On the Use of the Ring Test for Determining

Rate-sensitive Material Constants

A general technique is described of utilizing the explosive-ring test in conjunction with recently developed analytical techniques to determine the material parameters governing strain hardening or perfectly plastic rate-sensitive flow

by Nicholas Perrone

ABSTRACT--An explosive-ring test is reappraised in light of recently developed material-behavior models and analytical predictive techniques. It is demonstrated in Complete detail how this test may be utilized to deter- mine the uniaxial flow laws of rate-sensitive perfectly plastic and strain-hardening materials.

Introduction M a n y structural metals possess dynamic material properties tha t differ appreciably from their static counterparts. As one might expect, experimental determination of these dynamic properties is an extremely difficult task. Of course this information must be provided to the analyst before he can at- t empt to predict the dynamic response of rate- sensitive structures.

For the vast major i ty of structural metals, appropriate material constants are unknown for strain rates up to the order of thousands of inches per inch per second. Moreover, no s tandard test is available for their determination.

A brief review of available test procedures should prove helpful. For a more exhaustive survey of studies on changes in physical properties with strain rate, the reader is referred to a paper by Henriksen, et al. 1

Manjoine conducted a series of constant-velocity tensile tests to determine the rate sensitivity of mild steel. ~ While this testing procedure is ade- quate for modest s t ra in rates, it suffers from the influence of wave-propagat ion effects which are significant at higher strain rates.

Hopkinson pressure bars have been utilized to

Nicholas Perrone is Acting Head, Structural Mechanics Branch, Office of Naval Research; also, AdjUnct Professor of Mechanics, The Catholic University of America, Washington, D. C.

determine dynamic material properties.~, 4 Al- though promising, pressure-bar techniques are not without limitations. Mushrooming of the speci- men wafer may introduce complicating two-dimen- sional effects. 5 In addition, inertia and frictional effects also present considerable difficulties. 6

A novel experimental approach of approximately assessing rate-sensitivity effects by indentation with

DIRECTION OF OBSERVATION

/ EXPLOSIVE (COMP. C-5)

, , ~ ~ DE TON ATOR

EXHAUST TUBE

~ ~ SUPPORT ROD

Fig. 1--Ring and core assembly o

232 I May 1968

conica l i n d e n t e r s is w o r t h no t ing . 7 O t h e r nove l t e chn iques are be ing s t u d i e d which are, a t t h i s t ime , s t i l l in a p r e l i m i n a r y i n v e s t i g a t i v e s t a g e 3

A c lever m e t h o d for d e t e r m i n i n g d y n a m i c m a t e - r ia l p r o p e r t i e s was u t i l i zed b y J o h n s o n , S te in , a n d D a v i s 2 T h e t e s t spec imen is a c i r cu la r r ing which is s n u g l y f i t t ed a r o u n d a core (Fig. 1). T h e core is ho l lowed, filled w i th an exp los ive m a t e r i a l , a n d ign i ted . A r a d i a l wave is s en t ou t wh ich t r a n s - m i t s a u n i f o r m pulse to t he r ing spec imen , which in t u r n m o v e s r a d i a l l y o u t w a r d . T h e m o t i o n of t he r ing is r e c o r d e d b y a h i g h - s p e e d camera , f r o m which the c i r c u m f e r e n t i a l s t r a i n r a t e m a y be d e t e r m i n e d . T h e d y n a m i c flow s t ress is c a l c u l a t e d f rom the e q u a t i o n of mo t ion . One o f t h e diff icul t ies en- c o u n t e r e d b y Johnson , S t e in a n d D a v i s was in t h e d e t e r m i n a t i o n of t he r ing dece le ra t ion . T h e y e s t i m a t e d t h a t t he n u m e r i c a l c o m p u t a t i o n of second d e r i v a t i v e s of e x p e r i m e n t a l l y o b s e r v e d d i sp lace - m e n t - t i m e cu rves i n t r o d u c e d e r ro r s of + / - 1 0 per- cen t on the ave r age and in e x t r e m e s i t u a t i o n s u p to a b o u t 20 percen t .

A l t h o u g h an a c c u r a c y p r o b l e m exists , t h i s m e t h o d does h a v e the def in i te a d v a n t a g e of n o t be ing seve re ly l im i t ed b y w a v e - p r o p a g a t i o n effects, even for v e r y h igh s t r a i n ra tes . I n a d d i t i o n to u t i l i z ing th i s m e t h o d in t he m a n n e r sugges t ed in Ref . 9, a modi f i ed a p p r o a c h is poss ib le which w o u l d enab l e a d i r ec t d e t e r m i n a t i o n of a r a t e - s e n s i t i v i t y law. N o a t t e m p t was m a d e b y J o h n s o n and his asso- c ia tes to co r re l a t e t he i r e x p e r i m e n t a l o b s e r v a t i o n s wi th an a n a l y t i c a l m o d e l of a r a t e - s e n s i t i v e flow law.

A t t e n t i o n is focused in t he p r e sen t p a p e r on the p romis ing e x p e r i m e n t a l t e c h n i q u e d i scussed in Ref . 9. T h e a p p l i c a t i o n of th is a p p r o a c h to d e t e r m i n e the m a t e r i a l c o n s t a n t for pe r f ec t l y p las t i c or l inea r s t r a i n - h a r d e n i n g m o d e l s is d i scussed in de ta i l .

Perfectly Plastic Model One m o d e l of m a t e r i a l b e h a v i o r which r ep re -

sen ts a r e a sonab l e c o m p r o m i s e b e t w e e n p h y s i c a l r e a l i t y and m a t h e m a t i c a l t r a c t a b i l i t y is t he pe r f ec t l y p la s t i c one w i th r a t e - s e n s i t i v e y ie ld s t ress . A n e x a m p l e of such a flow law is as fol lows (Fig. 2) :

~/~o = 1 + (~/D) ' / " (])

where

n,D = m a t e r i a l c o n s t a n t s = s t r a i n r a t e = d y n a m i c y ie ld s t ress

ao = s t a t i c y ie ld s t ress

A n u m b e r of i n v e s t i g a t o r s h a v e u t i l i zed success fu l ly t he s t r e s s - s t r a i n r a t e l aw of eq (1) to p r e d i c t t he r e sponse of i m p u l s i v e l y l o a d e d c a n t i l e v e r beams.~0, ~

Shou ld a m a t e r i a l obey a s t r e s s - s t r a i n r a t e l aw c o n t a i n i n g two m a t e r i a l c o n s t a n t s [such as n a n d D of eq (1) ], t hese would be d e t e r m i n a b l e i f t he s t ress a n d s t r a i n r a t e were k n o w n for a n y two po in t s on t h e curve . I n Ref. 12, the a u t h o r has shown t h a t

%

4.0

N U

3.0 o

Q 2 . 0

>-

I,O 0.1

[ I

, / , +

Z TE L

I I I I0 I 00 I 0 0 0

STRAIN RATE ( S E C - ' ) I0,00

Fig. 2--Yield s t ress -s t ra in rate law

t h e f inal d e f o r m a t i o n of a n i m p u l s i v e l y l o a d e d r ing of a p e r f e c t l y p l a s t i c r a t e - s e n s i t i v e m a t e r i a l g iven b y eq (1) is a c c u r a t e l y g iven b y the fo l lowing ex- p ress ion :

x ' = p ~ / ( 2 a ) (2)

where

x ' = f inal s t r a i n of m e d i a n r a d i u s p = m a s s d e n s i t y

= i n i t i a l r i ng v e l o c i t y = in i t i a l d y n a m i c s t r ess

T h e bas is for t he d e r i v a t i o n of eq (2) is t h e ob- s e r v a t i o n t h a t t he s t r a i n r a t e is s u b s t a n t i a l l y con- s t a n t d u r i n g t h e d o m i n a n t p o r t i o n of p l a s t i c f low (see Fig . 3 which is a r e p r o d u c t i o n of a t y p i c a l t e s t f rom Ref . 9). Hence , t he y ie ld s t ress m a y p r o p e r l y be t a k e n to be a c o n s t a n t a s soc i a t e d w i t h i t s in i t i a l wtlue. 12

B y m e a s u r i n g the in i t i a l r i ng ve loc i ty , f inal r ing d e f o r m a t i o n , a n d u t i l i z ing eq (2), t h e in i t i a l d y - n a m i c y ie ld s t ress ~ a s soc i a t ed w i t h t he i n i t i a l o b s e r v e d s t r a i n r a t e ~ for a g iven m a t e r i a l can be

~ 6 Z Lal ~ 5 rY h i

v 4

I-- ff)

2 [

~_'___ ~ 0 ~ ~ .~. 99.99% ALUMINUM

, . / ' K i I I i i I I I I I I h I I I I I I I 0 5 I0 15 20

TIME (FRAME NUMBER)

Fig. 3 - -Response of impuls ively loaded ring s

25

Experimental Mechanics 233

~ 2 . r

N

$

LC

w

determined. Hence, two experiments with differ- ing initial ring velocities should suffice to determine the two material constants. Relative to the mea- surements required in Ref. 9, experimental deter- minat ion of the final ring deformation and the initial ring velocity should be comparat ively simple tasks. Moreover, experimental errors will not be amplified nearly as much as in Ref. 9 since only one (the initial) first derivative of the experimental de- formation time plot is required rather than a con- t inuous spectrum of second derivatives.

We proceed now to demonstrate in detail how two tests may be utilized to find n and D for the spe- cific flow law ofeq (1).

Assuming the material obeys eq (1), we obtain for the two tests

~l/Cro = 1 -~- ( t l / D ) 1In (3a)

~2/~o = 1 + ( t . . /D) ~/~ (3b)

where 1 and 2 subscripts correspond to the two tests, t corresponds to the initial strain rate.

From eq (2) and the definition of the strain rate, we have :

~, = p~t~2/(2xt ') ~.,. = ps ') (4a)

t,, = ,a~/~ t2 = '~2/~ (4b)

where ~ = initial median radius. Of course, the r ight-hand sides of eqs (4) and

hence, ~ and t terms are experimentally deter- minable. Therefore, eqs (3) represent two equa- tions in two unknowns, n and D. Eliminating D and making use of equation (4b) we find

ln(~,/~.2) n = ln{ [(p(t12/2x~'ao) - 1] /[ (p( t22/2x2 'ao) - 1]} (5)

Subst i tut ion back into eqs (4) and (3) gives D the other material constant.

"Experimental" Example for Perfectly Plastic Material

To illustrate more pointedly the use of the fore- going equations to determine the rate-sensitive

J J j J

x "EXPERIMENTAC POINTS I

DER,VEO CURVE: .~'= I * t ~

O.I 1.0 IO I00 I000 STRAIN RATE, ~ (I/SEC)

Fig. 4--Example of calculated stress-strain rate law-- perfectly plastic material

material constants, let us consider a f i c t i t i o u s e x p e r i - m e n t with rings of 1-in. radius. As outlined earlier, two ring experiments are performed from which the initial ring velocity and final ring deformation are recorded as follows:

~ = 2000 ips x~' = 0.015 = l i n .

~2 = 1000 ips x2' = 0.004

Substi tut ion of these "experimentally observed" quantities into eq (5) immediately provides the first material constant: n = 6.245. Subsequent use of eqs (4b) and (3) yields the second constant D = 181.5.

A plot of the associated yield stress-strain rate law is given in Fig. 4 with the "experimental" points clearly noted.

Strain-hardening Model Should strain hardening be deemed significant,

the following modified rate-sensitivity law is pro- posed:

a/Zo = [1 + (~ /D) ~/'1 (1 + co) (6)

where c is a strain-hardening coefficient. The above strain-hardening rate-sensitive flow

law const i tutes a special case of a general product- type law previously suggested.l~ Linear hardening is chosen for mathematical expediency.

In the flow law of eq (6), the hardening coefficient is taken to be independent of strain rate; for the limiting static case, the flow law reduces to the familiar uniaxial form

~I~o = 1 + ce (7)

On the other hand, for the limiting perfectly plastic case with no strain hardening (i.e., c = 0) eq (6) reduces to eq (I).

Recent work by Lindholm 4 suggests that , for lead and aluminum, the strain hardening is approxi- mately linear and independent of the strain rate. However, other experimental results 14 suggest tha t the strain hardening for some materials may be in- fluenced by strain rate. Should this be the case, the localized nature of the flow in each strain rate zone should permit the use of piece-wise constant values of the hardening coefficient which may vary from one zone to another. ~3 Of course, for any specific problem, c would be a constant appro- priate to the zone of the initial strain rate of flow.

To calculate the dynamic value of c, one more experiment than in the perfectly plastic case would be necessary. Although a minimum of three ex- periments would be required to determine the rate sensitivity and strain-hardening constants, clearly a greater number of experiments should produce more accurate values of the material constants.

In order to predict the response of an impulsively loaded strain-hardened ring which obeys eq (6), a modification of a previous rate-sensitive perfectly plastic analysis ~2 is necessary. For definiteness the following parameters and variables are specified:

234 I M a y 1968

= i n i t i a l d y n a m i c s t r ess

= c u r r e n t d y n a m i c s t ress

= in i t i a l r i ng m e d i a n r a d i u s

r ' = c u r r e n t r ing m e d i a n r a d i u s

r = r ' / ~ = d imens ion les s c u r r e n t m e d i a n r a d i u s

x = r - 1 = d imens ion les s d i s p l a c e m e n t or s t r a i n

x l = f inal d imens ion les s d i s p l a c e m e n t

t = t i m e

p = m a s s d e n s i t y

~{ = in i t i a l r i ng c ross - sec t iona l a r ea

u = d r ' / d t = r ing ve loc i t y

= in i t i a l r i ng v e l o c i t y V ~ U 2 / U 2

= 2~ / (p~ ~)

Gene ra l l y , b a r r e d va lues refer to in i t i a l cond i - t ions .

E n f o r c e m e n t of N e w t o n ' s s econd l aw re su l t s in t he fo l lowing e q u a t i o n gove rn ing r ing m o t i o n :

p~,4 d2r' ~ A (8) dt 2

Fo l lowing the a r g u m e n t s set f o r t h in Refs . 12 a n d 13, k ine t i c e n e r g y is t r a n s f o r m e d in to p l a s t i c work before s ign i f i can t ve loc i t y changes t a k e p lace .* E x p e r i m e n t a l o b s e r v a t i o n s conf i rm t h a t t h i s is in- deed the case, even for s t r a i n - h a r d e n i n g ma te r i a l s . 8

I n o t h e r t e rms , we a s s u m e d p las t i c flow com- mences wi th t he pe r f ec t ly p l a s t i c d y n a m i c y ie ld s t ress , ~, a n d conc ludes a t a h ighe r s t r a i n - h a r d e n e d s t ress a s soc i a t ed wi th the final s t r a i n v a l u e ( A B in Fig . 5).

= ~ (1 q- ee) (9a)

= ao [1 + (~ /D) '/"] (9b)

B y u t i l i z ing the i n c o m p r e s s i b i l i t y cond i t i on , eqs (9), a n d an i n d e p e n d e n t v a r i a b l e c h a n g e ( f rom t to r) , eq (8) t a k e s t h e fo l lowing d imens ion l e s s f o r m

dv o~ [1 + c ( r - l ) ] (10)

dr - r

wi th in i t ia l c o n d i t i o n

v(1) = 1

A fu r the r , b u t s t r a i g h t f o r w a r d , v a r i a b l e c h a n g e f rom r to x t r a n s f o r m s eq (10) to t he fo l lowing

dv a (1 + cx) (11)

dx l + x

I n t e g r a t i o n o f eq (11) b e t w e e n a p p r o p r i a t e l i m i t s

f f f f o x~ ~ ( l + c x ) d x dv = - (1 + x)

y ie lds t he fo l lowing so lu t i on

= 1 / [ c x l + (1 - c) ln(1 + x l ) ] (12)

W h e n c van i shes , eq (12) r educes to t he r e s u l t in Ref . 12 for a p e r f e c t l y p la s t i c ma te r i a l . S u b s t i t u t i o n

~ T h a t is much less than an order of magnitude.

~ D

%

A8 IDEALIZED FLOW

ACD ACTUAL FLOW C E

STRAIN RATE

Fig. 5 - -S t ra in -harden ing rate-sens i t ive f low mode l for impu ls ive load

for a, in eq (12) r e ve a l s a r e l a t i o n b e t w e e n the in i t i a l flow s t ress a n d the f inal d e f o r m a t i o n .

= p~2/2[cxs + (1 -- c) In (1 + xl) ] (13)

A t l ea s t t h r e e t e s t s a re n e c e s s a r y to d e t e r m i n e the m a t e r i a l c o n s t a n t s n, D, a n d c. T h e p e r t i n e n t gove rn ing e q u a t i o n in which the t h r e e se ts of t e s t va lues are s u b s t i t u t e d , in t u rn , is f o u n d b y e q u a t i n g eqs (13) a n d (9b)

1 q- ( ~ / D ) 1In = p(t~2/(2~o[CX~ + (1 -- c) X in (1 + x~)]) (14a)

1 + (~2/D) ~/" = pft22/(2ao[CX2 -{- (1 -- c) X In (1 + x2)]) (14b)

1 + ( & / D ) ~/~ = p ( ~ 2 / ( 2 ~ [cx~ + (1 - - c) X In (1 + x~)]) (14c)

where s u b s c r i p t s re fer to t e s t n u m b e r s . E l i m i n a t i o n of n a n d D b e t w e e n t h e t h r e e e q u a -

I.O x "EXPERIMENTAL" POINTS !

" I r 3 8 8 8 j ~ER,VED co.vEs: ~ : I,+,o.o .I [, +(~.--~ -)

f 1

~ LO IO IOO Iooo STRAIN RATE, ~ (I/SEe)

Fig. 6 - -Examp le of ca lcu la ted s t ress-s t ra in rate law- - s t ra in -harden ing mater ia l

/

/

7 ~

I0,000

Experimental Mechanics t 235

tions results in a single transcendental equation in C:

In (~1/~) n =

In [(1 -- pgZl~/{2~o[CX~ A- ( 1 - c ) In ( l + X l ) ] } / (1 -- pfi2~/{2~o[cx2 -4- (1--c) In (1--x~)]})]

In (~1/~3) In [(1 - p~2/{2ao[cx~ A- ( 1 - c ) In (1+x~)]})/

( 1 - p~3~/{2ao[CX~ -? (1 - c ) In (1 ~-x3)]})] (15)

By making use of the relation between strain rate and velocity [eq (4b)], we m ay recast eq (15) into the following more useful form:

In (~1/fi2) n =

In [(1 - pftl~/{2ao[CXl + ( 1 - c ) In (1 A-x1)]})/ ( 1 - p~2~/{2ao[cx2 + ( l - c ) I n ( l+x2)]}) ]

In (al/a3)

In [(1 - p~2/{2~o[cXl -4- ( 1 - c ) In (1-~xl)]})/ ( 1 - p~t32/{2~o[cx~ -? (1 - c ) In (1 ~-x3)]} )]

(16)

The value of c can be found by solving eq (16) by any s tandard numerical approach. Subsequent determination of n and D from eqs (13) and (14) is straightforward.

"Experimental" Example for Strain-hardening Material

For the determination of the rate-sensitive strain- hardening material parameters, we proceed as in the earlier perfectly plastic case to conduct a series of fictitious experiments. More specifically, three "experiments" are run on 1-in.-radius rings from which the initial velocity and final ring deformation are recorded as follows:

z~, = 4000 ips xl ' = 0.0800 ~2 = 2200 ips xJ = 0.0259 ~3 = 1000 ips xJ = 0.0059

Substi tut ing these "experimental" data into eq (16) and solving for the hardening coefficient c, we find

c = 10.0 (17a)

From eqs (15) and (14) we can solve for n and D, respectively

n = 3.888 (17b)

D = 7475 (17c)

We may demonstrate the derived material-be- havior law by utilizing eq (17) in eq (9) and plotting the results for a few constant strain level curves, Fig. 6. Also shown in the same figure are the "experimental" points which led to the derived flow law.

Discussion In this paper, a general technique is described of

utilizing the explosive-ring test in conjunction with recently developed analytical techniques to deter-

mine the material parameters governing strain hardening or perfectly plastic rate sensitive flow. The technique suggested is illustrated by using hypothetical test results.

The ring test is especially useful at high strain rates where other methods break down for various reasons. The only essential restriction on the use of the method is tha t the proposed rate-sensitivity law should be a relatively slow one, tha t is describ- able on a logarithmic strain-rate plot. Of course, if a different rate-sensitivity law were postulated, equations similar to those presented here should be derived from which the material constants could be calculated.

Since the initial velocity of the ring is the only experimental quant i ty of importance (in addition to the final ring deformation), a simplified experi- mental approach may be possible to calculate this initial velocity. For example, a very light mass may be at tached to the ring and its initial motion clocked with a simple high-speed camera.

In view of the fact tha t most of the plastic flow takes place in the initial or peak-strain-rate zone, it may be desirable, for greater accuracy, to determine different material constants for the various zones.

Finally, it should be mentioned tha t the product type of rate-sensitivity hardening law illustrated by eq (6) and discussed in Ref. 13, appears to offer great possibilities of potential application. The significant virtue of this approach is tha t rate-sensi- t ivi ty effects a~d strain-hardening effects are un- coupled when use is made of the localized nature of the flow law in a particular strain-rate zone.

References 1. Henriksen, E. K., Lieberman, I., Wilkin, J. F., and McPherson,

W. B., "Metallurgical Effects of Explosive Straining," A S T M Special Technical Publication No. 336, Symposium on the Dynamic Behavior of Materials, 104-164 (1963).

2. Manjoine, M. J. , "Influence of Rate of Strain and Temperature on Yield Stresses of Mild Steel," Jnl. Appl. Mech., 11, Trans. A S M E , 66, A-211 (1944).

3. Hauser, F . E., Simmons, J . A., and Dorn, J . E., "'Strain Rate Effects in Plastic Wave Propagation," Response of Metals to High Velocity Deformation. Edited by P. G. Shewmon and V. F. Zackay, Interscience Publishers, New York, 93-109 (1963).

4, Lindholm, U. S., "Some Experiments with the Split Hopkinson Pressure Bar/" Jnl. Mech. Phys. Solids, 12, 317-336 (1964).

5. See discussion of 3 by D. Wood in same volume, p. 110. 6. Davies, E. D. H., and Hunter, S. C,, "'The Dynamic Compression

Testing of Solids by the Method of the Split Hopkinson Pressure Bar," Ibid., 11, 155-179 (1963).

7. Davies, C. D., and Hunter, S. C., "Assessment of the Strain-Rate Sensitivity of Metals by IndenE&tion with Conical Indenters," Ibid., 8, 235-- 254 (1960).

8. Bodner, S. R., and Humphreys, Y. S., "Determination of the Rate- dependence of the Yield Stress from Impulse Testing of Beams," Part IV , 33rd Shock and Vibration Bulletin, 141 (March 1964).

9. Johnson, P. C., Stein, B. A., and Davis, R. S., "Measurement of Dynamic Plastic Flow Properties Under Uniform Stress," A S T M Special Technical Publication No. 336, Symposium on the Dynamic Behavior of Materials, 195-207 (1963).

10. Bodner, S. R., and Symonds, P. S., "Experimental and Theoretical Investigation of the Plastic Deformation of Cantilever Beams Subjected to Impulsive Loading," Jnl. Appl. Mech., 29, Trans. A S M E , 82, Series E, 719 (1962).

11. Ting, T. C. T., "'The Plastic Deformation of a Cantilever Beam with Strain-Rate Sensitivity Under Impulsive Loading," Ibid., 31, Trans. A S M E , 86, Series E, 38-42 (1964),

12. Perrone, IV., "On a Simplified Method for Solving Impulsively Loaded Structures of Rate-Sensitive Materials," Ibid., 32, Trans. A S M E , 8"1, Series E, 489--492 (1965).

13. Perrone, IV., "A Mathematically Tractable Model of Strain Harden- ing Rate Sensitive Plastic Flow," Ibid., 33 (1) 210-211 (1966).

14. Clark, D. S., "'The Behavior of Metals under Dynamic Loading," Trans. Am. Soc. Metals. 46, 34 (1954).

236 I May 1968