The effect of plastic anisotropy on flange wrinkling behaviour during sheet metal forming

^hweiwe

THE COLLEGE OF AERONAUTICS

C R A N F I E L D

THE E F F E C T O F P L A S T I C ANISOTROPY ON F L A N G E WRINKLING BEHAVIOUR DURING S H E E T M E T A L FORMING

b y

CoA Note Mat. 12

THE COLLEGE OF AERONAUTICS

DEPARTMENT OF MATERIALS

The e f f e c t of p l a s t i c a n i s o t r o p y on f l a n g e

w r i n k l i n g b e h a v i o u r d u r i n g s h e e t m e t a l foiming

by

-H. N a z i r i , D.A.E. and Roger P e a r c e , B . A . , B . S c , F . I . M .

Page 2 , E q u a t i o n s ( 3 ) and (k) s h o u l d r e a d :

0.1+6 {^f ^ ^ ^ 0.6U ( | ) ^ ( 5 )

1.65 ( f ) ^ N « 2 . 5 3 ( f ) W

Page 6, paragraph 2, line 2 should read: (he+ht)

The mean cup-height -^—-—- where h and h are the heights of the ears

Page S, equation (12) shoiild read:

°x /2r+2

THE COLLEGE _0 F AERONAOTICS

DEPAE-MENT OF MATERIALS

The e f f e c t of p l a s t i c a n i s o t r o p y on f l a n g e

w r i n k l i n g b e h a v i o u r d u r i n g s h e e t m e t a l fonaing

b y

-H. Naziri, D.A.E. and Roger Pearce, B.A., B . S c , F.I.M,

S U M M A R Y

During the drawing of sheet metal between a die and a blankliolder, compressive hoop-stresses are developed which attempt to thicken or i^nrinkle the flange. Previous work on t h i s behaviour has ignored any effects due t o normal or planar p l a s t i c anisotropy. I n t h i s paper i t i s shown that the blanliliolder pressure necessary to suppress i/rinkling increases with decreasing normal anisotropy ( r ) and increases with incieasing planar anisotropy (Ar). The approximate plane s t r a i n conditions {de^ = O) operating in the flange can be simiilated by an edge-notched t e n s i l e specimen and t h i s simiiLatlon demonstrates the effect of t e x t u r e hardening and softening upon flange wrinlcling beliaviour. The r e s u l t s obtained can be i n t e r p r e t e d qToalitatively by the use of a n i s o t r o p i c p l a s t i c i t y theory.

Tlie speed of drawing also effects wrinld.ing, in g e n e r a l , t h e number of wrinlvles decreases with increasing drawing speed.

C o n t e n t s

Page No.

Summary

1. Fundamental t e s t s ( u n i a x i a l t e n s i l e and plane s t r a i n

[de = o3 notch) 5

2. Simulative tests (flange wrinkling) k

Discussion 7 References 10 Tables

Tlie sheet metal fonriing process i s u s u a l l y terminated by the onset of i n s t a b i l i t y , which can be t e n s i l e or conpressive. In t h e t e n s i l e case, and in s o - c a l l e d deep-drawing, which i s characterised generally by f a i l u r e on or near the punch-profile r a d i u s , f a i l u r e occurs i n plane s t r a i n (d^y = O) (Figure l ) and the p l a s t i c s t r a i n r a t i o , r , i s probably the most important m a t e r i a l p r o p e r t y . I n s t r e t c h forming where de^, dCy, dGg V 0, a d u c t i l i t y parameter such as uniform or t o t a l elongation, or the work-hardening

exponent, n, seems to be the parameter which c o r r e l a t e s most closely with maximum s t r a i n i n s t a b i l i t y . Bott and Pearce^^''have shoim t h a t r also has an effect in t h i s s i t u a t i o n .

The onset of compressive i n s t a b i l i t y can also be a cause of f a i l u r e , e i t h e r because the shape requirements of the pressing are not achieved, or because the foldlng-up of the metal - an advanced form of conpressive i n s t a b i l i t y - r e s u l t s in the development of high forming loads which caiise f r a c t u r e elsewhere in the p a r t . Compressive i n s t a b i l i t y i s variously termed buckling, folding, crinld.ing and vTrinkliruj. The teim 'buclilirg' i s used - often expanded to ' body buckling' - to describe bad shape in the body of a p r e s s i n g , w h i l s t wrinkling i s confined to the flange behaviour. I t i s t h i s term in t h i s connotation which w i l l be used in the present paper.

Wrinkling has been t r e a t e d by previous workers ( 2 - 7 ) , both t h e o r e t i c a l l y and experimentally. The f i r s t treatment was by Geckeler (2) who considered t h e o r e t i c a l l y the case of drawing without a blankholder and t r e a t e d the r a d i a l c r o s s - s e c t i o n of the flange as a bar of length equal t o the mean flange circumference, which, under the induced hoop s t r e s s , collapsed i n t o a s e r i e s of r a d i a l l y disposed wrinJcles. He e v a l u a t e d :

-(1) Tlie c r i t i c a l compressive hoop-stress (cr ) which 'vrould cause t h i s to occur, v i z . ,

E t^

a = 0.46 ~ 2 - (1) c b^ ^ '

where E is the buckling modulus, t the metal thickness and b the width of the fJange.

(2) The number of waves ( N ) into which the flange would collapse, viz.,

N = 1.65 f (2)

where a i s the mean flange r a d i u s .

The f i r s t experimental i n v e s t i g a t i o n was t h a t of Howald and Baldwin (3) using copper, 70/5O b r a s s , varioios tempers of aluminium and a r i m - s t e e l and t h i s work was l a t e r checked and expanded by Senior ( 5 ) ( 6 ) , 1*10 r e j i a c e d the r i m - s t e e l of the f i r s t i n v e s t i g a t i o n with an aluminium-stabilized s t e e l .

2

-Howald and Baldwin found good agreement between t h e i r experiments and

Geckeler*s theory, and concluded t h a t the p l a s t i c p r o p e r t i e s of the material had l i t t l e effect on wrinkling, except t h a t aluminium and brass were s l i g h t l y more prone t o wrinkling than was p r e d i c t e d . Senior found a wider spread

in r e s u l t s and introduced a t r a n s i t i o n region between wrinkled and wrinkle-free conditions, v i z .

OM ( | ) % f ^ 0.61+ ( i ) ^ (5) o

Even though the number of wrinkles formed for various blank diameters was in reasonable agreement with the r e s u l t s of Howald and Baldwin, Senior afforded a similar treatment to Geckeler's second equation, v i z . ,

1.65 ( f ) ^ N< 2.33 ( f ) . (h)

He further investigated constant clearance and constant load blank-holding and derived equations for N for those two conditions which f i t t e d well with the experimental res\iLts. An anomaly in S e n i o r ' s work is that

i n the case of the aluminium-stabilized s t e e l s , the theory overestimates a^, i . e . , g r e a t e r reductions are possible than those predicted, before wrinkling ensues. The work of MiV7agawa (7) i n general confirms t h a t of previous workers, and does not introduce anything s i g n i f i c a n t l y new.

I n g e n e r a l , these authors conclude t h a t material p r o p e r t i e s , even e l a s t i c and tangent modulus, play l i t t l e part in flange-wrinlcli'-g behaviour i n comparison with geometrical and dimensional f a c t o r s . However, a s w i l l be shown l a t e r , conditions e x i s t where experimental r e s u l t s do not agree with t h e o r i e s based simple on geometrical and dimensional factors and material p r o p e r t i e s must be considered.

I n t h i s work the effect of anisotropy on flange wrinkling has been

studied. Isotropy in a l l sheet metal whether commercially or experimentally produced i s the exception r a t h e r than the r u l e . One parameter for the

c h a r a c t e r i z a t i o n of a n i s o t r o p i c p l a s t i c p r o p e r t i e s i s the s t r a i n r a t i o de

X de

z

measured in u n i a x i a l t e n s i o n , and usually termed the r - v a l u e * The r - v a l u e v a r i e s in the plane of the sheet and a parampter Ar i s used as a measure of planar anisotropy. Ar i s r e l a t e d t o the earing behaviour of a metal ( 8 ) . The present work was i n i t i a t e d by empirical observations of the apparent r e s i s t a n c e to flange wrinkling of c e r t a i n s t e e l s with average r-values g r e a t e r than u n i t y . A systematic study was then made of the behaviour of m a t e r i a l s a range of r eind Ar values and the r e s u l t s i n t e r p r e t e d using a n i s o t r o p i c p l a s t i c i t y theory ( 9 , 1 0 ) .

* ï" + 2r^^ + r„^

- o 45 90

Materials

The materials used in this investigation were as follows: 1. Steel

Copper-containing extra-deep-drawing steel, 0.036-in.,thick 'Jouvcncel' ex- S.A. Metallurgique d'Esperance-Longdoz, ex- Copper Development Association. 2. Aluminium

Commercial-purity aluminium 0.036-in. thick coil, ex Alcan Industries Ltd. 5. Zinc

0 . 2 5 - i n . Commercial-purity zinc p l a t e , r o l l e d to 0.036-in. thickness before drawing, ex RTZ Ltd.

h. Titanium

0.036-in. t h i c k titanium 115 ex M I (Kynoch) Ltd.

Experimental Procedure

Most of the apparatus used in the t e s t s i s standard equipm nt and has been previously described ( l ) .

The experimental procedure can be conveniently subdivided i n t o two p a r t s :

1. Fundamental t e s t s (liniaxial t e n s i l e and plane s t r a i n [de = o] notch)

Duplicate t e n s i l e t e s t specimens having o v e r a l l lengths of about 6.5 i n . and a width of 0.5 i n . over a p a r a l l e l length of 2.5 i n . were used. To observe d i r e c t i o n a l i t y the t e n s i l e - t e s t specimens were cut with t h e i r axes a t 0 ° , 1+5° and 90° to the d i r e c t i o n of r o l l i n g of the s h e e t . Fran these specimens proof s t r e n g t h , t e n s i l e strength and s t r a i n r a t i o (see Table l ) were determined.

The required vee-notches were cu'' in the edges of 0 . 7 5 - i n . wide s t r i p s ; the notch dimensions are quoted in Table 2. From these specimens values of the flow s t r e s s a t vaiious s t r a i n s (see Table 2) were obtained. A Hounsfield extensometer was used for the s t r a i n measurement in t h i s p a r t of the

1+

-2 . Simvilative t e s t s ^flange wrinlcLing) Deep drawing p r e s s and t o o l i n g .

Individual flat-bottomed punch and die s e t s were attached t o a 35-ton-maximum-load, Hille-Engineering p r e s s , equipped with an X-Y recorder to record load versus pionch t r a v e l . Since t h e press was not capable of providing a blanlc-holder pressure (constant load) of l e s s than 6o p . s . i . a hydraulic p i s t o n and cylinder u n i t was attached to the blankholder pressure t r a n s m i t t i n g u n i t on the machine. With t h i s attachment i t was

possible t o obtain blankholder pressures as low a s 25 p . s . i . by the application of dead weights. The die set and flat-bottom punch gave a nominal 2 - i n .

diameter cup, s e l f - c e n t e r i n g fingers were used to l o c a t e the blanks.

The constant-load blankholder pressure was hydravilically transmitted onto the blank surface by means of a pressure p l a t e , which, v*ien the

blankholder (constant load) was not employed, acted as a constant-clearance blanldiolder up t o a maximumclearance of 0.25 i n . This could be reduced, i f

r e q u i r e d , such that there was zero clearance between t h e d i e and pressure p l a t e . The depth of partially-foi-med cups was measiired by means of a d i a l gauge ( l e a s t count 0.001 i n . ) , and another d i a l gauge ^•ras appropriately mounted t o i n d i c a t e any increase in clearance caused by the d e f l e c t i o n of

the tools during the drawing operation. When no blaiikholder was required the pressure p l a t e 'v/as removed. The blanks could then be placed on the punch, and pressed into the die t o the required depth of punch t r a v e l .

Simulative flange-wrinkling t e s t s .

To standardise the blanlc diameter, i t was necessary t h a t 8.1 the metals under consideration could be drawn successfully into a flat-bottomed

c y l i n d r i c a l cup. Preliminary t e s t s showed t h a t the l a r g e s t blank which could be drawn successfully for z i n c , the lowest r-value metal in t h i s study, was k i n . Thus, in a l l the flange-wrinkling t e s t - , 1+ i n . diameter blanks were used. The blanks were cut oversized and machined to s i z e -using a slow cutting-speed to minimize any edge work-hardening e f f e c t s . A l l blanlcs were O.O36 i n . ±0.002 i n . t h i c k n e s s . A punch speed of O.6I ft/min. was used

i n most cases, but a speed of 59-^ f t / n l n . was a l s o uoed. Measurement of ear and trough h e i g h t s .

The simple s e t - u p shown in Fig. 2 was used to measure the ear and trough h e i g h t s of the cupped blank. The cup was i n s e r t e d into the ring r e s t i n g on

a gauge p l a t e , and the three Allen screws on the peripheiy of the ring were used to c e n t r a l i s e and clamp the cup. The V-ended spindle of a d i a l gaiige ( l e a s t count 0.001 i n . ) was brought i n t o contact with the rim and the r i n g , with t h e cup, r o t a t e d . Readings of the d i a l gauge were noted to evaluate the ear and trough h e i g h t s .

Detection of the ccmmencement of wrinkling without a blankholder.

mounted on the f i x t u r e shown in F i g . 3- The blanks were r o t a t e d by means of the spindle and the wave amplitude noted. The blanks were then drawn by increments of 0.01 i n . and a f t e r each draw, the partially-drawn blank re-mounted on the f i x t u r e and r o t a t e d to determine the magnitude of the wrinkling.

Lubrication.

As f r i c t i o n between the blank and the tool surfaces can produce considerable variatlcxis in the r e s u l t s of simulative t e s t s , i t was

necesssary to adopt a standard l u b r i c a t i o n procedure in t h i s i n v e s t i g a t i o n . Polyethylene film, 0.0015-ln thick v/as eraplcyed on both sides of the

blank, cut away t o allow t h e punch nose t o remain in contact with the sheet m e t a l . Thie l u b r i c a t i o n system, by eliminating metal to metal contact between the die and blankholder and the blank prevented any

v a r i a t i o n s in the surface f i n i s h of the sheet metal from having an effect upon the experimental r e s u l t s . By allowing contact between the blank and the punch n e g l i g i b l e s t r a i n i n g occurred over the punch nose and punch p r o f i l e r a d i u s , and so a condition akin t o pure deep drawing was obtained. F i n a l l y , t h i s set of conditions allowed the successful drawing of four-inch

diameter z i n c - b l a a k s , which would have been d i f f i c u l t under other l u b r i c a t i o n conditions .

The r e s u l t s of the i n i t i a l t e s t s , c a r r i e d out t o determine the simulative t e s t i n g procedures, showed that no increase in drawing load occurred with increasing blankliolder pressure over the range 6O-850 l b s . , though an Increase in the ironing component i s observed (Figure k). The reason for t h i s i s t h a t the e x t r e m i t i e s of the e a r s which, l a t e on in the draw, must protrude s l i g h t l y from the polyethylene film, are nipped between the d i e ^nd the

blankholder, and thinned. Examination of the cups r e v e a l s Increased thinning with increasing blanliholder pressure, consequently an Increased load i s recorded. Figure 5 shows the effect of l u b r i c a t i o n on cups drawn a t 300 lbs blankholder press\ire. An Increase in cup-rim i r r e g u l a r i t y without l u b r i c a t i o n i s

demonstrated in Figure 6, while a study of the cup wall snows t h a t a more p a r a l l e l wall i s achieved without l u b r i c a t i o n ; c l e a r l y , f r i c t i o n i s ' s t i c k i n g '

the blank to the punch and providing a better-shaped p a r t . Figure 5 also shows a f a l l in the maxim^om drawing loaü with the a d d i t i o n of TSD 996 - a mineral o i l - to both sides of the polyethylene film. However, with the l a c k of any systematic i n v e s t i g a t i o n of the e f f e c t of o i l on polythene l u b r i c a t i o n , dry polyethylene film, applied as previously described, was used.

ResiiLts

The u n i a x i a l p r o p e r t i e s and the blanlcholder pressure required to

suppress wrinkling are given in Table 1. The modulus values are taten from various sources - the proof strengths and t e n s i l e strengths are not anomalous and do not require comment. The s t r a i n r a t i o s agree in general with

previously measured v a l u e s , though in the case of titanium, r Is seen t o be strain-dependent. However r shows a value of approximately 5 for the

t h r e e s t r a i n s used. The loads required to suppress vzrlnkling are quoted in the l a s t colijmn; with t h e exception of z i n c , t h i s l e a d decreases with increasing r .

6

-Effects due to normal anisotropy

The effect of d i f f e r e n t depths of piinch penetration upon wrinkle-formation in zinc and s t e e l i s sho\ni in Figure 7* In t h e case of s t e e l ( r « 1.50) the blank was not f l a t a t the s t a r t of the t e s t , but the wrinkle-amplitude was v i r t u a l l y unchanged a t 0.ll+-ln. punch p e n e t r a t i o n ,

while zinc ( r = 0.35) which was i n i t i a l l y f l a t , wrinkled considerably. When a f u l l set of wrinkles had been developed, using a punch speed of 0 . 6 l f t / m i n the n-umber was invariably seven, in good agreement with Geckeler's p r e d i c t i o n of 6 - 8 for the tool and blank dimensions employed in the present

i n v e s t i g a t i o n . However, due to the experimental d i f f i c u l t i e s associated with lack of f l a t n e s s , t h i s approach was abandoned and ihe load required

t o suppress wrinkling i n t h e range of materials used was determined, the r e s u l t s are l i s t e d in t a b l e 1. The load decreases with increasing r

except in the case of z i n c . From other r e s u l t s ( l l ) the p l a s t i c deformation of commercial-purity zinc i s thought to be anomalous, and t h i s r e s u l t i s thus not unexpected.

The value of r has an e f f e c t on f i n a l cup-height as shown in Figure 8 The mean cup-height (hg-h^) where h and h a r e t h e h e i g h t s of t h e e a r s

—2 e

and the troughs r e s p e c t i v e l y , was p l o t t e d against r and i t was found t h a t , f o r a given blanlc diameter, the mean cup-height Increases with Increasing r .

Effects due t o planar anisotropy ( r = 1^Ar ^ O)

For round blanks of z i n c , aluraini-um, aluminium-stabilized s t e e l drawn without a blankholder at O.61 ft/min. seven wrinkles were observed as previously described for the r o t a t i o n a l l y sjonmetrical case. However, at the higher speed of 3 9 ' ^ ft/min a d i f f e r e n t effect was observed as i l l u s t r a t e d in Figure 9 in t h a t zinc showed two wrinkles, wh:il.e aluminium and s t e e l each shovred four. This i s remeniscent of the earing behaviour of these metals (same figure) and suggests t h a t , at higher drawing speeds^ wrinkling occurs a t p o s i t i o n s of low r - v a l u e . Sqiiare blanlis were a l s o dra-vm. For z i n c , r o t a t i o n of the blanlc r e l a t i v e to the r o l l i n g d i r e c t i o n through ^5° a l t e r e d the number of wrinkles from 2 t o 1+ (Figure lO) while cutting out t h e blank i n different o r i e n t a t i o n s had no e f f e c t on the wrinkles developed in s t e e l and aluminium both of which show four symmetrical ears (Figure l l ) . In the case of titanium four equal wr.Inkles developed when r = 5'3 'was put i n the centre of the s i d e s , but two deep and two shallow wrinkles r e s u l t e d from 14-5" r o t a t i o n (Figure 1 2 ) . This i s intermediate between the

two-wrinlcle case of zinc and the four-equal--vTrinkle cases of aluminium and s t e e l . The reason for t h i s is made clear in Figure 13 . Wrinliiing w i l l not occur on the diagonals, for the hccp s t r e s s i s zero a t the extreme corner and, t r a v e l l i n g inv;ards fron t h i s p o i n t , any othe^" point i s r e s t r a i n e d from buckling by the t r i a n g u l a r tab consisting of the corner of the square. Similar effects are observed with constant-clearance and constant-load blankholding (Figure 1I+).

in the total amount of cold reduction. In this way the effect of Ar on the blankholder pressure required to suppress wrinkling may be seen (Figiure 15); it is evident that, with the two metals tested, the greater Ar the greater the required blankholder pressure.

Discussion

The p r e s e n t work h a s shown t h a t f l a n g e wrinld.ing b e h a v i o u r depends on b o t h noa-roal and p l a n a r a n i s o t r o p y , i n c o n t r a s t t o p r e v i o u s work, i n which t h e e l a s t i c modulus and t h e t a n g e n t modulus were t h e on3.y m a t e r i a l p r o p e r t i e s t h o u g h t r e l e v a n t t o t h i s form of i n s t a b i l i t y . G e c k e l e r ' s e q u a t i o n s may be r e w r i t t e n : UE T 1+E T 'c = ^ o = ^ ' T^^^ = (VEWT )' m ' m and so va = k . m VEWT^

where E i s the e l a s t i c modulus of Tj^^ the tangent modulus.

(5)

P l o t t i n g Vac against VE or VT gives a curve of the foim shown in Figure 16 where 'the asymtote i s K T/E or 1C/T r e s p e c t i v e l y .

P l o t t i n g a should increase with Increasing E and increasing Tj^. Table 2 shows quoted values for E and measured values of the r a t i o t e n s i l e strength to 0 . 1 ^ proof strength ( T / Y ) for a l l the m a t e r i a l s t e s t e d . This l a t t e r i s a measure of the slope of strain-hardening portion of the load/extension curve and decreases with increasing n (where cr = Ae^) or Tj^^. Txus materials with high E and high Tj^ (or T / Y ) should give h i ^ values of cr^ and so increased r e s i s t a n c e t o wrinkling. A l t e r n a t i v e l y , the factor E x T/Y should be as

high as p o s s i b l e . I t can be seen that predictions based on t h i s model do not f i t the experimental r e s u l t s . However, Pearce ( l 2 ) has shown that the r a t i o Y/T i s important in the case of body buckling in ageing temper-rolled r i m - s t e e l s .

A b e t t e r model for wrinkling can be constructed using anisotropic p l a s t i c i t y theory and the concepts of t e x t u r e hardening and softening. The von Mlses y i e l d - c r i t e r i o n p r e d i c t s yielding in the special case of an

i s o t r o p i c m.aterlal ( r = 1, Ar = O) when:

(az-ax)^ + ( V % ) ^ "• ( V z ^ ^ = 2X2 (6) where X i s the yield s t r e s s in u n i a x i a l tensl;:n.

This r e l a t i o n s h i p i s u s u a l l y shown g r a p h i c a l l y for the plane s t r e s s condition, a^ = 0, where i t p l o t s as an e l l i p s e of the form

a^-aa+a^^X^. (7)

Hill' s generalization for the anisotropic case predicts yielding when

F(a -0 )2 + G(a -a ) ^ + H(a -a ) ^ = 1, (8)

^ y z' ^ z X ^ X y' ' ^ '

where F , G and H are anisotropic s t r e s s parameters, and t h i s reduces t o :

^ ^ - ( S T ) ^ a + a 2 = X-

(9)

X * r + l ' X y y ^^^

for the case of r o t a t i o n a l symmetry about the z a x i s , and has been applied in the present work.

Recently Hosford (13) has proposed a more general r e l a t i o n s h i p for p l a n e - s t r e s s loading, namely:

P(R+l)a 2 - 2KPa cr, + R(P+l)a ^ _ P ( R + I ) X 2 (lO)

where R can be equated t o t h e average s t r a i n r a t i o ( r ) as defined in t h i s paper ( l O ) .

Equation (6) p l o t s as an e l l i p s e having the major axis a t 1+5° t o the o r i g i n of the orthogonal co-ordinates on which i t i s constructed, while equation (7) with increasing r - v a l u e s , s t r e t c h e s out the e l l i p s e s in the tension - tension quadrants and p u l l s in the e l l i p s e s in the tension

compression quadrants, preserving t h e i r symmetry with regard to the o r i g i n a l axes. The s t r e t c h i n g out and p-ulling i n of y i e l d l o c i by anis-'trop/ i s r e s p e c t i v e l y teimed t e x t u r e hardening and softening. For the more general case (eq. lO) w l t h A r VO o r , as s t a t e d by Hosford R x P, the major axis of the e l l i p s e i s r o t a t e d away from 1+5° t o the o r i g i n .

Any attempt t o draw a cup from sheet •r.etal without applied blanlcholder pressure w i l l r e s u l t i n e v i t a b l y in wrlnl\ling and can be therefore considered as a s e r i e s of bending operations with ds.^ = 0. This plane s t r a i n s i t u a t i o n i s represented in the f i r s t quadrant by the loading path

T+1

and thus increasing r values w i l l give r e s i s t a n c e to bending and hence

wrinkling. \Ihen t h e flange of a cup i s confined between die and blankholder i t i s under a s t r e s s system giving r i s e to plane s t r a i n with de^ = 0,

ignoring the tendency t o thicken a t the rim. The loading path i s represented by CH = - 1 in the second quadrant of the plane s t r e s s l o c i of figiore

7-Again, a high revalue m a t e r i a l , which here possesses a lower flow s t r e s s than a low r value m a t e r i a l , w i l l r e s i s t v/rinkling by defoimlng more readily in the sheet plane due to texture softening. C r y s t a l l o g r a p h i c a l l y , a high r - v a l u e material i s one in which the majority of the operating s l i p systems l i e in the plane of the s h e e t . The r e s i s t a n c e to bending and -tTrinkling

represented in the tension - tension quadrant and t h e deformation i n the plane of tlie sheet under imposed blankholder pressure i s thus simply

explained. The tendency t o wrinkle w i l l be g r e a t e r in low r - v a l u e m a t e r i a l , and a g r e a t e r blanldiolder pressure w i l l be required to suppress i t . I n the p r e s e n t work the r e s u l t s for aluminium and titanium should be compared. A similar l o c a l i z e d effect would be expected p r e f e r e n t i a l l y i n the regions of low r - v a l u e in a metal e x h i b i t i n g planar anisotropy, and tills was found in the present work. I t w i l l be noticed in Figure l8 that the zinc blanlc has vrrinkled in the low-r regions long before any significant geometrical changes have ensued which might be invoked as an a l t e r n a t i v e explanation for t h i s phenomenon.

Hosford and Backofen have calculated various values of the l a r g e r p r i n c i p a l s t r e s s r e l a t i v e to the u n i a x i a l y i e l d s t r e s s , and those relevant t o the flange are given below. For t h e s t r e s s r a t i o a = - l

^x r + 1

X " T i r T T

a

so for increasing r, — decreases, i.e. the flow stress in the sheet plane

A

decreases. For the case of an edge-notched s t r i p , which r e l a t e s also t o the strength of material in a cup-flange deforming in plane s t r a i n , the r e l a t i o n s h i p i s

a

X VL^r -I- 1 ^ ''

a a wh.L'.-h gives twice the value for ~ though producing a curve of rr- v s . r

of s.ijnllar form. S t r i c t l y , the loading p-sth here does not l i e in Figure 17 for rr, is now zero and there is a s t r e s s r _ . The measurea values of

a

X

rr- in t h i s work, however, f i t b e t t e r with the l a t t e r r e l a t i o n s h i p , so t h i s

A

i s used for the curves in Figure 19^ which are p l o t t e d from the data in Table 2, which gives the range of notch dimensions t r i e d . A notch depth of O.I63 i n s . with anotcii angle of 90° proved the most s a t i s f a c t o r y and the s t r a i n s developed are c l e a r l y similar in the two cases.

Conclusions

1 . Previous work on wrinkling which ignores anisotropy does not explain the observed phenomena.

2 . For anisotropic m a t e r i a l s ( r j^ 1, Ar = 0, Ar f. O) which a r e deformed slowly, Geckeler"s equation p r e d i c t s the observed number of wrinkles.

10

-number of wrinkles equal to the -number of ears which would f i n a l l y form on a drawn cup; t h i s number represents the low r - v a l u e regions of the blank and i s not predicted by Geckeler's equation.

k. Blankholder pressure required to suppress wrinkling increases with decreasing r v a l u e .

5. Blankholder pressure to suppress wrinkling increases with Increasing Ar, suggesting the minimum r-value as the c r i t i c a l parameter.

6. Texture hardening and softening provide an adequate q u a l i t a t i v e explanation for a l l these phenomena.

a

7. The experimental r e l a t i o n s h i p between — and r for an edge-notched X

X

t e s t piece i s shown to give a reasonable f i t with the t h e o r e t i c a l r e l a t i o n s h i p a X 2 ( r + l ) 'V2rfl References 1. B o t t , C.H., and Pearce Roger

The effect of varying s t r a i n - r a t i o on the hydraulic bulging behaviour of aluminium

sheet.

The College of Aeronautics Note Mat. No. 9' 2 . Geckeler, J.W. 3 . Howald, T . S . , and Baldwin, W.M. h. Sanders, J . S . , and Loxley, E.M. 5. Senior, B.W. 6. Senior, B.W.

P l a s t i c folding of the walls of hollow cylinders and seme other folding phenomena in bowls and s h e e t s .

Z angew. Math. Mech. 1928, 8 3^1-Foldirig i n the cupping equation. Trans.'A.S.M. I9I+7 ^

VI-Wrinkling during c y l i n d r i c a l drawing o p e r a t i o n s .

BISRA report M W / E / 3 7 / 5 2 .

Flange wrinkling in the cup drawing operation with no blankholding. BISRA r e p o r t MW/E/1+5/53.

Flange wrinkling in deep drawing operations. J . Mech. Phys. Solids 1955-57 i t r i 2 3 .

8. Wright, J.C. 9. Hill, R. 10. Hosford, W.F. Jr. and Backofen, W.A. 11. Williams, D.A.C, 12. Pearce Roger 1 3 . Hosford W.F. J r .

The phenomenon of earing in deep drawing. Sheet Metal I n d u s t r i e s I965 h2 &lk.

Mathematical theory of p l a s t i c i t y . o . u . p . 1950.

Strength and p l a s t i c i t y of textured m e t a l s . Proceedings of the ninth Sagamore ordnance m a t e r i a l s research conference.

Syracuse University Press 1965»

The Investigati on of a d u c t i l i t y phenomenon i n commercial-purity z i n c .

The College of Aeronautics Thesis 1967* Use of the Swift-cupping press for the

assessment of s t e e l sheet for cold pressing, Sheet Metal I n d u s t r i e s I96O 2 1 61+7.

Texture strengthening.

TABIE 1 Metal Zinc ccmmercial-p u r i t y Aluminium c o n m e r c i a l -p u r i t y Aluminium -s t a b i l i -s e d E t e e l Copper r i m -s t e e l Titaniiffli, commercial p u r i t y Modulus of E l a s t i c i t y (E) X 10* p s i 13 10 28 15 Proof Strength ( r ) a t 0 . 1 ^ e l o n g a t i o n x 10= p s i Ï 0 Ï*B ^ - ° \ 1 2 . 2 12.8 13-1 1 2 . ? 3-1 5-3 3-2 3.2 2 0 . 6 2 2 . 6 22 2 1 . 7 2 0 . 1 2 1 . 1 2 1 . 0 2 0 . 7 3 7 . 3 1*3.8 1*7.9 '^3.20 T e n s i l e Strength ( T ) X l O ' p . s . i . T^ T4S T90 \ 17.5 17-8 1 8 . 0 1 7 . 7 9•l^ 9-7 9 . 7 9.6 39.2 1*3-5 1*3-3 1*2.0 39-1* '*0.o 3 9 . 3 39.6 51*.0 5i*-8 55-8 5l*-9 1.1* 3-0 £-.0 1.5 m 1.8 30.0 51* 19 s t r a i n R a t i o r 2 0 , 2 0 _ ' 20r 2 0 -^0 T^ ^go r -275 .335 -390 .35"^ .690 .610 .660 .61*2 1.35 1-55 2.00 1.61* 1.61* 1.33 1.6-7 1.50 3-37 3-37 6 05 l*.Ol* 10_ l O r ' lOj. 1 0 -- 0 ^45 90 r 5.39 3-1*1 5-63 3.97 15r„ 1 5 r ' 15r 15= ^0 ^4S 9o -3-83 3.29 5.52 3.90 BlanMiolder P r e s s u r e Required to Suppress Wrinkling ( p . s . i . ) 55 60 55 1*0 35 = (where a = measured a t p e r c « i t e l o n g a t i o n b = angle t o the d i r e c t i o n of r o l l i n g )

Dimensions Angle ( 9 ° ) W i n 60 0.250 60 0.1+25 90 0.250 90 0.1+25 Zinc commercial p u r i t y .0025^ .005^ •Ol'/j 2.15 2.35 2.1+0 2.15 2 . 3 5 2.1+0 1.95 1-92 1.87 Aluminium - S t a b i l i s e d S t e e l .0025^0 .005^ .Olfo 1.31+ 1.36 1.35 1.375 1.^0 1.50 1.50 1.1+2 1.38 T i t a n i u m Commercial p u r i t y .0025^ . 005^5, . 0 1 ^ 2 . 0 0 1.85 1.50 1.1+0 1.22 1.375 1.280 1.160

dez=o

F I G . i i . EFFECT OF BLANKHOLDER PRESSURE ON PUNCH LOAD,

-s

6 Bnch Lood (^ Div-IOOlbs) Scale 11, Blankholder Pressure=300ps.i. Stabilised Steel

No Lubrication

_ ^ " —• _ , - . — ^~"*~<1.^ ••"- ~~. ' • " ~ ^ ^ . ^ _ — - - " - " ^ — " ^ •— ~ — , — _ _ ' — - w 1 1 1 1

i

EDO RIM STEEL

FIG. 7. WOVE AMPLITUDE AT DIFFERENT PUNCH TRAVELS.

MEAN CUP HEIGHT/ STRAIN RATIO I

I

1 ' z - ,

aS'

: > *

o 1-b A A •^. -*. ro LL CM

Ö

Blank holder pressure/(ear trough-height) 160-Q . UJ Ê

s

-- EDO RIM srecL

- Z I N C 'OS .12 -M ( h e - h t Kin.) .18 RG.m y<jc V E or-/Trn FIG. 16. ^ Quadrant I (Tension -Tension)

0 ^

<^-i. Spherical pressure vesseL

- t - . Plane s t r a r , dgy=0,

Sheet rolling, cupvwall and drawing, S h E T B&DNG, face notched tensile strip.

oi = i , Will of cylindrical pressure vessel o<=0 Pure in plane o£.=1,Plane strain, d £ j = 0 ^^-^z^ CUP FLANGE IN ^-""^ DRi^WING. Edge notched >«• tensile strip. Ouadrent I I (Tension -Compression) -CTv

X b II a> X 2 —

ii

ii

= 5

I