A general forming limit criterion for sheet metal forming.pdf

International Journal of Mechanical Sciences 42 (2000) 1}27

A general forming limit criterion for sheet metal forming

Thomas B. Stoughton*General Motors Research and Development Center, Warren, MI 48090-9057, USA

Received 9 June 1998; received in revised form 20 October 1998

Abstract

The forming limit of sheet metal is de"ned to be the state at which a localized thinning of the sheet initiatesduring forming, ultimately leading to a split in the sheet. The forming limit is conventionally described asa curve in a plot of major strain vs. minor strain. This curve was originally proposed to characterize thegeneral forming limit of sheet metal, but it has been subsequently observed that this criterion is valid only forthe case of proportional loading. Nevertheless, due to the convenience of measuring strain and the lack ofa better criterion, the strain- based forming limit curve continues to play a primary role in judging formingseverity. In this paper it is shown that the forming limit for both proportional loading and non-proportionalloading can be explained from a single criterion which is based on the state of stress rather than the state ofstrain. This proposed criteria is validated using data from several non-proportional loading paths previouslyreported in the literature for both aluminum and steel alloys. In addition to signi"cantly improving thegauging of forming severity, the new stress-based criterion is as easy to use as the strain-based criterion in thevalidation of die designs by the "nite element method. However, it presents a challenge to the experimentalistand the stamping plant because the state of stress cannot be directly measured. This paper will also discussseveral methods to deal with this challenge so that the more general measure of forming severity, asdetermined by the state of stress, can be determined in the stamping plant. ( 1999 Elsevier Science Ltd. Allrights reserved.

Keywords: General forming limit criterion; Sheet metal forming

Notation

p*

principal true stresse*

principal true strainp6 e!ective stresse6 e!ective strain

*Tel.: 810 986 0630; fax: 810 986 9356; e-mail: [email protected]

0020-7403/00/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 0 - 7 4 0 3 ( 9 8 ) 0 0 1 1 3 - 1

a ratio of minor to major true stresso ratio of minor to major true strainm ratio of e!ective stress to major true stressj ratio of e!ective strain to major true strainq normal anisotropy coe$cientl Poissons ratioK, n coe$cients used in a power law stress}strain relationA, B, p

0coe$cients used in a saturation-type stress}strain relation

E (&) proposed forming limit functionk, p

ccoe$cients used in the proposed forming limit function

1. Introduction

The ability to gauge the forming severity with respect to necking is critical to the analysis of thesheet metal forming process. The most commonly used method, based on the forming limitdiagram (FLD) developed by Keeler [1] and Goodwin [2], requires a comparison of the principalstrains to a curve in a plot of major vs. minor strain. The shape of this curve, such as the one shownin Fig. 1 for a particular mild steel is a characteristic of the sheet metal. The premise is that as longas the principal strains are signi"cantly below this curve in the diagram, then that region of themetal will be safe from necking and tearing.

Fig. 1. Conventional forming limit curve for a particular colled-rolled aluminum-killed steel. Half open circles representdata at strain states for which a neck initiates, through which the forming limit curve is drawn. Fully open and solidcircles are conventionally drawn for strain states just before and after a neck initiates, respectively.

2 T.B. Stoughton / International Journal of Mechanical Sciences 42 (2000) 1}27

Although the FLD method is proven to be a useful tool in the analysis of forming severity, it hasbeen shown to be valid only for cases of proportional loading, where the ratio between the principalstresses remain constant throughout the forming process. This condition is sometimes falselyequated to a condition of proportional straining, where the ratio between the principal plasticstrains is constant. Since the latter ratio is observed by both measurement and FEM prediction tobe nearly constant during most "rst draw forming processes, and this process is considered to bethe most critical with respect to formability, the path-dependent limitations of the FLD are oftennot considered.

The issue cannot be ignored in the analysis of secondary forming processes involving redrawingand #anging dies where principal strain increments are independent of the strains of the "rstforming process. In these cases, net strains far above the conventional FLD curve are sometimesfound to be safe from necking while in other areas strains far below the curve are found to neck.Motivated by this problem of multi-stage forming, Kleemola and Pelkkikangas [3] reported theforming limits of a mild steel, copper, and brass following uniaxial and equi-biaxial prestrain,noting the dependence of the FLD curve on the magnitude and type of prestrain. Interestingly, theauthors report that the observed shift of the strain-based forming limit can be calculated assumingthat the limit depends only on the state of stress, independent of strain path. Arrieux, Bedrin, andBoivin reported a similar "nding for prestrained aluminum [4].

Knowledge of the discovery of a path-independent stress-based forming limit is not widespreadnor is its signi"cance appreciated. This lack of awareness is due in part to the belief that theconventional FLD method is satisfactory when limited to analysis of the "rst draw die, whichgenerally receives the most attention. This belief is based on the assumption that a constant ornearly constant ratio in the plastic strain components implies proportional loading. This assump-tion is not only false but leads to a dangerous misinterpretation of forming severity. While it iscertainly true that the plastic strain ratio rarely deviates from near linearity in the "rst draw die, thestate of stress is rarely proportional in production applications.

As the metal is drawn into the die cavity and stretched over the tool surface, the local state ofstress can change rapidly. In applications involving pockets and channels which are formed late inthe forming process, as well as in deep draws, the change in the direction of stress can be abrupt anddrastic. These changes may result in stresses which instantaneously fall below the point of yielding.FEM analysis shows that such behavior is typical of "rst draw applications where stresses areobserved to drop below the yield surface part-way through the forming process. In some cases thestress continues to fall, though rarely on the same path as the origin loading. Other areas are foundto reload and approach the yield surface in a di!erent state of stress.

To illustrate the impact of dropping below the yield surface, consider two areas of the metal, bothstrained under near-proportional loading conditions to strain levels (A) just below and (B) justabove the line of safety such that by the conventional FLD method the analyst decides that area Ais safe and area B is not. Following standard engineering practice, the analyst attempts to reducethe strain level in area B by modi"cation of the die or forming process, spending time and otherresources. But if the stress drops below the yield point during the forming process after reachingthese strain levels, the measure of criticality assigned to these areas will be completely false eventhough both appear to be under proportional straining conditions. For example, area B mayunload and carry a stress level far below the actual forming limit. The analysts e!orts to reduce thestrain in this area would be an unnecessary waste of time and resources. In addition, area A may

T.B. Stoughton / International Journal of Mechanical Sciences 42 (2000) 1}27 3

have unloaded to a point and then reloaded to another state of stress at or near the yield surface.Since the shape of the yield surface and forming limit surface are distinct functions, the two surfacesintersect following high levels of prestrain, such as in area A. This means that while area A may besafe in the state of stress prior to it falling below the yield surface, it may be much more critical as itre-approaches the yield surface in another state of stress. In fact, if the forming limit surface is insidethe yield surface at this new state of stress, the material will tear before yielding. Therefore, labelingthis area as safe based on the plastic strain may be false even though the area remains underproportional plastic strain at all times.

Another factor which undoubtedly contributes to the apparent reluctance of the metal formingindustry to seriously consider a stress-based limit is that it is not practical to measure stresses ona deformed panel, yet strains are easy to measure. This argument was valid in the early applicationsof the FLD method which were primarily focused on analysis of physical panels. But FEM analysespredict the state of stress as well as strain. Since these analyses have become the dominant methodof evaluating and validating die designs, a stress-based forming criterion is no longer impractical.Nevertheless, the conventional FLD method continues to dominate the interpretation of FEManalysis.

It is from the latter school of thought based on the conventional FLD method that the problemof a general forming severity criterion is approached. This investigation leads to the rediscovery ofthe "ndings of Kleemola and Pelkkikangas, and Arrieux, Bedrin, and Boivin. It will be shown thataccounting for the strain path, all curves for data from experiments for steel and aluminum,prestrained in equi-biaxial, plane-strain, and uniaxial tension, map to the same curve in stressspace. This rediscovery introduces several factors which add to the credibility of this concept offorming limit behavior. First of all, the data on which these observations are made are based onpublished experimental works in which this author was not involved. The authors of these latterworks did not consider a stress-based explanation for their observations. Since the drawing of FLDcurves involves some degree of freedom, the fact that these curves are, without question, drawnwithout bias towards a common curve in stress space, makes the conclusion all the more believable.Secondly, the materials studied in this report are di!erent than those used in the earlier works,which suggests that the criterion is a universal characteristic of material behavior. Finally, anindependent discovery is always less disputable than a con"rmation of an earlier observation sincethe latter may be in#uenced by the power of suggestion.

Although the authors of the original discovery are undeniably aware that the stress-basedcriterion is applicable to the "rst forming process, the motivation for and the signi"cance of theirwork was presented as a solution to overcome the limitations of conventional FLD method in theanalysis of multi-stage forming or predeformed material. Given the ease and con"dence ofmeasuring and predicting strains in the "rst draw die, it is little wonder that the strain-basedcriterion continues to be exclusively used to determine formability in these applications. Asdescribed above, and should become more clear when a comparison is made of relation of thestress-based forming limit to the yield surface, is that the conventional FLD curve typicallymisrepresents how close the material is to the forming limit, and the stress-base forming limit is theonly valid determining factor of forming severity. This is true even in the case of the "rst draw diewhere the metal is under near proportional strains, but often not under proportional stress.

One of the earliest experimental demonstrations of the path dependent nature of the forminglimit was reported by Ghosh and Laukonis [5], who investigated strain path e!ects on the forming


limit of a cold-rolled, aluminum-killed steel of 0.89 mm thickness by prestraining the metal to truemajor strains of 3.1, 6.7, and 11.9% in equi-biaxial tension [6]. They measured the forming limitcurves for these prestrained specimens which are shown in Fig. 2a along with the conventionalforming limit curve for the as-received material. They observed not only a shift in the forming limitcurve but changes in its shape which become more signi"cant as the magnitude of the prestrain isincreased. Laukonis continued the investigation by pre-straining three sets of a similar material of0.84 mm to true major strains of 6.8, 9.1, and 14.0% in uniaxial tension [6]. Forming limit curveswere created for secondary forming along an axis parallel to and perpendicular to the major strainaxis of the uniaxial tension prestrain. These results are shown in Fig. 2b. For strains parallel to theprestrain, the forming limit curves shift to higher levels of strain, while strains perpendicular to theprestrain, lower the curves.

Graf and Hosford [7] reported strain-path e!ects for Al 2008 T4 pre-strained in uniaxial,equi-biaxial, and near-plane-strain tension. Forming limit curves were generated for the as-receivedmaterials, as well as the prestrained specimens. Fig. 2c shows the shift in the forming limit curve forbiaxial prestrains of 0.04, 0.07, 0.12, and 0.17 true strain. Fig. 2d shows the upward shift in the curvefor uniaxial prestrains of 0.05, 0.12, and 0.18 true strain parallel to the secondary strain axis, and thedownward shift in the curve for uniaxial prestrain of 0.04, 0.125, and 0.18 true strain perpendicularto the secondary strain axis. Finally, Fig. 2e shows the folding up of the curves near the point ofplane strain for plane-strain prestrains of 0.08 and 0.13 parallel to the secondary strain axis anda shift downward for plane-strain prestrains of 0.08 and 0.14 perpendicular to the secondary strainaxis.

Suppose that the stress state determines formability. This idea is reasonable since it is the state ofstress that determines other material behaviors, such as plastic yielding, buckling, etc. Supposefurther, that there is no stress-path dependency in the forming limit. In other words, suppose thatfor a given material, there is a single curve in stress space which represents the forming limit. If it istrue, then all four curves in Fig. 2a should map to the same curve in stress space, representing theforming limit of the steel used in the Ghosh}Laukonis experiment. All seven curves in Fig. 2b,should degenerate to a single stress curve representing the forming limit of the steel used in theLaukonis experiment. Finally, the ,ve curves in Fig. 2c, the six additional curves in Fig. 2d, andthe four additional curves in Fig. 2e, should also map to a single curve in stress space, representingthe forming limit of the aluminum used in the Graf}Hosford study.

Equivalantly, we should be able to predict the shape of the forming limit curves for all of thesestrain-path experiments from a single stress-based forming limit curve for each of the threematerials used in the studies. The "rst approach has the advantage in which it demonstrates thesimplicity of the stress-based forming limit, while the latter approach has the advantage of utilizingour greater experience with strain-space data due to the widespread use of the conventionalforming limit diagram with which we can better judge the signi"cance of the di!erences between thecurves. Both approaches are taken in this paper to demonstrate that this simple non-path-dependent stress-based criterion can explain the apparently complex path dependencies of thestrain-based limit curves.

Section 1 describes the equations required to translate the strain data from the above experi-ments to stress space taking into account the prestrains involved in the experiments. The speci"csof these equations depend on the shape of the yield surface as well as the stress}strain relation.Therefore, only the generic form of these transformation equations are given in this section with the


Fig. 2. (a). Path-dependent forming limit curves as determined by Ghosh and Laukonis for equi-biaxial prestrain ofa colled-rolled aluminum-killed steel. Curves 1, 2, and 3 are for prestrains of 0.031, 0.067, and 0.119 true strain, respectively.The solid curve is the conventional forming limit curve for this material in the as-received condition; (b) Path-dependentforming limit curves as determined by Laukonis for uniaxial prestrain of a colled-rolled aluminum-killed steel. Curves 1,2 and 3 are for prestrains of 0.068, 0.091, and 0.140 true strain parallel to the strain axis of the secondary forming process,respectively. Curves 4, 5, and 6 are the corresponding curves for prestrains perpendicular to the prestrain axis; (c) Path-dependent forming limit curves as determined by Graf and Hosford for equi-biaxial prestrain of 2008 T4 aluminum. Curves1, 2, 3, and 4 are for prestrains of 0.04, 0.07, 0.12, and 0.17 true strain, respectively: (d) Path-dependent forming limit curvesas determined by Graf and Hosford for uniaxial prestrain of 2008 T4 aluminum. Curves 1, 2, and 3 are for prestrains of 0.05,0.12, and 0.18 true strain parallel to the strain axis of the secondary forming process respectively. Curves 4, 5, and 6 are forprestrains of 0.04, 0.125, and 0.18 true strain perpendicular to the prestrain axis; (e) Path-dependent forming limit curvesas determined by Graf and Hosford for near-plane-strain prestrain of 2008 T4 aluminum. Curves 1 and 2 are forprestrains of 0.08 and 0.13 true strain parallel to the strain axis of the secondary forming process, respectively. Curves3 and 4 are for prestrains of 0.08 and 0.14 true strain perpendicular to the prestrain axis.


Fig. 2. (Continued)


speci"cs described in the appendix for commonly used forms for these two functions. Section 2shows that the all of the data for each of the three materials map to a single curve in stress spacewhich is a characteristic of the material and appears to be independent of the type or amount ofprestrain. This procedure is inverted in Section 3 to predict the location of the forming limit instrain space for each of the prestrain experiments using the stress-space forming limit calculatedfrom the data for the as-received material. These predictions are compared with the experimentalpath-dependent strain-based forming limit curves. Section 4 discusses the features of the stress-based limit function and how it can be used in "nite element modeling, as well as in our stampingplants and tryout facilities. The importance of this new criterion to the analysis of the results of the"rst draw die will also be stressed.

2. Transformation between stress and strain states

Using the notation of Hosford [7, 8] for analysis of plane-stress (p3"0) conditions, the ratio of

the minor true stress, p2, to the major true stress, p

1, is de"ned by the parameter

a"p2p1

. (1)

Similarly, the ratio of the minor true strain rate, eR2, to the major true strain rate, eR

1, is de"ned by the

parameter

o"eR 2eR1

. (2)

Plasticity theory de"nes an e!ective stress, pN , which is a function of the stress tensor componentsand a set of material parameters. For materials with in-plane isotropy or for cases with zero shearstress in a coordinate system aligned with the anisotropy, as is the case in the above experiments,the de"nition of the e!ective stress can be expressed in terms of the principal stresses

pN "pN (p1, p

2). (3)

This relation can also be expressed in terms of p1

and a,

pN "p1m(a), (4)

where m(a) is a function of material parameters. Using the e!ective stress as a potential function,plasticity theory de"nes the plastic strain rates from pN as follows:

eRij"eNR LpN

Lpij

, (5)

where eNR is the e!ective strain rate. This #ow rule leads to a relation between a and o which can beexpressed as

o"o(a) (6)


or

a"a(o) . (7)

The #ow rule also leads to a de"nition of the e!ective strain rate eNR which is a function of the straintensor rate components. As is the case for the e!ective stress, for a material with in-plane isotropyor in the absence of shear strains, the de"nition of the e!ective strain rate can be expressed in termsof the principal strain rates, eR

1and eR

2:

eNR"eNR (eR1, eR

2). (8)

This in turn can be expressed in terms of eR1

and o,

eNR"eR1j(o), (9)

where j (o) is a function of the material parameters. The speci"c form of the functions, m(a), o (a),a(o), eNR (eR

1, eR

2), and j(o) depend on and are derived from the equation used to de"ne the e!ective

stress, pN (p1, p

2). Examples of these functions are given in Appendix A for Hills quadratic theory

[9], in Appendix B for Hills non-quadratic theory [10], and in Appendix C for Hosfordsnon-quadratic theory [8]. All of these cases consider only materials with in-plane isotropy. We willalso consider the e!ects of in-plane anisotropy using Hills general quadratic theory given inAppendix D.

In addition to the above equations, we also require a relation between the e!ective stress and thee!ective strain which characterizes the work-hardening behavior of the material under plasticdeformation. The e!ective strain is de"ned by the time integral of the e!ective strain rate

eN"P dteNR . (10)The relation between the e!ective stress and e!ective strain can be written formally as

pN "pN (eN ) , (11)

and its inverse

eN"eN (p6 ) . (12)

The most commonly used representation of this relation is the power law

pN "KeN n, (11a)

where K and n are material constants. Another commonly used representation, which generally "tsbetter than the power law to data for aluminum alloys, and will be investigated in this study, isa saturation law

pN "p0(1!Ae~Be6 ) , (11b)

where p0, A, and B, material constants. We will assume that the selected relation between the

e!ective stress and the e!ective strain apply to all states of stress even though the parameters of therelation are often determined only in uniaxial tension.


We can de"ne the transformation from the strain states for each of the previously de-scribed experiments to the stress state using the above equations and the assumption that thematerial does not yield plastically during the second loading stage until the e!ective stressrises to the same level of stress attained at the end of the prestrain loading stage. First, we note thatin the absence of shear stresses, as is the case in the above experiments, the principal axes arealigned with the coordinate axis. This is advantageous because in this case the above equationsapply also to the case where the indices 1 and 2 are, respectively, associated with the rolling andtransverse direction of the sheet. This association is important because the de"nition of major andminor stresses (and strain increments) switch between the two loading stages in some of theexperiments.

If the prestrain results in a strain state (e1, e

2)"(e

1*, e

2*), where the index i denotes initial, and

the secondary stage results in a "nal strain state (e1&

, e2&

), then the principal stresses at the end of thesecondary stage are given by

p1"pN (eN (e1* , e2*)#eN (e1&!e1* , e2&!e2* ))

m (a(e2&!e

2*)/(e

1&!e

1*))

(13)

and

p2"aA

e2&!e

2*e1&!e

1*B p1 , (14)

where a (o) is a function given in Eq. (7), eN (e1, e

2) is equivalent to that given in Eq. (8), pN (e6 ) is the

function de"ned by Eq. (11), and m (a) is the function given in Eq. (3). Explicit de"nitions of a (o) andm(a) may be found in the appendices.

The above two relations allow us to map each point on the strain-based forming limit curves intostress space for each of the prestrained conditions, as well as for the as-received material(e1*"e

2*"0). We will use these relations to determine if indeed all of the curves map to a single

curve in stress space. In order to better judge the signi"cance of any di!erences in these curves wecan also invert this process. In other words, given a single curve in stress space which represents theproposed forming limit of the material, can we predict the forming limit in strain space for a givenprestrain? In this case a forming limit is de"ned by the locus of points (e

1&, e

2&) derived from a locus

of points in stress space (p1, p

2) and a prestrain of (e

1*, e

2*) given by

e1&"e

1*#e6 (pN (p1 , p2))!eN (e1* , e2*)

j(o (p2/p

1))

(15)

and

e2&"e

2*#(e

1&!e

1*)oA

p2

p1B , (16)

where o(a) is the function given in Eq. (6), e6 (p6 ) is the function de"ned by Eq. (12), e6 (e1, e

2) is

equivalent to the function given in Eq. (8), and j(o) is the function given in Eq. (9). Explicitde"nitions of o (a), e6 (e

1, e

2), and j(o) are also given in the appendices, depending on the de"nition of

the e!ective stress function, p6 (p1, p

2).


3. A simple stress-based forming limit criterion

The aluminum used in the Graf}Hosford study was reported to "t to a power law over a strainrange from 5 to 20% with K"539 MPa and n"0.285. The anisotropy coe$cients (r

0"0.58,

r45"0.48, r

90"0.78) yield an average value of rN"0.58. Using Eqs. (13) and (14) with a power law

representation of the stress}strain relation, and using Hills quadratic in-plane isotropic plasticpotential and associated equations as de"ned in Appendix A, we can map each of the 15independent curves given in Fig. 2c}e to the stress states shown in Fig. 3. Twelve of the 15 curvesare virtually identical to each other over the entire range of the data. The three curves which appearto be higher are apparently due to the di$culty of measuring the forming limit curve. Thisconclusion is justi"ed because one of the three occurs at an intermediate level of equi-biaxialprestrain. The curve for the highest level of equi-biaxial prestrain falls back to overlap with theother curves. Although the other two occur at the highest levels of prestrain in uniaxial tension,the shift for the non-coaxial prestrain is only half as large as the arguably erroneous shift in theintermediate equi-biaxial prestrain data. Furthermore, there is no evidence of a shift in the forminglimit curves for the two lower levels of prestrain for both the case of coaxial and non-coaxialalignment of the uniaxial prestrain. Since there is no evidence of a trend, the shift observed in thesetwo data are of questionable signi"cance.

The degeneracy of the strain-based forming limit curves to a single curve in stress space is notrestricted to aluminum as can be seen in Fig. 4a for the steel used in the Ghosh}Laukonisequi-biaxial study, and in Fig. 4b for the di!erent steel used in the Laukonis uniaxial study. For

Fig. 3. Transformation of the 15 independent curves given in Figs. 2c}e using a power law and Hills quadratic in-planeisotropic plastic potential. The stress is scaled by K. Twelve of the 15 curves overlap over the entire range of the data. Thedeviation of the other three-appear to the due to experimental di$culty as explained in the text.


Fig. 4. Transformation of the forming limit curves for two steel alloys using a power law and Hills quadratic in-planeisotropic plastic potential, in units of K. (a) Transformation of the as-received and three equi-biaxial prestrainedspecimens used in the Ghosh}Laukonis study shown in Fig. 2a: (b) Transformation of the as-received and six uniaxialprestrained specimens used in the Laukonis study shown in Fig. 2b. Transformation of the raw data from which theforming limit curve for the as-received material as shown in Fig. 1, is also shown in 4b.

comparison Fig. 4b also shows a mapping of the raw measurements taken from Fig. 1 from whichthe strain-based forming limit curve was generated. A comparison of the scatter of this data inrelation to the di!erences in the forming limit curves in stress space is another indication that thelatter curves are equivalent to each other.

Although the stress forming limit curves for the same material under di!erent prestrains arearguably identical, this conclusion may be challenged by the fact that at these levels of stress,relative variations in stress are approximately "ve times smaller than variations in strain for thesematerials. This reduced sensitivity to stress arises from the small exponent n in the stress}strainrelation of approximately 1

5for all three materials. However, the interpretation that the close

proximity of the stress-based curves is due to the saturation of the stress}strain relation is notconsistent with the observation that there are no systematic dependences in the shape of thestress-based curves with either the magnitude or type of prestrain, whereas the systematic trends ofthe strain-based curves with both the magnitude and type of prestrain are evident as seen in Fig. 2.A more conclusive proof that the scatter in the stress-based curves shown in Figs. 3 and 4 is entirelydue to experimental uncertainty is given in the demonstration that all of the strain-based curves canbe derived from a single stress based curve. This proof will be given in the next section of this paper.

4. Cause of the apparent path-dependent strain-based forming limit

Given that all of the curves for a given material appear to map to a single curve in stress-space, itfollows that the observed path-dependent nature of the forming limit is an artifact of our insistence


to interpret the data in terms of strain. In other words, when viewed in stress, there is no evidence ofany path-dependent behavior.

If the above interpretation of the data is true then we should be able to predict the path-dependent nature of the forming limit curves in strain space by mapping points on the stress-basedlimit curve to strain for a given prestrain condition using Eqs. (15) and (16). We will use the forminglimit for the as-received material to de"ne the forming limit in stress space since this curve isgenerally the most accurately determined for a given material. The results of these mappings areshown in Figs. 5}7 for the three sets of prestrained aluminum used in the Graf}Hosford study, andin Figs. 8 and 9 for the steel alloys used in the Ghosh}Laukonis and Laukonis studies, respectively.Generally the predictions match the experimental curves remarkably well. As noted in the previoussection, di!erences are observed at the highest levels of prestrain in uniaxial tension for thealuminum as seen in Figs. 6c and e,but there is no signi"cant di!erence at lower levels of prestrain.Interestingly, the minimum in the curve in Fig. 6c, as well as all of the parallel uniaxial prestrainedsteel specimens shown in Figs. 9a}c do not occur at the intuitively expected minor straincorresponding to the given prestrained state. This suggests that the experimental curves for thesecases may not be well de"ned, a possibility which implies no disrespect for the experimentalists,given the inherent di$culty of judging the location of the onset of necking which is required tode"ne the forming limit curve and that, until now, the shape of the curve and its dependence onprestrain was not constrained by any model or theory.

5. Discussion

In the previous sections it has been shown that a simple stress-based forming limit criterionexplains the apparent path-dependency of the strain-based forming limit. This demonstration used

Fig. 5. Transformation of the stress-based forming limit curve for the aluminum used in the Graf}Hosford studyassuming equi-biaxial prestrains of (a) 0.04; (b) 0.07; (c) 0.12; and (d) 0.17.


Fig. 6. Transformation of the stress-based forming limit curve for the aluminum used in the Graf}Hosford studyassuming uniaxial prestrains of (a) 0.05; (b) 0.12; and (c) 0.18 along the parallel axis, and (d) 0.04; (e) 0.125; and (f) 0.18along the perpendicular axis of the secondary forming stage.

Hills 1948 theory for the plastic potential of a material with in-plane isotropy and a simple powerlaw for the stress}strain relation. In the interest of brevity it is not practical to show thecorresponding results for other plastic potential functions or stress}strain relations. Although theshape of the stress-based forming limit curve depends on these two functions, the conclusions arethe same. For example, Fig. 10 shows the mapping of all 15 sets of data from the Graf}Hosford


Fig. 7. Transformation of stress-based forming limit curve for the aluminum used in Graf}Hosford study assumingnear-plane-strain prestrains of (a) 0.08 and (b) 0.13 along the parallel axis, and (c) 0.8 and (d) 0.14 along the perpendicularaxis of the secondary forming stage.

experiments using Hosfords plastic potential (a"8) and a saturation law for the stress}strainrelation (p

0"667, A"0.773, B"0.28). Since the authors only reported the parameters for

a power law "t to the stress}strain relation, the parameters for the saturation law were calculatedfrom a "t to data over the applicable range of the reported power law data. As can be seen, theshape of the forming limit in Fig. 10 is di!erent from that in Fig. 4. Nevertheless, the curve appearto degenerate into a single curve, even more than in the case of a power law and Hill potential.Similar conclusions are also seen using a Hill potential with a saturation law and a Hosfordpotential with a power law, with a scatter in the curves in these latter cases similar to that seen inFig. 4. Therefore, although the forming limit in stress space is dependent on the plasticity theory,the fact that a stressed-based forming limit exists is independent of the speci"cs of the theory.Fig. 11 shows the dependence of the shape of the forming limit for the steel used in the Laukonisstudy on the shape of the plastic potential using Hills non-quadratic function as well as his generalanisotropic theory. In the latter case the curve represents the forming limit for the case of zero shearstress with p

1representing the stress along the rolling direction of the sheet and p

2along the

transverse direction. Fig. 12a compares the expected forming limit in strain space followinguniaxial tension to 0.14 strain using Hills non-quadratic theory with m"1.6 to 2.4. The forminglimit for the quadratic theory (m"2) is between these two very close curves. The di!erences in thecurves are even smaller for lower levels of prestrain. Fig. 12b compares the expected shape of


Fig. 8. Transformation of the stress-based forming limit curve for the steel used in Ghosh}Laukonis study assumingequi-biaxial prestrains of (a) 0.031; (b) 0.067; and (c) 0.119.

forming limit using Hills in-plane isotropic theory with his general anisotropic theory showing theinsensitivity of the conclusions on the anisotropy in the plane of the sheet.

Irrespective of the dependence of the forming limit curve on the details of the plasticity theory, itis also observed that the stress-based forming limit appears to the functionally less complex thanthe strain-based limit. In particular, the strain limit curve is usually drawn to suggest a signi"cantslope discontinuity in plane strain as seen in Fig. 1. The curves in stress space, shown in Figs. 3 and4, suggest a much less complex dependence. Super-position of the raw data in Fig. 4b, gives littleindication of any discontinuity at plane strain, suggesting that the discontinuities typically drawnin strain space may be due to the di$culty of drawing a curve through the nearly linear behavior inthe negative minor strain domain and the non-linear data going in apparently the oppositedirection on the positive side. But there is no such di$culty in drawing a curve through the rawdata shown in Fig. 4b which may be approximated by a linear relation between p

1and p

2.

Furthermore, the aluminum and for the case of steel using Hills non-quadratic theory withm"2.4, the forming limit appears to be simply a limit on the maximum principal stress. Theseobservations are not meant to imply a speci"c functional form for the stress-based forming limit,nor to constrain the selection of the plastic potential or stress}strain relation, in order to simplifythe forming limit curve. Obviously, these functions must be selected on the basis of how well theycharacterize the plastic behavior of the material. Nevertheless, for the plastic potentials and


Fig. 9. Transformation of the stress-based forming limit curve for the steel used in the Laukonis study assuming uniaxialprestrains of (a) 0.068; (b) 0.091; and (c) 0.14 along the parallel axis, and (d) 0.068; (e) 0.091; and (f ) 0.14 along theperpendicular axis of the secondary forming stage.

stress}strain relations commonly used today, the nature of the stress-based forming limit is far lesscomplex than the strain-based limit.

Although the stress-based forming limit as described above can be used in analysis withoutfurther consideration, the apparent linearity of the curve is reminiscent of the elasto-plasticityrelations, suggesting an even more simple criterion. For example, the relations for the principal


Fig. 10. Transformation of the 15 independent curves given in Figs. 2c}e using a saturation law and Hosfordsnon-quadratic in-plane isotropic plastic potential, in units of K.

Fig. 11. Transformation of the strain-based forming limit curve shown in Fig. 1 for the as-received steel alloy used in theLaukonis study to stress space using a power law and Hills non-quadratic plastic potential for m"1.6, 2, and 2.4, andHills general anisotropic potential.

major elastic strain E(%)1

and plastic strain rate EQ (1)1

are given by

E(%)1"1

E(p

1!lp

2) (17)

and

EQ (1)1

"eNRpN Ap1!

r1#rp2B , (18)


Fig. 12. Transformation of the stress-based forming limit curves shown in Fig. 11 to strain for a uniaxial prestrain of0.119. (a) Comparison of Hills non-quadratic potential using m"1.6 to m"2.4; (b) comparison of Hills in-planeisotropic theory to Hills general anisotropic theory.

where E is Youngs modulus, l is Poissons ratio, and p1

and p2

are the major and minor stresses,respectively, eNR is the e!ective strain rate, pN is the e!ective stress, and r is the normal anisotropycoe$cient. The apparent linearity of the forming limit in stress space suggests the existence ofa forming limit &&strain E(&) which uniquely characterizes the material forming limit.

E(&)"1pc

(p1!kp

2), (19)

where pcis the true stress in uniaxial tension at the onset of necking and k is another material

constant whose determination depends on the stress}strain relation and yield function. Forexample, k"0.18 using a power law for the stress}strain relation and Hills 1948 theory fora material with in-plane isotropy with the Laukonis data. k"0.0 for Hills non-quadratic theorywith m"2.4.

The shape of the forming limit curve is distinct from the shape of the yield surface as illustrated inFig. 13 for the aluminum data using Hills quadratic yield function. This "gure shows the positionof the yield surface following equi-biaxial prestrain of 0.04, 0.07, 0.12, and 0.17. As can be seen inthis "gure, while the forming limit in the equi-biaxial state is well above the state of stress followingeach of these prestrains, this is not the case in other modes of deformation, particularly nearplane-strain for the highest levels of prestrain. In fact this "gure suggests that the material will splitbefore yielding when loaded under plane strain conditions following an equi-biaxial prestrain of


Fig. 13. Comparison of the stress limit curve with the expected yield surface following biaxial prestrains of 0.04, 0.07,0.12, and 0.17 for the Graf}Hosford aluminum described by a power law stress}strain relation and Hills quadratic yieldfunction. Principal stresses are in units of K.

about 0.12 or higher. This dramatic e!ect is mitigated by the fact that the aluminum yields ata lower stress than expected following equi-biaxial as reported by Graf and Hosford and shown inFig. 14. Furthermore, the precise relationship between the yield surface and forming limit dependson the equations used in the plasticity theory. In any case, although non-zero and subject to theassumptions of the theory, the ductility remaining in the plane-strain mode is far less than thatremaining in other states of stress. As discussed in the introduction, whenever the stress dropsbelow the yield surface, the distinct shapes of the yield surface and forming limit curves can result incritical forming conditions even though the plastic strains are well below the forming limit and thestrain ratios appear to be constant.

The possibility that there exists a single path-independent forming limit greatly simpli"es theproblem of assessing forming severity. The state of stress is predicted by "nite element methodsimulations of the sheet metal forming process at the same time that these simulations predict thestate of strain. Until now we have been using the state of strain, in comparison to the forming limitto judge formability, and either ignore the e!ect of strain path or shift it by some empirical rule. Itshould now be clear that we should ignore the state of strain in judging formability, and compareonly the state of stress to the new forming limit as de"ned in stress space.

Although the application of this new criterion is simple in the case of numerical simulation of thesheet metal forming process, its use in our stamping plants and tryout facilities may appear to beimpractical for two reasons. The most important is that it is not feasible to directly measure thestate of stress, whereas the state of strain is (relatively) easy to measure. The second challenge is toconvince the plant and tryout engineers that they should no longer consider the state of strain as an


Fig. 14. Stress}strain relation for uniaxial forming following equi-biaxial prestrain in comparison to the as-receivedmaterial for the Graf}Hosford material. The dashed line indicates the locus of yield points for the prestrained materialwhich is far below the expected yield for the as-received material.

indicator for forming severity. The di$culty of the latter challenge is that it has taken so long toconvince everyone that we should make the necessary investments to monitor the strain.

Such arguments against adoption of this new criterion are weak given the fact that the forminglimit is obviously not directly dependent of the state of strain. Nevertheless, it is necessary tomeasure the strain, and more particularly, the strain increments, to determine the state of stress. Bycomparing the state of strain at the end of the forming process with its state just before or with thatpredicted by numerical simulation, we can determine the "nal strain rate. This de"nes the value ofo which we can use to the de"ne the ratio between the principal stresses from the de"nition of a (o).We can also estimate the e!ective strain eN by integration of the strain incements from either panelbreakdowns or from numerical simulation. Then we can calculate the e!ective stress, p6 "p6 (e6 ), fromthe parameterization of the stress}strain relation, and solve for the principal stress componentsusing Eqs. (1) and (4). Although in principal this method requires a more detailed analysis of panelbreakdowns, and the reasonably accurate assessment of strain increments, which in turn requiresmore accurate strain measurements, the need for analysis of panel breakdowns may be reduced oreliminated by the simultaneous use of numerical simulation of the process, from which the path ofthe forming process might be derived.

Unfortunately, the derivation of the state of stress as described above, does not work if the stressis below the yield surface at the end of the forming process. In this case, a valid interpretation offormability can only be made through FEM analysis.

The determination of the state of stress has another bene"t. Given the stress-based forming limit,the engineer has direct knowledge not only of the degree of the forming severity, but alsoa procedure for determining what needs to be done to "x it. For example, if it is found that thestress in the die wall is 50 MPa over the forming limit in a direction normal to the punch openingline, then using back-of-the-envelop calculations for the e!ects of the die pro"le radius and other


pertinent geometry, the engineer can determine how much to modify the draw bead #ow stress toreduce the stress in the die wall by the desired amount. This type of analysis can greatly acceleratethe trial and error procedure currently used in numerical simulations, as well as during physicaltryout.

An alternative for the application of this improved criterion to physical applications is touse numerical simulation or die tryout panel breakdowns to determine both the critical areasof the panel and the strain-path in those critical areas. A strain-based forming limit can then becalculated for each critical area which is then passed on to production and used to assess andmonitor forming severity. In this application, only the "nal strains need be measured in theproduction plant.

6. Conclusions

In Sections 2 and 3 of this paper the evidence shows that all of the apparent path-dependente!ects on the forming limit vanish when properly viewed in stress coordinates. Most of these curvesmap to a narrow band in stress space whose width is comparable to the scatter observed in the rawdata used to generate the conventional forming limit curves. The relatively few curves which falloutside this band are not consistent with any systematic trend in the data. This suggests that theirdeviation probably arises from the inherent experimental di$culty of de"ning forming limit curvesdue to the subjectivity involved in judging the onset of a localized neck.

Although the shape of the forming limit in stress space is dependent on the speci"c stress}strainrelation and plastic potential function, the previous section shows that the degeneracy of thepath-dependent strain-based curves to a single curve in stress space is not. Furthermore, thestress-based forming limit appears to be functionally less complex than the strain-based limit.

Application of the stress-based forming limit extends the validity of our forming limit criterion toapplications involving non-proportional loading. This not only eliminates a critical obstacle to ourability to assess formability, but by forcing us to look at stress distributions, which can be moredirectly in#uenced by control variables such as draw bead #ow stresses and binder pressure, we canmodify the forming process to lower the stresses below the critical levels and determine the desiredprocessing conditions more quickly. This technique will bene"t validation of die designs by bothcomputer simulation and physical die tryout, although the latter will require more carefulmeasurements. Having identi"ed the critical areas on the panel either by simulation or tryout,area-speci"c strain-based forming limit curves may be de"ned and passed along to the productionplant for monitoring the forming severity.

Although the discovery of a stress-based forming limit was "rst reported by Kleemola andPelkkikangas, and Arrieux, Bedrin, and Boivin, its ulitity was promoted as solution to the analysisof multi-stage forming processes and has not been widely communicated. This independentdiscovery based on the experimental FLD work of others adds considerable weight to its validity.Furthermore, this work stresses the importance of the stress-based forming limit criterion in theaccurate determination of forming severity in the analysis of the "rst draw die. Given that stress,not strain, determines formability, and in FEM analysis stresses are predicted with as muchcertainty as strains, there is no excuse for continued use of the strain-based criterion in any sheetmetal forming process.


Acknowledgements

This work would not have been possible without the considerable work done and reported onstrain-path e!ects in the literature. Therefore, I owe a debt and wish to thank Dr. Joseph Laukonis,Prof. Amit Ghosh, Dr. Alejandro Graf, and Prof. William Hosford for their excellent work onwhich this study is based. Although at times my comments are critical of some of the curves, anyonefamiliar with the work involved in experimental forming limit diagram studies, will agree that thehigh degree of correlation between the predictions of this simple model and the experimentalresults, is as much a testament to the quality of the experiment as it is to the validity of the model. Infact it is more so in this case, given that the experiments were done without the aid of a theoreticalfoundation with which anomalies could be identi"ed.

I also thank Dr. Michael Wenner, Dr. G. Paul Montgomery, and Dr. Jerry Chen of theManufacturing and Design Systems Department, of the General Motors Research and Develop-ment Laboratories for their many suggestions during the development of this concept, and Mr.Lorenzo Smith of General Motorss Metal Fabrication Division for his help in evaluating andpromoting its utilization within GM, as well as for bringing to my attention the work of Kleemolaand Pelkkikangas, and Arrieux, Bedrin, and Boivin.

Finally, I note the use of the mathematica program to translate strains on the experimental FLDcurves into stresses, and then back into strains, as well as to draw all of the curves.

Appendix A. Hill:s quadratic normal anisotropic plastic potential

Hills quadratic normal anisotropic plastic potential is a special case of his general anisotropicpotential given in Appendix D. In this limit the potential is a function of the principal stresses andthe normal anisotropy coe$cient, r. The plastic potential or e!ective stress function given by

pN "Sp21#p22!2r

1#r p1p2 . (A.1)

The e!ective strain rate function is

eNR" 1#rJ1#2rSeR 21#eR 22#

2r1#r eR 1eR 2 . (A.2)

The ratio between the e!ective stress and major stress is

m"S1#a2!2r

1#r a . (A.3)

The ratio between the e!ective strain and major strain is

j" 1#rJ1#2rS1#o2#

2r1#r o . (A.4)


And the relationship between o and a is

o"(1#r)a!r1#r!ra (A.5)

and its inverse

a"(1#r)o#r1#r#ro . (A.6)

Appendix B. Hill:s non-quadratic normal anisotropy plastic potential

Hills non-quadratic normal anisotropic plastic potential is a generalization of the normalanisotropic potential given in Appendix A. In this case the potential is a function of the normalanisotropy coe$cient, r, and an additional material constant m. It reduces to the quadratic case form"2. The plastic potential or e!ective stress function given by

pN "A1

2(1#r) ( Dp1#p2 Dm#(1#2r) Dp1!p2 Dm)B1@m

. (B.1)

The e!ective strain rate function is

eNR"[2(1#r)]1@m2 A DeR 1#eR 2 Dm@(m~1)#A

DeR1!eR

2Dm

(1#2r) B1@(m~1)

B(m~1)@m

. (B.2)

The ratio between e!ective stress and major stress is

m"A1

2(1#r) ((1#a)m#(1#2r)(1!a)m)B1@m

. (B.3)


j"[2(1#r)]1@m2 A(1#o)m@(m~1)#A

(1!o)m(1#2r)B

1@(m~1)

B(m~1)@m

. (B.4)


o"(1#a)m~1!(1#2r) (1!a)m~1(1#a)m~1#(1#2r) (1!a)m~1 , (B.5)

and its inverse

a"[(1#2r)(1#o)]1@(m~1)![1!o]1@(m~1)[(1#2r)(1#o)]1@(m~1)#[1!o]1@(m~1) . (B.6)


Appendix C. Hosford:s non-quadratic normal anisotropic plastic potential

Hosfords non-quadratic normal anisotropic plastic potential is a special case of a class ofnon-quadratic yield functions proposed by Hill. In this case the potential is a function of thenormal anisotropy coe$cient, r, and an additional material constant a. It reduces to Hillsquadratic potential for a"2. The plastic potential or e!ective stress function given by

p6 "A1

(1#r) ( Dp1Da#Dp2 Da vert p1!p2 Da)B1@a

. (C.1)

The e!ective strain rate cannot be expressed as a simple function of the strain tensor components,as it can in the case of the other plastic potentials. Instead we must use the de"nition of the plasticwork rate which yields the following equation:

eNR"1pN

(p1eR1#p

2eR2)"eR 1

m(1#ao). (C.2)

The ratio between the e!ective stress and major stress is

m"A1

(1#r) (1#DaDa#r D1!aDa)B1@a

. (C.3)


j"1m

(1#ao). (C.4)


o"aa~1!r (1!a)a~11#r(1!a)a~1 . (C.5)

The inverse relation, a"a(o) cannot be given explicitly but must be numerically solved for eachvalue of o using the equation o"o(a). There are seven solutions to this equation for a"8, as is thecase for the aluminum used in this study. However, only one of the seven solutions is real.

Appendix D. Hill:s quadratic generally anisotropic plastic potential

Hill originally proposed a fully anisotropic plastic potential which is a quadratic function of thestress tensor components expressed in a coordinate system aligned with the axes of the asymmetry.In this generalization the potential is a function of the normal anisotropy coe$cients measuredalong the rolling, transverse and diagonal directions of the sheet (r

0, r

90, r

45, respectively). The

plastic potential or coe$cient stress function as originally de"ned by Hill is given by

pN "JF (pyy!p

zz)2#G(p

zz!p

xx)2H(p

xx!p

yy)2#2p2

yz#2Mp2

xx#2Np2

xy, (D.0)


where F, G, H, N, , and M are material constants. In the case of in-plane stress this function can beexpressed in terms of the normal anisotropy coe$cients

pN "SHA1#r

0r0

p2xx#1#r90

r90

p2yy!2p

xxpyy#r0#r90

r0r90

(1#2r45

)p2xyB , (D.1)

where H is an arbitrary material constant. H is sometimes equated to r0/(1#r

0) which scales the

e!ective stress to be equal to the true stress is uniaxial tension along the rolling direction. Using thestress invariants, p

xx#p

yy"p

1#p

2and p

xxpyy!p2

xy"p

1p2, it can be shown that this potential

reduces to the case given in Appendix A for r0"r

90"r

45"r. The e!ective strain rate function is

eNR"S1H A

r0r90

1#r0#r

90A1#r

90r90

#eR 2xx#1#r0

r0

eR 2yy#2eR

xxeRyyB#

4r0r90

(r0#r

90) (1#2r

45)eR 2xyB . (D.2)

The ratio between the e!ective stress and major stress is complex when the shear stress, pxy

isnon-zero. However, in the experiments discussed in this paper the shear stress is always zero. In thiscase it is most convenient to replace the major and minor principal stresses and strains with thestress and strain components along the x- and y-axis, respectively. For example, we rede"ne a to bethe ratio of p

yyto p

xxand m to be the ratio of eNR to eR

xx. With these de"nitions

m"SHA1#r

0r0

#1#r90r90

a2!2aB . (D.3)The ratio between the e!ective strain rate and eR

xxis

j"Sr0r90

H(1#r0#r

90) A

1#r90

r90

#1#r0r0

o2#2oB . (D.4)The relation between the o and a is

o"a(1#(1/r90))!11#(1/r

0)!a , (D.5)

and its inverse

a"o (1#(1/r0))#11#(1/r

90)#o . (D.6)

References

[1] Keeler SP, Backhofen WA. Plastic instability and fracture in sheet stretched over rigid punches. ASM TransactionsQuarterly 1964;56:25}48.

[2] Goodwin GM. Application of strain analysis to sheet metal forming in the press shop. SAE paper, No. 680093,1968.


[3] Kleemola HJ, Pelkkikangas MT. E!ect of predeformation and strain path on the forming limits of steel copper andbrass. Sheet Metal Industries 1977;63:591}599.

[4] Arrieux R, Bedrin C, Boivin M. Determination of an intrinsic forming limit stress diagram for isotropic metalsheets. Proceedings of the 12th Biennial Congress IDDRG, 1982:61}71.

[5] Ghosh AK, Laukonis JV. The in#uence of strain-path changes on the formability of sheet steel. 9th BiennialCongress of the International Deep Drawing Research Group, Sheet Metal Forming and Energy Conservation,ASM Publication, 1976.

[6] Laukonis JV. private communication.[7] Graf AF, Hosford WF. Calculations of forming limit diagram for changing strain paths. Metallurgical Transactions

A 1993;24:2497}2501.[8] Logan R, Hosford WF. International Journal of Mechanical Sciences 1980;22:419.[9] Hill R. Proceedings of the Royal Society 1948;193A:281}97.

[10] Hill R. Mathematical Proceedings of the Cambridge Philosophical Society 1979;85:179.


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