computer aids in sheet metal engineering

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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 parts are characterized by a complicated shape; the strains involved in forming rarely exceed 20': but the displacements are large. ditions can cause large changes in failure rates. while the limitation of useful straining in sheet materials can be reasonably modelled, the determination of strain distri- butions in complex parts can not yet be determined in an accurate and economic fashion. gested that the use of these ideas leads to tractable. approxiniate computer design aids for comp?ex shapes which can be used effectively by experienced designers. Sheet forming operations are conducted close to a failure limit and therefore small changes i n con- The various techniques employed as a basis for computer modelling of sheet forming are reviewed and it i s concluded that The use of idealizations of deformation processes, materials, forming operations and shapes i s introduced and it i s sug- INTRODUCTION Sheet metal engineering encompasses material selection, pro- cess design, tool design and the setting-up of press lines for high volume production o f sheet metal components. There are many specialized branches of sheet metal engineering but the areas con- sidered here are in the automotive, appliance, and packaging in- dustries. Parts are produced in quantities greater than 5,000 per day and the price of the finished product is typically a few dol- lars per kilogram rather than many hundreds of dollars per kilo- gram as is often the case in the aircraft and electronic indus- tries. The forniing process i s characterized by a complicated final part shape and although the sheet undergoes large displacements during forming the deformation or strain imposed i s often quite small. Studies at Toyota [l], f o r example, show that most areas of sheet i n an autobody are deformed less than a few percent and higher strains, which rarely exceed 204, are confined to relative- ly small volumes of material. large amounts of money are expended on tooling while other manu- facturing costs per part are quite small. It is often considered that there is a high utilization of material in forming processes. This is not always true and sheet metal plants rarely convert more than 70% o f the incoming sheet into a final product. Most of the scrap loss comes from blanking and trisming around the final part rather than breakage. Surpris- ing as it may seem, however, the objective in press forming i s not to avoid breakage altogether but rather to run the process so close to the limit that some breakages do occur. then clearly, the material i s too good, the presses could be run faster, less lubricant could be used or various other savings achieved. The consequence of operating very close to failure is that small material or process changes can have very large effects on scrap rate and further that failure analysis must be conducted on a probabilistic basis. The extent to which the workpiece i s controlled during form- ing must also be considered. The tool designer aims a t maximum control but invariably the sheet is sliding over partially lubri- cated surfaces, sticking to the tool in some regions, constrained i n others and i n certa;n areas of the die suspended freely in air inviting wrinkling and other forms of instability. geometry i s changing at every instant in the forming stroke and clearly one could not expect to develop a simple mathematical model o f such a process. The fascination of sheet metal engineering lies in the broad physical phenomena encountered, the inherent uncertainty of the process and the fact that it cannot be reduced to a simple set Of rules. methods of sheet metal engineering are being replaced only at a gradual rate by computer-aided methods. no less computer-oriented than others and hopefully they are just as intelligent, but the basic process depends critically on many variables and does not invite simple mathematical solutions. There are enormous incentives to improve the efficiency Of sheet metal manufacture i n a l l areas - i n the design process, in the utilization of stronger but less formable materials, in the lowering o f scrap rates and in reducing manufacturing costs. In this paper we consider how computer-aided design and computer graphics are contributing towards this improvement. trol of presses and stamping plants is also important but i s not discussed here because the fundamentals involved are no different from those i n other branches of manufacturing. The discussion i s divided into three parts; two of these concern analysis, one Of the forming process and the other of the response of sheet metals to deformation. In the final section we address the subject Of whether the existing concentration on analysis is really appro- priate when in fact the basic problem is in the area of design both of the process and of the details of tooling. Various exam- ples are presented where the investigation of idealized processes has been more profitable than the detailed analysis of existing forming operations. ANALYSIS OF SHEET FORMING From a process point o f view, very If no parts fail, The actual These attributes are also the reason why traditional Sheet metal engineers are Computer Con- There are two distinct parts to the analysis of a sheet form- ing process. The first is to predict or inodel the distribution of strains and show how these develop as forming proceeds. strain distribution i s determined predominantly by the geometry of the part, the tooling and the blank and also by friction and the mechanical properties of the sheet. It i s required to determine the magnitude of strain and the strain path. ing of an element is considered to follow a simple proportional or linear path; this is not necessarily true but in many cases the assumption i s reasonable. Figure 1 i s schematic representation of successive strain envelopes, which have been experimentally deter- mined from the defonnation of selected elements in a blank during the deep drawing of a square cup. The second part of the analysis is to determine the extent to which the material will deform t.?fore its ability to distribute the strain becomes exhausted. formation history. stresses, then the material limits can also be described in the strain space. The well known forming limit diagram, FLD, defines the useful limits of formability, as a function of straining path, based on the criterion of the onset of localized necking. A form- ing limit curve i s shown schematically in figure 2. There are other competing modes o f failure, which may intersect the FLD, and these can also be plotted on Figure 2. buckling failure which can occur before the onset of localized necking, other possibilities are stress controlled or strain con- trolled fracture loci. The l e f t hand hatched curve i s suggested from a maximum shear stress criterion, while the right hand locus is based on a competition between the continuing deformation of a localized groove and eventual fracture or fracture preceding (or sometimes coinciding with) the development of a localized groove. The combination of the process and material diagrams i n Fig- ures 1 and 2 permit the prediction of overall limits. the strain envelope i n Figure 1 cannot go beyond any limit curve in Figure 2 although there are exceptions. small radius bending, localized necking i s prevented by tooling constraints and the strain envelope can exceed the necking curve. The probabilistic nature of sheet foning must also be remembered and the lines in Figure 2 should more properly be considered as mean curves. This overall view of sheet metal forming analysis was dis- cussed in a previous contribution [2] and limit curves are re- viewed more deeply in a review of failure maps [3]. the numerical techniques which are used to detemine both the strain distributions during forming and the strain limits which can be sustained by the materials are discussed. The Usually the strain- This i s also dependent on the de- If the sheet i s not subject to large surface One is a wrinkling or In general, I n some cases, such as In this paper, ANALYTICAL TECHNIQUES FOR SHEET FORMING PROCESSES The analytical study of any forming operation requires a union of a model of the material behaviour and of the process: the model must also provide for realistic boundary conditions and frictional effects at the interface of the tools and material. Since the process may be performed hot or cold, and at either a fast or slow speed, these aspects must also be considered. tletal forming operations can be broadly classified into two main groups: either bulk or sheet forming processes. The former may be truely three dimensional in character with both the stress and strain components varying from point to point throughout the body, while in the latter operations it i s often reasonable to assume that at any location there is no variation in physical quantity across the thickness o f the sheet. Due to the complexity of most forming processes an exact so- lution i s usually unattainable. Therefore before embarking upon any analytical study of a metal working operation it i s prudent to ask what i s required from the eventual solution. Often a rate in- dependent, rigid-plastic analysis i s suitable if the requirement i s a reasonable estimate o f a load or pressure to execute a form- ing operation, or if it is required to enquire how the load i s affected by changing certain process parameters. In addition the actual process may follow closely either plane stress, plane strain or axisynmetric deformation. modes, in particular, when coupled with a rigid-perfectly plastic material model have formed the basis of many solution procedures for bulk forming processes. lower and upper bound approaches [6, 7, 81 and slip line field The l a t t e r two deformation Slab or force balance methods [4, 51, Annals of the ClRP Vol. 30/2/1981 541

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Page 1: Computer Aids in Sheet Metal Engineering

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

Page 2: Computer Aids in Sheet Metal Engineering

(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-

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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

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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

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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.

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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

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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.

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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.

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

SIMPLE TENSION

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

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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 .

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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.

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Page 6: Computer Aids in Sheet Metal Engineering

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

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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.

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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

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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.

546