Stretch forming under fluid pressure

STRETCH

FORMING

UNDER

FLUID

PRESSURE

By P. B. MELLOR Department

of Mechanical

Engmeering,

(RKdord

mk

The University

of ShetY&ld

.JN&. I 959)

b given of xn experkncntal inveat@tion into the hydrostatic bulging ol c~nakr metal dirphrymr. TLC~velopment d drain and of instubdity ir &ad&d, and o strmim-&&ning

An scours

cbmctddk ia derived for each of the matenab tented. Expenmental rauita OR combted imt&Uity. und in the came of half-hard aluminium with theoretical withthea?&Ap?edbtbnrof pwdktbna of the general &rain geometry of the bulge.

1.

INTHODWTION

THE stretch formmg of a thin circular sheet of metal by fluid pressure provides a means of studymg strains under biaxial tensile conditions and also, if measurements are taken over the strained surface, of obtaining a strain-hirrdening characteristic of the material. This manner of stretch forming sheet metatis one of the simplest to be found in practice and the m&t amenable to mathematical solution. A large volume of experimental work on this subject was carried out by American mvestigators (BROWN and SACHS 1948; BROWN and THOMPSON1949 and others) during and after the war, but no detailed comparison was made between strainhardening characteriafics d the same material obtained by different methods of straining, nor wem mathematical solutions based on the theory of plasticity available at that time. HILL (1950)developed a theory which predicts the strain development across the bulge for a material with a linear strain-hardening characteristic, and SWIFF (1%%?)obtained a solution for the strain at instabihty. In the present paper experimental results are correlated with these theories. The purposes of the present investigation

have been :

1.

To examine the credentials of a potentially new means for obtaining strain-hardening characteristic for sheet metal.

2.

To examine the validity, over its applicable the general strain .geometry of t.he bulge.

8.

To &certain to what extent theory is capable of predictiiig of ‘instability under hiaxial tension.

The following agamptions e;xperimental results : I.

The normal

principal

and limitations stress

for

the condition

have been made in interplrcting the

is negligible 41

range, of HILL’S theory

a

compared

with the principJ

P. B. MICLLO~

42

stresses m the plane of the sheet. under conditions of plane stress.

In other words straining

takes place

2.

The flexural effect of the peripheral clamping stiffness of the sheet is negligible.

may be neglected, i.e. the

8.

The material is isotropic initially and remains so throughout bon.

4.

The material is incompressible, i.e. there IS no volumetric change during straining.

the deforma-

apparatus (Fug. 1) was designed to produce a 10-m *meter bulge on street metal of 20 s.w.g. and to w&h&and working pressures up to 1500 lb. per sq. tn. The design was intended to allow a small thickness-to-diameter ratio, thus reduemg the effects of flexural stiffness and keeping the experimental presawea as low as possible for a given gange of material. Pmcautkms were taken in the design of the apparatus and m the methods of measurament, to ensure pm&ion and especially consistency in the results, since their stgnillcance depefrds on smaI1 increments, and the standards of measurement in some of the earlkr American work were sear&y commensurate with ita purpose For thusmason the methods of measurements am here dea&bed in some detail. The

apprwmnately

Fig. I.

Layout of apparatus.

The die is of simple but robust construction.

The upper portion eonatstx of 8 hollow steel

girder&,!+ ini+v41 W#qrasThe nu$erbd to be tested rs seoureQ chmaw betwean this upper cylinder and the steel base-piate by means of twelve l-in.-@ameter a+& and bolts. Oil is supplied, under pressure from a displacement cylinder, through a hok in the centre of the base~~,Pnd~Dil_YBbtjointirensuredbya~~rinSMthe~~ematiDggmave jnthe~~~y. ~e~~~apush~t~theb~y~d~ap~le~~~~.~ whrh was found to be the suitabk mInimum radius which woukl obviate shearing on the pm& efthedieofallthemetalstute&

St&&

forming under fh&d preeaure

4?3

fnp~~~thcoifandcrpnru~auarppticdfromahrbeh~~rn~ t?lruugItalc&dngvalve,and8mlauet(p&FauPtrding) type of_ mriaducLtdintbsou~lint~~tbc~vrJveudtbctatI&cttocopblcpn&termin&i em&ant preuuti to he maintained for a eonvement kqfth of time. It wmafamd that even St bubnwch mmmbtthm the klmml to indua fallurc. -vL! crssp oocumd. This & cleady ren horn Fig. 2, which abornrthe c&et of creep. with tb duration of&ad sp@katkm, on the bulge profile of a half-hani aluminium specimen arbjected to a of 186 lb. per rl. in., which ir 75 per cent of the load known G induce lkilure.

25minutes

under pmsure

I

0~8~

I

1-o

CUM-entradius pii.

2.

to

2-O pof=ticlc

I f

: in. ’

Eifect of creep w bulge profile.

In order to rrduu emep to 8 minimum and to allow closrr eontrot of the oii feed, the large hydmulie acoumuhtor wu d@eti with and a amail oil-dlplacement cylindq, Atted into a universal multikver testing machine. used in ita rtead. With thie a rmngement the oil pnrun, for a even rate of tentmg, depended entirely on the resistance of the sheet metal to deformation. An oil flow of 11.8 cu. in. pr mmute was chosen as givmg u convenient period for the completion of the teat. allowing time for aimultauemu readmg of oil premun and polar height, and beh comporabk with the possible rate of draining in the tenalle testa on the material. The ~WrmtolhcturronyrpecimenthnnioredepndcdonlyonthcmPtcnicrLwryingf~m nilal2r&nutea lo? half-krd b tacQtyooe miamtee for a&uud ataiqka rbel. F@ra@aof~taw?.remwkintbkwork: oilprenamq ~-=utJal~hoop)~, thioiqaawainacmaatbtbu&udpr4Je. ~laatinckUkdradiuaanddepthfkomtbecentre orp&~tbe+dnbenk B7dore,myMuurefnent8weretakenthedtpmpnwaa&eaeed‘Tbb ~~to~w~rrtthoutfesrofcrap,~itrurBornby~~ ofeou&enaMed

44

I’. 5s. MRWR

~forbout~n~~d~~~torrtathptitmpdeno~~~mt~ ?=snb. Theoilprcesure aacrareentndb~Pp~~~u~1Rs.n~tothtdie.ToeBRlte~~~ readings of p-m four eGbm&d gauge weee uaed to -er the raw O-1,SfNJlb- per aq. ia The datum hnes for the hoop strains were coneeutric ink ~rcies drawn from the oeutre of the blank at n&al mtewdaof approximately O-2 in. Speclsf.ore w taken in loevting the geometric centre of the blank when mnunted in the apparatus, and Y patch of perspex was used to pro&et this cen& from in&nt&ion while the rtrcks were being drawn.

The surfatrs of the blanks were rlenned with ~ricblarethylene, and It WLL~ found that, d the cleaned surface was firmtly smeared w&h mk, the mk c~clerr could he drawn e8sdy on this base. Tbls simple method pfuduced satlsfartory results, und smce the total number of blanks required was not large, it was considered unneccmry to have recoumc to the photognd method @?HEWRR and Gwssco 1911 ; Jnclrso~ I@&%) ff Q is the mittid horizontal distance of a point on the test piece from the rentre and r tls tinal dtstance. then the I~r~thtI~~ hoop &ram B deArted as Q = la (r/r,). Hoop-&ram measurements rtt any stage of bu&png were &ken over the upper surface of the bhuzk by means of tl tntvellmg mzc-p-e, graduated to IO4 m f posrtroned by dowels OR the top s&ace of the upper p&ion of the dte. ~f~su~~ut~ were taken on the out&e edges of the ink &&es along two dxameters at right angles, one drsm&er fying along the diffetron af mlhng. Measurements were taken OR the original blank bet’ore stminmg and at BRmany mtermedrute stages durmg bulging as reqmred The four sets of readrug* were averabwd. to nnnlmrze any amsotroplc effect, and the mean CUFVM were plotted for half the bulge, a typic81 example bemg shown m Fig 4. Smce the bulge M intrmsically symmetrrcrl about the centre, the hoop&rain curve should cut the prime ordmate of the graph orthogonallv. l’se is mrlde of thrs Pa& m extnrpolatron of the hoop strains to the pole. Measurements of tbirknt%s were made by means of a simple dial-gauge micrometer $8 gr&uated to X)4 m. The mean thickness of the undeformed blnnk was found by takmg eight readings at points around the blamk ; the drfferenee between readmgs on any sheet never exeeeded 0 Ouos in. To measure the tbicknese strains at a gzven stage of bulgmg the preq were removed from the apparatus and cut into four qUadrants, and measurements were taken 8long the same two dtanteters 8s for hoop &ram Thtekness s&a& gs defined as co = - in ff&) where +, is the mitud and f the current thlekness of the sheet (where at w mom convement the sbsohzte msgnrtude of the thickness stram fs plotted). It will be seen that e set of thickness measurements rt any Intermediate stage of bulging involves the destruction of the sheet under test. The polar heqht and proflle of the bulge were measured with the ald of a depth nucrometer mounted on a sad& wluch could be traversed across the bulge by means of a 20 t.p.i. mlcmmater screw. When tha pointer made electrical contact with the bulge eurfaee an Indicator lamp ~~18 ht and this obviated any damage to the test spenmen. weight recording8 were taken acthe bulge, at horzmntal lntervais of 0 2 in., along the ssrne two perpendtrulsr diameters w the i&nun nleasuremeH8, To measure the rsdms of curvature *it the pole *ad tu show the trend of n&h of curvature it other pom&, the gene& method used by BRQWNand SACI~U (1W) wsa adopted. Average radii Of CUrV8tU~ Wt?P3 i’fSSf+Ud for Wl’kZUSchord lengths, and t% r&&s at the pole was obt&ed by extrqolation, &ch of these radii was calculated from the chord length and &s eorrespondiq sag&a, neglectiug any devi&son of the are from B eimle. If d upthe vet&d distance of a point from the pole snd t it43bo&zontd dist4meefrom the pole, tkeu it kpeasi@ shown tbst the &W of the circle peasing through the pole and the given point and having fta antre on the sxicr of symmetry is p = (9 -t_&)/2d (Pi. 5)

Stntteh formmg under fluid pressure

45

readings of polar hegght and oil pressure were taken. The results are shown plotted in Fig. 8 and give some idea of the wide range of physical properttes of the materials tested, Instability of the material IS said to occur If strammg m any region is able to continue at constant or decreasing 011 pressure. From FIN. 8 It is seen that only theannealed materials and half-hard copper developed \urh a condition of instabihty; the other materials fractured while the hydroqtatlc pressure was stdl mcreasmg.

Polar height

h

tn

@‘lg.3. 011 preaeure - polar height dagrams : continueus testing to fracture. For an isotropic blank of unrform thickness the maximum thinning would he at the pole of the bulge, and it would be expected that fracture would occur at this point, In praet+e the fracture was found to occur within a region up to I-5 in. from the pole, and to extend m the direction of rolling for approximately equal distances on either srde of the pole. Some of the copper and brass pressings exhibited a pin-hole fracture+ the cause of whleh was ascribed to initial defects in the material. Where such failures took place the results were ignored and repeat tests were made on other specimens. (2) Di&ibution of Straiw. Typlcal curves showing the dlstrlbution of hoop strains for half-hard brass are plotted in Fig. 4. Measurements ape taken up to a radial distance of approximately 8.6 in. from the pole, and it will be noticed that the &rain mcreases from the die edge and is maximum at the pole. Al$o, the curves an of the same general form at all stages of the test. In a limited number of cases hoop and thickness measurements were made across the entlre pressing. The ct&vta of results for annealed copper, killed steel, and lead, present a similar appeme.

Y. B. l+hsmR

46

It was observed that for all these metals there was a~ eppreeieble thickness strain in the region of the die. These results differed from those for a severely coldworked material such as half-hard aluminmm (Fig. 18) where the thickness strain at the die is exceedingly small. Considerable scatter was, found in the thickness-strain measurements for the lead bulge, because the metal developed an ‘ orange peel ’ effect during the straining. This roughening of the surface, which was not obtained with killed mild steel and soft copper, was reflected in the consistency of their thickness strains. U’J

I

Elulge

test: half-hard

broRs

-

I. -

bin, lin. in. n. -

b I.

. I

I

I

1-o

0 Fig. 4.

2.0 Currant fWius

Dstributmn

1

3-O

1

to porticle r

of hoop strain for hdf-had

i” in.

brnm.

Because of the symmet~e~ nature of the bulge there is a state of balanced biiial tension at the pole, which means in effect that the hoop strain erris eqw9 to the radial strain cp and to half the thickness strain, - z#. Therefore, taking into account the scales on which the hoop and thickness strains are plotted, each psir of curves in Fii. 18 should coincide at the pole (i.e., when r = 0). Part of the discrepancy is no doubt due to experimental error in determining the strains. but

stretchforming under fluid pn%oure

47

it must be remembered that. the effect of 8nisotropy may not h8ve been taken fully into scoount by averaging of the strains in different directions. 8. 9% &om&q of the BuZge.The shape of 8 hydrostatic bulge depends m8inly on the rmtio between metal thickness and die diameter, but it may he mod&A by the profile radius of the die, which has the effect of slightly decreasing the effective diameter*of the blank as the polar height increases. From the plot of the profile it would appear that, except in the proximity of the die edge, the bulge is spheric81 in shape, but by taking values of computed radii of curvature it is possible to detect a divergence in shape from the truly spherical. This is shown in Fig 5 for the c&se of half-hard 70/80 brass. It is seen that at small strains the curves have negative slopes, indicating that the radius of curvature decreases from the pole. With increasing strain the slope gradually appro8ches zero, indicating uniform curvature, and with still further strain the slope of the curves increases in a positive direction, showing that the radius of curvature is least at the pole. This behaviour is identical for all materials and may be caused by the clamping conditions. Sinular results were noted by BROWN and THOMPSON (1949).

cwrantmdius to partiek Fig.5.

Within

&t.fTmiIMtiOn

a distance

OfpO~SrradiuS

P

in.

OfCUrV8tUfC fOrhallhdbr8&t.

of Z-0 in. from the pole the derived radius of curvature is

constant for a given height, and the polar radius of curvature could ‘he estimated with, some eon&knee by extrapolating the measuremtnts in this region to xero radius. On wnds d consistency it is thought that the v&es obtained for polar

w&i of curv8m

less than 10 in: are correct to within o-8 in.

1’. B. WELLoR

48

(4) Stmio-hardming (‘hnractfri8iics. At the pole of the bulge the relationship between the internal pressure p, the hoop stress u,,, the radius of curvature p and the current thickness 1 is given by p L= 20, t ‘p where p is measured dim&Iy, and p and 1 are derived from the profile and hoop stram measurements respectively. By correlating Hence the biaxial stress u, ran be derived. strain based on measured hoop and thickness strams hardening characteristic for the material in the vicinity By adopting strain-hardenrng

an appropriate characteristics

this with a representative we can obtain of the pole.

a strain-

system of representative stress and strain the obtamed in the above manner can be compared

with strain-hardening characterlsttcs of the same maternal obtamed under different stress systems. It has been suggested (SWIFT 1946) that all stress-strain data should be evaluated in terms of q, the root-mean-square shear stress, and #, its natural strain analogue defined by the relations q* =- 23 If the

principal

stresses

z

(u, -

us)*,

are taken

in descending

algebraic

order,

in plane

stress

0s = 0 and uB = sui where the stress ratio s is a proper fraction. In cases where r IS constant during strainmg (e.g in a tensile test to necking or at the pole of a circular bulge) the assumptions of constancy of volume and the L&y-Mists law Icad to the followmg

relations

between

the total

strains

:

In the normal tensile test q = u, 6 and (G = cl 2/(8/g) where u1 is the longitudinal stress and li the longitudinal logarithmic strain. Under biaxial tension at the pole of a circular bulge q = CT,,l/ii and (G = zh t/(3 ‘2) where u,, is the hoop stress and E,, the hoop strain, which is equal to half the thickness strain E#; i.e. 9 = 54(3,8). In the case of cold rolling with inhibited spread (that is, deformation under plane strain) q cannot be determined directly, but 9 = pi 2/2 where E, is the longitudinal strain (equal to the thickness strain - ts). With this method of stress-strain representation the strain-hardening

charac-

teristics of 70/30 brass, copper, and aluminium have been obtiamed from the ‘ bulge test ’ and are shown m Figs. 6,7 and 8. The strain-hardening characteristics for the half-hard tempers were superimposed on the curves for the respective annealed materials and, making allowance for the nutial strain, the two curves have in each case the same general form. This sigmfies that the plastic properties of these

materials

at a given

yield-stress

are the same

whether

they

have

been

cold-worked by rolling or biaxial tension. These curves are compared, m the same diagrams, with strain-hardening chararterlstics obtained from tensile tests taken to the pomt of neckmg. It is \een that, although the tensile tests on the annealed materials show a very similar rharacterirtic to stretch forming. the necking point m simple tension occurs at a In order to obtam some cornfar lower strain than IS the case m stretch forming.

partson beyond the pomt of tensile necking, the materials were first strain hardened by rolhng and then tested m tension (FORD 1948). The technique adopted was strips of the material were cut from the parent sheet in Rh follows . S-m.-w& the direction of rolling, and rolled successively m approximately 16 per cent

Stretch fmmmy

under fluid pnssure

49

“0 40, ;; .5

.d

3,

+

p

20

10

I

0

Fig. 6

C

Stram-hardenmg from bulge test.

6

I

C 5

chanwterlstws for 70180 bruss. 0 -soft brass, X - hulf-bard Full-line curves obtained by rolling and testing in tension.

brwa,

I-

5-

3-

1 5-

>

Fig. 7.

Stmiu-luudcnmg fkom bulge test.

c~ica Full-line

curves

for copper. O-soft copper, X-half-bard obtained by rolling and testing in tension.

copper,

Y. B. Mma

50

reductlon3, one strip bemg taken out of the batch after every reduction. (It wa\ ascertained that the lateral spread of the strip was negligible.) TenGle \pecnnens were then cut from the centre of the strip and tested in the normal manner. at least two specimens being tested for each temper. It will be seen from Figs. 6, 7 and 8 that there is quite good agreement between the stress-&ram relations ohtanred by biaxial tension and plane compression plu\ tenrlon for 70 ‘30 brass. copper. and alummium.

Fii. 8. Strain-hardening clwactenstics aluminium from bulge test.

for alumimum. 0 - soft alumuuum, X - half-hard Full-line curves obtained by mlling and testmg m tension.

For the purpose of easy analysis It is convenient to fit some empirical equation to the curves. A suitable equation, suggested by Swwr (1952), has the form q = c (a + I,%)”where q is the root-mean-square shear stress and (G its natural strain analogue. In this expresslon for strain-hardening c \ets the general scale of the relation, n is an inverse measure of the decrement of the rate of hardening, and a depends on the initial state of the material, its value increasing with previous The method of fitting the equation q = c (a + #)” to a stress-strain overstrain. curve is an approximate one intended to give the best result over the whole range of stiining. A comparison of an experimental and empirical curve is shown in Fig. 9 for 70,430 brass. The greatest discrepancy is in the case of annealed copper, which has a well-rounded stress-stram curve. Empirical equations for other metals are shown in Table 1. In the case of a fully annealed material, a = 0 and we then have the better-known equation to a strain-hardening characteristic q = cp. By plotting q and 4 on logarithmic scales (Fig. 10) the empirical equations m the annealed form can be represented simply as straight lines of slope 11. The instability of plastic strain under plane (5) i%perime&al Instability Strains. stress is of importance in stretch-forming operations because it determines the

51

P

P. B. MaLLoR

52

degree of thinning which the sheet can undergo before fracture. Values of polarthrtkness strain at mstabihty can be determined from a graph of this strain agamst oil pressure. However, since this method gives only approximate values, a graphical solutton proposed by BROWNand SAWS (1938) was adopted. As stated earlier, the conditions at the pole are grven by the equatron p = Yuht,‘p. The instability condrtion dp = 0 then yields the relation

‘10, --

ub

-

dp

lit

T-+t=O

which may be written

Frg. 11.

Experimental

mnstabd&ty straim

from bulge test.

*soft

copper,

0

soft uluul;niunl.

where f, is defined as - In (f,, t). If the two sides of the last cquatlon are plotted m terms of Edthe mtersection will give the value of the instability strain. Such curves are shown for annealed copper and aluminium in Fig, 11. The mstabihty strain 1%shown to be 0.593 for copper and o 575 for aluminium. Failure of the (‘urvcs to interseet before fracture would of course mean that no instability had occurred. This was w for annealcd and half-hard 79,~~ brass and half-hard aluminium. Values of the experimental urstabrhty stram for all materials tested are shown tn Table I. The strain IS represented as the ratro of t, the current thickness of the sheet at the pole, to te, the uutial thickness of the sheet, The experrmental thn*knes\

St&A

formiq under

58

fluid premure

ratios at fracture are also shown tabulated. By comparing these ratios at instability and fracture it will be noticed, as might be expected, that there is more straining after instability with the annealed materials than with the tempered materials (also, of course, in the tensile test after the start of necking). 4.

CORRELATION

OF RIWULTS WITH THEOILY

SWIM (1952) has put forward a theory predicting the instability strains in a deformed circular membrane and has demonstrated that the instability occurs at the pole of the bulge. The stram-hardening characteristic adopted was of the form Q = c (a + $), It was concluded that the polar strain es at instability was the appropriate root of the equation

The corresponding reduction in thickness at the pole is then given by In (t/Q = 21s The equation has been solved graphically for all the materials tested, their stressstrain curves having been represented by the empirical relation proposed above. A comparison between the experimental and predicted critical thickness ratio is given in Table 1. In all cases the theoretical values are higher than the experimental values, so that the theory seems to underestimate the stretch forming capabilities of the materials. On the other hand, the theory does place the materials in the correct order of merit.

ThihWMtiO q = c (a + 9)”

Metal

q q q q q q

= = = = = =

28,QoO (0.01 f gp’* 28,900 (Oa7 + Jr>D* 54,000 Q” 51,000 (odk%#+ #p uI 10,800 p 10,800 @ISa + $pts

q = 41,000 p-9 q = 115,000(0~0x f #)O~

Indrr#liq(

pnrdunr

r/r,

t/to

o-555 0.59

030 o*sa

No-y No instability O-565 No imt&Iity om

om

060

0.51 0.48 0-W 048

0450

In discussing this lack of correlation, it should he kept in mind that, in order to apply the theory, it is necessary to adopt some analytical expression for the strain-hardening characteristic, which has in fact been chosen to cover the whole useful range of deformation. NOW the condition of instability is concerned solely with the value of the %loI~ of the cWae&s& at a particular point, aud not otherwise with the 8cngral form ~of this characteristic. A f&ted equation which would give aeeurate values for the slope of an empirical curve at all points involves difllculties of a higher order than are necesspsry to reproduce the general form of

Current m&al distance Fig. 12. PmMe of bulge for half-hard aluminium. _-____ experimental potnts.

P

in.

theoretical curve,

3 m

s yc .Y

I____

1*46in.

I

f

11% Cornnt rodiol distance r l thkknees strain, 0 boop Fig. IS. Strain dintribution for half-hard aluminium. bulge test. - - - - - - - theoretical thicknena &rain.

Stretch forming under fluid presmwe

65

such a curve, 8nd it may well be that the wsnt of closeness in agnement ia in fact due men to imperfection of the empirical strain-herdening expmssion than to any error in the theory itself. HILL (1950) obtained a general solution for the geometry of the bulge and the strsin distribution in the case of a metal whose stress-strain curve can be approximated by the linear relation o/Y = 1 + Hc, where Y is the initial yield stress and H is 8 nondimensional constant. Such 8 relation is approximated by a met81 in a work-hardened state, but of the materials tested it can only be applied to half-hard aluminium whose strain-hardening characteristic can be expressed in the form u/15,460 = 1 + 0.76 E where the stress is measured in lb. per sq. in.

)-

)-

)-

L

3

1.0

2.0 Palar height ‘h’

c 3 in.

Fig. 14. Theontical and experimental results for the variation of oil preaaurewith polar heiit ex&zimental points, for half-hard ahuninium. theo@lcal curve ; -O-O-Ocontinuous t(st ; -X-X-X-

experimental points, interrupted test.

Comprrrison between experiment and HILL’S theory are given in Figs. &e to 14. The eom%tion Or cults for the proflle of the bulge (Fig. 12) is good except in the region df thk cismped +e. The thickness strains are more sensitive than the profile to the value of II, and it will be seen from Fig. 13 that the theoretical straine are gnater $&LII the experimental thickness strains and teqd to lie rather closer to the experimental hoop strains (which are plotted to twice the scale). There is also fair correlation between the experimental and theoretical values of oil pressure and polar height (Fig. 14) except near maximum pressure. The theory

P. B. M~LLCX~

56

predicta an instability at a polar height of 2.6 in., but in fact the material fractured under ineraasing pressure at 8 poh height of apprOXkI8tiy 2.5 in. Following the work of HILL, THOMAS (1954) bar taken the particurlar case of soft copper and has obtained a solution by a method of successive approximations which is in fairly good agreement with the experimental work. But he concluded that’ the numerical integmtion entailed in the solution is so lengthy that it is unsuitable for frequent use. 5.

CONCLCYIONb

The extent of general thinning before the point of mstablhty is reached in the bulge test is so much gre8ter than in simple tension that it can be used to obtain a strain-hardening characteristic to far higher strains than is possible in the tensile test, and it is therefore more useful under conditions of severe plastic deformation. However, the procedure IS lengthy and tedious, since the oil pressure, the polar strain, and the polsr radius of curvature must all be measured to obtam a single point on the strain-hardening characteristic. In the c8se of half-hard aluminium good experimental correlation has been obtained with HILL’s theory for the development of straining and for the geometry of the bulge, and THOMAS(1954) has shown that it is possible to obtain as good a correlation for soft copper. Thus, this particular stretch-formmg operation appears now to be understood sufficiently well to serve as an introduction to the more general problems such 8s stretch-forming with a solid punch and deep-drawing operations where friction between the tool and the blank has to be taken into account. ACKNOWLEDGMENTS The author wishes to thank Professor H. W. SWIFT for his guidance and advice throughout the coum of this work, Mr. W. JOI~NSON for his help in preparing the paper and BISRA for which Association this work was carried out.

Bnew~a, C. A. and Gm, FL B. BIWWN, W. F. and SACHS,G. Bnow~, W. F. and TEOMPE~N,F. C. CHOW, C. C., DANA, A. W. and SACIM, G. FORD, H. GL~UUL, A. HILL,R. LANIWAb. W. T. and Low, J. R. Mosmw, W. Swrm, Tnoras,

H. W. D. G. B.

1941 lw8 1949

J. Aero. Sci., 9. Tmna. A.S.M.E. Trans. A.S.M.E.,

1949

J. Metab, 1, 49. Proc. I. Meeh. E., 159, 115. Trans. A.S.M.E., 70, 288. Phal. Ma&, 41, 1188. Tmna. Am. In& Min. 1146.Emgra., i 71,674. Pkuiic Dk#omMio*,+wkpmiwm@d & H. H. HMJSNEZL, chap. 6,187 (tiphton Howe. NY). Engwwxring, 162,881. J. Mach. Php. Solids, 1. 1. R. Z.S.R. A. Rcrcarch ZteportMW/B/I/M.

1948 19js 193 1947 1948 19-i& 1952 1931

70,241. 71. 57s.