-
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
4 T.B. Stoughton / International Journal of Mechanical Sciences
42 (2000) 1}27
-
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
T.B. Stoughton / International Journal of Mechanical Sciences 42
(2000) 1}27 5
-
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.
6 T.B. Stoughton / International Journal of Mechanical Sciences
42 (2000) 1}27
-
Fig. 2. (Continued)
T.B. Stoughton / International Journal of Mechanical Sciences 42
(2000) 1}27 7
-
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)
8 T.B. Stoughton / International Journal of Mechanical Sciences
42 (2000) 1}27
-
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.
T.B. Stoughton / International Journal of Mechanical Sciences 42
(2000) 1}27 9
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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).
10 T.B. Stoughton / International Journal of Mechanical Sciences
42 (2000) 1}27
-
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.
T.B. Stoughton / International Journal of Mechanical Sciences 42
(2000) 1}27 11
-
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
12 T.B. Stoughton / International Journal of Mechanical Sciences
42 (2000) 1}27
-
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.
T.B. Stoughton / International Journal of Mechanical Sciences 42
(2000) 1}27 13
-
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
14 T.B. Stoughton / International Journal of Mechanical Sciences
42 (2000) 1}27
-
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
T.B. Stoughton / International Journal of Mechanical Sciences 42
(2000) 1}27 15
-
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
16 T.B. Stoughton / International Journal of Mechanical Sciences
42 (2000) 1}27
-
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
T.B. Stoughton / International Journal of Mechanical Sciences 42
(2000) 1}27 17
-
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)
18 T.B. Stoughton / International Journal of Mechanical Sciences
42 (2000) 1}27
-
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
T.B. Stoughton / International Journal of Mechanical Sciences 42
(2000) 1}27 19
-
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
20 T.B. Stoughton / International Journal of Mechanical Sciences
42 (2000) 1}27
-
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
T.B. Stoughton / International Journal of Mechanical Sciences 42
(2000) 1}27 21
-
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.
22 T.B. Stoughton / International Journal of Mechanical Sciences
42 (2000) 1}27
-
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)
T.B. Stoughton / International Journal of Mechanical Sciences 42
(2000) 1}27 23
-
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$cient, r, and an additional material constant m. It reduces to
the quadratic case form"2. The plastic potential or e!ective stress
function given by
pN "A1
2(1#r) ( Dp1#p2 Dm#(1#2r) Dp1!p2 Dm)B1@m
. (B.1)
The e!ective strain rate function is
eNR"[2(1#r)]1@m2 A DeR 1#eR 2 Dm@(m~1)#A
DeR1!eR
2Dm
(1#2r) B1@(m~1)
B(m~1)@m
. (B.2)
The ratio between e!ective stress and major stress is
m"A1
2(1#r) ((1#a)m#(1#2r)(1!a)m)B1@m
. (B.3)
The ratio between the e!ective strain and major strain is
j"[2(1#r)]1@m2 A(1#o)m@(m~1)#A
(1!o)m(1#2r)B
1@(m~1)
B(m~1)@m
. (B.4)
And the relationship between o and a is
o"(1#a)m~1!(1#2r) (1!a)m~1(1#a)m~1#(1#2r) (1!a)m~1 , (B.5)
and its inverse
a"[(1#2r)(1#o)]1@(m~1)![1!o]1@(m~1)[(1#2r)(1#o)]1@(m~1)#[1!o]1@(m~1)
. (B.6)
24 T.B. Stoughton / International Journal of Mechanical Sciences
42 (2000) 1}27
-
Appendix C. Hosford:s non-quadratic normal anisotropic plastic
potential
Hosfords non-quadratic normal anisotropic plastic potential is a
special case of a class ofnon-quadratic yield functions proposed by
Hill. In this case the potential is a function of thenormal
anisotropy coe$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)
The ratio between the e!ective strain and major strain is
j"1m
(1#ao). (C.4)
And the relationship between o and a is
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)
T.B. Stoughton / International Journal of Mechanical Sciences 42
(2000) 1}27 25
-
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
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[2] Goodwin GM. Application of strain analysis to sheet metal
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26 T.B. Stoughton / International Journal of Mechanical Sciences
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[3] Kleemola HJ, Pelkkikangas MT. E!ect of predeformation and
strain path on the forming limits of steel copper andbrass. Sheet
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