workability theory and application in bulk forming processes*

Chapter 12

Workability Theory and Application inBulk Forming Processes*Howard A. Kuhn, Consultant

WORKABILITY, as described in previouschapters, is not merely a property of a materialbut a characteristic of the material/process sys-tem. The role of the material is measured by asimple test or tests and should be expressed in aquantitative form that is applicable universally.This measure is taken to be a basic property ofthe material composition and structure, and it re-flects the macroscopic outcome of the micro-mechanisms of plastic flow perturbed by suchinhomogeneities as voids, inclusions, and grainboundaries. Such phenomena are dependent notonly on the material structure but also on theprocess parameters (strain rate and temperature),which define the role of the process in determin-ing workability.

The micromechanisms of ductile fracture inbulk forming processes are strongly influencedby the stress and strain environment imposed bythe process. As shown in Fig. 10 of Chapter 2,overall measure of strain to fracture can be re-lated to an overall value of hydrostatic stress inthe material during processing. In the spectrumof processes (extrusion, rolling, forging, andwire drawing), the average hydrostatic stress be-comes increasingly tensile, and the strain tofracture progressively decreases. Within each ofthese processes, however, the stress and strainstates can be considered on a localized basis inthe specific regions in which fractures initiate.These localized conditions are controlled by thegeometry of the workpiece, die design, and fric-tion at the die/workpiece interface. These threefactors, in addition to the strain rate and temper-ature parameters already mentioned, embody therole of the process in determining workability.

This article focuses on the effects of mechan-ical plasticity on workability; that is, processcontrol of localized stress and strain conditionsto enhance workability. First, the nature of localstress and strain states in bulk forming processesis described, leading to a classification schemethat facilitates the application of workabilityconcepts. This defines testing procedures and

specific process measurements for an experi-mental approach to workability evaluation.Theoretical models and fracture criteria are thendescribed and compared with experimental re-sults. Finally, the application of workability con-cepts to forging, rolling, and extrusion processesis discussed.

Stress and Strain States

Forging, extrusion, and rolling processes gen-erally are considered to involve the applicationof compressive force to material to impart achange in shape and dimensions. On close ex-amination, however, it is clear that deformationresulting from the applied load causes secondarystress and strain states that vary from point topoint throughout the deforming workpiece.These stress and strain states may include ten-sion, so fracture can occur at certain locations ina material even though the primary (applied)load is compressive.

To explore this possibility further, it is usefulto review briefly the von Mises yield criterion,which is the foundation of plasticity theory forisotropic materials. It gives the relationships be-tween normal and shear stress components atyielding. From this, relationships between stressand strain components are derived:

(Eq 1)

where e denotes strain, � denotes stress, and thesubscripts 1, 2, and 3 designate the three direc-tions in an orthogonal coordinate system. Here,l is a proportionality factor dependent on thedeformation history and flow stress curve of thematerial. The resulting strain in a given directionis affected by the stress in all three coordinatedirections. In addition, these relationships sat-isfy the volume constancy condition, e1 � e2 �e3 � 0.

To illustrate these relationships, it is useful toconsider a two-dimensional case of plane stress,s3 � 0:

(Eq 2)

Referring to Fig. 1 and Eq 2, a reduction inthickness (compressive strain in the 2-direction)

e l s s

e l s s

1 1 2

2 2 1

1

2

1

2

= -ÈÎÍ

˘˚

= -ÈÎÍ

˘˚

e l s s s

e l s s s

e l s s s

1 1 2 3

2 2 3 1

3 3 1 2

1

2

1

2

1

2

1

2

1

2

1

2

= - -ÈÎÍ

˘˚

= - -ÈÎÍ

˘˚

= - -ÈÎÍ

˘˚

*This article is revised from Howard A. Kuhn, Workability Theory and Application in Bulk Forming Processes, Forming and Forging, Vol 14, ASM Handbook, ASM International, 1988, p 388–404.

Similarity of deformation under (a) horizontal tension and (b) vertical compressionFig. 1

Handbook of Workability and Process Design G.E. Dieter, H.A. Kuhn, and S.L. Semiatin, editors, p 172-187 DOI:10.1361/hwpd2003p172

Copyright © 2003 ASM International® All rights reserved. www.asminternational.org

Chapter 12: Workability Theory and Application in Bulk Forming Processes / 173

can be accomplished either by a compressivestress, P2 (or �s2), or by a tensile stress, s1.Although the nature of the deformation is thesame in both cases, accomplishing that deforma-tion in the latter case could risk fracture becauseit involves a tensile stress.

For more general cases, Eq 2 can be re-arranged to determine the following stresses:

(Eq 3)

Therefore, the stresses depend on the localizedstrains e1 and e2 that result from metal flow.Equation 3 is a more convenient representationof plasticity relationships for workability study.

For example, the cylindrical surface of a com-pression test undergoes various combinations ofaxial and circumferential strains, depending onthe aspect ratio and the friction at the die contactsurfaces (Fig. 2). When no friction exists, theratio of circumferential strain to axial strain ise1/e2 � �1⁄2. According to the first part of Eq 3,s1 � 0 for this case. The deformation in thiscase is referred to as homogeneous compression,because the only stress acting is s2 and it is uni-form throughout the specimen. Therefore, thehomogeneous compression test is suitable formeasuring flow stress.

When friction exists at the die contact sur-faces, material at these surfaces is constrainedfrom moving outward, while material at the mid-plane is not constrained. As a result, bulgingoccurs, as shown in Fig. 2(a). Under these con-ditions, the circumferential strain e1 (�0) in-

sl

e ele

ee

sl

e ele

ee

s

1 1 2 12

1

2 2 1 12

1

3

4

3

1

2

4

31

1

2

4

3

1

2

4

3

1

2

0

= +ÈÎÍ

˘˚= +

È

ÎÍ

= +ÈÎÍ

˘˚= +

È

ÎÍ

˘

˚˙

=

creases, and the localized axial compressivestrain e2 (�0) decreases. From Eq 3 as e1/e2 be-comes more negative (��1⁄2), s1 becomes morepositive. Therefore, increasing bulging due tofriction during the compression of a cylinder in-creases the secondary tensile stress s1 and en-hances the likelihood of fracture.

Similarly, at the edges of bars during rolling(Fig. 3), the elongation strain e1 is determinedby the overall reduction in area. The localizedcompressive vertical strain e2, however, dependson the shape of the edge. Greater convexity andsharpness of the edge decrease the compressivevertical strain for a given reduction, which, fromthe first part of Eq 3, increases the secondarytensile stress s1 at the edge. Therefore, edgecracking during rolling is also due to secondarystress states.

More complex cases of the same type of sec-ondary tensile stress states occur in forging (Fig.4). During the forging of a hub shape, for exam-ple, the top surface of the hub is subjected to bi-axial tension because of friction during flowaround the die radius (Fig. 4a). This is identicalto the conditions present at the nose of a billetthat is being extruded or rolled. Similarly, duringforging, the top surface of a rib undergoestensile strain in the direction of curvature, andessentially no strain occurs along the length di-rection of the rib (Fig. 4b). In both cases, local-ized secondary tensile stresses are generated thatmay cause fracture. These stresses can be calcu-lated from measured strain values using Eq 3.

Figures 2 to 4 show examples of plane stress,with the stress normal to the free surface beingzero. Other regions of workpieces in bulk defor-mation processes, however, are subjected tothree-dimensional stress states. For example,material at the die contact surfaces in forging,extrusion, and rolling (Fig. 5) is subjected to

strains e1 and e2 in the plane of the surface, as inFig. 2 to 4. In Fig. 5, however, this surface is alsoacted upon by pressure P3 normal to the plane.Similarly, at internal locations of the workpiecesin such processes as forging or wire drawing(Fig. 6), a material element of the central longi-tudinal plane is subjected to strains e1 and e2 andstress normal to the plane.

In Fig. 5 and 6, the material element can bethought of as the plane-stress elements in Fig. 2to 4 with s3 acting normal to the plane. If e1 ande2 are taken to be the strains in this plane, thenEq 3 becomes:

In other words, for the same deformation (that is,the same values of e1 and e2 as in Fig. 2 to 4), thestresses s1 and s2 in Eq 3 are biased by s3. Thestress normal to the surface, then, increases thehydrostatic stress component of the stress stateby s3. This reflects the basic concept that hydro-static stress does not affect yielding or plastic de-formation. As shown in Fig. 10 of Chapter 2,“Bulk Working Behavior of Metals,” however,the hydrostatic stress has a significant effect onfracture. In Fig. 5, the compressive stress (diepressure) P3 would increase the strains at frac-ture, but in Fig. 6 the internal stress s3 may betensile and decrease the strains at fracture.

The preceding discussion of stress and strainstates in various regions of workpieces suggestsa convenient method of classifying cracking de-fects. Figures 2 to 4 in this article illustrate thelocations at which free surface cracks occur inforging and rolling, as well as at the nose ends ofbillets in extrusion or rolling. Figure 5 in this ar-ticle shows examples of circumstances in whichcontact surface cracking occurs—for example,the fir tree defect in extrusion or the longitudinalsurface cracks in rolled plates.

Empirical Criterion of Fracture

The stress and strain environments describedin the previous section in this article suggestthat a workability test should be capable of sub-jecting the material to a variety of surface straincombinations. A capability for testing undersuperimposed normal stress would also bedesirable.

When considering workability tests, it is im-portant to recognize that fractures initiate in lo-calized regions where interaction between thestress and strain states and the material structurereaches a critical level. Orientation, shape, andvolume fraction of inclusions and other inhomo-geneities have a dominant effect on the fractureprocess. Therefore, it is critically important thatworkability test specimens contain material hav-

sl

e e s

sl

e e s

s s

1 1 2 3

2 2 1 3

3 3

4

3

1

2

4

3

1

2

= +ÈÎÍ

˘˚+

= +ÈÎÍ

˘˚+

=

Localized strains on (a) the bulging cylindrical surface of an upset test and (b) their variation with aspect ratioand friction conditions. Source: Ref 1

Fig. 2

174 / Process Design and Workability

Localized strains at the edges of bars during (a) rolling and (b) their variation with edge profile. Source: Ref 2Fig. 3

Strains at the free surfaces of forgings. (a) Axisymmetric hub. (b) Rib-web forgingFig. 4


ing the same microstructural features as the ma-terial in the localized, potential fracture regionsof the actual process.

Specifically, when evaluating a workpiece forsurface fractures, specimen surfaces must con-tain the as-received surface of the workpieceunder consideration because it may contain laps,seams, a decarburized layer, and so on, which af-fect fracture initiation. By the same argument,evaluation of material for internal fractures suchas central burst must involve test specimenstaken from the middle of the workpiece, where,for example, segregation of second phases mayhave occurred. Because of possible anisotropyeffects, orientation of the critical stresses withrespect to any inclusion alignment must be thesame in the test specimens as it is in the actualprocess and material of interest.

The compression test has become a standardfor workability evaluation. As shown in Fig. 2, arange of strain combinations can be developed atthe cylindrical free surface simply by altering

friction and geometry conditions. The influenceof friction and consequent bulging on circumfer-ential tensile stress development is clearlyshown in Fig. 7. Compression with friction pro-duces circumferential tension that leads to frac-ture, while frictionless compression preventsbarreling, tension, and cracking as described inFig. 2 and Eq 3.

Alterations of the compression test geometryhave been devised to extend the range of surfacestrains available toward the vertical, tensile

strain axis, e1 (Ref 3). Test specimens are artifi-cially prebulged by machining a taper or aflange on the cylinders (Fig. 8). Compressionthen causes lateral spread of the interior mate-rial, which expands the rim circumferentiallywhile applying little axial compression to therim. Therefore, the tapered and flanged upsettest specimens provide strain states consisting of

Compression tests on 2024-T35 aluminumalloy. Left to right: undeformed specimen, com-

pression with friction (cracked), compression withoutfriction (no cracks)

Fig. 7

An element of material at the center of a forging undergoing (a) double extrusion and (b) at the center ofwire being drawn. The element undergoes strains e1, and e2, as in Fig. 2 to 4, with stress s3 normal to the

1–2 plane.

Fig. 6

An element of material at the die contact sur-faces during (a) forging, (b) drawing or extru-

sion, and (c) rolling

Fig. 5

(a) Flanged and (b) tapered prebulged compres-sion test specimens. Lateral spread of interior

material under compression expands the rim circumferen-tially while little axial compression is applied (see Fig. 9).

Fig. 8


small compressive strain components. Eachcombination of height, h, and thickness, t, givesa different ratio of tensile to compressive strain.The strain states developed at the surfaces ofstraight, tapered, and flanged compression testspecimens are summarized in Fig. 9.

The variety of strain combinations availablein compression tests offers the possibility formaterial testing over most of the strain com-binations that occur in actual metalworkingprocesses. A number of samples of the same ma-terial and condition are tested, each one underdifferent friction and geometry parameters. Testsare carried out until fracture is observed, and thelocal axial and circumferential strains are meas-ured at fracture. Figures 10 to 12 give some ex-amples of results for AISI 1045 carbon steel,2024-T351 aluminum alloy at room tempera-

ture, and 2024-T4 alloy at a hot-working tem-perature. In some cases, the fracture strains fit astraight line of slope �1⁄2; in others, the data fit adual-slope line with slope �1⁄2 over most of therange and slope �1 near the tensile strain axis.Similar data have been obtained for a wide vari-ety of materials. In each case, the straight linebehavior (single or dual slope) appears to becharacteristic of all materials, but the height ofthe line varies with the material, its microstruc-ture, test temperature, and strain rate.

The nature of the fracture loci shown in Fig.10 to 12 suggests an empirical fracture criterionrepresenting the material aspect of workability.

The strain paths at potential fracture sites inmaterial undergoing deformation processing(determined by measurement or mathematicalanalysis) can then be compared to the fracturestrain loci. Such strains can be altered by processparameter adjustment, and they represent theprocess input to workability. If the processstrains exceed the fracture limit lines of the ma-terial of interest, fracture is likely. Other ap-proaches to establishing fracture criteria, as wellas applications of the criteria, are given in thefollowing two sections of this article, “Theo-retical Fracture Models and Criteria” and“Applications.”

Fracture locus for AISI 1045 cold-drawn steelFig. 10

Fracture locus for aluminum alloy 2024-T351 at room temperatureFig. 11

Range of free surface strain combinations forcompression tests having cylindrical (Fig. 2),

tapered, and flanged (Fig. 8) edge profiles. The rangesshown are approximate and they may overlap a smallamount.

Fig. 9

Fracture locus for aluminum alloy 2024-T4 atroom temperature and at 300 °C (570 °F).

e � 0.1 s�1

Fig. 12


Theoretical Fracture Modelsand Criteria

Fracture criteria for metalworking processeshave been developed from a number of view-points, as described in Chapters l and 2. Themost obvious approach involves modeling of thevoid coalescence phenomenon normally associ-ated with ductile fracture. Another approach in-volves a model of localized thinning of sheetmetal that has been adapted to bulk formingprocesses. In addition to models of fracture, cri-teria have been developed from macroscopicconcepts of fracture. The Cockcroft criterion isbased on the observation that both tensile stressand plastic deformation are necessary ingredi-ents, which lead to a tensile deformation energycondition for fracture. The upper bound methodhas been used to predict fracture in extrusionand drawing. Other approaches are based on thecalculation of tensile stress by slip-line fields.

Void Growth Model. Microscopic observa-tions of void growth and coalescence alongplanes of maximum shear leading to fracturehave led to the development of a model of holegrowth (Ref 4). Plasticity mechanics is appliedto the analysis of deformation of holes within ashear band. When the elongated holes come intocontact, fracture is considered to have occurred(Fig. 13). When the McClintock model is evalu-ated for a range of applied stress combinations,a fracture strain line can be constructed.

Figure 14 shows the calculated results fromthe McClintock model in comparison with theexperimental fracture line. The predicted frac-ture strain line has a slope of �1⁄2 over most of itslength, matching that of the experimental frac-ture line. Near the tensile strain axis, the slope ofthe predicted line is �1, matching that of actualmaterial results shown in Fig. 10 and 12.

Localized Thinning Model. In sheet form-ing, the observation that a neck forms beforefracture, even under biaxial stress conditions in

which localized instability cannot occur, hasprompted consideration of the effects of inho-mogeneities in the material. For example, amodel of localized thinning due to a small inho-mogeneity has been devised (Ref 5). Beginningwith the model depicted in Fig. 15, plasticitymechanics is applied to determine the rate ofthinning of the constricted region tB in relationto that of the thicker surrounding material tA.When the rate of thinning reaches a criticalvalue, the limiting strains are considered to havebeen reached, and a forming limit diagram canbe constructed. The analysis includes the effectsof crystallographic anisotropy, work-hardeningrate, and inhomogeneity size.

This model was applied to free surface frac-ture in bulk forming processes because of evi-dence that localized instability and thinning alsoprecede this type of ductile fracture (Ref 6). Twomodel geometries were considered, one having agroove in the axial direction (Z-model) and theother having a groove in the radial direction (R-model) as shown in Fig. 15. Applying plasticitymechanics to each model, fracture is consideredto have occurred when the thin region, tB, re-duces to zero thickness. When these fracturestrains are plotted for different applied stress ra-tios, a fracture strain line can be constructed. Asshown in Fig. 16, the predicted fracture linematches the essential features of the experimen-tal fracture lines. The slope is �1⁄2 over most ofthe strain range and approximately �1 near thetensile strain axis. Again, this model is in gen-eral agreement with the dual-slope fracture locishown in Fig. 10 and 12.

Cockcroft Model. The Cockcroft criterion offracture is not based on a micromechanicalmodel of fracture but simply recognizes themacroscopic roles of tensile stress and plasticdeformation (Ref 7). It is suggested that fractureoccurs when the tensile strain energy reaches acritical value:

s ee

*0

fd CÚ =

McClintock model of void coalescence by shear from (a) initial circular voids, through (b) growth, and (c)void contact

Fig. 13

Fracture strain locus predicted by theMcClintock model of void growth. The

shaded area represents typical experimental fracture locisuch as Fig. 10 to 12.

Fig. 14 Models for the analysis of localized thinningand fracture at a free surface. (a) R-model. (b)

Z-model. Source: Ref 6

Fig. 15 Fracture strain locus predicted by the model oflocalized thinning. The shaded area repre-

sents typical experimental fracture loci.

Fig. 16


where s* is the maximum tensile stress; e is theequivalent strain; and C is a constant determinedexperimentally for a given material, tempera-ture, and strain rate. The criterion has been suc-cessfully applied to cold-working processes. Ithas also been reformulated to provide a pre-dicted fracture line for comparison with the ex-perimental fracture strain line. Figure 17 showsthat the fracture strain line predicted by theCockcroft criterion is also in reasonable agree-ment with experimental results. This criteriondoes not show the dual-slope behavior of theprevious models and some actual materials. Thequestion of why some materials exhibit the dual-slope fracture locus and others only a singleslope remains the subject of speculation, discus-sion, and further study.

The upper bound method of plasticity analy-sis requires the input of a flow field in mathe-matical function form. The external workrequired to produce this flow field is determinedthrough extensive calculation. This value for ex-ternal work is an upper bound on the actual workrequired. Through optimization procedures, theflow field can be found that minimizes the cal-culated external work done, and this flow field isclosest to the actual metal flow in the processunder analysis. The upper bound method hasbeen applied to a number of metalworkingprocesses (Ref 8).

In consideration of metal flow fields, pertur-bations can be incorporated that simulate defectformation. In some situations, the external workrequired to create the flow field with defects isless than the work required to create the flowfield without defects (sound flow). According tothe upper bound concept, defects would occur inthese situations.

The upper bound method has been appliedto the prediction of conditions for central burstformation in extrusion and wire drawing. Asshown in Fig. 18, for die angles less than �1,sound flow requires less drawing force than flowwith central burst formation. Above �2, extru-sion with a dead metal zone near the die requires

lower drawing force. In the range of die anglesbetween �1 and �2, central burst is energeticallymore favorable (Ref 9).

Repeated calculations using the upper boundmethod provide the combinations of die angleand reduction that cause central burst (Fig. 19).Friction at the die contact surface affects the re-sults. If operating conditions are in the centralburst region, defects can be avoided by decreas-ing the die angle and/or increasing the reductionso that the operating conditions are in the safezone. This example is a clear illustration of therole of process parameters (in this case, geomet-ric conditions) in workability. An application ofthis method is given in the section “Appli-cations” in this article.

It should be pointed out that the upper boundmethod for defect prediction gives only a neces-sary condition. The strain-hardening and strain

rate hardening characteristics of the materialhave been included in the analysis (Ref 10), butthe microstructural characteristics have beenomitted. Therefore, when operating in the cen-tral burst range illustrated in Fig. 19, fracturecan occur; whether or not it will depends on thematerial structure (voids, inclusions, segrega-tion, and so on). When operating in the safe areashown in Fig. 19, central burst will not occur, re-gardless of the material structure.

Tensile Stress Criterion. The role of tensilestress in fracture is implicit yet overwhelminglyclear throughout the discussion of fracture andfracture criteria. The calculation of tensilestresses in localized regions, however, requiresthe use of advanced plasticity analysis methodssuch as slip-line fields or finite-element analysis.One result of slip-line field analysis that haswide application in workability studies is dis-cussed here.

Double indentation by flat punches is a classi-cal problem in slip-line field analysis (Fig. 20).

Variation in mode of flow with die angle inwire drawing. The mode requiring the small-

est force at any die angle is the active mode. This is aschematic for one value of reduction. Source: Ref 9

Fig. 18

Fracture strain locus predicted by theCockcroft criterion

Fig. 17

Upper bound prediction of central burst inwire drawing. Increasing friction, expressed

by the friction factor, m, increases the defect region of themap. Source: Ref 10

Fig. 19

Slip-line fields for double indentation at different h/b ratios. Source: Ref 11Fig. 20


The boundaries of the deformation zone changeas the aspect ratio h/b (workpiece thickness-to-punch width) increases. For h/b � 1, the slip-line field meets the centerline at a point, and forh/b � 1, the field is spread over an area nearly aslarge as the punch width.

This tooling arrangement and deformationgeometry approximates several other metal-working processes, as shown in Fig. 21. For sim-ilar h/b ratios in these processes then, the stressesthroughout the deformation zone of the processcan be approximated by those calculated fromslip-line field analysis for double indentation.

Most of the work on double indentation hasfocused on the calculation of punch pressure andthe extrapolation of these results to other

processes—for example, those in Fig. 21. In avery detailed study of double indentation, re-markable agreement was found between experi-mental results and slip-line field results (Ref 12).

For workability studies, however, it is neces-sary to locate and to calculate the critical tensilestresses. It can be shown that the hydrostaticstress is always greatest algebraically at the cen-terline of the material and that this stress is ten-sile for h/b � 1.8. The calculated results forpunch pressure and centerline hydrostatic stressare given in Fig. 22. Therefore, it is necessary tospecify die and workpiece geometric parameterssuch that h/b � 1.8 in order to avoid tensilestress and potential fracture at the centerline ofthe processes shown in Fig. 21.

For example, in extrusion, h/b is approxi-mated by:

where � is the die half-angle and R is area re-duction. Taking h/b � 1.8, the relationship be-tween � and R that produces tensile hydrostaticstress at the centerline can be calculated. The re-sult is given in Fig. 23, which is shown to besimilar to the relationship predicted by upperbound analysis (Fig. 19). The correlation is re-markable in view of the dissimilarity in dieshape between extrusion and double indentation.Furthermore, the similarity emphasizes that theflow mode for defect formation in the upperbound method is physically equivalent to the de-velopment of tensile hydrostatic stress.

As in the case of the upper bound method, theexistence of a tensile hydrostatic stress does notensure fracture, but it is a necessary ingredient.The material structure must be considered inconjunction with the tensile stress. In otherwords, the upper bound and tensile stress crite-ria are useful for defining approximate deforma-tion limits and successful process parameters,but more detailed criteria or models are requiredto provide more exact values. An experimental-analytical approach in which an experimentalvalue reflecting the inherent material ductility isdetermined first would be most useful (this valueis used to define a point on the fracture strainlocus), followed by development of the rest ofthe fracture-limit line, as in Fig. 10 to 12, 14, 16,and 17.

Applications

The fracture criteria discussed previously inthis article can be used as tools for troubleshoot-ing fracture problems in existing processes orfor designing/modifying processes for newproducts. In either case, graphical representationof the criteria permits independent considerationof the process and material parameters in quan-titative or qualitative form.

An example is the bolt-heading processshown in Fig. 24(a). If it is required to form abolt head diameter D from the rod of diameter d,the required circumferential strain is ln (D/d ),indicated by the horizontal dashed line in Fig.24(b). The strain paths that reach this level,however, depend on process parameters, asshown previously in Fig. 2(b), and the fracturestrain loci vary with material, as shown in Fig.10 to 12. Referring to Fig. 24(b), if the strainpath labeled a describes the strain state at the ex-panding free surface for one set of processingconditions and the material used has a forminglimit line labeled A, then, in order to reach therequired circumferential strain, the strain pathmust cross the fracture line, and fracture is likelyto occur. As shown, one option for avoiding de-

h bR

R/

– /

=+ ( )[ ]a 1 1 1 2 2

Slip-line fields for (a) rolling, (b) drawing, and (c) side pressing. These fields are similar to those for dou-ble indentation shown in Fig. 20.

Fig. 21

Variation of the normalized indentation pres-sure (P/Y where Y is the yield strength) and the

normalized centerline hydrostatic stress (�h/Y ) with h /bratio as calculated from slip-line field analysis

Fig. 22 Prediction of central burst in wire drawing bythe tensile stress criterion and slip-line field

analysis of double indentation. The range of predictionsby upper bound analysis (Fig. 19) is shown by dashedlines.

Fig. 23


fects is to use material B, which has a higherfracture limit. Another option is to alter theprocess so that strain path b is followed by thematerial. The latter option represents a processchange, which in this case involves improved lu-brication, as shown in Fig. 2. This procedure hasbeen quantified and implemented in a computer-ized tool for upsetting process design (Ref 13).

In more complex cases, other means areavailable for altering the strain path, such asmodification of die design, workpiece (preform)design, and redistribution of lubricant. Examplesof application to powder forging preform designand other metalworking processes are given inRef 14 and Ref 15.

The workability concept presented earlier inthis article provides a useful supplement to theexperience and intuition of the die designer, be-

cause it presents a graphical and quantitative de-scription of the relationship between materialand process parameters. Some examples of theapplication of the workability analysis proce-dures already described follow.

Bar Rolling. As shown in Fig. 3, the strains atthe edges of bars during rolling are similar tothose at the bulging free surface of a cylinder dur-ing compression. It should be possible, then, topredict fracture in bar rolling from compressiontests on the alloy of interest. This is pertinent incurrent attempts to roll ingots of high alloy con-tent into bar form. The complete workabilitystudy of bar rolling includes physical modeling ofbar rolling to obtain the strain states at the edgesof the bar, compression tests to obtain the mate-rial fracture limits, and comparison of the two setsof results to establish roll pass reduction limits.

Such a study is illustrated by the analysis ofcracking during the rolling of 2024-T351 alu-minum alloy bars. The intent was to roll squarebars into round wire without resolutioning.Rolling was done on a two-high reversible barmill with 230 mm (9 in.) diam rolls at 30 rpm(approximate strain rate: 4s�1). The roll groovegeometry is shown in Fig. 25. Defects occurredprimarily in the square-to-diamond passes (1–2and 3–4), but the two diamond-to-square passes(2–3 and 4–5), the square-to-oval pass (5–6),and the oval-to-round pass (6–7) also were ex-amined for completeness.

Lead was used as the simulation material forthe physical modeling of bar rolling. Pure(99.99%) lead was cast and extruded into 25 mm(1 in.) round bars and then squared in the boxpass (step 1, Fig. 25). Grids were placed on thelateral edges of the bars by an impression tool,and the grid spacing was measured before andafter each pass for calculation of the longitudi-nal, e1, and vertical, e2, strains. Different reduc-tions in area were achieved by feeding variousbar sizes and by changing roll separation dis-tances. A transverse slice was cut from the barsafter each pass for measurement of the cross-sectional area and calculation of the reduction.

Results of the strain measurements are sum-marized in Fig. 26, in which tensile strain isplotted simultaneously with the compressivestrain and reduction. As expected, the square-to-diamond passes involve the least compressivevertical strain, and the square-to-oval pass hasthe greatest compressive strain. The tensilestrain versus reduction plot is the same for allcases, reflecting volume constancy.

Compression tests were performed on the2024 aluminum alloy at room temperature and at250 °C (480 °F) at a strain rate of 4s�1 to deter-mine fracture limit lines. Straight, tapered, andflanged specimen profiles were used. Results aregiven in Fig. 27. Superposition of Fig. 26 ontoFig. 27 gives the rolling deformation limits.

To test the workability predictions, aluminumalloy bars were rolled at room temperature andat 250 °C (480 °F). Grid and area reductionmeasurements were made for the square-to-diamond passes. Figure 28 shows the measuredstrains at room temperature, which agree with

Upsetting (a) of bar diameter d to head diameter D. (b) Material fracture strain limits are superimposed onstrain paths reaching the final required strain. Strain path b (low friction) prevents fracture for both materi-

als. Material B avoids fracture for either strain path.

Fig. 24

Roll groove geometry for rolling square bars into round wire. Dimensions given in millimetersFig. 25


those measured in lead bars for the same pass(Fig. 26). Open circles indicate fracture, andclosed circles indicate no fracture. The fractureline for the aluminum alloy at room temperatureis superimposed as the dashed line. It is clearthat edge cracking in bar rolling conformed withthe material fracture line, and the limiting reduc-tion is approximately 13% for this combinationof material and pass geometry. Similarly, at 250

°C (480 °F), there was conformance betweenfracture in bar rolling (Fig. 29) and the fractureline of the alloy (Fig. 28). In this case, the limit-ing reduction is approximately 25%.

Extrapolating the preceding results for coldrolling, the limiting reduction for diamond-to-square passes would be approximately 15%; forthe oval-to-round pass, approximately 20%; andfor the square-to-oval pass, approximately 25%.

Similarly, in the hot rolling of this aluminumalloy, the reduction limit for diamond-to-squarepasses would be approximately 27%; for oval-to-round passes, approximately 30%; and forsquare-to-oval passes, approximately 40%. Thelatter two are beyond the reduction normallyused because of fin formation, so cracking oc-curs rarely in such passes.

Example 1: Preform Design for a BallBearing Race. A low-load high-torque ballbearing outer race was cold forged from a low-alloy steel powder preform (Fig. 30). The pre-forms were compacted from 4600 grade powderwith carbon added to give 0.20% C in the sin-tered material. The sintered preforms were 80%of theoretical density.

Initial efforts led to cracking through the pre-form along a diagonal beginning at the point ofcontact between the punch and preform (Fig.31). The large shear stress developed by the con-tact was beyond the fracture limit of the porouspreform, and two solutions were considered thatwould avoid such stresses (Fig. 32). The first so-lution was the use of a flat preform that involvesback extrusion flow into the outer rim, and thesecond was the use of a tall, thin-wall preformthat involves radial inward flow into the innerflange. The first option was rejected because itwould generate circumferential tension thatwould most likely cause fracture. The secondoption is desirable because compression is ap-plied at the top face; this option was pursuedthrough physical modeling.

The primary concern with the second option(Fig. 32b) was the large amount of radially in-ward deformation required to form the innerflange. As a result, this option was examined byphysical modeling. Model preforms were pro-duced from sintered 601AB aluminum alloypowder and gridded on the inside surface (Fig.33a). Grid displacements were measured aftereach of several increments of deformation, andthe calculated strains were plotted along with thefracture line of the material (Fig. 33b). It is clearthat both the axial and circumferential strains arecompressive throughout the process and do notexceed the fracture line. Some wrinkling of theinside surface occurred, but this was smoothedout when the surface contacted the mandrelunder pressure.

Actual production of straight-wall preformsas in Fig. 33(a) was not feasible, because theheight-to-thickness ratio is too large for com-paction. A compromise was developed in whichthe preform angle was 17° instead of 30°, asused in the original preform, or 0°, as used in thephysical model (Fig. 33b). This ensured initialpunch contact at the top face of the preform andgenerated compressive strains on the inside sur-face, as in Fig. 33(b). Cold-forging trials onthese preforms produced no cracks, developedthe desired full density in the ball path region,and showed the added benefit of a smooth ballpath surface that did not require grinding.

Example 2: Back Extrusion of CopperAlloy. A low-ductility dispersion-strengthenedcopper alloy was back extruded into a cup shape,

Measured localized strains during the rolling of lead bars. Left side shows longitudinal tensile strain versusvertical compressive strain. Right side shows longitudinal strain versus cross-sectional area reduction at

room temperature.

Fig. 26

Fracture strain lines for 2024 aluminum alloy in the T351 temper, measured by compression tests at roomtemperature and at 250 °C (480 °F)

Fig. 27


strain is ln (0.75/0.625) � 0.18. Workabilityanalysis would then require only measurementof the material fracture line and comparison withthe required circumferential strain 0.18.

Fracture strains were measured in flange com-pression tests as shown in Fig. 36, giving a min-imum circumferential strain of 0.2, which is suf-ficiently above the required strain for avoidanceof fracture. A hydraulic press was used (giving astrain rate of approximately 0.5 s�1), on the as-sumption that there is no strain rate effect atroom temperature. Because the workabilityanalysis showed that fracture should not be a

Ball bearing outer race that was cold forgedfrom sintered powder preform of 4620 low-

alloy steel

Fig. 30

Cracks initiated at the point of contact be-tween the punch and preform in the original

preform design. The preform had a taper of 30° on theinside diameter.

Fig. 31

as shown in Fig. 34. The deformation was car-ried out at room temperature on a mechanicalpress. Crack formation on the rim caused highrejection rates. The original slugs for thisprocess were smaller in diameter (16 mm, or0.625 in.) than the die inside diameter (19 mm,

or 0.75 in.), as shown in Fig. 35(a). Deformationof such preforms involves circumferential ex-pansion strain (equal to ln (D/d ), where D is thedie bore diameter and d is the slug diameter)along with very little compressive strain at therim (Fig. 35b). For this case, the circumferential

Superposition of fracture line (dashed) on measured strains during rolling of 2024-T351 aluminum alloy barsat room temperature. Solid line represents the strain path measured during rolling of the lead model mate-

rial shown in Fig. 26.

Fig. 28

Superposition of fracture line (dashed) on measured strains during the rolling of 2024-T351 aluminumalloy bars at 250 °C (480 °F). Solid line represents the strain path measured during rolling of the lead

model material shown in Fig. 26.

Fig. 29


problem, the effect of strain rate was exploredfurther.

Tests were performed on the same alloy usingcontrolled strain rate servohydraulic test equip-ment at strain rates of 5, 10, and 15 s�1; the thirdstrain rate given is close to that in the production

mechanical press. Figure 37 shows the surpris-ing result that the fracture limit line decreaseswith increasing strain rate. In particular, the min-imum circumferential strain falls below the re-quired value of 0.18 for successful forming ofthe rim; this explains the occurrence of fracture

on the production press. The problem wascorrected by using slugs of larger diameter todecrease the circumferential tension and by pre-forming a taper on the top face (Fig. 38), whichproduced some axial compression in the mate-rial at the rim. The strain path then avoidedcrossing the fracture line (Fig. 39), and the re-jection rate during production on the mechanicalpress was nil.

Contact Surface Fracture and InternalFracture. All of the previous applications andexamples involved free surface fractures andcould be treated directly by the fracture line.Consideration of contact surface fractures (Fig.5) and internal fracture (Fig. 6), however, re-quires modification of this approach or use of anew approach. In the following, an example isgiven of the application of the upper bound andtensile stress criteria to central burst in extru-sion. The empirical workability concept de-scribed previously is then modified for applica-tion to contact surface fracture as well as centralburst.

Example 3: Central Burst During Extru-sion. Central burst can occur in extrusion whenlight reductions and large die angles are used(Fig. 19), and it is encountered in the productionof shafts for transmissions and suspension sys-tems. A test of the central burst criterion wascarried out by processing shafts from hot-rolled1024 steel bars 22 mm (7⁄8 in.) in diameter (Ref16). The processing sequence consisted of initialdrawing followed by three extrusion steps in aboltmaker:

Process Reduction, % Die half-angle, degrees

Drawing 8 9Extrusion 22 22.5Extrusion 23 22.5Extrusion 16 22.5 or 5

All passes are in the central burst area of Fig. 40,except for the last pass with a 5° die angle.

A total of 1000 shafts were processed with the22.5° die, and 500 shafts were processed withthe 5° die. All shafts were tested ultrasonicallyfor internal defects. Central bursting was de-

Part that was back extruded from copperFig. 34

Preform alternatives for forging the ball bearing outer race shown in Fig. 30. (a) Back extrusion. (b)Compression and radial inward flow

Fig. 32

(a) Physical model for the second option (Fig. 32b) and (b) the measured strains during forging of the pre-form. The heavy line is the material fracture line. It is clear that the strain path never crosses the fracture

line and that defects are prevented.

Fig. 33


tected in 4.5% of the shafts extruded with the22.5° die, and no defects were detected in theshafts extruded with the 5° die. These resultsshow that the upper bound central burst criterionis a necessary condition. It was further shown inRef 16 that central burst was avoided in otherheats with slightly different compositions be-cause their strain-hardening coefficients werelarger than the original heat. This confirmed thepredicted results in Ref 10.

Modified Empirical Criterion. It was shownpreviously in this article that measured free sur-face strains at fracture fit a linear or bilinear linethat constitutes a fracture locus for the materialtested (Fig. 10 to 12). This is a convenientrepresentation of the complexities of ductile

fracture that are controlled by stress and defor-mation. The experimental fracture locus is alsoreproduced by several theoretical fracture crite-ria (Fig. 14, 16, and 17).

For contact surface and internal fractures,however, the surface on which the strains can be

Back extrusion of the cup shape shown in Fig. 34. (a) The preform slug was 16 mm (0.625 in.) in diameter,the die was 19 mm (0.75 in.) in diameter. (b) Strains at the cup rim where fracture occurred consist of cir-

cumferential tension to a value of 0.18 and very little compressive strain. The heavy line is the material fracture strainline.

Fig. 35

Modified preform slug for the back extrusionof the cup shape shown in Fig. 34. The slug di-

ameter is 18.8 mm (0.74 in.) and has a 5° taper on the topsurface.

Fig. 38Comparison of measured strains at the cuprim during back extrusion of the modified

preform slug shown in Fig. 38. The strains do not exceedthe material fracture strain line for low or high strain rateforming.

Fig. 39

Fracture strain line for copper alloy, deter-mined by flanged and tapered compression

tests. Specimen geometries used for each test are shownalso.

Fig. 36

Decrease in circumferential tensile strain atfracture with increasing strain rate for the cop-

per alloy tested in Fig. 36. Results are for thin-flangedcompression specimens, which have the lowest fracturestrain.

Fig. 37

monitored is subjected to stress normal to thatsurface. It was shown in Eq 3 that stress statesleading to a given set of surface strains differonly by a hydrostatic stress component, and thiscomponent is equal to the applied stress normalto the surface on which the strains are moni-tored. Experience shows that this hydrostaticstress affects fracture, and it should also affectthe fracture strain locus. It should be possible,then, to use the theoretical fracture criteria topredict the effects of hydrostatic stress on thefracture strain locus.

The simplest criterion described previously inthis article is that due to Cockcroft; therefore, itwas modified to predict the effects of stress nor-mal to the plane (Fig. 5 and 6) on the fracturestrains e1 and e2. The result (Fig. 41) shows thatsuperimposed pressure (P � 0) increases the


height of the fracture strain line and also in-creases its slope slightly. Superimposed tension(P � 0) decreases the height of the fracture line,decreases its average slope, and gives it a slightdownward curvature. It is clear that the increasein strains to fracture due to additional pressure isunlimited as pressure increases, but the strains tofracture due to additional tension are limited byzero as tension increases. This result is discussedwith regard to internal fracture and die contactsurface fracture in the following paragraphs.

Central Burst in Forgings. Internal fracturesalong the centerline of extruded or drawn barswere discussed earlier (Fig. 18, 19, 23, and 40).Similar fractures are observed in forged shapessuch as that shown in Fig. 42 for heat-treated6061 aluminum alloy. Here, as the outer region

is compressed between dies, material flows radi-ally inward and then vertically into the opposedhubs. This develops a hydrostatic tensile stressstate at the center (Fig. 6a), and fracture is astrong possibility.

Through visioplasticity analysis on split, grid-ded specimens (Fig. 42), the strain and stressesat the center of the workpiece were calculatedfor several increments of deformation. Thehydrostatic stress state at the center of the spec-imen is not always tensile; initially, it is com-pressive, and then it reverses, becoming tensileas the flange thickness is reduced and flow intothe hub occurs. Meanwhile, the strains at thecenter are increasing monotonically as deforma-tion progresses. This is illustrated in Fig. 43 bythe steps 0-1-2-3. As deformation proceeds, thestrains at the center increase, but the hydrostaticpressure is also increasing, so the fracture linemoves upward. Then, as the flange thicknessapproaches one-half of the hub base diameter

(die orifice diameter), the hydrostatic stress be-comes tensile, so the fracture line decreases inheight. The strains at the center continue to rise,however, and cross the fracture line, leading tothe central burst. The calculated hydrostatic ten-sion at fracture was 0.3Y. This approach couldbe used for predicting central burst in drawingand extrusion to provide a material-dependentcriterion, as opposed to the more simplisticupper bound and tensile stress criteria describedpreviously.

Die Contact Surface Fracture. Frequently,cracks occur during forging on surfaces that arein contact with the dies (Fig. 5). One commonlocation of such defects is the vicinity of a die orpunch corner. From the observation of a varietyof such defects, it appears that a common char-acteristic is an abrupt change in frictional sheartraction distribution in the region of the crack.High friction to retard metal flow in advance ofthe crack location is one method for preventingsuch defects.

A technique for studying die contact surfacecracks was developed by means of a disk com-pression test and dies having a rough surface inthe central region and a smooth surface in theouter region. Figure 44 shows the top view of a6061 aluminum alloy disk compressed betweensuch dies. In the transition region between therough central die surface and the smooth outerregion, radial cracks initiate and propagate out-ward. Such cracks occurred at approximately30% reduction when the smooth outer regionwas lubricated with Teflon. The cracks occurredat approximately 45% reduction when grease lu-brication was used in the outer smooth region.No cracks occurred even for very large reduc-tions when the smooth outer region was not lu-bricated.

Location of process conditions on a theoreti-cal central burst map. For an angle of 22.5°,

central burst occurred in 4.5% of the extruded shafts. Fora die angle of 5°, no central burst occurred.

Fig. 40

Movement of the fracture strain line due to su-perimposed hydrostatic stress. Applied stress

is represented in terms of multiples of the yield strength,Y. Negative values of P indicate hydrostatic tension.Calculations are based on a modification of the Cockcroftcriterion.

Fig. 41Internal fracture during the double-extrusionforging of aluminum alloy 6061. Grid defor-

mations on the middle longitudinal plane are shown.Stress and strain states are defined by Fig. 6(a).

Fig. 42Progression of surface strains and fracture lineat the central internal location of the double-

extrusion forging shown in Fig. 42. The fracture line risesfrom 0 to 1 to 2 as internal pressure increases and thenfalls to point 3 as the internal stress becomes tensile.

Fig. 43


Grid marks placed on the die contact surfaceof the disks were used to measure the distribu-tion of surface strains in the radial direction.Figure 45 gives an example of such measure-ments. In the rough central region the strains arezero, while in the smooth outer region the strainsare equal and constant. In the transition, how-ever, the circumferential strain, e, jumpsabruptly from zero to its constant value in theouter region, and the radial strain, er, overshootsto a very high value before returning to its con-stant value in the smooth outer region.

The strains shown in Fig. 45 were the sameregardless of the friction condition in the smoothouter region. Therefore, fractures in the transi-tion region occur because of the combination oflarge tensile surface strains and low hydrostaticstress state. This explains the occurrence ofcracks at low reduction when Teflon is used, andno occurrence of fracture when no lubricant isused. The Teflon, having a near-zero friction co-efficient, results in very low radial back pressureon the transition region, while grease and nolubricant provide progressively larger backpressures.

By means of visioplasticity analysis, thestresses were determined at the contact surfacein the vicinity of the transition region. The re-sulting normal die pressure plus the surface ra-dial and circumferential strains define the stressand strain states in the transition region and canbe illustrated on a forming limit diagram. Figure46 shows the change in surface strains and theincrease in the fracture line due to increasing

normal pressure during compression of a diskwith grease lubricant in the outer region (indi-cated by the increments of reduction to 45%).The fracture line increases at a slower rate thanthe strains increase with increasing pressure, andat 45% reduction the strain path exceeds thefracture line and cracks are observed. For Teflonlubricant, the crossover occurs at about 30% re-duction, while in the case of no lubricant thefracture line moves progressively away from thestrain path.

Example 4: Fir Tree Defect. The criterionfor contact surface fracture can be applied qual-itatively for interpretation of the fir tree defect inextrusion. Such defects occur on the surfaces ofextruded bars as well as in localized areas offorgings containing ribs. In a section of a rib-web forging from a preform of sintered alu-minum alloy powder, small cracks formed witha regular spacing on the rib surfaces (Fig. 47).

Such defects occurred only when the thicknessof the rib was greater than approximately one-half the web length; that is, extrusion reductionwas less than one-half. Each crack formed byshear at the corner as material flowed from theweb into the vertical rib.

The stress and strain state on a surface mate-rial element is shown in Fig. 48. As material inthe web is compressed, the surface element ex-periences tensile strain in the direction of flow,and the strain increases as the element ap-proaches the die corner. At the same time, thereis a compressive stress from the die onto the sur-face element. This pressure diminishes, how-ever, as the element nears the die corner andalmost disappears as the element moves aroundthe corner.

(a) Top view of aluminum alloy 6061 diskcompressed between dies. (b) Cracks form at

the transition region between rough and smooth areas ofthe die.

Fig. 44

Radial variation of contact surface strains after30% compression of the disk shown in Fig. 44

Fig. 45

Progression of surface strains and fracture lineat the transition region between rough and

smooth zones of the compressed disk shown in Fig. 45.Points 1, 2, 3, and 4 represent 10, 20, 30, and 45% re-duction, respectively.

Fig. 46

Die contact surface cracking during forging extrusion of aluminum alloy powder compact. (a) Cross sec-tion. (b) Normal to vertical rib surface. Note also the cracks at the top free surface. Stress and strain states

are defined in Fig. 4(b).

Fig. 47


vide a method of predicting the draft angle inorder to prevent fracture.

REFERENCES

1. H.A. Kuhn, P.W. Lee, and T. Erturk, AFracture Criterion for Cold Forging, J. Eng.Mater. Technol. (Trans. ASME), Vol 95,1973, p 213–218

2. H. Ortiz and H.A. Kuhn, Physical Modelingof Bar Rolling for Workability Study, Physi-cal Modeling of Bulk Metalworking Proc-esses, Conf. Proc., ASM International, 1987

3. E. Erman and H.A. Kuhn, Novel TestSpecimens for Workability Measurements,Compression Testing of HomogeneousMaterials and Composites, STP 808,ASTM International, 1983, p 279–290

4. F.A. McClintock, J. Appl. Mech. (Trans.ASME), Vol 90, 1968, p 363

5. Z. Marciniak and K. Kuczynski, A Model ofLocalized Thinning in Sheet Metalforming,Int. J. Mech. Sci., Vol 9, 1967, p 609

6. P.W. Lee and H.A. Kuhn, Fracture in ColdUpset Forging—A Criterion and Model,Metall. Trans. A, Vol 4, 1973, p 969–974

7. M.G. Cockcroft and D.J. Latham, Ductilityand the Workability of Metals, J. Inst. Met.,Vol 96, 1968, p 33–39

8. B. Avitzur, Metal Forming—Processes andAnalysis, McGraw-Hill, 1968

9. W.M. Evans and B. Avitzur, “Die Design forDrawing Extrusion,” Paper MF67–582,Society of Manufacturing Engineers, 1967

10. B. Avitzur, “Strain Hardening and StrainRate Effects in Plastic Flow through Con-ical Converging Dies,” Paper 66-Prod-17,American Society of Mechanical Engineers,1966

11. R. Hill, On the Inhomogeneous Deform-ation of a Plastic Lamina in a CompressionTest, Philos. Mag., Vol 41, 1950, p 733

12. J.F. Nye, Experiments on the PlasticCompression of a Block between RoughPlates, J. Appl. Mech. (Trans. ASME), Vol19, 1952, p 337

13. J.J. Shah and H.A. Kuhn, An EmpiricalFormula for Workability Limits in ColdUpsetting and Bolt Heading, Proceedings of13th NAMRC, Society of MechanicalEngineers, 1985

14. C.L. Downey and H.A. Kuhn, Applicationof a Forming Limit Criterion to Design ofPreforms for Powder Forging, J. Eng.Mater. Technol. (Trans. ASME), Vol 97H,1975, p 121

15. H.A. Kuhn, Deformation Processingof Sintered Powder Materials, PowderMetallurgy Processing, H.A. Kuhn andA. Lawley, Ed., Academic Press, 1978, p 99

16. Z. Zimerman, H. Darlington, and E.H.Kottcamp, Jr., Selection of OperatingParameters to Prevent Central Bursting dur-ing Cold Extrusion, Mechanical Workingand Steel Processing, The MetallurgicalSociety, 1970, p 405

Increase in strain e1, and decrease in die contact pressure P on a surface element as it moves from theweb into the rib section of the extrusion forging shown in Fig. 47.

Fig. 48

Progression of strain e1, and decline of frac-ture line due to decrease in pressure P for an

element moving from the web to rib section during theforging extrusion shown in Fig. 48. At stage 3, the strainexceeds the fracture line.

Fig. 49

This can be illustrated schematically on thefracture strain diagram shown in Fig. 49.Because the deformation is in a state of planestrain, the strain path is represented as a vectorof increasing length along the vertical axis.Meanwhile, the fracture line decreases inheight because the pressure acting normal tothe element is progressively decreasing. Whenthe strains cross the fracture line, fractureoccurs.

This phenomenon does not take place whenextruding ribs of small thickness, because theextrusion reduction, and therefore the pressure,is larger, which maintains the fracture line at ahigh level. For thick ribs, two solutions wereconsidered. One approach is to increase frictionalong the rib walls by roughening the die surfaceor avoiding lubrication of the die rib. This pro-duces greater back pressure at the die corner, el-evating the fracture line and preventing crack-ing. Such an approach is difficult to implementand can be used only with segmented dies be-cause the formed rib cannot be removed fromthe die. The second approach is to use a draftangle on the rib, which has the same effect as in-creased friction. An angle of 10° prevented frac-ture in the current case, but other alloys may re-quire a smaller or larger angle. A quantitativeanalysis combining the pressure effect on thefracture line and plasticity analysis would pro-

workability theory and application in bulk forming processes*

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