications now present in the first-diameter w a v e d e v e l o p m e n t , t h a t w h e n b o t h specimens are linear-elastic, the v e l o c i t y ratio is n o t appreciably altered, i.e. VH/V~ = 1.417. The n o n - s y m m e t r i c a l i m p a c t series in a l u m i n u m and copper described here were carried out in this l a b o r a t o r y for t h e writer b y C. Jeffus, p r i m a r i l y for t h e purpose of s t u d y i n g t h e a n t i c i p a t e d complications of first-diameter b e h a v i o r in this n o n - s y m m e t r i c a l situation, as well as for s t u d y i n g t h e effect of the n o n - s y m m e t r y u p o n t h e unloading behavior. To s t u d y t h e l a t t e r behavior, as will be shown below, t i m e of specimen c o n t a c t and separation velocities after i m p a c t were measured. As will also be shown below, the a n t i c i p a t e d i m p a c t face and first-diameter effects of n o n - s y m m e t r y axe observed in t h e a l u m i n u m vs. copper situation. Before describing these n o n - s y m m e t r i c a l data, it is i m p o r t a n t to re-emphasize t h a t s y m m e t r i c a l experiments of annealed copper w i t h annealed copper, and annealed a l u m i n u m w i t h annealed a l u m i n u m , at the same projectile v e l o c i t y of a p p r o x i m a t e l y 2700 cm/see (as in n e a r l y all t h e nons y m m e t r i c a l experiments) p r o v i d e reproducible a g r e e m e n t w i t h t h e finite a m p l i t u d e w a v e t h e o r y g o v e r n e d b y the respective parabolic stress-strain fmlction of e q u a t i o n (1), b o t h for a r r i v a l t i m e s below the strain of the m e a n energy g and for the m a x i m u m strain. As for s y m m e t r i c a l experiments, at 300°K for copper, r = 4 with fl = 49.9 kg/rnm ~, and for a l u m i n u m , r = 2 with f~ = 39.4 kg/mm%

The dynamic plasticity of non-symmetrical free-flight collision impact

645

As m a y be seen in Fig. 8, the only two symmetrical experiments in copper performed at the hitter velocity of v~ = 2699 cm/see provide averaged data in very close agreement with prediction, as do the two symmetrical experiments in a l u m i n u m at a velocity of 2674 cm/see, also shown in Fig. 8. These a l u m i n u m measurements at two different distances from the impact face are from a series of earlier experiments at this velocity. For these experiments, of course, the m a x i m u m particle velocity is the same in each specimen, vR/2.

COPPER P O L Y C R Y S T A L S , 9 9 - 9 % P u r i t y Average of Tests 9 8 8 & 9 8 9 E 0.025

theorehcat Ernax--~ r=4 O

Hitter v~ilocity = 2699 cm/sec

0.020

Grating at 2"

0

0

0

A L U M I NUM P O L Y C R Y S T A L S 99.2% Purity

o E

0.015

0,015

Test94, grating at I" Test97 j grating at 31/2" theoreticol Ernax r=2 .ee~o°eeo ,

OO

0,010

0,010

0

o

o

o

"

o

Tes

0.005

0

0

J

5O

0'005 F

I I00 150 TIME, p. sec

I 200

J

~..G,~,

50

,/ -

Average hitter velocity= 2674 cm/sec I

f

I00 150 TIME, ~ sec

I

.

200

FIo. 8. The diffraction-grating-measured strain-time profiles (circles) for two annealed copper polycrystals in symmetrical impact are compared with prediction (solid lines). Diffraction-grating measurements (circles) for two symmetrical free-flight a l u m i n u m experiments at the designated positions are compared with prediction (solid lines). The projectile velocities for these four symmetrical experiments in a l u m i n u m and copper are approximately the same as those for the eighteen non-symmetrical experiments of Fig. 9.

Shown in Fig. 9 are eight experiments in which a copper specimen struck a n a l u m i n u m specimen, and ten experiments in which a n a l u m i n u m specimen struck a copper specimen. All these experiments were conducted at approximately the same projectile velocity, a n d all diffraction grating measurements, whether in the a l u m i n u m or copper struck specimen, were made at the same position of two diameters, or 5 era, from the impact face. Unlfl~e the symmetrical free-flight impact situation shown in Fig. 8 and unlike the non-symrnetrical impacts with identical mass densities described above, there is a great a m o u n t of scatter in these data.

646

JAMES F. BELL

The h e a v y solid line in Figs. 9(a) and (b) shows t h e theoretical s t r a i n - t i m e curve, including b o t h the arrival t i m e and m a x i m u m strain for each situation. These predictions result f r o m introducing the respective parabola coefficients into equations (13), (9) and (10) to d e t e r m i n e m a x i m u m particle velocities, arrival times and m a x i m u m strain. The dashed line is the average of t h e widely varied e x p e r i m e n t a l data, which, while agreeing fairly well w i t h t h e arrival time, provide m a x i m u m strain values of 1.42 per cent instead of t h e p r e d i c t e d v a l u e of 1.2 per cent for the copper specimen, and 2.28 per cent instead of the p r e d i c t e d value of 2.08 per cent for the a l u m i n u m struck specimen.

COPPER

E 0'030

( Bell B deffus )

----,-ALUMINUM test no.

,- ..............

953

spec. hitter I0"

10"

16" /....::;~....-.,""~'.~1050 12" ~i;::';................... 1063 16" ,~f" /AVERAGE iif[ / . . . . . . , ~ 9 5 6 t0 ~// ,7 :~:::=:::"::::::~'- 049 2"

8" 8" 8"

i.../.,.....................~ 1 0 8 8

0.025

]/i '//

o.ozo

~!f ': i

'ii / d~ _ . ~/: : ; ii//:

,.~973 ~" ............ ....... 959

test no. hitter spec.

IO"

8" ~THEORETICAL

/ / / ~

I0" I0"

I0" I0"

"

0.005

"'!//

o L/-fl/'/ 0

TIME,

(o)

/ ,~1117 ," . . , . - S ~ 9 5 1 ........... 95z

{6"

W//~,................

992

~I/.........---"'. .... /// v \ ~979

0.005

'

'

200

300

~ sec

0

S"

8"

to"

to"

,o"

,o"

//i///" /AVERAGE / ///'~':C'....',.~.~991 2" 8" //,~/~'~i\:::.'""~ 993 12" 8" /.~f,f/~,~/,fd/ " ~ 9 8 0 I0" I0" b ? ~ ~THEORETICAL M,C--~~sr~ ,o" Io"

0.010

Average Hitter velocity= z ~ z ~ ~m/,.c

I00

16"

/. /- -/ 0.015

it!4!/ I] { i I

,..............~1105

/

0.020

/// o.o,o

COPPER

ALUMINUM

E 0"085

~ 0

!

I00

8"

I0"

I0"

Average H i t t e r Velocity=

: i/ / ~

12"

2676 cm/sec -

J 200

300

TIME, /zsec

(b)

F i e . 9. Diffraction-grating s t r a i n - t i m e profiles for eight experiments, each h a v i n g an annealed polycrystalline copper projectile specimen and a n annealed polycrystalline a l u m i n u m struck specimen; and ten experiments, each h a v i n g an a l u m i n u m projectile specimen and a n annealed polycrystalline copper struck specimen. The a v e r a g e d d a t a in each instance (dashed lines) are c o m p a r e d w i t h theoretical prediction for this n o n - s y m m e t r i c a l i m p a c t (solid lines). A t first glance, t h e grouping of these d a t a seems to suggest t h e possibility t h a t m o d e indices other t h a n t h e r = 2 of a l u m i n u m a n d the r = 4 of copper from the s y m m e t r i c a l free-flight situations m i g h t be operable. I t is easy to show t h a t this is n o t the situation, since t h e arrival times of the s t r a i n - t i m e curves w i t h low m a x i m a , a n d therefore, possibly lower values of the m o d e index, r, should be less t h a n the arrival times of the tests w i t h t h e high strain m a x i m a and, therefore, possibly larger m o d e indices. A n inspection of Fig. 9 shows t h a t n o t only are arrival times just the opposite of such a conjecture of v a r y i n g m o d e indices, b u t t h a t t h e averages of all the d a t a p r o v i d e a fairly close a g r e e m e n t w i t h the s y m m e t r i c a l m o d e indices of r = 2 and r -- 4.

The dynamic plasticity of non-symmetrical free-flight collision impact

647

When one realizes (a) t h a t the complex three-dimensional first-development of the finite amplitude waves in the two materials must occur differently in the two specimens; (b) t h a t an interchange of energy in this non-symmetrical situation involving different mass densities m a y be expected to occur across the impact face; and (c) t h a t even though large deformation is involved in both projectile and struck specimens, the differences in the magnitude of the strains introduce an interface friction situation : it is then obvious t h a t the origin of the observed variability is the instability of the first-diameter finite-wave formation. Such first-diameter phenomena have been described in detail by the writer elsewhereS. 10-12 and need not be repeated here. In view of the general agreement of non-symmetrical impacts for equal mass densities described above, one might dismiss the non-symmetrical aluminum copper data of Fig. 9 with a warning regarding non-symmetrical impact tests with large mass ratios on the ground of experimental difficulty, were it not for the fact t h a t the stability properties of the first diameter, which might be expected to produce such a distribution, are of major interest. To pursue further this interesting initial behavior, additional experimental data were obtained in the form of surface-angle measurements, time of contact T c measurements and coefficient of restitution measurements for alumintun vs. copper non-symmetrical impact. The diffraction grating measurements of finite strain always are accompanied by the simultaneous determination of the angle of the surface normal to the incident light beam.l, s, 11, 20 I f perfect axiality is achieved in the impact of two cylindrical rods, the entire change in surface angle is produced by the lateral swelling of the rod as the dispersive finite-amplitude wave propagates along the axis. I f the impact is non-axial, then the observed surface angle includes a combination of the finite-wave contribution and the surface angle associated with the non-symmetrical collapse of the initial peak stress at the impact face. The first and second situations are easily separated, since, for an axial impact, the final surface angle at m a x i m u m strain returns to zero. Assuming incompressibility and a plane wave over the entire specimen cross-section, the writer in 1960 showed 2° that, taking Poisson's ratio as ½ for the relatively low range of finite strain considered and neglecting the small change in the predeformation radius R 0, the surface angle variation from the wave front alone is: Our Re ~e a--~ = - E ~"

=

(21)

N. Cristescu* has pointed out to the author t h a t from purely geometrical considerations, assuming only a plane wave and incompressibility, one obtains the more exact relation: R 0 ae 1 a = 2 ~x (1--e)l"

(22)

In 1961, Filbey, ~1 from similar geometric considerations, also had provided a small correction to equation (21) in terms of the measured strain e:

a = ~Re

~e (1 +~Smeas).

(23)

F o r the range of surface angles and strains considered here, i.e. a < 0.004 and e < 0.03, equation (21) is suffleiently accurate. Since wave speeds Cv are a known function of strain, we m a y write Ox -- Cv"

(24)

Thus equation (21) becomes =

* P r i v at e communication.

Re

-~ 0~"

(25)

648

JAMES F . BELL

Equation (25) applies, whether or not the wave front proceeds from an infinite step at the impact face. Since R 0 a n d C~(£) are known and ~ can be measured from the finite s t r a i n time profile, as the writer has shown earlier8, 82, ~s it is possible to compare diffraction grating surface angle data with prediction and thus justify the plane wave and incompressibility hypotheses. Such correlations have been remarkably close for both symmetrical experiments and elastic-plastic boundary experiments. ~4 If, as for strains below g, the finite amplitude waves do proceed from an infinite step at the impact face, then a special case of equation (25) m a y be written. For such a n infinite step initiation where d x / d t = x/t, differentiating equation (8) with respect to time provides: = 4C~ e

(26)

X

Substituting equation (26) in equation (25) furnishes 2R0 e a = - X

(27)

Equations (25) and (27) m a y be used separately to study the axiality and initiation behavior. I f neither equation applies, non-axiality is expected. I f equation (25) applies, b u t not equation (27), the phenomenon is a plane finite-amplitude wave front of incompressible deformation, not originating from a n infinite step at the impact face. I f both apply, not only is there a n incompressible plane wave front over the entire specimen crosssection, b u t the wave initiation is such that an infinite step at the impact face m a y be assumed. To consider whether or not the variability of the non-symmetrical impact data of Fig. 9, in contrast to the reproducibility of the symmetrical free-flight impact data, is produced b y non-axiality, the diffraction grating measured surface-angle data for the five measurements in a copper specimen with the highest m a x i m u m strain are compared with t h e Fig. 10 averaged surface-anglo data for the five diffraction-grating strain measurements with the lowest m a x i m u m strain. For these experiments, the projectile specimen was polycrystalline aluminum. The opposite situation also is shown in Fig. 10 for the averaged surface-angle data in four a l u m i n u m specimens with the highest m a x i m u m strain, compared with the four measurements in a l u m i n u m with the lowest m a x i m u m strain. I n these experiments, the projectile specimen was the copper polycrystal. One notices that in the heavy copper specimen, the variation of surface angle and thus impact axiality, is fairly small, whereas in the very much lighter a l u m i n u m specimen, there is a big variation in averaged surface angle between the high and low measurements indicating some degree of impact non-axiality in the experiment. However, spark photograph measurements such as those described earlier by the writer s, 2o have revealed that at the instant of impact the two specimens in the non-symmetrical free-flight impact situation are as accurately aligned as in the symmetrical free-flight impact situation. Therefore the lack of reproducibility in the data of Fig. 9 must be attributable to the far greater complexity of the wave initiation phenomenon in the non-symmetrical impact of solids with widely different mass densities. A new form of instability is introduced, in addition to that already observed in the symmetrical situation, in that a major transfer of energy across the impact face can occur during wave initiation, i.e. during the collapse of the initial peak stress in either specimen. The summation of calculated particle velocities for the extreme strain maxima of one material, and minima of the other, approximate the projectile velocity vB in both types of impacts of Fig. 9. To further examine that situation, the averaged surface angle a of all ten copper specimens with a l u m i n u m projectile specimens is compared with predicted values calculated from equations (25) and (27) in Fig. 10. Also in Fig. 10 is a similar comparison of the averaged surface angle of the eight aluminum specimens which have been struck b y copper projectiles. One notes that in both instances no part of the wave propagates as if from an infinite step at the impact face, despite the close correspondence of the theoretical curves (solid lines) of Fig. 9, determined from the syrmnetrical free-flight impact experiments for the same m a x i m u m particle velocities, with the averaged strain-time curves (dashed lines, Fig. 9).

T h e dynamic plasticity of non-symmetrieal free-flight collision impact

649

At this point it should be made clear that the magnitude of the surface angles determined b y diffraction gratings was exceedingly small, and the degree of accuracy of the measurement, as has been shown in the axial symmetrical free-flight impact situation, is very high. Thus, the slight differences between the solid and dashed lines of Fig. 9 are sufficient to produce the observed variation from the averaged surface-angle data shown b y the dashed lines in Fig. 10. The initial coincidence for the lower part of the strain indicates that the surface-angle variations observed in Fig. 10 do not arise from mechanical non-axiality alone, but from some form of instability in the wave initiation. The study of this phenomenon, of course, was the main reason for having performed these experiments. COPPER ~ALUMINUM

ALUMINUM~COPPER a~rad

0.004 t~

0.003

/

_

Lowest \

0.002

/

0.001

~ "~----AVERAGE ~

i

l

AVERAGE ~ for HEST..Emax TESTS

for

',,'=~---AVERAGE ~ for ~ LOWESTEmaxTESTS

i

LOwest Emax Tests 1049,956,973,959

977. 980

// J-

~ H I G

9,2,979

f-'A\~

0

Highest 6max Tests I050~ 953 t I088,1063

1105, 1117,'951t" 952,993 EmaxTests 99l

Highest 6moxTests

i

/ ./

~AVERAGE C~for LOWEST ,,I ,--E.,ox iTESTS ,

o J/"/ / 0

0

\

Q,rad

0.004 -/~

/

0.003

PREDICTED ~ = 2ROE

= 2RoE

x

I

/ ~ /.......~

--

I'1~ //I

0"00 l

/.~

\.:

~ . . . . . i. . .

50

i \'~' ' ,

I00 150 TIME , /.L sec

\~

PREDICTED

\!

J/I I/

~' /

I/

AVERAGE (~ i TESTS

0,002

l

during wove

/ P x

I

~

\ ..... ": ",.

P~ ~

,,

PREOICTEO" : Ro~__ 2 Cp ,

200

0

•~ , ,

i

50

i

~

"""..."~ .........

I00 ~ TI ME, ~L sec

I

200

FIQ. 10. Diffraction-grating-measured surface angle vs. time data for the designated non-symmetrical impacts. Upper two diagrams compare the averages for the higher half of the tests with the lower half. Lower two diagrams compare the total average with surface-angle prediction from equation (25) (dotted line) and equation (27) (dashed line).

A second series of six experiments enabled the author to study the overall effects of non-axiality and first-diameter wave initiation phenomena in terms of the time of contact To, i.e. t h e l e n g t h of time in which the two specimens remained in contact following impact. I n e a r l i e r p a p e r s , e, 3=, =s, 35 the writer described an experiment in which a uniform light beam in the horizontal plane of the specimen diameter was interrupted by the projectile specimen, causing the amount of light as viewed in a photo-multiplier tube at the opposite side of the specimen from the light source to decrease until, at contact, complete extinction occurred and continued until the specimens separated ; after separation, the observed light increased. By carefully calibrating a very uniform light field 6, 33, 23, 35 the ratio of the slope of the decreasing light and the increasing light provides a measure of the overall coefficient

650

JAMES F. BELL

of restitution, e, and t h e period of extinction provides a measure of the t i m e of specimen c o n t a c t during the impact. Two such e x p e r i m e n t s are shown in Fig. 11. I n one test, 995, t h e projectile specimen is an a l u m i n u m polycrystal and t h e struck specimen is a copper polycrystal. I n the o t h e r e x p e r i m e n t , test 996, the reverse is t h e situation, i.e. t h e copper polycrystal is t h e projectile specimen a n d t h e a l u m i n u m polycrystal is t h e struck specimen. Irrespective of the m a t e r i a l of t h e two specimens, one

l,

Tc

-I ~----

TEST

995

AI~Cu .......... T E S T

\

, Tc = 3 4 6 , 2

/.L sec

996

Cu~AI

. Tc = : 3 5 0 . 8 / J , sec

\ \

\

I 200

I 400

600 ~

800

TI M E , /.L sec

FIG. 11. Two n o n - s y m m e t r i c a l , optically determined, time-of-contact and coefficient-of-restitution experiments, one w i t h a n a l u m i n u m projectile specimen (dashed line) and one w i t h a copper projectile specimen (dotted line). All specimens are 25.4 cm long and 2.50 cm in diameter. observes t h a t t h e t i m e of c o n t a c t and coefficient of restitution are nearly identical. I n earlier papers, 8, 23, 2a the writer has shown t h a t it is possible to d e t e r m i n e t h e t i m e of c o n t a c t Tc w i t h great a c c u r a c y at all i m p a c t velocities, f r o m v e r y low values to v e r y h i g h values, b y m e a n s of e q u a t i o n (28) : L L Tc = ~.+~.

(28)

This e q u a t i o n has been shown to be in close correlation w i t h e x p e r i m e n t a l d a t a for b o t h t h e s y m m e t r i c a l free-flight i m p a c t situation of identical specimens and for the nons y m m e t r i c a l collision of a l u m i n u m a n d copper polycrystals w i t h h a r d elastic bars. The a v e r a g e e x p e r i m e n t a l t i m e of c o n t a c t for two n o n - s y m m e t r i c a l e x p e r i m e n t s shown in Fig. 11 is Tc = 348.5/zsec. As one w o u l d expect, this t i m e of c o n t a c t is i n d e p e n d e n t of which m a t e r i a l is used for t h e projectile, i.e. for t h e a l u m i n u m projectile specimen, T c = 346.2 tzsec; for t h e copper projectile specimen, T c = 350.8/~sec. F o r t h e s y m m e t r i c a l collision of 10 in. long a l u m i n u m specimens w i t h the same m a x i m u m particle velocity, t h e p r e d i c t e d t i m e of c o n t a c t from e q u a t i o n (28) is T c = 302/zsec, while for t h e s y m m e t r i c a l collision of 10 in. long copper specimens at t h e same m a x i m u m particle velocity, e q u a t i o n (28) furnishes T c -- 430/~sec, neither of which agrees w i t h the n o n - s y m m e t r i c a l value. I f one includes the i m p e d a n c e m i s m a t c h at t h e interface for t h e linear-elastic unloading w a v e t r a n s m i t t e d into t h e copper specimen, t h e calculated u n l o a d i n g t i m e is T c -- 342/zsec, which does agree w i t h experiment. I f the ratio of t h e l e n g t h of the a l u m i n u m specimen to the length of t h e copper specimen is sufficiently increased, t h e s y m m e t r i c a l t i m e of c o n t a c t calculation, e q u a t i o n (28), once again describes t h e behavior. Thus for a v/~ = 2719 cm/sec, 16 in. long a l u m i n u m projectile specimen and an 8 in. long copper

The dynamic plasticity of non-symmetrical free-flight collision impact

651

struck specimen, the non-symmetrical experiment 1096 provided Tc = 479.9/zsec. The symmetrical aluminum prediction for the velocity v= is T c = 486 ~sec, and for copper, Tc = 349 tzsee. The predicted symmetrical aluminmn value now agrees with the nonsymmetrical experimental observation. As a further check on the behavior, the projectile and struck specimens were reversed and the projectile velocity was reduced from 2700 cm/sec to 984.5 cm/sec. The experimentally measured non-symmetrical Tc is 358/+see. The predicted symmetrical value from the 16 in. long aluminum specimen is T c = 365/~see, which once again agrees with observation. I n Table I are shown the time of contact and coefficient of restitution data and the time of contact predictions for all six of the experiments referred to above. TABLE 1

Test No.

Projectile specimen (in.)

Struck specimen (in.)

Tc (experimental)

Tc (calculated)

vH

vI

(cm/sec)

(cm/sec)

994 995 996 997 1095

10A1 10A1 10 Cu 10 Cu 8 Cu

10 Cu 10 Cu 10hl 10 AI 16A1

333 346 350 357 358

342 342 342 342 365

2691 2704 2706 2715 985

241 257 --736

0.111 0.143 0.154 0.114 0.125

1096

16A1

8 Cu

480

486

2712

824

0.024

In all thirty-one of the copper vs. aluminum experiments here described, both the initial projectile specimen velocity and the final struck specimen velocities were measured by means of an electrouie chronograph. These data are tabulated in Table 2 and are grouped in terms of specimen length. The aluminum final velocities are reproducible at each specimen length, with the value decreasing as the specimen length increases. The copper final velocity data obviously are not reproducible, although a number of individual values are seen to be repeated. I n view of the reproducible final velocity of aluminum after impact, this variability of the copper specimens' final velocities is curious indeed, unless one again reflects upon t h e instability properties of the dual first-diameter finite-amplitude wave initiation situation on either side of the non-symmetrical impact face. The coefficient of restitution data of Table 1 shed some light upon this problem. The averaged coefficient of restitution for the four 10 in.-10 in. experiments is e = 0.130, again, independent of which metal was used for the projectile specimen. For the averaged projectile velocity, 2710 cm/sec, the separation velocity after impact as determined from the coefficient of restitution would be 353 cm/see, which, subtracted from the final averaged velocity of the struck aluminum specimen of 2260 cm/sec, provides a final velocity for the copper specimen of 1920 cm/sec; this is almost precisely the same as the m a x i m u m interface particle velocity of 1900 cm/see during the initial phases of impact. Thus the copper specimen is unloaded without acquiring any additional velocity from the unloading wave reflection at the free end of the specimen. On the other hand, if this same coefficient of restitution is used for the 10 in.-10 in. situation with the aluminum projectile specimen, the predicted final velocity of the copper specimen would be 806 cm/sec, which is much higher than an y of the observed 10 in.-10 in. measurements in Table 2. F o r the 16 in.-8 in. test 1096 described above for which the coefficient of restitution is 0.024, the experimental final value of the struck copper specimen is 824 cm/see, which is precisely the interface m a x i m u m particle velocity during wave propagation predicted from equation (13) from the projectile velocity, vH = 2712cm/sec. The calculated separation velocity in terms of a coefficient of restitution of e ---- 0.024 is 65 cm/see, proriding for the aluminum struck specimen a predicted final velocity of 1835 cm/sec. This is in close agreement with the averaged measured value of 1825 cm/sec for the six 8 in. copper vs. 16 in. aluminum experiments of Table 2 (with an averaged V H of 2856 em/sec).

2691 2700 2697 2695

2839

2818 2888 2833 2859 2882 2856

2760 2414 982

953 956 959 973

1049

1067 1063 1061 1066 1088 1062

1097 1094 1095

T e s t No.

1192 733 736

1857 1857 1845 1832 1778 1781

2040

2319 2254 2254 2220

8 8 8

8 8 8 8 8 8

8

10 10 l0 10

16 16 16

16 16 16 16 16 16

12

10 10 l0 10

Length of struck specimen L2 (in.)

2679 2610 2700 2645 2690 2712 1168

1105 1117 1096 981

2672 2691

2706 2618 2699 2665

992 991 993

951 994

977 952 980 979

T e s t No.

Initial projectile velocity v17 (em/sec)

313

1110 1100 824

945 491 215

243 241

685 671 541 536

10

16 16 16

12 12 12

l0 l0

10 10 l0 10

Length of projectile specimen LI (in.)

10

8 8 8

8 8 8

10 l0

10 10 10 10

Length of struck specimen Ls (in.)

Final struck specimen velocity vl (era/see)

Length of projectile specimen L1 (in.)

Final struck specimen velocity vl (cm/sec)

Initial projectile velocity vR (cm/sec)

A l u m i n u m p r o j e c t i l e s p e c i m e n vs. c o p p e r s t r u c k s p e c i m e n

Copper projectile s p e c i m e n vs. a l u m i n u m s t r u c k s p e c i m e n

TABLE 2

The dynamic plasticity of non-symmetrical free-flight collision impact

653

All the final copper struck specimen velocities of Table 2 (with vH ~ 2700 cm/sec) have the averaged value of 633 cm/sec, which is considerably below the value predicted from the coefficient of restitution measurements of 824 cm/sec. This variability of the copper specimen final velocities, when compared with the reproducibility and predictability of the a l u m i n u m specimen final velocities, provides further experimental evidence that the nonreproducibility of individual tests in the strain-time data of Fig. 9 for the rising loading wave is associated with variations in the initial velocity partition at impact. All the diffraction grating data above were for the same approximate projectile velocity of 2700 cm/sec. The problem of describing this non-symmetrical situation with large mass ratios is possibly even more difficult when one observes the increase in the arrival time delays when the projectile velocity is lowered. This may be seen in the comparison of the diffraction-grating-determined finite strain-time profile (circles) with prediction (solid lines) of Fig. 12 at the lower projectile velocity of vH --- 1290 cm/sec. COPPER

~ALUMINUM

Test 982

6 0"008

Grating position 2"

e e

ee

r~ o o

0'006

® theoretical r=2

e

ee

R t i Ema x

0"0O4 o

0'002 H H t e r v e l o c i t y = 1290cm/sec

oJ 0

I 50

I I00

TIME,

I 150

l 200

/_t sec

FIG. 12. Diffraction-grating strain-time profile (circles) for a copper projectile specimen and an a h u n i n u m struck specimen is compared with prediction (solid line) for the designated lower impact velocity. For all of the finite strain data of Fig. 9 at vii --- 2700 cm/sec, the 0.005 in. long, 30,720 lines-per-in, diffraction gratings were ruled at two diameter lengths from the impact face, i.e. at 5.08 cm. As seen below in Fig. 13, three diffraction grating experiments at one, two and three diameters from the impact face, for projectile velocities decreasing in the approximate ratio of the strain maxima b u t in the same range as those of Fig. 9 (vH = 2200 cm/sec), exhibit the finite-amplitude wave nature of the non-symmetrical impact. The thirty-one experiments described in this section of the paper were carried out in order to study the first-diameter wave initiation phenomena when the instability associated with the initial peak stress collapse differed between the projectile and struck specimens. The complications of this non-symmetrical impact situation when the mass density ratio is high should suggest caution to those engaged in oversimplified interpretations of interface experiments involving different materials. The experiments described above certainly provide a n excellent opportunity for further study of this interesting wave initiation behavior. I f the observed instability of the finite-amplitude wave initiation arises from the variability of the first-diameter wave initiation in the two soft specimens for nonsymmetrical impacts with specimens having a large mass ratio, then one would expect that this type of instability would not be observed for a non-symmetrical impact of a single soft specimen with a hard elastic bar having the same large mass ratio. That this is the precise situation m a y be seen from the experiment of Fig. 14 in which a 10 in. long, 1 in. diameter

654

JAMES F. BELL E 0"025 rr--

COPPER

- ALUMINUM _~

Test 1067

'~";- ; " ~ Test 956 0"020

./"

-

o o

/

0.00

/

/,e/

test number

velocity hitter spec. grating (cm/sec) length length posit/on

/ / d/

0

~ •0

109,4.

I 50

1 100

24.20

I 150

,0.

,0.

8"

t6'

I 200

3'

I 250

T I M E , .,u.sec

FIG. 13. T h r e e d i f f r a c t i o n - g r a t i n g s t r a i n - t i m e profiles (circles) for a polycrystalline copper projectile specimen and a polycrystaUine a l u m i n u m s t r u c k s p e c i m e n for t h e d e s i g n a t e d l e n g t h s . T h e s e d a t a were o b t a i n e d a t one, t w o a n d t h r e e d i a m e t e r s , respectively, in t h e s e 2.50 c m d i a m e t e r specimens. T h e m a x i m u m s t r a i n s are i n a p p r o x i m a t e l y t h e s a m e r e l a t i o n s h i p as are t h e i m p a c t velocities. Test 647,

o-t psi

ANNEALED

12,000

COPPER--~

HARD

ALUMINUM

r-Theoretical O-max

8,00C

~

"

~,,.,......

4,00C

predicted unloading ifrom free end of 16foot aluminum ~ hard ba r

0 Cpredicted from symmetrical impact )

- 4,000

I I

- 8,000

0

I

I

I

I

I

I

I00

200

300

400

500

600

TIME,

].2. sec

700

I

I

800

900

FIG. 14. S t r e s s - t i m e d a t a c o m p a r e d w i t h p r e d i c t i o n [ e q u a t i o n (16)] for a n a n n e a l e d c o p p e r projectile s p e c i m e n s t r i k i n g a h a r d a l u m i n u m s p e c i m e n for t h e large m a s s r a t i o of 3.3.

The dynamic plasticity of non-symmetrical free-flight collision impact

655

annealed copper polyerystalline specimen having a parabolic stress-strain function with the usual r = 4 for t h a t material is in non-symmetrical impact with a 6 ft long, 1 in. diameter, hard linear-elastic aluminum bar. The mass ratio still has the relatively large value of 3"3. Substituting the appropriate parameters in equation (16) provides the close agreement between prediction from the copper parabolic stress-strain function and the observed m a x i m u m stress measured in the hard aluminum bar. One notes t h a t although the predicted time, T z, of the initial unloading from the free end of the copper specimen is the same as for the non-symmetrical impact, the impedance mismatch for the tmloading wave arriving at the impact face in the copper specimen extends the time of contact, To, for the non-symmetrical impact to such an extent t h a t the actual unloading occurs from the linear-elastic reflection from the free end of the 6 ft long hard aluminum bar. A second experiment demonstrating the non-symmetrical impact predictability for large mass ratios when either the projectile or struck specimen is a linear-elastic solid m a y be seen in Fig. 15. I n this experiment, a 20 in. long, 1 in. diameter, hard linear-elastic

HARD A L U M I N U M • 7 0 - 3 0 CZ- BRASS E (Hartman) Test 1170 :500°K Predicted 0,055 Grating position 3.03" unloading strain ® ~-°--ff-O--OO-~

0.030 0'025

t°i ...... ! °/

--

°1 Predicted time _/of unloading

0'020-

theoretical ~ ~ / r=4 e /

0.015 -

/

o,oio

/ 0.005

Hittervelocity =

/

5148 cm/sec

i", ®~ e O ° e ® 0 ~ ° e e , ~ . . . . . / I e° I 0 50 I00 150 £Z, rad 0.007o . o o 6 -

0"005

I 250

I 300

.

vv

e R_~ ~ •.........

-

2Ro6

---a:---q-

0'004 -

0.003

I 200

"

r~

I/i

:~:/ /i//

-

!

~..

',

-

~

Measured surface

angle

0.002

0.001

-~

-

o ~.- - ~ - ~ = ~ 0 50 . Ioo

/

i i 150 200 T I M E , /_L sec

FIG. 15. Diffraction-grating-measured

I

250

i

300

finite strain-time

and

surface

angle-time data (circles) compared with theoretical prediction from equation (15) for a hard aluminum projectile specimen striking an annealed a-brass specimen with the mass ratio of 3.2. 44

656

JAM~S F. B ~

aluminum bar is in non-symmetrical impact with a 17 in. long, 1 in. diameter, annealed 70-30 s-brass struck specimen governed by a parabolic stress-strain function having the mode index r -- 4. The mass ratio for s-brass and aluminum is 3.16. One notes from Fig. 15 that the calculated arrival times from equation (9) (solid line) are in close agreement with the diffraction-grating-measured finite strain data (circles). Because of the relatively short length of the hard elastic aluminum projectile, unloading from its free end arrives at the 3-03 in. diffraction-grating position in the annealed s-brass bar before the maximum strain is reached. One observes that the arrival time of this unloading wave is in very close agreement with prediction. Shown in Fig. 15 is a comparison of the diffraction-grating-measured surface angle (circles) with the calculated behavior based upon the plane wave and incompressibility hypothesis [equation (25)], derived solely from knowledge of wave speeds and measured strain-time slopes and on the assumption of an infinite step at the impact face [equation (27)]. One sees that although the wave initiation in this large mass ratio non-symmetrical impact situation is not that of an infinite step at the impact face, nevertheless, the rising portion of the surface angle up to a strain of approximately 2 per cent is in close agreement with the surface-angle behavior prediction based upon the plane wave and incompressibility hypothesis for a wave front which has undergone the three-dimensional distortion in the first diameter. Thus one sees from the experiments of Figs. 14 and 15 that the observed close agreement with prediction from the appropriate parabolic and linear-stress-strain functions demonstrates that the observed variability of the high mass ratio of the non-symmetrical impact data of Fig. 9 does indeed arise from instability in the non-symmetrical dual wave initiation phenomenon in the two first diameters. CONCLUSIONS The test of a n y new generalization is its continuing ability to a c c u r a t e l y predict behavior p a t t e r n s in each new situation which arises. M a n y experimentalists a n d some theorists, still speculating on the possible i m p o r t a n c e of a viscous influence in d y n a m i c plasticity, base their conjectures upon measurem e n t s from the dubious extended quasi-static i m p a c t experiments, upon posti m p a c t m e a s u r e m e n t s from load bar experiments in which finite-amplitude w a v e fronts are n o t examined, a n d u p o n the analysis of s t r a i n - t i m e profiles measured b y electric resistance gages which have been k n o w n for two decades to contain m a j o r errors for the high strain rates of d y n a m i c plasticity, even at strain levels well below 2 per cent. I n the present paper, the non-viscous, linearly t e m p e r a t u r e - d e p e n d e n t generalized parabolic stress-strain function discovered without the prior a s s u m p t i o n of either a w a v e - p r o p a g a t i o n t h e o r y or constitutive equation, from accurate diffraction-grating m e a s u r e m e n t s of finite strain-time profiles in s y m m e t r i c a l free-flight impacts, is shown here to be extendable to the nons y m m e t r i c a l free-flight collision i m p a c t situation. The n o n - s y m m e t r i c a l i m p a c t s described here (which include projectile a n d s t r u c k specimens at v e r y different a m b i e n t temperatures, single crystal striking p o l y c r y s t a l a n d vice versa, h a r d bars in axial collision with b o t h annealed polycrystals a n d single crystals, a n d n o n - s y m m e t r i c a l impacts between different elements in various metallurgical states) have further d e m o n s t r a t e d the wide range of physical and metallurgical conditions for which the non-viscous finite-amplitude wave t h e o r y is applicable and for which the writer's parabolic generalized stress-strain f u n c t i o n is the governing high strain-rate constitutive equation. I t is t h o u g h t t h a t these d a t a provide a m a j o r extension of the field of d y n a m i c plasticity.

The dynamic plasticity of non-symmetrical free-flight collision impact

657

Acknowledgements

This research was sponsored by the United States Army, Army Research Office {Durham). The author wishes to express his appreciation to research assistants, J o h n Gottschalk, Elizabeth Blumenthal and F a i t h Paquet. He would particularly like to t h a n k Research Associate C. Jeffus for work pertaining to the a l u m i n u m vs. copper data. REFERENCES 1. J. F. BELL,J. appl. Phys. 27, 1109 (1956). 2. G. I. TAYLOR,British Ministry of Home Security, Civil Defence Research Committee Report, RC 329 (1952). 3. T. y o n KARMAN, NDRC Report A29, OSRD 365 (1942). 4. K. A. RAKHMATULIN,Sov. J. Appl. Math. Mech. (Prikl. Mat. Mekh.) 9, 91 (1945). 5. M. P. WHITe, and L. GRIFFIS, NDRC Report A72, OSRD 742 {1942). 6. J. F. BELL, J. appl. Phys. 32, 1982 (1961). 7. J. F. BELL, J. appl. Phys. 34, 134 (1963). 8. J. F. BELL, The Physics of Large Deformation of Crystalline Solids, Springer Tracts in Natural Philosophy, 14. Springer-Verlag, Berlin, Heidelberg, New York (1968). 9. W. F. H~a~TMAN, Ph.D. Thesis, The Johns Hopkins University (1967). 10. J. F. BELL, J. appl. Phys. 31, 2188 (1960). 11. J. F. BELL, I U T A M Symposium, Brown University (1963), p. 166. Springer-Verlag, Berlin {1964). 12. J. F. BELL, Proc. Symposium on the Mechanical Behavior of Materials under Dynamic Loads. Springer-Verlag, New York (1967). 13. J. SPEm~ZZA, Proc. 4th U.S. Natn. Congress Appl. Mech. 2, 1123 (1962). 14. J. F. BELL, Phil. Mag. 10, 107 (1964). 15. J. F. BELL, Phil. Mag. 11, 1135 (1965). 16. G. I. TAYLOR, J. Inst. Metals LXII, 307 (1938). 17. J. F. W. BISHOP and R. HILL, Phil. Mag. XLH, 1298 (1951). 18. W. BOAS and E. SCHMID, Plasticity of Crystals. F. A. Hughes, London (1936). 19. W. J. GILLICH, Phil. Mag. 15, 659 (1967). 20. J. F. BELL, J. appl. Phys. 31, 277 (1960). 21. G. L. FrLBEY, Ph.D. Thesis, The Johns Hopkins University (1961). 22. J. F. BELL, J. Mech. Phys. Solids 9, 1 (1961). 23. J. F. BELL, J. Mech. Phys. Solids 9, 261 (1961). 24. J. F. BELL, J. Mech. Phys. Solids 16, 295 {1968). 25. J. F. BELL and J. H. SUCKLING, Proc. 4th U.S. Natn. Congress Appl. Mech. 877 (1962).