Key-Note- Papers

Computer Aids in Sheet Metal Engineering

J. L. Duncan (1) and R. Sowerby; Department of Mechanical Engineering, McMaster University, Hamilton, OntarioKanada

SUWItARY

Sheet metal par ts are character ized by a complicated shape; the s t ra ins involved i n forming r a r e l y exceed 20': but the displacements a re la rge . d i t i o n s can cause la rge changes i n f a i l u r e rates.

wh i le the l i m i t a t i o n o f useful s t ra in ing i n sheet mater ia ls can be reasonably modelled, the determinat ion o f s t r a i n d i s t r i - but ions i n complex par ts can no t y e t be determined i n an accurate and economic fashion.

gested t h a t the use o f these ideas leads to t rac tab le . approxiniate computer design aids fo r comp?ex shapes which can be used e f fec t i ve l y by experienced designers.

Sheet forming operations are conducted close t o a f a i l u r e l i m i t and there fore small changes i n con-

The various techniques employed as a basis fo r computer model l ing o f sheet forming are reviewed and i t i s concluded t h a t

The use o f i dea l i za t i ons o f deformation processes, mater ia ls , forming operat ions and shapes i s introduced and i t i s sug-

INTRODUCTION

Sheet metal engineering encompasses mater ia l se lec t ion , pro- cess design, t oo l design and the sett ing-up o f press l i n e s f o r h igh volume product ion o f sheet metal components. There are many spec ia l i zed branches o f sheet metal engineer ing bu t the areas con- s idered here are i n the automotive, appliance, and packaging i n - dus t r ies . Par ts are produced i n quan t i t i es greater than 5,000 per day and the p r i ce of the f i n i shed product i s t y p i c a l l y a few do l - l a r s per k i logram ra the r than many hundreds o f d o l l a r s per k i l o - gram as i s o f ten the case i n the a i r c r a f t and e lec t ron i c indus- t r i e s .

The forniing process i s character ized by a complicated f i n a l p a r t shape and although the sheet undergoes la rge displacements dur ing forming the deformation or s t r a i n imposed i s o f ten qu i te small. Studies a t Toyota [l], f o r example, show tha t most areas o f sheet i n an autobody are deformed less than a few percent and higher s t ra ins , which r a r e l y exceed 204, are conf ined t o r e l a t i v e - l y small volumes o f mater ia l . la rge amounts of money are expended on t o o l i n g wh i le o ther manu- fac tu r i ng costs per p a r t are qu i te small.

I t i s o f ten considered tha t there i s a h igh u t i l i z a t i o n of mater ia l i n forming processes. This i s no t always t rue and sheet metal p lan ts r a r e l y convert more than 70% o f the incoming sheet i n t o a f i n a l product. Most o f the scrap l oss comes from blanking and t r i sm ing around the f i n a l p a r t ra the r than breakage. Surpr is - i ng as i t may seem, however, the ob jec t i ve i n press forming i s no t t o avoid breakage a l together bu t ra the r t o run the process so c lose t o the l i m i t t h a t some breakages do occur. then c lea r l y , the mater ia l i s too good, the presses could be run fas te r , l ess l u b r i c a n t could be used o r var ious o ther savings achieved. The consequence o f operat ing very c lose t o f a i l u r e i s t h a t small mater ia l o r process changes can have very l a rge e f fec ts on scrap r a t e and f u r t h e r tha t f a i l u r e ana lys is must be conducted on a p r o b a b i l i s t i c basis.

The ex ten t t o which the workpiece i s con t ro l l ed dur ing form- i ng must a lso be considered. The too l designer aims a t maximum cont ro l bu t i nva r iab l y the sheet i s s l i d i n g over p a r t i a l l y l u b r i - cated surfaces, s t i c k i n g t o the too l i n some regions, constrained i n others and i n certa;n areas o f the d i e suspended f r e e l y i n a i r i n v i t i n g wr ink l i ng and o ther forms o f i n s t a b i l i t y . geometry i s changing a t every i ns tan t i n the forming s t roke and c l e a r l y one could no t expect t o develop a simple mathematical model o f such a process.

The fasc ina t ion o f sheet metal engineer ing l i e s i n the broad physical phenomena encountered, the inherent uncer ta in ty of the process and the f a c t t h a t i t cannot be reduced t o a simple set Of ru les . methods o f sheet metal engineer ing are being replaced on ly a t a gradual r a t e by computer-aided methods. no l ess computer-oriented than others and hopefu l l y they are j u s t as i n t e l l i g e n t , but the basic process depends c r i t i c a l l y on many var iab les and does no t i n v i t e simple mathematical so lu t ions .

There are enormous incent ives t o improve the e f f i c i e n c y Of sheet metal manufacture i n a l l areas - i n the design process, i n the u t i l i z a t i o n o f stronger bu t l ess formable mater ia ls , i n the lowering o f scrap ra tes and i n reducing manufacturing costs. I n t h i s paper we consider how computer-aided design and computer graphics a re con t r i bu t i ng towards t h i s improvement. t r o l o f presses and stamping p lan ts i s a lso important bu t i s no t discussed here because the fundamentals invo lved are no d i f f e r e n t from those i n o ther branches o f manufacturing. The discussion i s d iv ided i n t o three par ts ; two o f these concern ana lys is , one O f the forming process and the other o f the response o f sheet metals t o deformation. I n the f i n a l sec t ion we address the subject O f whether the ex i s t i ng concentrat ion on ana lys is i s r e a l l y appro- p r i a t e when i n f a c t the basic problem i s i n the area of design bo th o f the process and o f the d e t a i l s o f t oo l i ng . Various exam- p les a re presented where the i nves t i ga t i on o f idea l i zed processes has been more p r o f i t a b l e than the de ta i l ed ana lys is of e x i s t i n g forming operat ions.

ANALYSIS OF SHEET FORMING

From a process po in t o f view, very

I f no par ts f a i l ,

The actual

These a t t r i b u t e s are a lso the reason why t r a d i t i o n a l

Sheet metal engineers a re

Computer Con-

There are two d i s t i n c t par ts t o the ana lys is o f a sheet form-

i ng process. The f i r s t i s t o p red ic t o r inodel the d i s t r i b u t i o n o f s t ra ins and show how these develop as forming proceeds. s t r a i n d i s t r i b u t i o n i s determined predominantly by the geometry o f the par t , the too l i ng and the blank and a lso by f r i c t i o n and the mechanical p roper t ies of the sheet. It i s required to determine the magnitude o f s t r a i n and the s t r a i n path. i ng o f an element i s considered to fo l l ow a simple p ropor t iona l o r l i n e a r path; t h i s i s n o t necessar i l y t rue bu t i n many cases the assumption i s reasonable. Figure 1 i s schematic representat ion o f successive s t r a i n envelopes, which have been exper imental ly de ter - mined from the defonnation o f selected elements i n a blank dur ing the deep drawing o f a square cup.

The second p a r t of the ana lys is i s t o determine the ex ten t t o which the mater ia l w i l l deform t .?fore i t s a b i l i t y t o d i s t r i b u t e the s t r a i n becomes exhausted. format ion h i s to ry . stresses, then the mater ia l l i m i t s can a lso be described i n the s t r a i n space. The we l l known forming l i m i t diagram, FLD, def ines the use fu l l i m i t s o f f o rmab i l i t y , as a func t ion o f s t r a i n i n g path, based on the c r i t e r i o n o f the onset of l oca l i zed necking. A form- i ng l i m i t curve i s shown schematical ly i n f i gu re 2. There are other competing modes o f f a i l u r e , which may in te rsec t the FLD, and these can a lso be p lo t ted on Figure 2. buck l ing f a i l u r e which can occur before the onset of l oca l i zed necking, o ther p o s s i b i l i t i e s are s t ress con t ro l l ed o r s t r a i n con- t r o l l e d f rac tu re l o c i . The l e f t hand hatched curve i s suggested from a maximum shear s t ress c r i t e r i o n , wh i l e the r i g h t hand locus i s based on a compet i t ion between the cont inu ing deformation o f a l oca l i zed groove and eventual f rac tu re o r f rac tu re preceding ( o r sometimes co inc id ing w i th ) t he development o f a l oca l i zed groove.

The combination o f the process and mater ia l diagrams i n Fig- ures 1 and 2 permit the p red ic t i on o f ove ra l l l i m i t s . the s t r a i n envelope i n Figure 1 cannot go beyond any l i m i t curve i n Figure 2 although there are exceptions. small rad ius bending, l oca l i zed necking i s prevented by t o o l i n g cons t ra in ts and the s t r a i n envelope can exceed the necking curve. The p r o b a b i l i s t i c nature o f sheet f o n i n g must a lso be remembered and the l i n e s i n Figure 2 should more proper ly be considered as mean curves.

This ove ra l l view o f sheet metal forming ana lys is was d is - cussed i n a previous con t r i bu t i on [2] and l i m i t curves are re- viewed more deeply i n a review o f f a i l u r e maps [3]. the numerical techniques which a re used t o de temine both the s t r a i n d i s t r i b u t i o n s dur ing forming and the s t r a i n l i m i t s which can be sustained by the mater ia ls are discussed.

The

Usual ly the s t r a i n -

This i s a lso dependent on the de- I f the sheet i s n o t subject t o l a rge surface

One i s a wr ink l i ng o r

I n general,

I n some cases, such as

I n t h i s paper,

ANALYTICAL TECHNIQUES FOR SHEET FORMING PROCESSES

The ana ly t i ca l study o f any forming operat ion requ i res a union o f a model o f the mater ia l behaviour and o f t he process: the model must a lso provide f o r r e a l i s t i c boundary cond i t ions and f r i c t i o n a l e f fec ts a t t he i n te r face o f t he too l s and mater ia l . Since the process may be performed hot o r cold, and a t e i t h e r a f a s t o r slow speed, these aspects must a lso be considered. t le ta l forming operat ions can be broadly c l a s s i f i e d i n t o two main groups: e i t h e r bu lk o r sheet forming processes. The former may be t r u e l y th ree dimensional i n character w i t h bo th the s t ress and s t r a i n components varying from po in t t o p o i n t throughout the body, wh i le i n the l a t t e r operat ions i t i s o f ten reasonable t o assume t h a t a t any l oca t i on there i s no v a r i a t i o n i n phys ica l quan t i t y across the thickness o f the sheet.

Due t o the complexity o f most forming processes an exact so- l u t i o n i s usua l ly unattainable. Therefore before embarking upon any ana ly t i ca l study o f a metal working operat ion i t i s prudent t o ask what i s required from the eventual so lu t ion . Often a r a t e i n - dependent, r i g i d - p l a s t i c ana lys is i s su i tab le i f the requirement i s a reasonable est imate o f a load o r pressure t o execute a form- ing operat ion, o r i f i t i s requ i red t o enquire how the load i s a f fec ted by changing c e r t a i n process parameters. I n add i t i on the actual process may fo l l ow c lose ly e i t h e r plane stress, plane s t r a i n o r axisynmetr ic deformation. modes, i n pa r t i cu la r , when coupled w i t h a r i g i d - p e r f e c t l y p l a s t i c mater ia l model have formed the basis o f many so lu t i on procedures f o r bu lk forming processes. lower and upper bound approaches [6, 7, 81 and s l i p l i n e f i e l d

The l a t t e r two deformation

Slab o r fo rce balance methods [4, 51,

Annals of the ClRP Vol. 30/2/1981 541

(s.1. f . ) analyses, o r t he method of cha rac te r i s t i cs [7-111, have a l l been employed t o study such operat ions as drawing, extrusion, r o l l i n g . indent ing, upsett ing, fo rg ing and the l i k e . An overview of some o f these methods i s a lso t o be found i n Refs. [12, 131, and the b ib l iography t o each o f these a r t i c l e s c i t e s many app l i - cat ions. The ana ly t i ca l techniques mentioned above have been in- troduced i n ascending order o f mathematical soph is t i ca t ion , and w i t h i n the conf ines o f the assumed mater ia l model and plane s t r a i n deformation s.1.f. analysis i s mathematically r igorous and can a lso Drovide a qood reDresentat ion o f the deformation mode i n cer -

t e rna l force. Tension forces are t ransmi t ted through the sheet and the process can proceed provided the a b i l i t y o f the mater ia l t o sus ta in t h i s tension i s no t exceeded. the mater ia l may f a i l e i t h e r by necking o r by f rac tu re . Necking f a i l u r e may be approached by an i n s t a b i l i t y anal s i s If we con- s ide r the greatest tension ( force per u n i t widthf i n ' t h e sheet as

As already mentioned

T1 = 'lt (1 1 as i l l u s t r a t e d i n Fiqure 3 where l r i s the area tes t o r i nc ioa l .

I . tens i le , s t ress and t the cur ren t thickness;then the maximum ten- s ion occurs when

t a i n working process. '

The app l i ca t i on o f the same ana ly t i ca l techniques t o sheet metal forming processes i s much less widespread. Reference [8] Drovides some simole examDles o f the o l a s t i c col laose o f f l a t A T

p la tes by bending' i .e . the format ion o f a p l a s t i c hinge along cer - u'l uGl d t = t a i n l i n e s i n the surface o f a p l a t e t o form a mechanism. The 7 = T+F- same idea can be app l ied t o the p l a s t i c forming o f sheet metal , where a f i n a l shape can be achieved by bending along spec i f i ed curved o r s t r a i g h t l i n e s i n the surface o f t he sheet, see Refer- ences 114-161, and the discussion l a t e r i n t h i s tex t . S l i p l i n e f i e l d ana lys is can a i d i n the development o f t he best i n i t i a l blank shape when deep drawing i r r e g u l a r par ts [17]. symnetrical ear ing, which i s o f ten seen when drawing c y l i n d r i c a cups from s tee l d iscs. can be pred ic ted using an iso t rop ic s . l . f l [ l8] . Szczepinski [19] has used the method o f cha rac te r i s t i cs t o study both steady and non-steady deformation processes o f axisym- met r ic she l l s . such as tube drawing and tube s ink ing operat ions. He a lso provides the under ly ing theory and c i t e s some examples f o r the forming o f t h i n wal led she l l s o f a r b i t r a r y double curva- ture. It i s t o be noted t h a t w i th the s.1.f. analyses o f Refer- ences [17-19], i t i s usua l ly necessary t o reso r t t o numerical procedures t o e f f e c t a so lu t ion .

One of the major c r i t i c i s m s o f s lab, load bounding o r s.1.f. methods i s the use o f a r i g i d - p e r f e c t l y p l a s t i c mater ia l model. However, attempts t o account f o r s t r a i n hardening, and poss ib ly s t r a i n r a t e and temperature e f f e c t s de t rac ts f r o m the simp1 i c i ty o f the methods. a1 b e i t sometimes very approximate. answers t h a t i s the s t rength o f these p a r t i c u l a r ana ly t i ca l techniques. S t ra in hardening, s t r a i n r a t e e f f e c t s and the l i k e are b e t t e r handled by a l te rna- t i v e ana ly t i ca l procedures such as f i n i t e d i f f e rence and f i n i t e element techniques, w i th the l a t t e r method having grown i n promi- nance dur ing the l a s t decade o r so. Nevertheless, how we l l the cons t i t u t i ve equations used i n these codes re la tes t o actual ma- t e r i a l behaviour i s a moot po in t . A f i n i t e element method. some- times re fe r red t o as the mat r ix method, has been developed by Kobayashi and h i s co-workers f o r the bu lk forming o f a r i g i d - p las t i c , r a t e independent so l id . The processes studied have gen- e r a l l y been ones o f a x i a l symnetry and much o f t he work i s sum- marized i n References [20. 213. Alexander and Pr i ce [22] have provided a b r i e f review o f hot, bu lk metal forming, where the most c m n assumption i s i ne las t i c . incompressible behaviour. The flow i s regarded as non-Newtonian and the "v i scos i t y " i s re - l a ted t o l oca l s t r a i n ra tes and o ther e f f e c t s such as temperature and t o t a l s t ra in : see a lso the work by Zienkiewicz and h i s co- workers [23, 241.

It i s t o be noted t h a t cons i tu t i ve laws f o r i n e l a s t i c so l i ds genera l l y r e l a t e s t ress ra tes and s t r a i n ra tes as discussed i n the recent survey paper by H i l l [25]. The s t ress ra tes employed i n the equations shown vanish under r i g i d body r o t a t i o n i . e . the s t ress r a t e measure i s sa id t o be ob 'ec t i ve The choice o f s t ress r a t e has been discussed by, among b i a g e r [26] and Masur 1271. A r igorous formulat ion o f the e l a s t i c - p l a s t i c , s t ress ra te - s t r a i n r a t e r e l a t i o n s and i t s embodiment i n t o a f i n i t e element code i s the sub jec t o f some recent work by Lee [28]. Incorporat- i ng e l a s t i c e f f e c t s i n t o the mater ia l model adds a f u r t h e r degree o f real ism, and t h i s would be a necessary adoption i f spr ing back o r res idua l stresses were being studied. The examples c i t e d by Lee, op c i t , deal w i t h bu lk forming processes such as extrusion. A s i m i l a r formulat ion i s t o be found i n References [29-341 f o r t he ax isy rmet r ic hydrau l i c bu lg ing o r punch s t re t ch ing o f p e r i - phe ra l l y clamped sheet metal discs, based on membrane analysis. These analyses have shown reasonably good agreement w i th exper i - mental observations and have supported ce r ta in ca l cu la t i ons per- formed using f i n i t e d i f fe rence methods [35-371. A f i n i t e element code f o r t he axisymnetr ic sheet metal forming o f a m d - p l a s t i c mater ia l i s g iven i n Reference [a]. Undoubtedly progress has been made i n the a b i l i t y t o model f i n i t e s t r a i n and/or l a rge d i s - placement metal forming processes, b u t the r e a l impact o f such f i n i t e eletnent procedures as a design a i d w i l l n o t be rea l i zed u n t i l more complex p rac t i ca l geometries can be accounted f o r and computer t ime reduced.

Th is sec t ion has dea l t almost exc lus i ve l y wi th the model l ing o f the process, and l i t t l e has been sa id about the use fu l l i m i t s o f f o rmab i l i t y of the mater ia l i.e. deformat ion p r i o r t o the onset o f wr ink l ing , necking o r f rac tu re . It i s c l e a r t h a t these l i m i t s a re inf luenced no t on l y by the ma te r ia l s t ruc tu re bu t a lso by the deformation a r i s i n g dur ing the forming operat ion. An assessment o f the in f luence o f the mater ia l s t ruc tu re on the a b i l i t y t o en- hance the forming l i m i t i s a h i g h l y complex problem. I n the nex t sec t ion some discussion i s devoted t o t h i s top ic . I n most cases the models employed provide a very imprecise representat ion of s t ruc tu ra l e f fec ts , furthermore the evo lu t i on o f t he models has r e l i e d heav i l y on experimental data gathered from simple propor- t i o n a l deformation processes.

Four f o l d

It i s the a b i l i t y t o ob ta in r e l a t i v e l y rapid,

LIMIT CURVE PREDICTIONS FOR SHEET MATERIALS

In most sheet forming processes, deformation occurs i n a re - g ion which i s some distance from the p o i n t o f app l i ca t i on o f ex-

For a mater ia l obeying a s t r a i n hardening law,

(3 )

where n and are e f fec t i ve s t ress and s t r a i n and n the s t r a i n hardening index, the i n s t a b i l i t y cond i t ion o f equation (2) leads to an i n s t a b i l i t y s t r a i n

(4 )

where B i s the s t r a i n r a t i o i n the process, i . e .

8 = €$El (5)

This cond i t ion can be i l l u s t r a t e d i n the forming l i m i t diagram by the l i n e shown i n Figure 4. I n the region E~ I 0. i l l u s t r a t e d by the heavier l i n e i n t h i s f igure, t he cond i t ion i s i d e n t i c a l t o the l oca l i zed necking c r i t e r i o n o f H i l l [39]. I n 1947, Lankford e t a1 C401 obtained exper imental ly a forming l i m i t curve as shown d iagramnat ica l l y i n Figure 5. Fol lowing subsequent work, [41, 421 t h i s became known as the Keeler-Goodwin formin l i m i t curve. A mathematical model of t h i s necking p r o c e s s d e t i r e range of s t r a i n r a t i o s was provided by an imperfect ion ana lys is o f Marciniak and Kuczynski [43] i n 1968 and essen t ia l l y the con- cepts of t e n s i l e i n s t a b i l i t y presented here are those o f Marciniak C441.

The Marciniak ana lys is provided a basis f o r computing a neck- i n g l i m i t curve su i tab le fo r use in a computer-aided design sys- tem. years; these have included the incorpora t ion o f s t r a i n r a t e sen-

s i t i v i t y [45]. kinematic hardening 1461 and damage [47]. The re- s u l t s obtained are sens i t i ve t o the degree o f imperfect ion assumed f o r the mater ia l . I n order t o overcome t h i s degree o f a r b i t r a r i - ness, Stgren and Rice [48] introduced a b i f u r c a t i o n ana lys is u t i - l i z i n g a d e f o m t i o n theory o f p l a s t i c i t y . A forming l i m i t curve can a lso be ca lcu la ted on t h i s basis however, experimental j u s t i - f i c a t i o n f o r t h i s p l a s t i c i t y model i s l ack ing and a rb i t ra r i ness o f a d i f f e r e n t k ind i s s t i l l present.

c r i t i c a l l y dependent on the cons t i t u t i ve laws employed and the l i t e r a t u r e on forming l i m i t curves ind ica tes t h a t extreme care must be exercised i n ob ta in ing mater ia l p roper ty data. If t h i s i s done however. it i s apparent t h a t adequate theory arid computation- a l techniques e x i s t t o provide appropr iate curves f o r the onset o f necking i n simple sheet forming processes.

Fa i l u re by l oca l i zed necking i s more l i k e l y i n sheet forming than d u c t i l e f rac tu re , however i n severe drawing operat ions and i n forming h igh l y strengthened sheet, t he determinat ion of a f rac tu re l i m i t i s important. As w i t h l oca l necking, there appear t o be competing theor ies based on conventional p l a s t i c i t y incorpora t ing a damage law [49] and b i fu rca t i on ana lys is [50] using a deforma- t i o n theory o f p l a s t i c i t y . The former i s favoured by the authors as i t can lead t o an understanding o f the r o l e o f mic ros t ruc ture i n r e s i s t i n g f a i l u r e . I n terms o f p rac t i ca l app l i ca t ion . i t i s poss ib le t h a t a simple maximum shear s t ress c r i t e r i o n o f f a i l u r e may be s u f f i c i e n t f o r a design system [51].

Many refinements have been made t o t h i s model i n recent

As w i th most i n s t a b i l i t y analyses, t he r e s u l t s obtained are

1DEP.L SHEET FORMING PROCESSES

Enormous e f f o r t has been devoted t o the development o f mathe- mat ica l models which describe, step-by-step. the actual process dur ing the forming o f a part . To do t h i s the i n i t i a l cond i t ions such as the blank shape and too l geometry must f i r s t be speci f ied. When the model i s used as a design t o o l , t he ca l cu la t i on i s con- t inued u n t i l an unsat is fac to ry cond i t ion i s i d e n t i f i e d . The i n i - t i a l cond i t ions are then changed and the ca l cu la t i on repeated.

I n a simple system, t h i s approach may be q u i t e e f f i c i e n t but, a t l e a s t w i t h the present generat ion of computers, i t has n o t been a useful way o f analysing t yp i ca l sheet metal forming processes. It i s worth examining there fore o ther branches o f engineer ing t o see how complicated problems are handled. For example, p rac t i ca l thermal power generat ing p lan ts are extremely complex and wh i l e some model l ing o f ac tua l performance i s done, the design concepts a re a r r i ved a t n o t from the model bu t through the app l i ca t i on of concepts o f c lass i ca l thermodynamics. This subject deals essen- t i a l l y with idea l systems which can never e x i s t i n the rea l world. Theoret ical concepts such as the reve rs ib le process, the per fec t gas and the idea l thermodynamic f l u i d have l e d t o some very prac- t i c a l resu l t s . Without wishing t o imply any c lose analogy, i t i s suggested here t h a t some purpose i s served by examining idea l pro-

542

cesses i n sheet metal forming.

computer a ids based on idea l i zed o r h igh l y s i m p l i f i e d systems. Observation has shown t h a t wh i le researchers i n s c i e n t i f i c labora- t o r i e s have veered towards r igorous model l ing o f e x i s t i n g sheet forming processes, t oo l designers have concur ren t ly and indepen- den t l y developed computer a ids based on h igh l y s imp l i f i ed , inexact models. They have used t h e i r own knowledge t o ad jus t and modify the r e s u l t s of the approximate analysis. d iscussion of i d e a l i z a t i o n may promote some convergence o f the d i rec t i ons o f the groups.

The Idea l Deformation Process

element cannot be determined so le l y from i t s i n i t i a l and f i n a l shape, bu t depends on the pa th by which t h i s i s achieved. The whole u t i l i t y o f the forming l i m i t diagram discussed prev ious ly i s based on the s imp l i f y i ng assumption t h a t paths i n sheet forming processes are monotonic p ropor t iona l processes which can be repre- sented by a s t r a i g h t l i n e i n t h i s diagram rad ia t i ng from the o r i - gin. Provided the designer exercises h i s own judgement, t h i s sim- p l i f i c a t i o n i s appropr iate f o r most cases. There are however add- i t i o n a l imp l ied assumptions. The more obvious one i s t ha t , p a r t i - c u l a r l y f o r f rac tu re curves, the diagram on ly app l ies t o plane s t ress processes where the hydros ta t i c component o f s t ress i s de- pendent on l y on the in-plane stresses and s t ra ins . A more subt le r e s t r i c t i o n i s t h a t the diagram app l ies n o t simply t o l i n e a r s t r a i n paths bu t ac tua l l y t o "pure homogeneous" deformation. t h i s process, the p r inc ipa l axes do no t r o t a t e w i th respect t o the mater ia l element; a p r i nc ipa l element whose sides are al igned w i t h the p r inc ipa l planes remains orthogonal and deforms as shown i n Figure 6.

I f we assume t h i s idea l deformation mode app l ies , a very simple formulat ion o f l a rge s t r a i n s i s possible. [Z] t h a t i f a two-dimension orthogonal , non-pr incipal l i n e p a i r OA, OB deforms as shown i n Figure 7, then by l oca t i ng the re fe r - ence axes f o r the deformed element so t h a t €

p r i nc ipa l t o t a l s t ra ins and t h e i r o r i en ta t i on , f! are given by

Various s i t ua t i ons are discussed here and examples given o f

It i s hoped t h a t t h i s

P l a s t i c i t y i s complicated by the f a c t t h a t the s t r a i n i n an

tn

It may be shown

= Eyx as shown, the XY

2EX tan 21: = !. = x x - cyy

= tn(1 + El,*) "1 ,z These equations not on l y have the same form as the f a m i l i a r ones f o r small s t ra ins bu t they are exact f o r pure homogeneous deforma- t i o n and agree w i th Green's la rge s t r a i n tensor [52]. formation i s no t a pure homogeneous one. then there i s no simple expression for t o t a l s t ra ins .

The example i s introduced because i t demonstrates tha t if a p a r t i c u l a r idea l i zed deformation mode i s assumed, the determina- t i o n o f la rge s t ra ins i s simple and eas i l y understood.

The Ideal Sheet Metal

I f the de-

Authors have given l i t t l e a t ten t i on to the idea l sheet mate- r i a l f o r metal forming. One can however suggest a few of i t s a t - t r i bu tes .

I n forming, the sheet i s transformed from a plane t o a non- developable surface. Some deformation i s necessary bu t i d e a l l y t h i s should be done by shear d i s t o r t i o n ra the r than by th inn ing . Hence the sheet should have an i n f i n i t e s t rength o r res is tance t o deformation i n the through-thickness d i rec t i on . We would wish t o form the sheet wi thout g rea t fo rce hence, t he in-plane y i e l d s t ress should be zero. then f l ow t o any o ther shape if i t s in-plane s t rength was zero so we want some mechanism whereby the sheet suddenly gain s t rength a f t e r forming.

No rea l sheet performs i n t h i s idea l fashion bu t i t i s re - markable t h a t one o f the most formable ma te r ia l s we have, namely drawing q u a l i t y rimned s tee l , has some o f these a t t r i bu tes . The resistance t o through-thickness deformation i s measured by the ex ten t t o which the t e n s i l e r -va lue ( r a t i o of width-to-thickness s t r a i n s i n the t e n s i l e t e s t ) exceeds un i ty . r i m e d s tee l i s no t i n f i n i t y as i n the idea l mater ia l bu t it i s a t l e a s t subs tan t i a l l y greater than un i ty . s t ress i s n o t zero e i t h e r , bu t i t can be reduced by r o l l e r l e v e l l - i ng before forming so t ha t deformation forces are a lso reduced. The mater ia l then has the useful p roper ty of rega in ing t h i s s t rength by na tura l ageing a f t e r the forming process.

duced f o r forming each year and it i s a h igh l y developed product. With h inds igh t i t can be sa id t h a t t h i s development process has been co r rec t i n tha t i t aims a t the " idea l sheet". ment process was, i n fac t , an evo lu t ionary process bu t perhaps w i t h o ther mater ia ls the concept o f ideal behaviour might be he1 p f u l . Geometric Mode l l ing

The ana lys is o f forming o f the idea l sheet can be considered as an essen t ia l l y geometric problem. thickness" elements would deform by shear w i thout change i n area.

An examination of many o f the t r a d i t i o n a l d ie design ru les shows t h a t they are based on a s i m i l a r assunrption so tha t , a l -

Unfortunately the formed component could

The r -va lue f o r

The in-plane f low

M i l l i o n s o f tons o f drawing q u a l i t y r i m e d s tee l a re pro-

The develop-

If there i s r.3 "through

though i t i s an abstract ion, such a constant area process must provide a useful basis fo r t oo l and blank design. t ha t the geometrical ana lys is requ i red can on ly be done manually for qu i te simole cases. I t seemed there fore t h a t a very useful computer a i d Mould be t o design a mapping system which could transform elements from a f l a t sheet t o a given surface, permi t t - i ng each element t o deform wi thout change i n area and ensur ing t h a t con t inu i t y was s a t i s f i e d and each element f i t t e d together w i thout gaps o r overlapping.

developed by the authors i n conjunct ion w i th the Un ive rs i t y o f B r i t i s h Columbia and the Ford Motor Company. An e a r l y example [53] i s i l l u s t r a t e d i n Figure 8(a). on a surface represent ing the f i n a l p a r t shape and the nodal po in ts d ig i t i zed . The equal area transformat ion o f t h i s mesh i n t o the plane represent ing the i n i t i a l blank i s a lso i l l u s t r a t e d i n Figure 8(b). Computation times are so small t ha t t h i s mapping can e a s i l y be performed a t an i n t e r a c t i v e graphics computer t e r - minal. The model considers on l y the i n i t i a l and f i n a l shape and maps element-by-element. Much remains t o be done i n determining appropr iate cons t ra in t s and i n avo id ing numerical i n s t a b i l i t i e s i n the mapping process however the i n i t i a l resu l t s are encourag- ing . The ob jec t i s t o develop methods o f adjustment so t h a t the designer can guide the transfonnat ion and use the r e s u l t as an a id t o f i n d i n g an appropr iate shape fo r the i n i t i a l sheet blank.

Although t h i s i s no t a f u l l y developed method i t i s c lea r tha t the "geometr ical ly spec i f ied" problem can be solved w i t h minimal computation whereas the r e a l problem which must invo lve equ i l i b r i um and mater ia l considerat ions i s , a t present, i n t r a c - table.

Ideal Sheet Metal Shapes

mar tens i t i c s tee l cannot be formed i n processes which requ i re s i g n i f i c a n t s t re tch ing . lopable surfaces such as cones and cy l inders and fo lded t o g ive reasonably small rad ius ( s i x o r e i g h t t imes thickness) bends. Shapes produced i n t h i s way can be considered idea l sheet metal shapes and the vast range' o f products inc lud ing s tee l f u rn i tu re , appliance cabinets and bu i l d ing products, produced by fo ld ing , bending rnd r o l l forming demonstrate t h i s . It would be des i rab le t o extend t h i s range.

M u l t i p l e developable surfaces are usua l ly produced by c u t t i n g and j o i n i n g however, they can a lso be formed by bending along curved l i n e s as i l l u s t r a t e d i n Figure 9. A complete descr ip t ion o f t h i s process i n terms o f d i f f e r e n t i a l geometry has no t y e t been publ ished however aspects o f the process have been considered by var ious authors [14, 151 and the use o f these shapes w i l l increase when design methods are establ ished.

It i s i n te res t i ng t o observe t h a t sheet metal s t ruc tu res tend t o f i n d these idea l shapes w i thout the bene f i t o f the compu- te r . Examination o f a veh ic le a f t e r a ser ious c o l l i s i o n w i l l show many curved l i n e fo lds i n which the sheet has col lapsed l a r g e l y by bending ra the r than by in-plane deformation processes which absorb much m r e energy.

S imp l i f i ed Models o f Forming Processes

i c u l t , t oo l designers have developed simple models which permit the process t o be displayed i n computer graphics systems. example i s i l l u s t r a t e d i n Figure 10. d ie r i n g e i t h e r as a f l a t surface o r a developable surface. It i s assumed t h a t as the punch decends, t he sheet i s penetrated by the punch and a t any ins tan t , elements o f the sheet a re e i t h e r unmoved or they adhere t o the punch. s i m p l i f i c a t i o n o f the process b u t t he ana lys is i s no t d i f f i c u l t as i t i s simply an in te rpenet ra t ion exercise. I t i s however a useful a i d and the designer can ad jus t the i n i t i a l clamped shape and the o r i e n t a t i o n angle o f the punch t o obtain. i n the l i g h t o f h i s own experience, a su i tab le " foo t -p r i n t " o f the punch on the sheet [54].

I n another approach the sheet i s assumed t o span the t o o l i n g so t h a t it i s e i t h e r on the punch o r tangent ia l t o i t [55] as i l l u s t r a t e d i n Figure 11. about the s t r a i n i n g process and a method developed f o r ob ta in ing idea l i zed too l p r o f i l e s . Many o f these shapes remain p rop r ie ta ry bu t a p a r t i c u l a r example i s the so-cal led t r a c t r i x d ie used f o r deep drawing [56].

An invers ion o f t h i s approach has been success fu l l y app l ied i n model l ing the forming o f superp las t ic sheet metal [57]. Here the sheet i s assumed t o e i t h e r s t i c k t o the too l surface o r span the in te rven ing areas with a deforming, uni form thickness membrane having a c i r c u l a r p r o f i l e as i l l u s t r a t e d i n Figure 12.

The most su rp r i s ing aspect o f these var ious s i m p l i f i e d m d e l s i s n o t t h a t they should g ive useful resu l t s bu t ra the r t h a t they are employed w ide ly throughout the i ndus t r y w i thout theore t ic ians genera l l y being aware o f t h e i r existence.

The problem i s

This idea was the basis o f the "geometric imodelling" system

A g r i d o f elements was marked

F u l l y work hardened sheet and heat t rea ted mater ia ls such as

They can however be curved t o form deve-

Because the ana lys is o f actual forming processes i s so d i f f -

The sheet i s clamped i n the An

C lea r l y t h i s i s a gross

S imp l i f y i ng assumptions can be made

CONCLUDING REMARKS

A b r i e f review o f the methods o f analysing sheet metal form- i ng processes has been given and t h e i r app l i ca t i on as a bas is fo r computer a ids i n design discussed.

An attempt has been made t o i l l u s t r a t e some idea l i za t i ons about sheet metal forming and t o show how cu r ren t computer a ids depend on ra the r c lever s imp l i f i ca t i ons o f an i nhe ren t l y complex process. This leads t o computer models which g ive useful r e s u l t s provided the so lu t i on i s guided by the input of an experienced

543

designer.

a ids f o r the experienced designer appears t o be much more appro- p r i a t e than attempts t o provide a complete mathematical model o f the whole process which could be used by a person no t f am i l i a r w i t h t r a d i t i o n a l sheet metal engineering. This "black box" approach might be su i tab le f o r i nhe ren t l y s impler manufacturing processes bu t i t has been shown here t h a t sheet metal forming i s dependent on a la rge number o f mater ia l and process var iab les and furthermore, the operat ions are always ca r r i ed ou t very c lose t o the f a i l u r e l i m i t . Under such cond i t ions there i s a rea l need t o develop computer a ids which ass i s t the experienced designer i n exerc is ing h i s judgement. no t t o replace it.

A t the present time, the development o f i n te rac t i ve computer

The ob jec t i ve i s t o support experience,

ACKNOWLEDGEMENTS

The authors would l i k e t o thank t h e i r colleagues and students f o r t h e i r help i n prepar ing t h i s paper and i n pa r t i cu la r , M r . E. Chu f o r h i s help i n model l ing and M r . D. Wong fo r h i s he lp wi th i l l u s t r a t i o n s . They a lso thank the Natural Sciences and Engineer- i ng Research Council o f Canada fo r t h e i r support o f sheet metal research a t McMaster Un ivers i ty . The work on geometric model l ing i s ca r r i ed on j o i n t l y w i th O r . J. P. Duncan o f the Un ive rs i t y o f B r i t i s h Columbia and the Engineering Research Staf f of the Ford Motor Company, Dearborn, p a r t i c u l a r l y D r . S. K. Samanta. The authors a re pleased to acknowledge t h e i r con t r i bu t i on and thank the Ford Motor Company f o r permission t o pub l i sh t h i s work.

REFERENCES

1. Ish igak i , H., i n "Mechanics o f Sheet Metal Forming", (Ed. by

2. Duncan. J. L. and Altan. T.. "New Di rec t ions i n Sheet Metal 0. P. Ko is t inen and N-M. Warlg). Plenum Press, 1978, p. 329.

Forming", Annals CIRP. Vol. h,-1980,,,p. 153. ;;;a Duncan. J. L.. Formab i l i t y Maps", Annual

n a r k i n s , R. N. ,."The Mechanical Treatment o f Metals", A l l en and Unwin, London, 1968.

H i l l . New York. 1968. 6. Av i tzur , 8.. "Metal Forming Processes and Analysis", McGraw-

7. Johnson, W: and Kudo, H.. "The Mechanics o f Metal Extrusion",

8. Johnson, W. and Mel lo r , P. B . , "Engineering P l a s t i c i t y " , Van

9. H i l l , R., "Mathematical Theory o f P l a s t i c i t y " . Clarendon

Manchester Un ive rs i t y Press, Manchester, 1962.

Nostrand Reinhold, London, 1973.

Press, Oxford, 1950. 10. Rowe, G. W., "An In t roduc t i on t o the Theory o f Metalworking", Edward Arnold, London, 1965. 11. Johnson, W., Sowerby, R. and Venter, R. D., "Plane S t r a i n S l i p L ine F ie lds f o r Hetal Deformation Processes", Pergamn Press, London. 1982. 12. Al tan, T. and Lahot i , G. O., "L imi ta t ions . A p p l i c a b i l i t y and Usefulness o f D i f f e r e n t Methods o f Analyzing Forming Problems", Annals CIRP, Vol. 28, 1979. p. 473. 13. Johnson, W. and Sowerby. R., i n "App l ica t ion of Numerical Methods t o Forming Processes", (Eds. H. Armen and R. F. Jones).

14. Johnson, W. and Yu, T. X., "The Angle o f Fold and the P1;stic Work Done i n the Folding o f Developable F l a t Sheets o f Metal . AMO-Vol. 28, ASME, 1978, p. 1.

J. Mech. Eng'g. Sci., Vol. 22, 1980. p. 233. 15. Duncan, J. L., Duncan, J . P., Sowerby, R. and Levy, B. S . , "Curved-Line Folding o f Sheet Metal" , Sheet Metal Indus t r ies , Vol. 58, 1981. p. 527. 16. Resch, R. D., " P o r t f o l i o o f Shaded Computer Images", PE IEEE., Vol. 62. 1974. p. 496. n a s e k , V. V. and Lange. K., "The Use o f the S l i p LIne Method i n Deep Drawing o f Large I r r e g u l a r Shaped Components". Proc. 7 th North American Metalworking Conf., 1979, Ann Arbor, Michigan, p. 65. 18. Sowerby, R. and Johnson, W . , "Pred ic t ion o f Ear ing i n Cups Drawn from An iso t roo ic Sheet usina S l i o L ine F i e l d Theorv".

Tech; Rep. AFML-TR-79-4105. Univ. Ca l i f o rn ia . Grke ley , 1979. 21. Kobayashi, S . , i n "Engineering P l a s t i c i t y : Theory o f Metal Formin Processes Vol. " Ed. by H. Lippmann . CISM Courses and Le%ures No. i39. SArinier-Ver lag, 19!7, p! 41. 22. Alexander, J. M. and Pr ice. J. W. H . , o f Hot Metal Forming", Proc. 18 th Mach. Tool Des. Res. Conf., MacMillan Press, 1978, p. 267. 23. Zienkiewicz, 0. C. and Godbole, P. N., "Flow o f P l a s t i c and V iscop las t ic So l ids w i t h Special Reference t o Extrusion and Form- i n g Processes". I n t . J. Num. Meth. i n Eng'g., Vol. 8, 1974, p. 3. 24. Zienkiewicz. 0. C. , Jain. P. C. and Onate. E.. "Flow o f So l ids

F i n i t e Element Analysis

During Forming and Extrusion: Some Aspects o f .Numerical Solut ions", I n t . J. Sol ids,Structures. Vol. 14. 1978, p. 15. 25. H i l l , R.. 'Aspects o f Invar iance i n So l i d Mechanics". & Appl. kchs . , Vo!. 18, 1978. p. 1. 26. Praaer, W . , An Elementarv Discussion o f De f in i t i ons o f Stress

lec tu res No. 139. Springer-Verlag, 1977, p. 81. 29. Key, S. W., Kr ieg, R. D. and Bathe, K. ,I., "On the app l i ca t i on o f the f i n i t e element method to metal forming process", Meth. i n Appl. Vyh . an; Eng'g.. Vol. 17, 1979, p. 597. 30. Chu, C. C . , An and y s i s of l oca l i zed necking i n punch s t re tch- ing", I n t . J . Sol ids Structures, Vol. 16, 1980, p. 913. 31. Wang, N-M. anbi3i-T. , "Analysis o f sheet nletal stamp- ing by a f i n i t e element method", Trans. ASME. J. Appl. Mech., Vol. 45E, 1978. p. 73. 32. Kitagawa, H . , Nakamachi, E. and Tomita, Y., "S ta t i c and dyna- mic ana lys is o f la rge de f lec t ions o f an e l a s t i c - p l a s t i c t h i n

-

plate" , Proc. 6 th North American Metalworking Conf., 1978, p. ,236. 33. Takezono, S., Nakanwhi, E. and Yamaguchi, T . , "E las t ic /v isco- p l a s t i c ana lvs is o f t h i n c i r c u l a r o la tes under la roe s t ra ins and la rge deformations", Trans. ASME. , ' J . Appl. Mech.,-Vol. 47E, 1980, p. 741. 34. Clifi, A. S . , "An Incremental Complete So lu t ion o f S t re tch Formina and Deeo Drawina o f a C i r cu la r Blank Usinc a Hemisoherical Punch"; M . : M e c h . S&., Vol. 18, 1976, p. 23.- 35. Woo, D. M., "The St re tch Forming Tests", The Engineer, Vol 220. 1965. D. 876. - , - 36. 'Woo, 0. M., "On the Complete So lu t ion o f the Deep Drawing Problem". I n t . J. Mech. Sci . , Vol: 10, 1968, p. 83. 37. Wana. N-M.. "Larae o l a s t i c deformation o f a c i r c u l a r sheet caused by punch s te tch ing" , Trans. ASME., J. Appl. h c h . , Vol.

38. Kobayashi, S. and Kim, J. H., i n "Mechanics o f Sheet Metal Formin , (Ed. by D. P. Koist inen and N-M. Wang). Plenum Press. d: 341.

37E, 1970, pp. 431-440.

39. H i i l , R., "On Discontinuous P l a s t i c States w i th Special Refer- ence t o Local ized Necking i n Thin Sheets", J . Mechs. Phys. So l ids , Vol. 1, 1952, p. 19. 40. Lankford, W. T.. Low, J. R. and Gensamer. M.. "The P l a s t i c Flow o f Aluminum A l l o y Sheet Under Combined Loads", Trans. AIME., Vol. 171, 1947, p. 574. 41. Keeler, S. P., "Pred ic t ing Forming L im i t s - 4". Sheet Metal -- Indust r ies , Vol. 48, 1971, p. 5. 42. Goodwin. G. M., "AoDl icat ion o f S t ra in Analvsis t o Sheet Metal Forming Problems'jn the Press Shop", SAE baper No. 680093, Jan. 196R. - -. . . . - - -. 43. Marciniak, Z. and Kuczynski, K., " L im i t S t ra ins i n the Pro- cesses o f S t re tch Forming Sheet Metal", I n t . J.~Mech. Sci., Vol. 9, 1967, p. 608. 44. Marciniak, Z . , i n "Mechanics o f Sheet Metal Formin$'. (Ed. by D. P. Ko is t inen and N - M m ] F P l e n u m Press, 1978. p. 215. 45. Marciniak. Z.. Kuczvinski. K. and Pokora. T.. In f luence o f the P las t i c Propert ies of a Mater ia l on the Formina L i m i t Diaaram i n Tension", I n t . J . Mech. Sci.. Vol. 15, 1973, p.-789. 46. Tvergaard. V., "E f fec ts o f Kinematic Hardening on Local ized Necking i n B i a x i a l l y Stretched Sheets", I n t . J. Mech. Sci,. Vol. 20 1978 p 651. 47: JaliAie;, J. M., Schmitt, J. H.. Argemi. R . . Salsmann, J . L. and Baudelet, B., "D i f fe ren t Damage Behaviours and Their I n f l u - ence on Forming Processes", Memoires Scient. Res. Meta l l . , No. 3, March 1980, (Proc. 11 th I n t . Dee0 Drawino Res. Conf.). D. 313. 48. StGren, S. and Rice, J. R., "LocalizGd Neckina i n7T6 in Sheets'' J. Ekch. Phys. So l ids , Vol. 23, 1975,,,p. 421. 49. Marciniak, Z. and Kuczynski, K., Bending Processes", I n t . J. Nech. Sci., Vol. 21, 1979, p. 609. 50. Hutchinson, J. W. and Tveraaard. V.. "Surface I n s t a b i l i t i e s

The Forming L i m i t Curve f o r

on S t a t i c a l l y Strained P las t ic -So l ids" , Jn t . J. Mech. Sci., Vol. 22, 1980, p. 339. 51. Glover, G., Duncan, J . L. and Embury, J. D . , "Fa i l u re Maps f o r Sheet Metal". W t a l s Technoloa, Vol. 4,,,1977, p. 153.

and Duncan, J. L.. Determination o f 52. Sowerby, R . . C m Large St ra jns i n Metalforming". submitted t o J. S t ra in Analysis, 1981. 53. Chu. E., Sowerby, R., Duncan, J. P. and Duncan, J. L., "Pre- l i n i n a r v Studies o f Geometric Model l ino o f Sheet Metal Stamoinos". McMastei Un ive rs i t y Report, Contract W&k With the Ford Motor to. ; Dearborn, June 1980. 54. McCormick, H. O., "CAD/CAM fo r Automotive Design", SME Tech- n i c a l Paper MS77-768, 1977. 55. Duncan, J. L. and B i rd . J. E.. "ADDroximate Calculat ions f o r Draw Die Fonning and The i r App l ica t ions t o Aluminum", Sheet Metal Indus t r ies , Vol. 55. 1978. p. 1015. 56. May. 0.. German Patent, No. 658898, 1932. 57. Forster, J. A., Kleiner. H. J., Duncan, J. L. and Grach, J., "Thickness D is t r i bu t i ons i n Themformed Suoerolast ic Zn-A1 Sheet". Second Symp. on Appl. o f So l i d Mechs., McMaster Un ivers i ty , Hamilton, Ontar io, Canada. Vol. 1, 1974. p. 77.

Rate", rt A 1 Math., Vo1.-18, 1961. p. 403. 27. Mas- n the D e f i n i t i o n o f Stress Rate", Qrt. Appl. Math., Vol. 19. 1961, p. 160. n e e , E. H.. i n "Engineering P l a s t i c i t y : Theory o f Metal Form- i n g Processes, Vol. 1". (Ed. by H. Lippmann). C I S M Courses and

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Figure 1 Diagram of successive envelopes of strains in a blank drawn into a square cup showing assumed proportional strain paths.

Figure 3 Diagram showing the definition o f principal traction. T1, in a defonning sheet.

f, 7k%, \ \ \ rRD

Figure 2 Diagram of approximate failure limits due t o necking (FLD), fracture and w r i n k l i n g i n simple sheet forming processes.

€1

€ 2 MINOR SIRAIN

Figure 4 Locus of stratns a t maximum traction force i n a power- law hardening material.

SIMPLE TENSION

Figure 5

€ 3

Diagram o f early fonnlng limit measurements. Ref. 40, fi t ted by a maximum normal stress criterion of failure.

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--7f //

C C

Figure 6 Diagram o f an element having faces aligned with the principal planes and deforming i n a "pure hanogeneous" defonnat 1 on process .

Y V

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Figure 7 Diagram sharing the orientation o f the measuring axes ox, oy appropriate for a large strain "pure hano- geneous" process. The axes are chosen so t h a t Exy=Eyx.

545

Figure 9 Example o f mul t ip le developable surfaces connected by curve l i n e folds.

I I I

Figure 10 Cross-section o f a forintng operation modelled on the " interpenetrat ion" basis.

m PUNCH

Figure 11 Cross-section o f a Conning operation modelled on a "tangential contact" basts.

z f

Y

X

Figure 8(a) Mesh obtained by d l g i t i z f n g nodal points o f a g r i d drawn on the surface o f pa r t o f a typ ica l stamping o f complicated shape.

SPECIMEN

Y

- - x

Figure 12 Cross-section o f a pressure fonnlng operation modelled on a "c i rcu lar p r o f i l e " basis.

Figure 8(b) Constant area napptng o f the mesh i n Figure 8(a) on t o the base plane.

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