Development of Scaling Laws for Explosive Forming

Study indicates that a workable scaling law can be developed, so that a trial-and-error development can be conducted on a small scale in such a manner that the results can be deduced for application to full-scale process

by A. A. Ezra and F. A. Penning

ABSTRACT--Explosive forming of large sheet-metal shapes suitable for boosters or spacecraft is potential ly a fast and economical process. However, the development of a n opt imum technique with a part icular material is a tr ial-and-error process. Wi th reliable scaling laws, this tr ial-and-error process can be carried out conveniently and inexpensively on a smaU scale. This s tudy develops workable scaling laws and shows what similitude require- ments may be neglected in a scale-model test.

Introduction M e t a l shee ts a re explos ive ly fo rmed b y using the energy re leased b y an explos ive charge to force a b l a n k in to a su i t ab le die. T h e resu l t ing me ta l - fo rming r a t e s m a y reach severa l h u n d r e d feet pe r second in c o n t r a s t to conven t i ona l d rawing a n d ex t rud ing ope ra t ions t h a t a re n o r m a l l y conduc ted a t r a t e s of on ly 1 to 5 fps. Deve lop ing an o p t i m u m technique , or even d e t e r m i n i n g the feas ib i l i ty of ex- p los ive ly fo rming a g iven size a n d shape f rom a pa r t i cu l a r ma te r i a l , is a t r i a l - and -e r ro r process. F o r large sizes, th is can be p roh ib i t i ve ly expensive. W i t h re l iab le scal ing laws, t h i s t r i a l - and -e r ro r process can be c o n d u c t e d conven ien t ly on a smal l scale a t a r e l a t i ve ly low cost.

Th i s p a p e r p resen t s t he d e v e l o p m e n t o f scal ing laws for explos ive fo rming f rom d imens iona l a n a l y - sis, t o g e t h e r wi th t he resu l t s o f an ex pe r ime n t a l inves t iga t ion to d e t e r m i n e the r e l a t ive i m p o r t a n c e of those scale fac tors t h a t a r e e i the r inconven ien t or imposs ib le to sa t i s fy in a scaled m o d e l tes t .

Theory of Scaling Laws A phys i ca l p h e n o m e n o n y m a y d e p e n d on severa l

i n d e p e n d e n t var iab les , Xl, x2 . . . . x , in some un- known manner . I f a l l t he i n d e p e n d e n t va r i ab les a re known, i t is poss ible to p red i c t t he ou tcome y for a g iven set Xl, x 2 , . . , x, b y m e a n s of a p r o p e r l y scaled mode l tes t , even ff t he func t iona l r e la t ionsh ip be tween y a n d i ts i n d e p e n d e n t va r i ab les is unknown.

A. A. Ezra is Chief, Technical Development Section, Structures and Aero- physics Dept.; F . A. Penning is with Structures Section, Space Flight Laboratory, the Martin Co., Aerospace Division, Martin-Marietta Corp., Denver 1, Colo. Paper was presented at 1962 S E S A Spring Meeting held in Dallas, Tex., on May 16-18.

T h e fol lowing t h e o r y under l i e s the de r i va t i on o f t he scal ing laws. T h e func t i ona l r e la t ionsh ip be- tween y a n d al l t h e va r iab les on which i t depends can be expressed in genera l as

y = [ (xl, x2 . . . . x.) (1)

I f t he exac t n a t u r e of t he func t ion f were known, then th i s would be t he m a t h e m a t i c a l express ion o f t he phys i ca l law express ing the dependence of y on xl, x~ . . . . x,. Th i s phys i ca l l aw is i n d e p e n d e n t o f t he un i t s used in t he m e a s u r e m e n t , a n d appl ies j u s t as well to the mode l as i t does to t he p r o t o t y p e . Us ing the subsc r ip t s m to a p p l y to t he mode l a n d p to a p p l y to t he p r o t o t y p e , we have t h e re la t ion- ships

y ~ = [ (x ,~ ,x2 . . . . . x ~ ) (2)

y~ = / (x,p, x~p . . . . x~)

Since the va r i ab le s in a p h y s i c a l p r o b l e m can be expressed in t e r m s o f t he four bas ic d imens ions of mass , length , t ime , and t e m p e r a t u r e , t hen t h e n + 1 va r iab les (y, Xl, x2 . . . . x~) can be combined in to (n + 1) - 4 = n - 3 d imens ionless groups.

R e m e m b e r i n g t h a t the p h y s i c a l law is the s ame for mode l and p r o t o t y p e , eq 2 can t hen be r ewr i t t en

~l,~ = F (7r2~, m . . . . . ~r(,_ ~ ) (3) lrlp = F (~2p, m~ . . . . ~r(~_ a~)

where t he d imens ionless g roups ~ , ~r~p, con ta in t h e d e p e n d e n t va r i ab l e y, a n d t h e o the r d imens ion- less groups c on t a in t he i n d e p e n d e n t var iables . I f t he mode l is so cons t ruc t ed t h a t t he s imi l i tude r e qu i r e me n t s

~'2m ~- 7r2p

~r(~_ ~)m = r(n- 3)p (4)

a r e satisfied, t h e n i t m u s t fol low f rom eq 3 t h a t t he scal ing l aw for t he d e p e n d e n t va r i ab l e is

~1~ -- rip (5)

Since y~ (conta ined in r l~) can be measu red f rom t h e mode l tes t , t he d imens ionless q u a n t i t y rl~ = ~ is therefore d e t e r m i n e d expe r imen ta l ly .

T h u s knowing ~rlp, the n u m e r i c a l va lue of the

234 [ A u g u s t 1962

Fluid Surfoce

L ~-~ Explosive

Specimen (Metal Blank)

L

.IL

W

D

\ \ \

F

.~---Tonk

-[ !

Die

Fig. 1 - Assembly for explosive forming

physical quanti ty yp contained in 7rl, can be cal- culated. Equation 5 thus gives the desired scale factor for the dependent variable y~.

With four basic dimensions in the problem (mass, length, time, and temperature), four scale factors may be chosen arbitrarily to suit test require- ments. The remaining scale factors will then be expressed in terms of one or more of these four.

In general, if all the variables on which a par- ticular phenomenon depends are known, dimensional analysis will give the desired scaling law for the dependent variable and all the similitude require- ments necessary to ensure its theoretical validity. However, two important practical difficulties usually must be overcome to develop a workable scaling law. The first difficulty, not peculiar to explosive forming alone, is the practical impossibility of satisfying all the similitude requirements that are theoretically necessary (eq 4). An approximately correct scaling law, however, can still be obtained. Only by experimentation can the importance of the similitude requirements tha t are necessarily neglected be determined. In any case, empirical

corrections to the scaling law can usually be de- veloped. For example, if a particular similitude requirement, ~rkm = 7rkp, proves to be important, and yet cannot be satisfied, the difficulty can be overcome by determining experimentally how 7rz varies with 7rk, keeping all other 7r terms constant.

The second difficulty is that a complete knowledge of all the variables may not exist. Thus, the un- satisfied similitude requirement for each unknown variable may cause deviations from the scaling law.

Scaling Laws for the Explosive-forming Process

The assembly used for explosive forming in this investigation is shown schematically in Fig. 1.

Since the purpose of a model test in explosive forming is to predict the amount of explosive charge required to produce a desired deflection, the de- pendent variable in this analysis will be the amount of charge.

The known variables in the problem are listed

Experimental Mechanics I 235

below, together wi th their d imensions (mass M ,

l eng th L, t ime T, and t empera tu re 0).

DEPENDENT VARIABLE DIMENSION

e The energy of the explosive charge ML 2 T -2 INDEPENDENT VARIABLES

w The maximum permanent deflection of L the metal blank

Static yield stress of metal blank before explosive forming

D Diameter of die opening and maximum L diameter of explosively formed shape

X Diameter of cylindrically shaped charge L d Diameter of cylindrical container holding L

the fluid medium L Distance of charge from metal blank L H Total depth of water L t Thickness of metal blank L r Draw radius of die opening L B Size of metal blank L [ Total holddown force on metal blank MLT -= Pl Density of fluid ML -~ p Density of metal blank ML -~ # Viscosity of fluid M L - I T -1 M~ Mass of die M C r Velocity of sound in fluid LT -t C Velocity of sound in metal blank LT -~ C~ Velocity of sound in die material LT -1 3' Specific heat ratio of explosive gases ... Of Temperature of fluid ~ R 0 Temperature of metal blank and die ~ R

ML-t T-.~

I t should be no ted t ha t the explosive pressure exerted on the me ta l b lank is no t an i ndependen t var iable . Th i s pressure is de te rmined by the in- dependen t var iables l isted above.

Since all the 22 var iables listed, inc luding the dependen t variable, can be expressed in te rms of the four basic d imens ions of mass, length, t ime and tempera ture , the physical law re la t ing the dependen t to the i ndependen t var iables can be expressed in te rms of 18 ( 2 2 - 4 ) dimensionless parameters or groups of variables, ~rl, r 2 , . . . ~r~s, of which 7rl con ta ins the dependen t var iable e, so t h a t eq 3 becomes

~ = P (~m, ~ . . . . . . ~ ) ~v = F (~2~, ~r~v . . . . . ~s,) (6)

where: "~r 1 : e /cr t 3 ~ = w / D % = d / D ~4 = L / D �9 ~ = H / D ~r6 = D / t ~r7 = B / D % = D / r ~r~ = D / X

%0 = f / ~ t 2 ~n = ~r /pC 2 ~,~ = C ~ / C ~ 3 = C ~ / C ~

~1~ ~ p . f / p

~r~ = 0~/0 ~q7 = p B 3 / M ~ ~q8 = p C z D / t t , the Reynold's

Number

Since the physical law is the same for model a nd pro to type , eq 6 shows t h a t 7r~ = ~r~, provided we make r ~ = ~r2 . . . . . ~r~s~ = ~sm, t hus giving the scale factor ev = e ~ z ~ t ~ / a m t m ~. The remain ing scale factors can be derived f rom the other dimensionless pa-

rameters , each of which mus t be the same for bo th model a nd pro to type .

Fou r scale factors, nl to n4, can be chosen ar- b i t ra r i ly since the four basic d imens ions of mass, length, t ime and t empera tu re are involved. There- fore,

(1) Let n~ = Dr~Din , arbitrary scale factor L ~ / L ~ , (2) Let n2 = % / ~ , arbitrary scale factor M ~ L v - ! T ~ - ~ /

M , ~ L ~ - ~ T ~ - ~ , (3) Let n~ = pv/pm, arbitrary scale factor M ~ L ~ - a /

M~/.,-3, (4) Let n4 = 8~/0,~, arbitrary scale factor ~ R ~ / ~ R ~ .

These four scale factors de te rmine model and p ro to type relat ions such tha t

(5) wp = n l w m (14) ~p = 7,, (6) ep = nl~n2em (15) pip = n ~ p f ~ (7) dv = n~d~ (16) Ofp = n40f,~ (8) L v = n~L,~ (17) [p = n~n2f ,~ (9) H p = n ,H ,~ (18) Cv = (n~/n3) 1/2C~,

(10) tp = nl tm (19) CIv = (n2/n3)~12Cf~ (11) B ~ = n l B m (20) Cdv (n2/n~) ~/2Cdm (12) r~ = n~r~ (21) Ma~ = n ~ n 3 M a ~ (13) k~ = n~h~ (22) #v = n~(n2 n~)i/2#~.

I f no more t h a n the 22 variables l isted completely describe the process of explosively forming a meta l b l ank as shown in Fig. 1, their sat isfact ion will ensure perfect s imil i tude. F r o m eq 6, we can then deduce the dimensionless funct ional re la t ionship

~1 = ~(~2) (7)

provided 7r3, ~r4 . . . . ~rls are each kept constant . This can be in te rpre ted in the following manner .

Suppose specimens wi th a par t icu lar value of D (diameter of shape) are explosively formed, and the exper imenta l ly de te rmined values of ~rl are p lot ted agains t r2. Then , for specimens wi th a larger or smaller va lue of D, the i r exper imenta l ly de termined curve of ~rl vs. r2 will correspond exact ly to the first one, provided the values of 7r3 . . . . rlS are kept the same by using the appropr ia te scaling factors. Therefore, a small-scale tes t will deter- mine the re la t ionship be tween the a m o u n t of ex- plosive a nd the p e r m a n e n t deflection of the blank. Expressed nond imens iona l ly as a plot of 7rl vs. 7r.o, this c u r v e m a y be used to predict the outcome of a large-scale test.

However , difficulty arises when all the values of Ir3 . . . . ~r18 canno t be kep t the same for bo th model and proto type . W h e n this occurs, t h e dimension- less plot of ~rl vs. ~r2 m a y be different for various sizes, a nd if one or more of the values 7r3 , . . . rlS va ry from one size to the nex t in a n uncont ro l led man- ner, it m a y not even be possible to plot a curve of ~r~ vs. ~r2, for a n y size. The procedure used in this inves t iga t ion was to keep cons tan t as m a n y values of r ~ , . . . 7r18 as possible, and to de te rmine Vl ex- pe r imenta l ly as a func t ion of r2 for a range of sizes of die d iameter f rom 4.8 to 24 in. Devia t ions from the resul t ing curves revealed the impor tance of the neglected parameters .

The dimensionless paramete rs t h a t were no t control led in this inves t iga t ion are 7rlo = f / ~ t 2,

7r~7 = p B 3 / M d and ~r,s = p C f D / # . To make ~17 the same for all sizes of die d iameter would have re- quired scaling the mass of the die by the scale factor

236 ] A u g u s t 1 9 6 2

I0.0 - $~mbo! Die Die (i_r~ Tonk Held B~

* 4~ Weights 6.0 Weight~ 9 .6 We]ghls

O 7.7 Bolts 14.4 Bolts

o 16.3 Bolts ':': ~4,0 Bolts

@

/ i

-~ I.O

b

O,I I ] I I l I I I ] O,01 0.10

Fig. 2--Plot of 2014-0 aluminum specimens

/ e/o" t 5 = 95.6 (~"/D) 1"69 Index of Correlot ion = 0.94

Note: 24- in. diameter not used to fit the curve.

I _1 I I I l l L 0

21. This , however , was n o t done. To keep r ls cons tan t , i t would have been necessa ry to scale t he v iscos i ty of t he fluid su r round ing the explosive. Thus , each la rger size of die would r equ i re a fluid of a p r o p o r t i o n a t e l y h igher k i n e m a t i c v i scos i ty for comple te s imi l i tude . Dimens ionless p a r a m e t e r ~10 = f / a t ~ var i ed f rom one size to t he nex t because of the a r r a n g e m e n t shown in Fig. 1 where mos t of t he ho ld -down force on the b l a n k was p r o v i d e d b y the explosive pressure .

T h e tes t ma te r i a l s consis ted of m e t a l b l anks of 2014 a l u m i n u m al loy of th ree different t e m p e r s (0, T-4 and T-6) . T h e T-4 spec imens were ma in - t a i ned a t - 4 0 ~ F by pack ing t h e m in d r y ice un t i l t hey were p l aced on the die and explos ive ly formed. Seven different die d i ame te r s were used to explo- s ively fo rm m a t e r i a l of each t emper . T h e same t y p e of explosive was used for a l l t e s t s (25 pe rcen t Power to l No. 7). E a c h size of t es t was a scaled mode l of al l t he o ther sizes (only scale fac tors 17, 21 and 22 were no t sat isf ied). T h e p e r t i n e n t t e s t d a t a are given in Tab les 1 a n d 2.

Experimental Verification F r o m th icknesses of commerc ia l ly ava i l ab l e alu-

m i n u m sheet , t he series of seven die d i a m e t e r s was

evolved , each d i a m e t e r be ing 96 t i m e s the shee t th ickness . Since t h e d ie cons is ted of a s tee l p l a t e w i th a c i rcular hole, t h e m e t a l b l anks were fo rmed freely, t he i r f inal shape depend ing en t i r e ly on t h e i n t e r ac t ion be tw e e n t h e p ressure pu lse a n d t h e m e t a l b lank .

Con t ro l spec imens of t h e a l u m i n u m a l loy 2014-0 and 2014-T6 t e m p e r s were t e s t ed before explos ive fo rming to d e t e r m i n e the mechan ica l p r o p e r t i e s of t he m e t a l b l anks : tens i le y ie ld s t ress , u l t i m a t e tens i le s t ress a n d e longat ion . F o r t h e 2014-T4 t e mpe r , a t y p i c a l va lue for y ie ld s t ress of 42,000 psi was used i n s t e a d of t e s t ing con t ro l spec imens .

T h e m a x i m u m def lec t ion was m e a s u r e d a f t e r t he shapes were explos ive ly formed. T h e measu re - m e n t s were t a k e n in four d i rec t ions to f ind a good ave rage def lec t ion a n d to no t e a n y a s y m m e t r y in t he fo rmed shapes . N o a s y m m e t r y was found. M e a s u r e m e n t s were m a d e to o n e - t h o u s a n d t h of an inch. Af t e r averag ing , t he def lec t ion was r eco rded to t h e nea re s t o n e - h u n d r e d t h of a n inch. I t was no t a l w a y s poss ib le to d e t e r m i n e the m a x i m u m de- f lect ion before r u p t u r e . These tes t s showed t h a t t he scal ing law given b y d imens iona l ana lys i s was s t i l l a p p r o x i m a t e l y correc t w i th the s imi l i tude p r o v i d e d .

T h e d imens ionless p a r a m e t e r s ~1o = f / a t 2,

v~7 = p B ~ / M d , a n d ~ 8 = pC/D/ t~ were n o t ex- p e r i m e n t a l l y cont ro l led . However , i t was ver i f ied t h a t a func t iona l r e la t ionsh ip , eq 7, be tw e en t h e d imens ionless p a r a m e t e r s e / a t ~ a n d w / D could be de t e rmined , regard less of size. T h e fo rm of ~ f rom eq 7 was chosen such t h a t

. (;). ~t~ = ~ ( 8 )

where a a n d ~ are cons tan t s .

T h e energy of t he explosive charge was p r o p e r - t i ona l to t he weight of t he charge , s ince t he s ame explosive was used for al l tes ts . F o r t h e q u a n t i t y e (measure of energy) in eq 8, i t was found con- ven ien t to use t he weight of t he explosive, m e a s u r e d in grains, whi le a is in un i t s of p o u n d s pe r squa re inch, and t is in inches. T h e t o t a l a m o u n t of ex- p los ive charge inc luded the d e t o n a t i n g cap. Thus , t he t o t a l energy of t h e explosion was the s u m of t h e energies of t h e charge a n d the cap. T h e ene rgy of the cap a lone was equ iva l en t to a p p r o x i m a t e l y 8 gra ins of the explos ive charge.

T w o m e t h o d s were used to secure t he w a t e r t a n k to t h e d ie shown in Fig . 1. F o r t h e 4.8-, 6.0-, a n d

TABLE 1--DIMENSIONAL DATA FOR TEST SERIES (MATERIAL 2014 ALUMINUM ALLOY IN O, T-4 AND T-6 CONDITIONS)

Diameter of Th ickness of Draw Tank D iameter Depth of S tandof f fo rmed shape, meta l b lank, radius, d iam, Blank size, of charge, water , d is tance, Length

D, in. t , in, r, in. d, in. B, in. X, in. H, in. L, in. scale fac to r Series

4.8 0.050 0.20 9.6 11 x 11 0.8 9.6 3.2 1.00 A 6.0 0.063 0.25 12.0 14 x 14 1.0 12.0 4.0 1.25 B 7.7 0.080 0.32 15.4 17.6 x 17.6 1.3 15.4 5.1 1.60 C 9.6 0.100 0.40 19.2 22 x 22 1.6 19.2 6.4 2.00 D

14.4 0.150 0.60 28.8 33 x 33 2.4 28.8 9.6 3.00 E 18.3 0.190 0.76 36.6 42 x 42 3.0 36.6 12.2 3.81 F 24.0 0.250 1.00 48.0 55 x 55 4.0 48.0 16.0 5.00 G

E x p e r i m e n t a l Mechan ics I 237

~ K T

to

~-NN

• ~ ' N a

X

" o

0

a

to

o ~ o o ,~ o o .: ~ x

~ x T

CM

x7

~oxT

to

; . ~ ~ ~ ~ ~ ~ ~ ~ r

x I

t o

o'~ r

x7

t o

.--

e -

e-. �9

t o

. . , -

0 ~

e -

t o e -

e -

t o e -

e -

e -

t -" .

e -

e=

,- . .

t o t -

O

e'- .

If If ff II II [[ [I II II II II II It [l tl II

9,6-in. diam dies, the tank was secured by weights placed atop the tank with circumferential load vary- ing from 8.1 to 19.3 lb/in. For the 7.7-, 14.4-, 18.3-, and 24.0-in. diam dies, the tank was secured by bolting the tank to the die. There were minor differences in response for the two different fasten- ing methods but, as seen from the data points plotted in Figs. 2, 3 and 4, the resulting curves fit all points reasonably well.

For each temper, the curves of best fit were es= tablished without using the results from the 24.0- in. diam die, and are presented as Figs. 2, 3 and 4. The constants a and/~ from eq 8 are given, as well as the index of correlation.

The index of correlation is

./1 - ( s j y \ r~ /

where: S~ = the standard error of estimate r~ = the standard deviation

Sources of Experimental Error All data points did not plot on a smooth curve.

Possible sources of experimental error were partial wetting of the explosive, the shape of the charge, and variability in hold-down force on the water tank.

The plastic containers used to immerse the ex- plosive were too shallow for the larger charges and had to be spliced together. When the plastic con- tainers were used as a unit without splicing, the problem of sealing the upper end still remained.

The larger cylinders and those with bases dis- torted from use took longer to fill with water. Thus, some explosive charges were immersed for longer periods than were others.

The charge was of cylindrical shape for all tests, and its diameter was held at a constant relative value of D / 6 . For small charges, the explosive covered the bot tom of the container in a thin layer. With explosives detonated from one end, an optimum cone exists tha t permits use of all the explosive. Use of a cylinder of scaled diameter resulted in some variation in efficiency of charge used.

Examination of the distribution of data points in Fig. 4 for 2014-T6 shows that the larger speci- men sizes required larger values of e / a t ~ to achieve the same value of draw depth ratio, w / D . T h i s may be explained by first considering the yield stress of the T-6 temper. From the scaling law, it is evident tha t greater amounts of explosive were required for a given w / D , because of the higher yield stress. As a result of this, a larger separation ap- peared momentarily between the bot tom of the water tank (held in place either by weights or bolts) and the top of the specimen. Less blank deforma- tion was realized per grain of explosive because this temporary free surface relieved more of the re- flected pressure wave. This size effect is less evident in the data points in Fig. 3 for the 2014-T4 material (which has a lower yield stress), and is not at all

238 A u g u s t 1962

I0.01

" Symbol Die Dio (in.)Tonk Held By

+ 4~B Weights x 6,0 Weighls

- * 9,6 Weights O 7.7 Bolts t, 14,4 Bolts o 18.3 Bolts

e

~4

I [ I [ I [ I I o.o~ O.lO

Fig. 3--Plot of 2014-T4 aluminum specimens

/ o:

~x ~ : +~

X

I e/O" 13 = 86.7 (t~/0) 1.66

I Index of Correlotion = 0.87

I I i I I I I I

~ i.o

b

,.o

I Die Oio (in) Tonk Held By D ~//]** ~ * * * + + 4.8 Weights

r ' / ~ ' ~ - - : 6.0 Weights x) * 9.6 Weights

o x ~ o 7.7 Bolls x + ! 14.4 Bolts

~ / x 18.3 Bolts ::. 24.0 BOltS

J JNole; ~4.0-in. diameter not 1 [ - - used 10 fit the curve.[

l 1 ] L { I t ~ . _ _ t I 1 I I i LA oto

~/o

Fig. 4--Plot of 2014-T6 aluminum specimens

evident in the data points shown in Fig. 2 for the 2014-0 temper, which has the lowest yield stress of all three materials.

There was no significant difference between test results when the bolts were finger-tight, or were tightened with a wrench. The tests were conducted on a 7.7-in. diam die with 17.6 x 17.6 x 0.083-in. metal blanks of 2014-0 aluminum alloy, using the same amounts of explosive. Smaller deflections resulted when the bolts were in place but one-half inch free of the bracket, or when the bracket was not bolted (see Table 3).

The results also show that the following similitude requirements, which can easily be provided, are adequate for a workable scaling law:

1. Geometrical similitude must be provided; 2. The mechanical properties of the metal blank

before explosive forming must be the same for model and full scale,

3. The kind and shape of explosive charge must be the same for model and prototype.

Within the limits of these tests, no serious adverse effects were produced by not satisfying similitude requirements for the viscosity of the fluid medium,

TABLE 3---EFFECT ON DEFLECTION OF VARIOUS BOLT CONDITIONS

B o l t c o n d i t i o n

Maximum deflection

( in.)

Finger t ight 1.83 Tightened with wrench, very tight 1.80 In position, but 1/2-in. free of bracket 1,63 No bolts 1,42"

* C y l i n d e r j u m p e d a p p r o x i m a t e l y 4 f t .

the mass of the die, and the magnitude of the hold- down force on the blank. However, the relative importance of variations in hold-down force was due to the use of a relatively thick (D/ t = 96), large, square blank having side dimensions more than twice as large as the die opening. For relatively thinner and smaller blanks, the magnitude of the hold-down force becomes increasingly important. I t can be concluded, therefore, that a workable scaling law can be developed for the explosive form- ing process, so that trial-and-error development can be conducted on a small scale in such a manner tha t both qualitative and quantitative results can be deduced for application to the full-scale process.

Conclusions

The purpose of this investigation was to determine the feasibility of developing scaling laws for the explosive forming process, in spite of the physical impossibility of satisfying all similitude requirements for the independent variables.

The results plotted in Figs. 2, 3 and 4 show a definite functional relationship between the dimen- sionless quantities w / D and e/a t ~, provided D / t re- mained constant. The experimental scatter result- ing from the uncontrolled variables and lack of complete similitude was low enough to permit a workable scaling law for the dependent variable e/~t~. Since the results show that the functional relationship between w / D and e / z t 3 was approxi- mately the same over a wide range of sizes, it is therefore possible to determine from the results of a small-scale model test the amount of explosive needed to produce a desired deformation on a large scale.

Experimental Mechanics I 23