1

METAL FORMING

Betzalel Avitzur Lehigh University

I. Introduction II. Basic Concepts III. Typical Processes IV. Phenomena V. Replacing Brute Force VI. Approaches to Understanding Metal Forming VII. Summary

GLOSSARY

Drawing: Metal forming process whereby theworkpiece is a shaped longitudinal prismthat undergoes a reduction and change in itscross section area and shape while beingpulled through a shaped converging die.

Extrusion: Metal forming process whereby theworkpiece is placed in a chamber with anopening and is forced to escape through theopening, usually being pushed out by amandrel.

Forging: Metal forming process whereby theworkpiece is placed between an anvil and ahammer and subjected to compressive forcebetween them.

Friction: Resistance to sliding motion along theinterface between two solids.

Lubrication: Supply of substance on the inter-face between two sliding solids aimed at re-ducing the friction and/or the wear on theinterface.

Metal-forming processes: Processes that causechanges in the shape of solid metal articlesvia plastic (permanent) deformations.

Modeling: Procedure to present a physical re-ality by other means.

Perfectly plastic materials: Idealization of thecharacteristics of metals undergoing plasticdeformations.

Pressure-induced ductility: Increase in theability of metals to undergo plasticdeformations without fracturing, this abilitybeing enhanced by high environmentalpressure.

Rolling: Metal forming process whereby theworkpiece is a longitudinal prism, which isplaced between two opposing circular rollsthat rotate in opposite directions, drag theworkpiece along, and force it to reduce incross section.

Simulation procedures: Procedures thatprovide representation of a physical realitythrough a mathematical or physical model.

Metallic components can be shaped in amanner similar to the molding of pottery. Theraw material of a fundamental simpler shape isprovided by a primary process like casting,powder consolidation, earlier forming processes,or even by electric deposition. Metals deformvery much like soft clay or wax. Even in thesolid state, permanent changes in shape can beforced upon them by displacement of relativepositions between neighboring material points.To enforce these changes, external forces areapplied. While soft plasticine can be molded bytiny toddler’s fingers, for metal forming,specially constructed tooling, usually of hardmaterials, are manipulated, sometimes bycolossal machinery. A variety of processes, the equipment andtooling, and the concepts involved will bediscussed in this article. This will provide anunderstanding of the state of the art in metalforming, typical processes (not all), and basicphenomena and concepts involved.

I. Introduction

A. PRIMARY AND SECONDARY FORMING

PROCESSES

The ingots of a relatively large volume,coming as cast billets through solidification ofmolten metal, are usually shaped through plasticdeformations into intermediate shapes. Thisprimary shaping provides profiles that are closerto the profile of the final product and also causes

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a refinement of the crystal structure of the castingot. This refining of the structure, calledrecrystallization, occurs at elevated temperatures.Furthermore, metals are softer and more ductileat elevated temperatures. Thus, primary formingis done at elevated temperatures. In the process of extrusion (Fig. 1), a billet isplaced into a chamber with a shaped opening(called a die) on one end and a ram on the other.As the ram is forced into the chamber, theworkpiece is forced out through the die. Theextrudate, a long product (i.e., a rod), emergesthrough the die duplicating its cross sectionalshape. The flow lines indicate that a dead metalzone forms in the corner on the exit side of thechamber where the separated ring of a triangularcross section remains stagnant. The process of rolling, whereby the ingot isgripped by two rolls and squeezed between themis described by Fig. 2. The rolls are identical andthey are rotating in opposite directions so thatthey grab the ingot and drag it by friction into thenarrowing gap between them. The product maybecome thinner while passing through the rolls.Flat products are produced by cylindrical rolls,while profiles are provided by grooved rolls. The process of forging is performed on a pressor a hammer. Basically, the ingot is placedbetween two platens that are forced one againsteach other, squeezing the ingot between them(see Fig. 3). A variety of shapes can be producedbetween flat platens by manipulation of the ingotwhile the platens squeeze and release the

FIG. 1. Extrusion (a) and an assortment ofextrudates (b).

FIG. 2. Rolling. Friction forces: F1, drivingforce; F2, opposing force. Net driving force = F1-F2; v0 < Ů < v f and v f / v0 = t0 / t f.

workpiece repeatedly. Alternately, the platensmay be shaped with a cavity that imparts itsshape on the product.

FIG. 3. Forging: open die (a) and closed die(b).

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B. INTERACTION BETWEEN THE MACHINE,THE TOOL, AND THE WORKPIECE

A typical system for a metal-forming processis presented here through forging (see Fig. 4).The platens are manipulated by a hydrauliccylinder. The force applied to the workpiecethrough the piston and platens is contained bythe frame. The resultant force on the system iszero. However, the frame must be strong enoughto contain the forming forces. While the largestforging press during World War II was a 5,000ton press available only in Germany in limitednumbers, there are throughout the world today afew production presses of 50,000 to 80,000 tons.These presses are huge and expensive. Thepower supply needed for a press this size is animpressive system by itself. Not so long ago, thecontrol and manipulation of the workpiece andtools were manual. Today’s modern presses areautomated. The following description is the stateof the art in several of the most advanced designs(Lange, 1985). The shape of the product, together with otherinformation about the feed stock is given as theinput to an online computer that activates thepress and its accessories. The entire workpiece,tooling, and press manipulations schedules arecalculated by the computer. Workpiece afterworkpiece is automatically fed to the press fromits storage. An assortment of tools is stored on arack at the press, and automatic selection of thedesired tools at the proper portion of the cycle isaffected. The tools and workpiece aremanipulated in synchronization to shape theworkpiece to the proper design by repeatedforging actions. When forming of one workpieceis completed, the workpiece is removed to makeroom for the next one. On-the-spot automaticinspection is, on occasion, affected with possible

FIG. 4. Schematic of a hydraulic “C” clamppress.

closed loop feedback capabilities for automaticcorrective measures. Almost all of the disciplines of engineering attheir most advanced stage interact in providingthe present-day metal-forming system. Startingwith the workpiece, knowledge of metallurgyand mechanics combine to provide an insight toits behavior. Sliding occurs on the interfacebetween the workpiece and the tool, friction ismanifested, and lubrication is exercised. Thus,tribology (i.e., the study of friction and wear) isnecessary. The tools are made from hard metalsand nonmetals as well, and therefore the latestadvances in material science are immediatelyapplied. Furthermore, to optimize tool wear, thelatest in surface treatments, by coating, ionimplantation, and laser beam surface hardening,are all practiced. In the design of the machinetool itself, all disciplines combine. To mention afew, the frame is made of any material from castiron to plastic, which will reduce weight andnoise. Hydraulics and electronics with roboticscombine to provide motion inspection andvibration control. [See MANUFACTURINGPROCESSES TECHNOLOGY; METALLURGY,MECHANICAL; PLASTICITY (ENGINEERING);TRIBOLOGY.]

C. THE STATUS OF THE METAL-FORMING

PROCESSES AS TECHNOLOGY AMONG THE

INDUSTRIAL PROCESSES

Metal forming is normally performed after theprimary processes of extraction, casting, andpowder compaction and before the finishingprocesses of metal cutting, grinding, polishing,painting, and assembly. With few exceptions, thebulk of the products of the metal fabricationindustry are shaped by forming or a combinationof forming and other processes like metal cuttingor joining. Forming operations are classified asthose processes where the desired shape isachieved by imparting plastic deformation to theworkpiece in the solid state. Classification by (1)product, (2) material, (3) forming temperature,and (4) nature of deformation (sheet metal versusbulk deformation) can also be helpful. However,the boundaries between categories are notperfectly defined. For example, impact extrusioncan be classified as a forging process or as anextrusion. It is needless to say that any specificproduct can be made from a number of materials,by a variety of processes, and at a range oftemperatures.

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II. Basic Concepts

A. PERFECTLY PLASTIC MATERIAL

Metal deformations are introduced through theapplication of external forces to the workpiece,these forces being in equilibrium. With theapplication of load to the workpiece, internalstress and displacements are generated causingshape distortions. If the loads are low, then withthe release of the loads, the internal stresses willdisappear and the workpiece will be restored toits original shape. It is then said that the appliedloads were elastic and so were the stresses andstrains. Elastic strains are recoverable on releaseof the loads. When the loads are high enough, thechanges in shape will not disappear after the loadis released. The changes in shape and the strain,those that did not disappear, namely, thepermanent ones, are called plastic deformations.The loads causing plastic deformations are saidto have surpassed the elastic limit. During metalforming by bulk plastic deformations (to bedefined), the plastic deformations are muchlarger than the elastic deformations, which ingeneral are ignored. Thus, only plasticdeformations are considered. It is alsorecognized that plastic deformations do notinvolve volumetric changes. Thus, volumeconstancy is maintained. In metals, the load andthe intensity of the internal stresses at which(plastic) flow initiates are functions of thestructure, the temperature, the deformationhistory, and the rate of straining. It is fair toassume that at temperatures below therecrystallization temperatures at which newcrystal structures emerge the material strainhardens but is not strain-rate sensitive. That is,the strength of the material increases withincreased deformation levels. Above therecrystallization temperature, the material isstrain-rate sensitive. That is, its strength is higherwith higher rates of straining, but it does notstrain harden. The point in loading whereincipient plastic flow commences is called the“yield point,” to be defined mathematically inEq. (7). A perfectly plastic material is such that it doesnot strain harden and is not strain-rate sensitive.The strength is not a function of strains nor ofstrain rates. A deformable material is said to be aperfectly plastic material (Talbert, 1984), when:

1. The material is incompressible.2. There exists a material constant K with

dimension of stress.

3. The stress deviator (sij) depends on the

strain rate ( ije ) in a quasilinear way:

ijij es φ= (1)

where φ is a scalar invariant function of theprincipal strain rates,

),,,( 321 eeeKφφ = (2)

These assumptions have been shown to implythat there exists a function ),,(, 321 eeeFF ,

homogeneous of degree –1 in the principal strainrates ie , such that

ijij eeeeKFs ),,( 321= (3)

Moreover, the ideal materials defined abovehave a yield criteria of the general form

KsssF /1),,( 321 = (4)

where s1, s2 and s3 denote the principal stressdeviators. The same constitutive function F thusenters in the definition of the constitutiverelations and of the yield criterion. Dr. Talbert (1984) specifies the function F forfour popular and different perfectly plasticmaterials named after Tresca, Mises, and others.For the Mises perfectly plastic material

2/123

22

21321 )]/(2[),,( eeeeeeF ++= (5)

which in an arbitrary Cartesian coordinatesystem reads

ijij eeF2

1/1= (6)

so that the yield criteria becomes2

2

1 Kss ijij ≤or

2231

223

212

233

222

2112

1 )()( Kssssss ≤+++++ (7)

where sij are the components of the stressdeviator and K is a constant property of thematerial to be determined experimentally. Aslong as the scalar quantity on the left of Eq. (7)does not reach the value of K, the material doesnot deform. On reaching the value of K, plasticflow commences, The tensile test is a mostcommon test to evaluate K. Thus, if the loadstress in tension at the yield point is σ0, then K =±σ0/√3.

B. FLOW

A pattern of deformations is known whenspecified by a velocity field. A velocity field isdefined through the vector Ůi in space, confinedto the workpiece. In a Cartesian coordinatesystem of X1, X2, and X3, at any instant of time,

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the velocity vector is Ůi(X1, X2, X3) (where i=1,2,3). The strain–rates components can bederived from the velocity vector as follows:

)]/()/[(2/1 ijjiij XUXUe ∂∂+∂∂= (8)

and by Eqs. (3) and (8), the components of thestress deviator can be determined from thevelocity field and the strength constant K.

klklijij eeeKs2

1/= (9)

C. WORK AND POWER OF DEFORMATIONS

When the stress deviator components and thestrain rates components are known, the internalpower of deformations per unit volume can bederived as follows:

ijiji eσω = (10)

and since sij = σij - sδij where s = (σ11+σ22+σ33)/3and δij =1 if i = j and δij =0 if i ≠ j, by Eq. (10)

ijiji eeK2

12=ω (11)

Thus, the internal power of deformations isdetermined from the strain rates components asderived by Eq. (8) from the velocity field vectorand from the material constant K. Although the characteristics of metals aremore complex than those described here by aperfectly plastic metal, we will not expandfurther. The treatment becomes too complex, andthe bulk of the study of metal forming is servedwell with the simpler model.

D. FRICTION, LUBRICATION, AND WEAR

In all of the processes described in Figs. 1-3,there exits a sliding motion along the interfacesbetween the workpiece and the tools. Wheneversliding occurs between solids, a resistance to thesliding motion is observed. This resistance iscalled friction. Friction resistance isaccompanied by damage to the surfaces, which ismostly manifested by the wearing of thesurfaces. The resistance to sliding, measured as ashear stress per unit surface area of contact (τ), isa complex function of many parameters,including workpiece and tool materials, surfacesmoothness of the tool, and speed of sliding.Friction and wear are also controlled by theintroduction of lubricants between the interfacingsurfaces. The lubricant serves not only tominimize friction and wear but also to cool thesurfaces by removing the heat generated throughsliding. Most effective lubrication methods mayprovide a thin film of lubricant separating the

two surfaces completely. When full liquid filmseparation is created, a condition of hydrostaticor hydrodynamic lubrication prevails,minimizing the friction and wear. The energygenerated through sliding can be calculated by

vf ∆=τω (12)

where τ is the friction resistance to sliding, and∆v is the relative sliding speed.

III. Typical Processes

Several processes representing a diversespectrum of secondary and primary metalforming operations are covered in this section.Only forging and wire drawing are dealt with insome depth, covering also the press system. Theother processes are covered only briefly.

A. FORGING

Forging is a most popular production processbecause it lends itself to mass production as wellas to the production of individual sample parts.The origins of forging may be traced to theancient process of hammering of gold foil,between a rock, the anvil, and a stone, thehammer. In hammering, the inertia of the fastmoving hammer provides the requireddeformation energy and force, while in pressingthe force is static. Usually the final shape isimparted on the workpiece by manipulating theworkpiece between the flat anvil and the flathammer as the hammer hits the workpiecerepeatedly. Complex shapes can be hammered byskilled blacksmiths. A conical protrusion fromthe anvil, holes in the anvil, a variety of pegswith different cross sections, and auxiliary tools,including a large selection of shaped handhammers, may assist the blacksmiths and theirhelpers (see Fig. 5). Today, hand-held hammers are replaced bymechanical and hydraulic presses. When a largenumber of identical components aremanufactured, the open dies are replaced byclosed dies (Fig. 3), each with a shaped cavity toimpart its shape to the product. The workpiecedoes not have to be manipulated, and theoperator therefore does not have to be skilled;completion of the product can be achieved in onestroke and processing efficiency is high. Feedingthe blank and ejection of the product areautomated. Mass production of a technologicalage emerges out of the ancient art for ornamentand artistic values. Generally speaking, hydraulic presses (see

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FIG. 5. The tools of the blacksmith.

Fig. 4) are slower than mechanical presses orhammers (see Fig. 6). Furthermore, the largerpresses, carrying higher loads or longer strokesare hydraulic. Thus, the mechanical press is moresuitable for mass production. Contact timebetween the tool and the workpiece is shorter ona mechanical press, protecting the dies betteragainst heating during hot forging. Mechanicalpresses are recommended for open-die forging,and when used for closed-die forging, a flash isusually incorporated. The role of the flash andwhen it can be eliminated are to be discussedsoon. For closed-die forging without a flash, ahydraulic press is recommended. Mechanicalpresses for closed-die, precision flashless forgingas economic alternatives have recently beenintroduced in the market and proved successful.The largest forgings, for example, airplane wingframes, are forged on the largest presses (with upto and over 50,000 tons forging force), which arehydraulic presses. The schematic of the forging of a flatcomponent with flash with closed dies ispresented in Fig. 7. The cavity between the topand bottom dies dictates the shape of thecomponent. The original shape of the blank maybe a predetermined length of a rod with a squareor round cross section. It is also quite common todesign dies with several cavities for theproduction of several components in each strokeof the press. The blank passes, in several steps,through a succession of pairs of dies in which itgradually approaches the final nominal desiredshape of the product. The cavity between the pairof final dies is designed to be as close as possibleto the nominal size of the product, but notprecisely to the product’s dimensions. Thecavityis oversized for a number of reasons; themost important ones are the following.

FIG. 6. Schematic of mechanical presses:knuckle joint press (a) and crank press (b).

1. Sharp corners require very high forgingpressures to be fully filled. Since such pressurescannot be well tolerated by a die, the cavity ismade slightly oversized, and the “degree of fill”of the corners becomes controlled by the flash.The forging force and pressure start to climbfrom the moment that first contact andcompression are established between theworkpiece and dies. In the beginning of thestroke, the blank does not match the shape of thecavity. The area of contact and the pressureincrease as the dies approach each other. At theend of the stroke, the shape of the workpiececonforms to the shape of the cavity and pressureis at its peak. Shape corners can never becompletely filled, and they are usually roundedorunderfilled. Furthermore, rounded corners

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FIG. 7. Schematic of forging of a flat component with flash.

prolong tool life when compared to sharp cornersthat crack easily. The peak forging pressure isachieved when the top die reaches its lowestposition; this position can be adjusted and thusthe thickness or height of the flash can becontrolled. The thinner the flash, is, the higherthe forging force is. With the need or desire tofill sharper and sharper corners, a need arises forhigher forging pressures, which are obtainablethrough thinner and longer flashes.

2. Variations in the volume of the incomingblank have to be tolerated and compensated bythe fill of the flash.

3. Variations from one product to anotherare inevitable because of size changes of theblank, temperature and strength changes, etc.,and because of the elastic flexing of the press.

4. Some surfaces should be machined afterforging for improved surface finish, removal ofscale caused by hot forging, etc.

So far, it has been established that the fill ofthe cavity by the workpiece is promoted throughthe flow of excess material into the flash, sincethe blank is cut to a size larger than that of thefinal product. Variations in the volume of the

blank are minimized to a practical tolerance. Thesmallest permissible volume of the blank islarger than the product size to assure a minimumflash and thus a minimum value of the peakforging pressure, which is dictated by therequired corner radii. Note that excess volume of the blank does notby itself assure a fill of the cavity. If the spacingbetween the dies at the bottom of the strokeleaves a flash height that is too large (and thus aforging pressure too low), the cavity may not fill,in spite of the fact that a flash has been created.In this case, the flash will flow outward tooeasily, leaving empty spaces in the corners of thecavity. Other means to assure filling of the thin websand sharp corners with moderate loads call forisothermal forging of super plastic materials(Section IV, D) and forging in the mashy state(Section IV, D). Recently, forming to “near netShape” was introduced by flashless forging. Thematching of the top and bottom dies to the formof a cavity may be designed without provisionfor a flash. Such design changes require that,rather than facing each other at contact, one dieenters the other.

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By eliminating the flash, the followingadvantages are gained.

1. The volume of the blank is reduced tothe nominal volume of the finished product. Thisprecipitates savings in material.

2. The dimensions of the forging mayconform closely to the final dimensions of theproduct, eliminating subsequent machining.Better corner filling can be accomplished due tothe higher pressures associated with flashlessforging.

3. The strength of a product after flashlessforging is superior to that forged with a flash,since fibering flow lines in a flashless forgingconform to the shape of the product, whereas in aforging with a flash they do not.

4. Usually flashless forging is performed inone forging step, from a blank of uniform crosssection to a final shape through a single pair ofdies, eliminating intermediate forging and(sometimes) annealing steps.

On the other hand, by eliminating the flash,much stricter tolerances are imposed on theblank. Too small a blank and the cavity will notbe filled. Too large a blank and the press loadwill be excessive to the point of causing diebreakage. Better choices of tool material and ahigher degree of expertise in die design arerequired. Flashless forging is gaining popularityand every day new components not producedhitherto by the process are being added.

B. FLOW THROUGH CONICAL CONVERGING

DIES

Figure 8 represents a billet and die. A varietyof cross-sectional profiles can be produced;however, in the following simplification, thebillet is a cylindrical rod of radius R0; the rod is

FIG. 8. Flow through conical converging dies;σxf = f(R0/Rf, α, and m).

reduced to radius Rf by forcing it to pass throughthe conical converging die. Reduction ismeasured from the cross sectional area of thebillet at the entrance to the die (A0) to that at theexit (Af). Besides the choice of the material itself, threevariables (the independent process parameters)involved in the reduction process are noted atonce. They are, the reduction, the semiconeangle (α) of the die, and the severity of thefriction between the workpiece and the die. These three process variables-reduction, coneangle, and friction-are independent in that theprocess planner may exercise a degree offreedom in choosing their values. The severity offriction, for instance, is controlled, within limits,by choices of lubricant, die material and finish,speed, etc. The above three parameters are the primaryfactors affecting the process and their effect onthe first dependent parameter the drawing orextrusion force will be analyzed first. Otherindependent parameters play a role duringprocessing. For example, we will find that thedrawing or extrusion force is linearlyproportional to the flow strength of the material,but when inertia forces are neglected, it isindependent of the speed (when a Mises’material is considered). The power, on the otherhand, is linearly proportional to speed.Furthermore, one may consider isothermalprocessing, where temperature is not a factor,and then extend the treatment to handle adiabaticprocessing and temperature effects. Thus, at first,only the effect of the three independentparameters (r%, α, and friction) is considered. The force required for drawing or extrusioncan now be characterized. In Fig. 8 the drawingforce F (or drawing stress, σxf = F/Af) isobviously a function of reduction (largerreduction required higher force), cone angle, andfriction, and similarly for extrusion force F (orextrusion stress, σxb = F/A0). In short, themotivation force or stress causing the drawing orextrusion is a dependent variable, which is afunction of reduction, cone angle, and friction.Description of the drawing force, for example, asa function of these three independent variables,may be undertaken by either an experimentalapproach or an analytical approach. Figure. 9 illustrates the characteristics of therelative drawing stress (or extrusion pressure orforce) as a function of the semicone angle of thedie (abscissa) and of reduction (parameter). Therelative drawing (or extrusion) stress is themotivation force divided by the cross-sectional

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0.00

0.20

0.40

0.60

0.80

1.00

0.00 4.00 8.00 12.00 16.00

Semicone angle

Rel

ativ

e d

raw

ing

str

ess r=50%

45%

40%

35%30%25%20%15%10%

5%

0.00

0.40

0.80

1.20

1.60

0.00 10.00 20.00 30.00 40.00 50.00

Semicone angle

Rel

ativ

e ex

tru

sio

n p

ress

ure

m=0.0

m=0.2m=0.4

m=0.6m=0.8

m=1.0

05.0,0 === mLxbσ

(a)

0σ

bp 0%,30 == Lr

(b)

FIG. 9. (a) The effect of α and reduction onthe relative drawing stress. (b) Relative extrusionpressure versus semicone angle and constantshear factor.

area on which it acts and by the flow strength σ0

of the workpiece. With too small a cone angle,the length of contact between the die and theworkpiece is excessive and thus friction ispredominant and makes the force excessive. Asthe cone angle increases, friction drops verydrastically and so does the drawing force. Anoptimal angle is reached for the power. A furtherincrease in the cone angle causes largedistortions and excessive resistance to thisdistortion to offset what has been gained onfriction, and thereafter redundant work (causedby distortion) is a predominant factor, notfriction, A further increase in the die angleproduces a further increase in the total power.For larger reductions, as well as for higherfriction values (τ), the drawing force and theoptimal angle that minimize it are increased. Forthe definition of the friction factor (m) see IV, C. Flow through conical converging dies can beimposed by drawing on the emerging product (in

FIG. 10. A drawbench (a) and a bull block (b)for wire drawing.

a process called wire drawing), by pushing-inprocesses called extrusion, or by a combinationof the two. Only limited reduction is achieved indrawing in a single pass because the tensionpermitted on the emerging wire should notexceed the strength of the product or the wirewill tear. Wire drawing can be achieved instraight, short products or by pulling whilecoiling over a drum of very long wire (see Fig.10). Occasionally a long, straight product can beproduced by the equivalent of two, hand-over-hand pulls. A tandem arrangement of manyblocks, one after each other, may be used toaffect large total reductions. When a billet is pushed through the die in theprocess of open-die extrusion (Fig. 11), thereduction is limited, just as in wire drawing,because here the allowable driving force islimited or the feedstock will be upset betweenthe die and the driving force. For largerreductions, the process of extrusion through aclosed chamber as described by Fig. 1 is used. Inthe process of hydrostatic extrusion (Fig. 12), thebillet is pushed through the die by a pressurized

0σ

σxf

α

α

10

FIG. 11. Open die extrusion.

liquid. Occasionally liquid under pressure maybe introduced at the exit to affect a process ofpressure-to-pressure extrusion.

C. ROLLING

In the process of rolling, long products of avariety of cross-sectional shapes can be produced

FIG. 12. Hydrostatic extrusion.

by forcing the feedstock to pass through the gapbetween rotating rolls. The rolls transfer energyto the workpiece through friction (Fig. 2). In flatproducts (strip), the strip is dragged by the rollsinto the gap between them. It decreases inthickness while passing from the entrance to theexit. Meanwhile its speed gradually increasesfrom v0 at the entrance to vf at the exit. Underregular rolling conditions, the strip moves slowerthan the rolls at the entrance to the gap betweenthe rolls (v0 < Ůi) and faster than the rolls (vf >Ůi) at the exit, with a neutral point in between atwhich the speeds of strip and the rolls (Ůi) areequal (vn = Ůi). This neutral point is also calledthe no-slip point. It can be successfully arguedthat a no-slip region exists about the neutralpoint. The friction force acting along the surfacesof the rolls between the entrance and the neutralpoint (F1) advances the strip between the rolls,while the friction force acting between theneutral point and the exits (F2) opposes therolling action. The difference between thefriction on the entrance side and the friction onthe exit (F1 – F2 in Fig. 2) provides the necessarypower for rolling. The position of the neutralpoint is automatically determined by the powerrequired to deform the strip and to overcomefriction losses. In the conventional range ofreductions practiced, the larger the reductionattempted, the farther the neutral point movestoward the exit, so that F1 increases, F2

decreases, and the net friction drag forceincreases to supply the higher power demand.Larger reductions can be achieved until theneutral point reaches the exit (vf = Ůi). Then themaximum reduction possible is achieved and theprocess becomes unstable. If larger reductionsare attempted, the rolls will skid over the stripand the strip will stop altogether. Larger reductions also require higher pressureson the rolls. Large pressures on the rolls causemore and more flattening and bending of therolls. A limit on the amount of reduction that canbe taken is set by one of two causes. Whenexcessive pressure is limiting the maximumreduction, it is said that limiting reduction orlimiting thickness is reached. If the neutral pointreaches the exit and the rolls start to skid overthe strip, it is said that maximum reduction isreached. The process of rolling is effectivelycontrolled by the application of front (σxf) andback (σxb) tension to the strip on both ends of therolls. However, the process of rolling is affectedby the friction drag. An increasing number ofmetal forming processes were introducedrecently, whereby friction provided the

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motivation force. These processes are classifiedas friction aided processes (Avitzur, 1982, 1983),and because the motivation source is applieddirectly to the deformation region, theseprocesses are typified by their ability to imposeexcessive or unlimited reductions in a singlepass.

D. TUBE MAKING

Tubes and tubular products are madeessentially from all metal and by all metal-forming processes available. In Fig. 13 a tube oflarger diameter is reduced to a smaller one by theprocess of tube drawing, similar to wire drawing,which is called (free) tube sinking, withoutspecific control of the inner diameter of theproduct and by the processes of tube drawingwith a floating plug, whereby the inner diameteris controlled by a plug, In tube drawing with afloating plug, the plug is free to move axially,and its position at the throat of the die is

(a)

(b)

FIG. 13. Tube sinking (a) and tube drawingwith a floating plug (b).

maintained automatically by the balance of thefriction force and interface pressure with thetube. Special care in the design of the plug anddie geometry must be taken so that the plug willstay in position, effectively control the insidediameter of the tube, and prevent tube tearing.

E. CAN MAKING

Cans can be made by deep drawing or impactextrusion and then wall thinning can be affectedby the process of ironing. The process of deepdrawing uses a rolled sheet, from which aproperly contoured blank is stamped for theproduction of cans and other products. Bathtubs,kitchen sinks, and autobody components aretypical deep-drawn articles. However, onlycylindrical cans (also called “cups”), such ascartridges, aerosol cans, and beverage cans, willbe discussed here. Blanks for cylindrical cups arecircular disks. Here, the process of deep drawing is coveredand compared with impact extrusion and ironing.In the deep drawing of a cylindrical cup, a planardisk is transformed into a cup with a flat bottom,cylindrical walls, and an open top. As shown inFig. 14, the disk is placed over the opening in thedie and forced to deform by a moving ram (alsocalled the punch). As the ram moves downward,it pulls the flange toward the center. The flangeis held between the die and the blank holder,with the purpose of preventing the flange fromfolding upward. The blank holder is also calledthe “blankholder,” “pressure pad” or “hold-downring.” The flange moves inward radially while itsinner side bends over the rounded corner of thedie and transforms from a flat disk to a circulartube. The bottom is not deformed, while thecylinder is already deformed but is notundergoing further deformation also, the toroidal

FIG. 14. Deep drawing.

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FIG. 15. The pattern of deformations in deepdrawing.

section between the cylinder and the flange isbending, and the flange is undergoing plasticdeformation (see Fig. 15). The process of impact extrusion (also calledinverse extrusion, backward extrusion, orpiercing) is utilized to produce hollow shellsfrom solid rods or disks. Schematically (Fig. 16),a disk (slug) is placed in the cavity of a femaledie and then the ram (mandrel, punch, or tool) ispushed into the raw material. While the rammoves downward, the wall of the produced canmoves upward, escaping through the annular gapbetween the ram and the die. Because the wall ofthe product moves upward in the directionopposite to that of the downward motion of thetool, the process is sometimes called inverse orbackward extrusion. In most of themanufacturing practices involved, the product ismade on a fast mechanical press; and the name“impact extrusion” has resulted. A hexagonal

FIG. 16. Schematic of impact extrusion.

FIG. 17. Hexagonal cavity produced in boltmaking.

cavity produced in the head of a steel bolt isshown in Fig. 17 and an assortment of aluminumcontainers is shown in Fig. 18. The well-knownwhite metal toothpaste tube needs no illustration. Ironing is usually performed after either deepdrawing or impact extrusion when a thin-wallcup is required. This same process is oftenapplied in the thinning of tubes. Thispresentation is concerned with the ironing ofthin-wall cups, of which the beer can is a classicexample. The deep-drawing operation is moresuited for heavy or medium gauge cups ofrelatively restricted depths. For the production oflonger cups of thinner walls, ironing can be used.

FIG. 18. Variety of shapes possible by impactextrusion.

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A cup (Fig. 19) of inner radius Ri, wallthickness t0, and fairly small height H, is firstproduced by deep drawing. The thickness t0 isusually much less than the radius Ri. Then,during ironing the cup is forced to flow into aconical die of semicone angle α and inner radiusRf and is pushed downward at a velocity vf by apunch of radius Ri over which the cup ismounted. The gap between the die and the punch(Rf -Ri) is the thickness tf: this is the finalthickness of the cup, and tf < t0. As the punchadvances, the wall of the cup extrudes throughthe gap and its thickness decreases from t0 to tf

while the length H increases. The outer radius ofthe cup decreases from R0 to Rf while the innerradius remains constant at Ri. The punch force P is transmitted to thedeformation zone (Fig. 20) partly through thepressure on the bottom of the cup, further bytension on the wall, and partly through friction.As the friction between the punch and the innersurface of the cup increases, less tension isexerted on the wall, thus enabling ironing withlarger reduction. By differential friction (i.e., byhaving the ram friction higher than the diefriction) and proper choice of die angle,unlimited amounts of reduction can in principlebe achieved through a single die (Avitzur, 1983). As of recently, processing of polymers in thesolid state is performed by the same metal-forming process described in the precedingparagraphs. The molecular orientations imposedby this process enhance the strength properties ofthe product. The product is made into its finalshape with no machining (Austen et al., 1982).

FIG. 19. Deep drawn cup; t0 << Ri.

FIG. 20. Transmitting the punch force to thedeformation region.

IV. Phenomena

A. PRESSURE-INDUCED DUCTILITY

REVERSIBLE FLOW, AND METALWORKING

UNDER PRESSURE

The most significant factor controlling theapplication of metal forming as a manufacturingprocess is the ductility of the workpiece.Metallurgical aspects determine the ductility ofthe workpiece at standard room temperatureconditions. The most popular experimentalprocedure to determine ductility is the tensiletest. One traditional method to improve theductility of metals is heating, which causes mostmetals to soften and become more ductile. Thus,traditionally, heating was employed both toreduce the required forming forces and toincrease the amount of deformation possible. The indication that ductility, or the lack of it,is not an inherent and solely metallurgicalproperty, but a property that can be controlled bymechanical means (namely, environmentalpressure), was suggested by Bridgman (1949).He showed that the ductility of metals asmanifested by the stress-strain curve increaseswith the mean superimposed hydrostaticpressure. The terms mean stress, average stress,hydrostatic stress or pressure, and environmentalpressure are used interchangeably. Not only themetallurgical parameters of the workpiece butalso the processing parameter pressure dictatethe formability. This phenomenon, the increase

14

in ductility with the environmental pressure, iscalled pressure-induced ductility (PID). Assuggested by Bridgman, the mechanism of PIDis the restraining effect of environmentalpressure in inhibiting void initiation and growth.Since the growth and coalescence of voids areprerequisites to ductile failure, their arrestextends formability, thus increasing ductility.For the mathematical treatment of the effect ofpressure on the deformation and strength of atensile bar and on void formation and prevention(Talbert and Avitzur, 1977). This increase inductility by superimposed hydrostaticenvironmental pressure was confirmed byBobrowsky et al., (1964), Pugh and Green(1956), Alexander (1964-1965), and manyothers. With the renewed research into hydrostaticextrusion, a convenient tool was developed forthe investigation of PID and its implementationin metal forming, namely, metalworking underpressure (MUP). For example, by pressure-to-pressure hydrostatic extrusion, the environmentalpressure can be controlled as an independentprocess parameter, through the control of thereceiver pressure, separately from the reduction,die angle, friction, or temperature. Bridgman wasable to demonstrate the PID phenomenon byextruding marble, a brittle material by all counts,into a high-receiver pressure. The extrudate cameout as a sound product. A prevailing theorytoday in geology is that rocks deep undergroundare capable of undergoing plastic deformationsin a ductile manner because they are constrainedby high environmental hydrostatic pressure.MUP for many hard to deform metals or shapesis demonstrated through pressure-to-pressureextrusion and implemented in industry. WhileMUP had been sporadically employed earlier,Bridgman’s pioneering work gave thephenomenon an identity, and since then it hasbeen applied deliberately in many processes. Awide range of processes to which PID can beutilized in MUP are included in these fivecategories: (1) forming, (2) cropping andshearing, (3) bonding, (4) powder compaction,and (5) reversible flow from smaller to largercross sections (Avitzur, 1983).

B. HIGH ENERGY RATE FORMING

Up to this point, the processes described wereachieved through static loading, as in the forgingof a disk between two platens of a press (Fig. 1).We now examine how the same result (upsetting)can be achieved by a high-energy rate-forming

(HERF) process. (The process is also calledhigh-velocity or HVF). If, hypothetically, the disk is thrown with ahigh speed at the bottom platen, the entire kineticenergy of the rushing disk will be absorbed at themoment of impact with the platen. If theprojectile achieves bullet speeds, it may, onimpact, either penetrate the platen, like an armor-piercing bullet, weld to the platen, deform, orundergo two of the above simultaneously. Figure 21 represents the most common designfor the use of explosives in a HERF process. Ablank made of a plate or sheet metal is placedover a die cavity of the desired shape. A vacuummust be formed in the cavity below the blank byevacuating the air. The tank above the bank isfilled with water. An explosive charge is placedjust below the surface of the water, directlyabove the center of the blank. When the explosive charge is detonated, ashock wave moves through the water. Water is avery effective shock-wave-transmitting mediumthrough which the impact of the explosion istransmitted from the source to the workpiecetarget. The effectiveness of the energy transfer isdemonstrated by observing uses in other fields.For example, sonar under water is most efficientand sensitive. The destructive force of the shockwave has been used for centuries (now illegally)by fishermen to destroy (or to stun) all life in avast sea or pond space. Submarine warfaredemonstrates the sharpness by which the shockwave from a bomb hits the submarine, as if ithad been hit directly by a hammer. On reaching the blank, the shock wave hits itso hard that the blank rushes downward andconforms to the cavity. Once the shock wave hashit the blank and set it in motion, the rest of theoperation is performed by the inertia of the

FIG. 21. Schematic of explosive forming.

15

FIG. 22. Intermediate shape.

moving blank. The blank moves downward as aplane during forming. Halfway through theoperation the part would look like a flat-bottomed bowl with sides conforming to thecavity. This intermediate shape is shown in Fig.22. This shape would also result if the explosivecharge were insufficient to complete theoperation. For smaller parts the surge of energycan be provided through other chemicals or byan abrupt electrical discharge of energy from abattery of capacitors.

C. FRICTION AND LUBRICATION

One of the last frontiers in the understandingof metal forming is the friction phenomenabetween the tool and the workpiece. No matterhow much care is taken to form a smooth toolsurface, the surfaces of both tool and workpieceare irregular surfaces with peaks and valleys.Opposite peaks clash with each other, resultingin damage to both surfaces. Temperature risesdue to the rubbing action. A thin layer underboth surfaces undergoes severe plasticdeformation.

FIG. 24. Disciplines affecting friction andwear.

Models of the typical behavior of theasperities of the surfaces of two solidsinterfacing one another under pressure, andsliding with respect to each other, are describedin Fig. 23. Many more possible outcomes of theclashing of the asperities may occur. Onespecific behavior, described in part (b) of Fig.23, is the steady state flow of the asperity,identified as the wave motion. In the model ofthis motion, the “wave model” (Fig. 24), wedgesof the harder surface indent into the softersurface because of the applied pressure, thusproducing opposing ridges on the surface of thesofter component (Leslie, 1804). According toLeslie, the ridges are supressed down under thesliding wedges, only to rise again in front of themoving wedges. This perpetual supression anduprising of the ridges are motions similar to themotion of ocean waves.

FIG. 23. Several patterns of distortion of asperities.

16

The wedges and the ridges are the asperities.The gap between the opposing asperities is filledwith lubricating liquid, establishing boundarylubrication. As sliding is maintained, the ridgesare mobilized and an eddy flow is established inthe trapped lubricant (Avitzur, 1990). The eddyflow creates high shear within the lubricant. Thisshear generates power losses, heating, and liquidpressure. The shear, the power losses, and thepressure, all increase with increasing speed andviscosity of the liquid. The power required tomobilize the ridges and to establish eddy flow inthe lubricant is calculated, and thus the frictionresistance to sliding is determined. Simulta-neously, the pressure generated in the liquid as aresult of shear is also evaluated. It becomes clearthat the height of the ridge, due to indentation, isinversely proportional to the speed of sliding.The higher the sliding speed, the higher theliquid pressure that is countering the loadingpressure and the smaller the indentation. At highenough speeds the entire load is supported by thepressure generated in the liquid, indentation iseliminated. And hydrodynamic lubricationcommences. A classic presentation of the resistance tosliding as a function of interface load is shown inFig. 25. When the load (p) is low orintermediate, the resistance to sliding isproportional to the load and, as suggested byCoulomb (1785) and Amonton (1699), τ = µpwhere µ is the coefficient of friction (Bowdenand Tabor, 1954, 1964). With increasing load theresistance levels to reach a plateau, τ = mσ0/√3,where m is a constant friction factor. Both theproportionality factor and the plateau arefunctions of the irregularities of the surface andof the effectiveness of the lubrication (Wanheim,1973; Avitzur, 1984). In Fig. 25 m0 representsthe inverse effectiveness of the lubrication and αrepresents the steepness of the irregularities onthe surface of the die. During the metal formingthe interface pressures required to impose plasticdeformations on the workpiece are high, and theconstant friction resistance indicated by the flatportion of Fig. 25 is realized, unless filmlubrication comes into effect as described next. In processes like wire drawing or rolling, thedeforming workpiece continually passes throughthe tools. These processes (unlike other bulkprocessing, i.e., forging), classified as “flowthrough” processes, are executed at high speedsand thus are most efficient. Being continuous,they minimize manual handling and lendthemselves easily to mechanization andautomation. Rolling on a finishing mill may be

(a)

(b)

FIG. 25. Friction versus load; (a) with wedgeangle (α1) as a parameter, and (b) with thefriction factor (m0) as a parameter.

performed at 10,000 ft/min, and even higherspeeds are reached in wire drawing. At highspeeds, an entry (or inlet) zone developswhereby fluid from the entrance squeezes as awedge between the workpiece and the die. InFig. 26a for wire drawing, this wedge extendspartway through to the point defined by Ri,where R0 > Ri > Rf. The faster the drawing is, thesmaller are Ri and the contact zone between theworkpiece and the die. As long as Ri > Rf, theliquid dragged by the workpiece (and the rolls, inthe case of rolling) into the wedge cannot escapethrough the exit and must return to the entrance.The profile of the lubricant flow through anycross section is described in Fig. 26b, showingthat at the surface of the workpiece flow, speed

17

(d)

FIG. 26. Lubricant film: (a) entry zone, (b)velocity profile of lubricant, (c) eddy flow inentry zone, and (d) hydrodynamic lubrication.

is equal to the speed of the workpiece, and at thesurface of the tool, speed is equal to the speed oftool. The total volume rate of the liquid passingthrough any section is zero. Thus, in the outerannulus, closer to the surface of the die, liquidflows in the general backward direction. At someintermediate point, where a reversal of thedirection of flow occurs, the velocity componentis zero (Fig. 26b). Liquid at that point does notflow in or out. It does however flow into the wireor into the die as shown in Fig. 26c.

The loops in Fig. 26c show flow lines in theinlet zone. The eddy current flowing in a closedloop retains practically the same particles ofliquid and its contaminants. This circular motionis associated with high-speed gradients and shearstrain rates within the liquid. The temperature ofthis trapped liquid in motion may riseappreciably. A very thin layer of lubricant at thesurface of the wire maintains a sort of laminarflow with the wire. This layer, through thelabyrinth of voids between the workpiece and thedie escapes with the wire through the die exit.Being extremely fine, this layer does notconstitute hydrodynamic film separation. Sincemetal to metal contact decreases due to anincreasing liquid wedge as a result of increasingworkpiece speed, it follows that friction dropstoo with increasing speeds, as shown in Fig. 27. The range of the resulting changes in thepower consumed through the mobility of theridge, and through shear losses in the trappedlubricant due to the eddy flow, is wide, asdemonstrated by Avitzur (1990). The complexityof the characteristics of friction is evident fromthe calculated value of the global friction factorm, as presented in Fig. 27. The abscissa is theSommerfeld number (S), the ordinate is theglobal friction factor (m), and the parameter isthe normal load (p) on the interface between thetwo sliding bodies. The local friction factor is m0

= 0.6 while the asperity’s angle is α = 1°. For thelower load values (p = 2) the characteristicbehavior of the Stribeck curve (1902) isobserved. The static friction factor value of m ishighest when no sliding occurs. With increasingspeed or Sommerfeld number values, resistanceto sliding drops because the ridge size reducessharply. Higher-pressure values produce higherresistance to sliding. Note also that for higherpressures the height of the ridge is higher, andtherefore the thickness of the film of the trappedlubricant is thinner. Furthermore, increases inSommerfeld number values are not as effectivein reducing the height of the ridge, and thus forhigh pressures, the lubricant film remains thineven with increasing values of Sommerfeldnumber. An interesting point can be observed hereregarding the die wear, called the “ring” at theentrance. The eddy current of the trapped liquidin the wedge causes an excessive liquidtemperature rise and liquid contamination;together with the pressure rise due to the reversalof flow, it may erode the die in the same manneras the water flow in the river bend erodes itsbank.

18

FIG. 27. Global friction vs. Sommerfeld number, at high pressures.

With increasing speed, a critical value isreached at which Ri = Rf and the wedge extendsto the entire conical surface of the die. At thatpoint and beyond, a thin film of laminar flowcommences at the surface of the wire, This flowwill proceed through, from the entrance to theexit of the die. The film will separate the wireentirely from the die. This separation will existalong the bearing (land) of the die (Fig. 26d).The wedge of eddy flow may or may notdisappear while the liquid escapes from theentrance side of the die to the exit of the laminarflow. Full separation between the workpiece andthe die and hydrodynamic lubricationcommence. When hydrodynamic lubricationprevails, the friction is represented by the shearwithin the liquid in the following form:

τ = η(∆v/ε) (13)Where η is the viscosity of the lubricant, ∆v isthe sliding velocity between the workpiece andthe tool, and ε is the thickness of the lubricant’sfilm. The Sommerfeld (1904) number can bedefined as a function of viscosity, velocity, wiresize, and strength in the following manner: S = ηvf/(Rfσ0). When the Sommerfeld number reachesa critical value (Scr) and above, hydrodynamic

lubrication prevails. The higher the Sommerfeldnumber, above the critical value, the thicker thefilm becomes, separating the workpiece from thetools. For processes where high speeds cannot beattained (forging, deep drawing, etc.), a film oflubricant can be introduced between the tool andthe workpiece by externally pressurized liquid.Hydrostatic lubrication then prevails.

D. HOT VERSUS COLD AND WARM FORMING

AND IN BETWEEN

Up to World War II, only soft metals had beenextruded on a large scale. In normal operations,lead was extruded at room temperature,aluminum either cold or hot, and copper hot. Theextrusion of steel was severely limited bylubrication problems. Excessive friction alongthe die wore it out so quickly that a satisfactoryextrusion was impossible. On the other hand,even moderate friction along the chamber wallentailed a considerable increase in the requiredforce so that direct extrusion had to be limited tovery short billets. Of course, indirect extrusion,where the die is inserted in the movable ram andnot in the opposite end of the chamber, could

19

have been quite beneficial by offsetting thisincrease, but in production it was limited tospecial cases due to design difficulties. Its mainuse was in laboratory studies, where it isdesirable to eliminate varying friction, the betterto observe the process variables. The Ugine-Sejournet process (Sejournet,1955; 1966) is based both on the use of alubricant in a viscous condition at extrusiontemperature and on a separation between thelubrication of the chamber wall and that of thedie. A steel billet is heated to the extrusiontemperature and then rolled in a powder of glass.The glass melts and forms a thin film, 0.5 to 0.75mm (20 to 30 mil) thick, of viscous materialcoating the lateral surface of the billet andseparating it from the chamber wall. The relevantcoefficient of friction is thus so reduced that theforce required for the extrusion is practicallyconstant throughout the extrusion, whatever thelength of the billet and with the exception of astarting point. On the other hand, a thick solid glass pad, 6 to18 mm (0.25 to 0.75 in) thick, rests on the entryface of the die, which, for this purpose, is at leastpartly flat. The front face of the billet (Fig. 28)shapes this glass pad into a longitudinal contourcorresponding to the metal flow and, at the sametime, melts a thin layer of glass, which will driftalong with the outflowing metal and willlubricate its contact with the die. This meltingwill continue during the whole extrusion andensure a continuous supply of viscous lubricantbetween die and extruded product. There is nometal dead zone, and a shear effect occurs in theviscous lubricant. Note that the actual filmthickness of the lubricant is exaggerated in Fig.28. This process has been developed to such anextent that appropriate powdered glass can be

\

FIG. 28. Steel extrusion by Ugine-Sejournetprocess.

found for any specific temperature range. Thethickness of the glass layer on the finishedproduct is of the order of 1 mil, and after coolingit is easily removed. Initially devised for steel,the Ugine-Sejournet process has been extendedto practically all metals and alloys that have adeformation temperature either above that ofsteel or limited to a narrow range. Present-day trends in metal forming tendtoward the replacement of hot forming and othermanufacturing processes by cold forming. Someof the advantages are stronger products, betterdimensional precision, surface finish, andsavings in material waste. In the extrusion (andother forming operations) of steel, cold formingbecame feasible with the introduction ofphosphate coating, a development thatcomplements (and competes with) the Ugine-Sejournet process, When the steel surface iscoated, the spongy phosphate coat absorbs thelubricating liquid, which thus becomes highlyeffective in reducing friction and wear. One may say that both developments, theUgine-Sejournet process and phosphatelubrication, are breakthroughs that solvedfriction and wear problems. Without thesesolutions, the metal forming of steel would notbe where it is today. When forming is conducted at temperaturesabove room temperature but below therecrystallization temperature, it is called warmforming. Today, many forming processes areperformed warm to achieve a proper balancebetween required forces, ductility duringprocessing, and final product properties. Duringwarm forming of most steels, a specific range oftemperatures (where the steel hardens byprecipitation hardening) should be avoided. Withtoday’s sophisticated equipment for the controlof temperature and its distribution, the choice ofthe working temperature may be more preciselyfollowed to ensure optimal production, astypically demonstrated by precision closed-dieflashless forming. Hot forging is usually done with high-alloytools that can withstand elevated temperature.Tool-life considerations require as short a time ofcontact as possible between the tool and theworkpiece; thus mechanical presses that do notdwell at the bottom of the stroke arerecommended for hot forging. Hydraulic pressesare commonly used for cold forging, especiallyof large components; recently they have beenreplaced for smaller parts by the fastermechanical presses. So today, the choice ofmechanical versus hydraulic press may be

20

Table ΙΙΙΙ. Process Comparisons

ModeCriteria Hot Cold Warm Isothermal

Ductility Good Poor to Good Moderate IdealForming Loads Moderate High Moderate LowForming Rate Fast Fast Fast LowDimensional Precision Poor Good Moderate to Good GoodSurface Finish Poor Good Moderate GoodMaterial Conservation Poor Moderate Good GoodDie Cost Moderate Moderate High HighDie Life Poor Good Moderate Poor

decided, not only by the temperature of forging,but also by part size and production volume. The choice of one process over another, forany product, depends on many factors, includingthe material of the workpiece, the size, quality,and quantity required, and the producers’ likesand dislikes. Some of the criteria for this choiceare covered briefly by Avitzur (1983), and acondensed summary is given in Table I. Forming in the mashy state is an emergingtechnology. Observing the tensile properties ofmetals, especially those of metal alloys, it isnoted that the gradual drop in strength withincrease in temperature undergoes adiscontinuity in slope at the solidus line (see Fig.29). The solidus line represents the temperatureat which a solid metal alloy starts thetransformation into the liquid state. Thistransformation proceeds gradually on heating, sothat larger and larger portions of the specimenbecome liquid with increasing temperature untilthe liquidus temperature is reached, at whichpoint the entire specimen melts. The way alloysliquefy is unique. First, drops of liquid nucleate

FIG. 29. Strength characteristics in the mashystate.

at the grain boundaries. When the temperaturerises only slightly above the solidus temperatureand only a small percentage of the workpiece isliquid, the entire network of grain boundaries isalready liquid and each individual grain floats inliquid. This condition accounts for the drasticchange in strength at this temperature andexplains the different behaviors of metals duringforming in the mashy state (Kiuchi et al., 1979a;1979b). [See PHASE TRANSFORMATIONS,CRYSTALLOGRAPHIC ASPECTS.]

Plastic deformations in the mashy state occurmainly through solid grains sliding along theliquid grain boundaries. During compacting, thegrains themselves may simply be rearrangedrelative to one another like sand or powder. Theviscosity of the liquid metal and the thickness ofthe liquid layer determine the strength of theworkpiece. Furthermore, the rate of shear withinthe liquid layer [see Eq. (13) and also Avitzur,1979] determines the resistance to flow; thehigher the rate, the higher is the resistance. Thedashed lines in Fig. 29 indicate higher strengthsfor higher shear rates. In actual forming, thesehigher shear rates are caused by several factors.For example, they can be brought about byhigher forming speeds, as with higher ramspeeds in extrusion and forging. In the forging ofdisks, for a constant ram speed, the thinner thedisk, the higher are the shear rate in the liquidand the resistance to flow. In the application of forming in the mashystate, precautions must be taken to preventsqueezing of the liquid outward through thesurface. For example, in extrusion, the billet isusually heated to the mashy state and placed in apreheated chamber. When the extrusion takesplace, the extrudate may heat up due todeformation and friction, and thicker layers ofthe grain boundaries may then melt. Preventing

21

the liquid grain boundaries from squeezing out isof utmost importance and can be achieved bycooling the product at the exit. Closed-dieforging is advantageous for forming in themashy state because the confinement of theworkpiece in the die automatically safeguardsagainst liquid escape. In general it is observed that forming in themashy state at a higher temperature and liquidphase results in lower required pressures butwith products of inferior properties (i.e., lowertensile strength combined with cast dendriticstructure of lower ductility). Comparisonsbetween forging from the melt and forging in themashy state are made in Table II. Forging from the melt is a process thatcompetes with forming in the mashy state.Forging from the melt is accomplished bypouring molten metal into a mold, which servesas a die, and applying ram pressure whilefreezing progresses, (Ramati et al. 1978). As themolten metal gradually solidifies, its volumedecreases and the ram force must be appliedcontinuously. The surface of the workpiecefreezes first. A drastic volume drop is associatedwith the phase transformation on freezing fromthe liquid to solid phase. This volume changedue to phase transformation is vastly greater thanthe shrinking that occurs with the temperaturedrop in the solid state. Thus, as the interiorfreezes and shrinks, the ram advances into theworkpiece and (depending on the skintemperature) causes the skin, which is already

TABLE II. Comparison between Forging fromthe Melt and Forging in the Mashy State

Forging from the melt Forging in the Mashystate

On solidification, theworkpiece conforms tothe die cavity

Solidification starts atthe interface with thedie, while the interioris still liquid. Whenthe interior freezes, thevolume is reduced, thepunch (or top die) goesdeeper, and thepreviously frozen skinwrinkles.

Upsetting to any degreecan be incorporated.The billet’s shapediffers from thecavity’s

The liquid phase isuniformly spreadthrough the workpieceon the grainboundaries, permit-ting easy shifts ofgrains throughout theworkpiece.

solidified, to fold over, wrinkle, and even crack.Skin defects are a major obstacle in theapplication of forging from the melt.Nevertheless, aluminum automobile wheels aremass produced by this process, as well as byforging in the mashy state. The two processes arecompeting for the same products. Presently, workers in the field on productionproblems prefer the process of forging from themelt, also called “forge squeeze,” “moltensqueeze,” or other similar terms, over forging inthe mashy state. However, progress in the workon forging in the mashy state suggests somepotential advantages, especially in extrusion ofcomposites containing filamentary hardwhiskers. A solution is still being sought for thebest mode of mixing the matrix alloy with thewhiskers.

E. SOFT TOOLING

In a typical metal-forming operation, the shapeof the product is imposed by the tools. Thus, thetool is required to have a higher strength thanthat of the workpiece. For our purposes, toolsinclude dies or rolls in drawing, extrusion,forging, and rolling; they also include thechamber and ram or punch in extrusion or deepdrawing. However, there are techniques wherethe tool is softer than the workpiece. For example, during the process ofconventional ram extrusion, the ram or punchthat pushes the billet through the die does notcontrol the shape of the extrudate (product). Theshape of the product is controlled solely by theshape of the opening in the die. The hard ram ofconventional ram extrusion is replaced by a softtool, the liquid, in the process of hydrostaticextrusion. Components, mainly those produced fromsheet metal or from tubing, can be shaped by ahard tool on one surface only. The contour ofthat surface will control the contour on theopposite side without the use of a hard, shapedtool on that side. Such components may becandidates for soft tooling. Brake-press bending of a strip into an angleforming a channel is shown in Fig. 30. In the toppicture rigid tooling is shown, while in thebottom picture, the female die is replaced by anelastomer. During rigid tool forming, the regionof deformation in the vicinity of the bent corneris in contact with the ram only and is free on theother side. When bent with a female die of a softyielding material, the bending corner is confinedby the hydrostatic pressure imposed by the soft

22

FIG. 30. Brake press: (a) hard tool and (b)elastomer pad.

tool, with beneficial effects on ductility in thedeformation region. The hydrostatic pressure caneasily be controlled through the strength of theelastomer material, the geometry of theelastomer, and its support. The examples shown in Fig. 30 pertainbasically to sheet-metal bending processes and tothin-wall tube forming. Even in this restrictedarea, only a small selection of shapes has apotential for soft tooling. The main incentives for the use of soft toolingare the low cost of tooling and the ductility of theworkpiece affected by the hydrostatic componentof pressure that the soft tool exerts. Usually the male component of the die is thehard tool, and the female component is the softtool. There is no shaping of the soft tool, and thedie cost is less than half the cost of hard male-female die set. The soft tool may have to be replaced moreoften than the hard tool, but the elastomer pad iseasier and cheaper to replace than a machined,heat-treated, and polished tool-steel die. Theelastomer pad of Fig. 30b can be placed in fouralternative positions before it is replaced. Themost usual elastomer material is rubber orurethane of controlled hardness. Avitzur (1983)presents alternative designs in which liquid

under pressure replaces the elastomer. Thusreasonable and sometimes unlimited lifetimescan be expected of the soft tool. A major problem in sheet-metal forming,especially in a brake press, is springback causedby the residual (elastic) stresses. For example, ifa bend with an included angle of 90° is desiredfor the channel, the ram might have to bedesigned with an 85° included angle. Anyreasonable change in the design of the ram iscompatible with the basic design of the softcounterpart. Brake-press bending with the hard tools mayleave tool marks, which are eliminated when softtools are employed. On the other hand, whenhard tools are used, a smooth product surfacewith duplication of intricate designs can beachieved, while the surface finish achieved bysoft tooling is rough and intricate designs can beduplicated only on the side facing the hard tool. Earlier in this section, the beneficial effect ofthe environmental hydrostatic pressure of softtooling on the ductile behavior of the workpiecewas mentioned. For example, in Fig. 31a, thedeep drawing of a cup with soft tooling isdescribed. When a cup is drawn with a spherical-nosed mandrel without the soft tool opposite tothe mandrel, the depth of draw is limited bystretching, thinning, and consequent tearing ofthe workpiece over the mandrel. When a soft toolpresses the workpiece against the mandrel,sliding between the workpiece and the tool isarrested and stretching and thinning areminimized, deterring cup tearing and extendingthe possible depth of draw. Furthermore, softtooling can be adapted to a standard press. Replacement of the elastomer in Fig. 31a witha pressurized fluid is described in Figs. 31b andc. While the design of the liquid pressurechamber can be adapted for a standard pressdesign, hydroforming and hydromechanicalforming are performed on specially designedpresses with the pressure chamber and pressuresupply mechanisms as integral parts of the press.The processes of hydroforming are also called“rubber-diaphragm-forming.” In Figs. 31b and c (hydroforming), thepressure chamber is sealed by a diaphragm,separating the workpiece from the fluid. Duringhydromechanical forming the workpiece is indirect contact with the liquid, with the seal at theblank holder to prevent liquid leakage. As thepunch advances, the pressure in the chamberrises, causing the workpiece to wrap around thepunch. Raising the pressure too early may lead

23

(a)

(b)

(c)

FIG. 31. Deep drawing with an elastomer die.(a) Deep drawing with an elastomer pad; (b) &(c) deep drawing with pressurized fluid; (b)before, (c) after.

to tearing, while too little pressure at the earlystages leads to wrinkling. When a very thinblank is to be hydroformed, it is customary toback the blank with a dummy blank of iron orcopper. The sandwich material is formed inunison, so that wrinkling is avoided. The dummyblank is called a “water sheet.” One fluid pressure chamber may serve for theproduction of a variety of components in a rangeof sizes. When a changeover to another producttakes place, only the punch need be changed.The liquid pressure can be programmedindependently of the position of the punch. Thus,changes in the pressure versus time can beeffected to maximize the drawability of thecomponent. In contrast with the use ofelastomers, these pressure changes can beeffected without tool modification. The

advantage of controlling the pressure sequenceindependently is in higher drawability. The use of soft tooling and high temperaturecombine together in the implementation of metalforming with super plastic flow. The phenomenaof superplasticity have been observed in thecharacteristics of many metal alloys. Withcertain small-grain structures and in certainranges of elevated temperature, these alloys mayflow practically without limit under very smallloads but at very low rates of straining. A tensilespecimen made of a superplastic metal whentested at the proper temperature at a low enoughrate will undergo over 1000% elongation andwill stretch like viscous glass into a fiber beforeseparation. The superplastic material in the stateof superplasticity is highly sensitive to the rate ofstraining. Any section that temporarily remainsslightly larger in cross section will instantlyproceed to stretch at higher rates than the smallersections because the strain rate is inverselyproportional to the area. Thus, the stretchedportion of the specimen is automaticallymaintained at a uniform diameter. When thesame tensile rod made of superplastic alloy istested at room temperature. The superplasticbehavior vanishes. Such a test will exhibit higherstrength and a stress-strain curve (Avitzur,1983). Thus, a superplastic alloy may be formed withsimple tooling and very low loads when it isbrought up to its superplastic temperature. Whenthe desired shape is obtained and the temperatureis lowered to room temperature, the partpossesses high strength and can serve as a load-carrying structural component. However, somesuperplastic materials will creep slowly and getout of shape if loaded for a long time. Titanium alloys can be forged in thesuperplastic range into a rotor of a turbine enginewith the blades intact. Another typicalsuperplastic material is the alloy of zinc withaluminum and minute additions of copper,magnesium, and other elements. A popularsuperplastic zinc alloy is 78% zinc by weightwith 22% aluminum and is called 78-22 alloy. Atroom temperature, the strongest zinc alloysexhibit ultimate strength of about 4000 kg/cm2

(60,000 psi) and a reasonable ductility. At 260°Cthis material is in its best superplastic state. Therange over which this material exhibits its highformability is between 250°C (480°F) and 275°C(527°F), which is a narrow range. Objects suchas the carrying case of a typewriter are made ofthis alloy or from a polymer. In Fig. 32 a mold made of two halves split

24

FIG. 32. Pressure-aided deep drawing.

parallel to the axis of symmetry is delicatelydecorated on its interior. A thin-wall, deep-drawn cup is heated and placed in the dieassembly. The opening is sealed and air or argongas at low pressure (above atmospheric) ispumped into the cup. Very slowly the cupexpands and fills the mold perfectly to reproduceevery detail of curved design. Alternately, a flat sheet may be placed overthe cavity of the mold. Vacuum may beintroduced in the interior, sucking the sheet intothe cavity. If desired, the operation may beinitiated by vacuum and completed by pressure. While the above process is slow, theequipment is simple and inexpensive and so arethe dies. Very low loads are exerted, and the dielife is unlimited. Even when a battery of dies isengaged to increase production rates, the overalleconomy of the process is retained because ofthe low cost of tooling and equipment. Becausethe processing is conducted at a constanttemperature, it is also termed isothermalprocessing.

V. Replacing Brute Force

As larger and larger components of strongerand stronger metals are in growing demand forforming processes at room temperature, even thelargest available equipment of conventionaldesign, for bulk deformation by static loading, isno match for the job. Finesse must replace bruteforce. Some of the measures are old and stillusable today. Inertia forces may replace staticloading as in hammering. Today the repeatedbeatings may be applied through ultrasonicexitations of the tooling. Another concept callsfor the application of the load locally, to affectminute changes at a spot, and transversing thespot throughout the entire workpiece, severaltimes over, if needed. One class of processes iscalled rotary forming, and spinning is theexample presented here. The local load is muchsmaller and therefore so is the equipment.Usually the processing time per part is higher,but large components are not produced in asmany numbers as smaller ones. The mechanicsof the process of spinning are similar to those ofpottery-making a clay vase. Only the materials ofboth workpieces and tools are different. A mandrel shaped to conform to the interior ofthe product is clamped to the rotating head of aspinning machine (See Fig. 33). For ages a stickof hard wood, with or without a copper tip, hasbeen manipulated against the blank, back andforth, progressively laying the thin gauge blankagainst the mandrel. Holes, at intervals on thebed of the machine, are pegged to provide

FIG. 33. Conventional manual spinning.

25

support and leverage for the operator. Whennecessary the leverage mechanism is made morecomplex. The operator may strap himself to theframe of the machine, like a window washer on atall building. Two operators may assist oneanother, and heavier gauges of harder materialcan be worked by heating the workpiece. Thefriction between the tip of the stick and theworkpiece is large. In later models, especially inpower-mechanized spinning, the solid tip of thestick is replaced by a roller to minimize frictionlosses, replacing sliding by rolling friction. The proper manipulation to achieve goodresults was an art gained by experience. Becauseof springback (which is a major factor inspinning with manual tool manipulations) andthe closeness of the operator to the operationwith no safeguards, all “old-timers” in spinningcarry scars caused by close encounters with theedge of the rotating blank. Cooling pots and frying pans of aluminum andcopper and like products were made byconventional manual spinning. They still are insome developing and under developed regions ofthe world. Precision of size and wall thicknessand repeatability are not critical for suchproducts and cannot be achieved by handspinning. The process is slow, labor-intensive,and unsuitable for mass production. But thetooling can be made simple and inexpensive. Occasionally prototype development on alimited production basis is produced by handspinning, even today, and even in advancedtechnological societies. Changes are easy andinexpensive to introduce. Some experiencedspinning machine operators can produce vesselsof complex shapes without the support orbacking of the mandrel to define the desiredshape. Changes in shape and narrower sectionscan be improvised on the spot. Next, spinning of cones and vessels will bepresented, and then tube spinning. Whenmechanized, spinning is more economical for theproduction of small numbers of pieces than deepdrawing because of the low setup time and costsof spinning. The pattern, for example, can oftenbe made of wood or aluminum, thus savingtooling expense. For the last thirty years the trend has beentoward mass production with power spinning.The advantages of replacing manpower bymechanical power are the same for spinning asfor any other industrial process. However,mechanical power requires a control system, andcontrols require prediction of the forces andmotions to which the machine has to be set. In

spinning operations, as they are performed today,this means the following things: (1) the tool isusually no longer manipulated back and forth,but performs the deformation in one pass; (2) therotational speed, the feed, and the head-inpressure have to be fixed and preset beforespinning is started. Most recently, numerically controlled (NC)machines have been offered. The followingadvantages are claimed:

1. They have the ability to produce andexecute extremely complex programs to makeextremely complex parts, impossible hitherto.

2. They have the ability to make minorprogram adjustments during initial runs foroptimization of machine and material utilization.

3. They have the ability to make short runsand store programs for reuse.

4. There is a high degree of repeatability.

In the early days of mechanized spinning, themanual application of the tool to the workpiecewas replaced by two hydraulic piston andcylinder assemblies pushing the tool carriageagainst the workpiece and in the feed direction.The path of the tool was dictated directly by atemplate. With the further advance of hydrauliccontrol systems, the guidance of the tools wasdelegated to a closed-loop feedback systemwhereby a stylus that followed the templateactivated, through valves, the pistons thatcontrolled the motion of the tools. The templatecould be scaled up or down, and in moresophisticated methods a drawing could replacethe template. The spinning could be performedby several passes of the tool, changing theposition of the template, or the drawing, for eachpass. With the introduction of the stylus control,manual manipulation of the tool could bereintroduced. While the power is suppliedthrough the hydraulic pistons, the positioning ofthe tool can be delegated back to manualmanipulation and the tool can be appliedrepeatedly as shown in Fig. 33. Brute force,applied by the machine, is delicately controlledmanually. While this option is feasible, it isinfrequently applied, because the last generationof skilled manual operators is fast disappearing.In Fig. 34b, for power spinning of a cone, thetool(s) advance only once, in a motion parallel tothe surface of the mandrel from top to bottom,pushing the flange ahead downwards whilelaying the workpiece under the roller flat ontothe mandrel.

26

FIG. 34. Multiple spinning heads.

The strong head-on pressure applied duringpower spinning with a single tool causes highbending moments on the mandrel andasymmetric loads on the machine. Mechanizedspinning machines are designed with twosymmetric tool heads (see Fig. 34) andsometimes, in vertical tube-spinning machines,with three heads. The individual heads areusually arranged in tandem. For example, in tubespinning (Avitzur, 1983) one head may take halfof the reduction in thickness while the secondhead follows by several revolutions and takes thesecond half of the reduction. In tube spinning (also called “cylindrical flowforming”), a heavy-gauge tube is mounted over arotating mandrel. The tool holder, with the rollerpressing against the tube, advances slowly in theaxial direction as the workpiece rotates under thetool. The wall thickness decreases locally underthe pressure of the tool while the tool graduallyadvances through the entire tube surface. Duringtube spinning the thinning of the wall results inelongation of the tube in the axial direction withno change in the nominal diameter.

VI. Approaches to UnderstandingMetal Forming

A. ANALYTICAL VERSUS EXPERIMENTAL

APPROACH

When developmental work is conducted withproduction equipment in the plant, it disruptsoperation of the plant and causes intolerableexpenses. It is common practice to establishseparate research and development activities.The full spectrum of such an activity and thealternative approaches, complementing oneanother, are presented in Fig. 35. Short of doing developmental work on aproduction scale, all other methods can beclassified as modeling or simulation. In Fig. 35,the pie is divided into formulation and a physicallab of simulation methods or into mathematicaland physical modeling sections. Mathematicalmodeling is further divided into analytical andnumerical modeling. The old, but still inexistence, method of developing and usingempirical expressions is overlooked here.

B. PHYSICAL MODELING; SCALED-DOWN

MODELS AND VISIOPLASTICITY

The most common physical modelingprocedures are modeled by scaling down of sizeor by experimenting with modeling materials. Ineither case the equipment may be highlyinstrumented with sensors and recording andcontrol capabilities that are not used onproduction equipment. When the proper tooling

FIG. 35. Classification for planned activities inmetal forming.

27

and procedures to produce the desired producthave been identified by physical modeling, thetransition to the production mode is usuallysmooth with no surprises. Scaling down of the size of a component withwhich the experimental work proceeds bringssavings in materials, tooling, equipment size,space requirements, and operating costs as wellas expediting the development stage. Thediscussion of mathematical modeling will showthat for metal forming at room temperature thescaling up is automatic, and conditions, forexample, that prevent failure on small-scalemodels are identical to those required for theproduction size. The loads employed to impose plastic flow inthe scaled-down model are proportionally lowerthan those required for the production size. Thus,the measured loads from the experiment providethe information regarding loads to be needed forproduction runs. Production equipment can be replaced in thelaboratory by less expensive means when theactual workpiece is modeled by a softer material.Lead was used to study flow patterns for manyyears. Plasticine and wax are very popular today.Since the loads are smaller, expensive diematerials can be replaced by aluminum, wood, orPlexiglas. The equipment itself need not have astrong frame or a large power supply. It can bebuilt at a fraction of the cost of productionequipment and is inexpensive to operate. Severallaboratories have been built for modelingmaterial simulation by wax. To make the model realistic, waxes can bemixed with additives so that their properties mayresemble those of the workpiece material. Strain-hardening properties and strain-rate behavior canbe controlled by those additives. Even toolmaterials can be chosen so that tool rigidity canbe made to match the strength of the wax in thesame manner that the real tool will relate to theworkpiece. Friction conditions can be controlledby means of lubrication to best simulate thefriction conditions during actual production(Wanheim, 1978-1979). The main purpose ofstudying through modeling materials is toobserve the flow patterns and thus to developdies that produce a sound product. A picture of a cross section along the axis ofsymmetry of a multilayer wax forging isreproduced here as Fig. 36. The originalcylindrical workpiece was constructed from twocolors of wax, checkered in coaxial cylinders andflat planes normal to the axis of symmetry. Eachrectangle represents a ring of rectangular cross

FIG. 36. Wax models. (From Wanheim, 1978-1979).

section. The boundaries of the rectangles form arectangular network of grid lines. These billetscan be forged, extruded, rolled, or formed by anyother conceivable metal-forming process. A fullbillet can be formed by using a complete set ofdies. On completion of the forming process, thebillet can be sliced and the deformation patternstudied. On the other hand, half billets can beformed and the flat surface may be exposed toview through a Plexiglas window. Pictures of thedeforming specimen can then be taken atpredetermined intervals. The positioncoordinates of each corner of the checkered gridcan be determined at each interval. Thisprocedure is defined as “visioplasticity.” Thedata is then fed into a computer with a programthat will determine the following: (1) adisplacement field, (2) a velocity vector for eachpoint at each moment. Thus, when the velocityfield is fully determined, (3) a strain-rate field iscalculated from the velocity field; (4) a stressfield is calculated from the strain-rate field, thehistory of deformation, and the characteristicstress-strain relationship for the workpiece; (5)the stress load on the surfaces of the die iscalculated from the stress field, and (6) themeasurement of the total load during forming isdetermined. By integration of surface stresses onthe die over its entire surface, the load iscalculated and compared with the measured totalforce. In parallel with the study of the distorted gridin the workpiece, the deformations of the dieitself are occasionally measured by the lightphotoelastic method. In those cases, it is possibleto compare the stresses in the die, including sitesof stress concentration and their levels, withthose derived from the workpiece.

28

The far-reaching program described here isstill in its developmental stages. Nevertheless,the results achieved so far leave no doubt aboutthe contributions already made and the potentialvalue of this approach. Preliminary tooling fornew products and even for new processes can bedesigned reliably by using a modeling materialslaboratory at a fraction of the cost of runningdevelopmental work on production equipment.

C. ANALYTICAL AND NUMERICAL METHODS

OF MODELING

Exact solutions are not available for problemsin metal forming. Approximations andsimplifying assumptions are inevitable, andmany approaches-slug equilibrium, slip-linetechniques, and others-have been partiallysuccessful. Limit analysis (Avitzur, 1980) is a promisingapproach and is being used with increasingfrequency. In this approach, as applied to thestudy of drawing or extrusion force, twoapproximate solutions are developed. One, theupper-bound solution (Kudo, 1960, 1961)provides a value that is known to be higher thanor equal to the actual force; the other, the lower-bound solution, provides a value that is known tobe equal to or lower than the actual force; theactual force thus lies between the two solutions.For example, in Fig. 37 with the drawing stressas ordinate and the semicone angle of the die asabscissa, upper- and lower-bound solutions areplotted for several reductions together withcorresponding measured values of the actualstress. Even when experimental results are notavailable, it is expected that the actual stress andthe exact solution, if these were available, wouldlie between the upper and lower bounds asobtained analytically. Thus, by limit analysis, anapproximate solution is given with an estimate ofthe maximum possible error. The gap betweenupper- and lower-bound solutions may benarrowed by providing several upper bounds,choosing the lowest, and by providing severallower bounds, choosing the highest. Upper- andLower-bound solutions are obtained only byfollowing strict rules (including the requirementof the proper description of friction behavior andmaterial characteristics). A full illustration oflimit analysis is given by Avitzur (1983). A large number of analytical approachescomplement the still existing empiricalexpressions. Each approach has some advantagesand shortcomings, and until a perfect approach isavailable, they all complement and assist each

FIG. 37. Relative drawing stress versussemicone angle and percentage reduction in areaduring wire drawing.

other. With further progress, and as eachapproach gets closer to the exact solution, theyalso get closer to each other to the extent thatsome initially thought to be conflicting rivalapproaches turn out to be identical. The upper-and lower-bound solutions for some metalforming processes, for example, disc and stripforging (Avitzur, 1983) are so close that whenplotted together, both curves are almostoverlapping. The expressions for drawing inplane strain, obtained by the upper bound areidentical to those obtained by a force equilibriumon a rigid triangular element (Westwood, 1960).The same similarities can be observed betweenthe upper bound and slipline solutions to otherproblems in plane strain. The popular sliplinetechnique identifies surfaces within theworkpiece, along which shear stress is at itspeak. Thus, the material along these surfacesshears producing plastic flow and the shaping ofthe workpiece. The slip line technique, basedon the stress equilibrium approach, identifiessurfaces along which maximum shear stressprevails. In a comparable manner the upperbound approach, based on deformation patterns,offers the stream line technique and theassociated stream function. See Talbert and

29

Avitzur (1996). Here a flow line along the pathof a particle through the deformation regiondetermines a function f (x, y, z and ε) = Constant,where x, y, and z are coordinates in space, andε identifies each specific flow line. When thisfunction and its derivatives are set to conform tothe geometrical boundary conditions imposed bythe tooling, a stream function is derived. Such astream function leads to the determination of thestrains and strain rate components and to thederivation of an upper bound solution for theenergy required. In practice this method isapplied to two-dimensional conditions.Although this method is applicable to non-steadystate flow, it is more easily applied to steadystate flow. In the free slug equilibrium approach bulkelements of the workpiece are considered and theforces acting on them are set in equilibrium. Ashas been shown, with proper adaptation, the freeslug approach provides the following expressionfor the relative drawing stress in wire drawing,which is very close to the upper bound solutionpresented in Fig. 38.

α

ασ

σ

σ

σ

tan33

4

0lncot3

2)0ln(2

00

+

++=fR

Rm

fR

Rxbxf

(Avitzur, 1980).

With the computer revolution, severalnumerical methods emerged as computationaltools to study metalforming. In the proceduresdescribed next the workpiece is divided into afinite number of elements. Each elementexperiences a continuous plastic deformationpattern, which may also be a rigid body motion.The elements also experience sliding motionwith respect to each other and along their contactsurfaces with the tools. The elements arebordered (restricted) by each other, and by thesurfaces of the tools, or they are free, exposedsurfaces. The total energy required to imposeplastic flow of the entire workpiece is calculatedpiece by piece as the sum of energies expendedin each of the elements and along their surfaces. The numerical methods mentioned can becompiled into three major groups, i.e., the“Upper Bound Elemental Technique” (UBET),its offspring, the “Spatial Elementary RigidRegion” (SERR) Method, and the “FiniteElement Method” (FEM). UBET was introduced

by Prof. Kudo (1960, 1961) as an extension tohis analytical upper bound solutions tofundamental elementary shapes. Any workpieceof complex shape undergoing plasticdeformations can be constructed as an assemblyof a number of the elementary shapes. TheUBET computerized procedure then determinesthe total deformation energy. The initial UBETprocedures were restricted to axisymmetric andplain strain deformations. The basic elementswere rings and cylinders of rectangular andtriangular cross-sections. The extension to three-dimensional problemsby the SERR method was made by theintroduction of the Tetrahedral element thatundergoes a linear rigid body motion. Aworkpiece of any shape can be constructed froman assembly of tetrahedral elements where thecurved surfaces interfacing the tools areapproximated by plane surfaces. The plasticdeformation of the workpiece is thenaccommodated by the sliding of the elements,one with respect to each other, and along thesurfaces of the tool. See Avitzur (1993),Azarkhin and Richmond (1992), and PrafullaKumar Kar (1998). The tetrahedrons themselvesact as rigid bodies and consume no energy ofdeformations. Energy is consumed only alongthe surfaces of the tetrahedral elements wheresliding occur. The potential for the SERR method isunlimited. The linear rigid body motion of theelements may eventually be allowed toexperience the more general rigid body rotationalmotion where the plane surfaces of thetetrahedral element are replaced by morecomplex curved surfaces. Any shape of theworkpiece will be automatically divided to aminimal number of tetrahedral elements, and themotion of each element will be automaticallydetermined. Better precision will be reachedwhen each large element will be subdividedautomatically to (twelve) smaller ones. Today the most popular of the numericalmethods is the FEM, which proved itself in otherfields such as electricity, heat transfer, and fluidflow. In the FEM the workpiece is divided into afinite number of elements that may undergoplastic deformations. In one approach, for eachelement the differential equations of equilibriumof the stresses are replaced by an equilibrium ona set of finite differences of the stresses. Material properties may be introduced in anycomplexity desired to reflect real behavior. Theprocedure to obtain flow patterns and tool loadsis iterative in nature. The more complex the

30

geometry and the material, the longer the time ittakes; and larger computers are then required fora single solution. There is no doubt that with theadvancement in computer technology and betterfinite element procedures more and more of thetooling design for metal forming will be assignedto the FEM method.

D. PROS AND CONS OF THE DIFFERENT

MODELING PROCEDURES

The construction of production tools for largecomponents may be rather expensive. Ideally,the first tool for a new product should be the bestone possible. In practice, however, the first toolmay fail to perform as expected, resulting in adefective product or a fractured tool. Thepossible modes of failure are numerous, andsome of them are most common. Modeling byscaling down, by modeling materials, and bymathematical models makes the road to thechoice of best production tooling shorter.Comparisons and guidelines for the choice of themost suitable modeling method for eachcircumstance will be attempted. In comparing experimental procedures ofsmall-scale or material modeling on the one handwith mathematical modeling on the other hand,one notes that the development of amathematical model is much more time-consuming. In the study of flow through conicalconverging dies (Fig. 39), many flow patternsare observed that lead to failure. The choices ofdie angle, reduction, and friction that lead to

those failures are presented by the inserts. Forexample, the development of the criterion for theprevention of central burst took about 18 months.This derivation was based on years of work inthe general area of flow through conicalconverging dies and the application of the upperbound approach to metal-forming processes ingeneral. Such work requires a specializedexpertise, which is quite scarce at present.Furthermore, the original criteria, althoughuseful, may need further study. The specific problem that prompted thecentral-burst study (Avitzur, 1968) could havebeen solved by a trial-and-error, experimentalprocedure in a few weeks or months bypersonnel without an analytical background. As amatter of fact, experimental procedures wereemployed successfully for the elimination ofcentral burst, when discovered, prior to thedevelopment of the criteria. Every newappearance of the phenomenon called for a newstudy, but the resolution of the problem did notadd to the understanding of the causes of thedefect or to a solution to the problem when itappeared again. The analytical criterion, on theother hand, is universal; it applies anywhere andto any material. When an analytical solution isapplied successfully to a new problem, thetechnique itself advances further, making thenext solution for another problem easier andmore reliable. The potential for the application ofanalytical methods is unlimited.

FIG. 38. Upper-bound solution to flow through conical converging dies.

31

FIG. 39. Criteria for failure in flow through conical convergine dies.

However, when a specific problem arises, thepressure of time may dictate experimentalmethods. For example, the study of flow patternsfor complex forging may one day be conductedanalytically or by numerical procedures. Atpresent, however, experimental work is the onlypractical solution. For the study of die design toprevent unfilled corners and cracks due tofolding, modeling materials are most helpful. Onthe other hand, complex deep-drawing studiescannot be made with plasticine, wax, or lead butcan be conducted successfully on a scaled-downmodel using the real material. Also, studies todetermine tool life have not yet been conductedmathematically or through modeling materials;even scaled-down models are not very useful.Die design, including the choice of die material,is only improved through practice on theproduction line with full-scale equipment.

VII. Summary

The desire to make large parts, take largereductions, use high-strength materials, work tonet shape (precision flashless closed die forging),and use cold forming to obtain certain physical

properties all require high forming forces. Theseforces may be larger than the capacity of existingmachinery. Some possible solutions are:

1. design larger machines,2. use inertia forces to allow smaller

machines to do the job,3. take many small reductions, as in

superimposition of ultrasonic vibrations,4. impart localized deformations,5. use high-energy-rate forming, such as

explosive forming, and6. use modeling methods to identify the

optimal condition to minimize forming forces.

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