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C. Hari Manoj SimhaDepartment of Mechanical Engineering,
University of Waterloo,Waterloo, ON, Canada, N2L 3G1
Javad GholipourInstitute for Aerospace Research,
National Research Council,Aerospace Manufacturing Technology
Center,
5145 Decelles Avenue,Campus of the University of Montreal,
Montreal, PQ, Canada H3T 2B2
Alexander Bardelcik
Michael J. Worswick1e-mail: [email protected]
Department of Mechanical Engineering,University of Waterloo,
Waterloo, ON, Canada, N2L 3G1
Prediction of Necking in TubularHydroforming Using an
ExtendedStress-Based Forming LimitCurveThis paper presents an
extended stress-based forming limit curve (XSFLC) that can beused
to predict the onset of necking in sheet metal loaded under
non-proportional loadpaths, as well as under three-dimensional
stress states. The conventional strain-basedFLC is transformed into
the stress-based FLC advanced by Stoughton (1999, Int. J.Mech.
Sci., 42, pp. 127). This, in turn, is converted into the XSFLC,
which is charac-terized by the two invariants, mean stress and
equivalent stress. Assuming that the stressstates at the onset of
necking under plane stress loading are equivalent to those
underthree-dimensional loading, the XSFLC is used in conjunction
with finite element compu-tations to predict the onset of necking
during tubular hydroforming. Hydroforming ofstraight and pre-bent
tubes of EN-AW 5018 aluminum alloy and DP 600 steel are
con-sidered. Experiments carried out with these geometries and
alloys are described andmodeled using finite element computations.
These computations, in conjunction with theXSFLC, allow
quantitative predictions of necking pressures; and these
predictions arefound to agree to within 10% of the experimentally
obtained necking pressures. Thecomputations also provide a
prediction of final failure location with remarkable accu-racy. In
some cases, the predictions using the XSFLC show some discrepancies
whencompared with the experimental results, and this paper
addresses potential causes forthese discrepancies. Potential
improvements to the framework of the XSFLC are alsodiscussed. DOI:
10.1115/1.2400269
Introduction
The strain-based forming limit curve FLC, introduced byeeler and
Backofen 1 and Goodwin 2, is widely used toredict the onset of
necking in sheet metal forming. These curvesre determined through
plane stress experiments that subject sheetamples to load paths
that are nominally proportional in principaltrain space and strain
paths that range from uniaxial to biaxialtress. The strain state at
the onset of necking in the sample,haracterized by the in-plane
true minor and major principaltrains 2 ,1 for a given load path, is
recorded and a formingimit curve in strain space is determined. For
linear load paths,his curve represents the limit of formability for
the as-receivedheet.
However, for load paths that are non-proportional in strainpace,
the formability curves shift with respect to the FLC of
thes-received sheet and may change shape. A non-proportional
loadath is one in which the principal loading directions vary.
Forxample, Ghosh and Laukonis 3 prestrained
cold-rolledluminum-killed steel to several levels of prestrain
under equi-bi-xial tension. They measured the forming limit of the
prestrainedamples and found that the forming limit curve shifted
downwardith increasing prestrain and also changed shape. Shifting
and a
hange in shape of the formability limit curve was also
observedor samples that were subjected to uniaxial prestrain. Graf
andosford 4 measured the FLCs of Al 2008-T4 samples subjected
o prestrain under uniaxial, biaxial, and plane strain loading.
Theyound the FLC of the prestrained sheet to be different from
thatf the as-received FLC. It is pertinent to mention that the
pre-
1Corresponding author.Contributed by the Materials Division of
ASME for publication in the JOURNAL OF
NGINEERING MATERIALS AND TECHNOLOGY. Manuscript received
September 17, 2005;
nal manuscript received August 9, 2006. Review conducted by Somnath
Ghosh.
6 / Vol. 129, JANUARY 2007 Copyright 2
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strain imposed in both these reports were tensile. The reason
forthis comment will be clear when the hydroforming of bent tubes
isconsidered in a later section.
In light of the above, the use of the FLC is restricted to
casesin which the sheet necks due to load paths that are linear in
strainspace. If the load path deviates from linearity, necking
along thenew path cannot be predicted without a knowledge of the
FLCdue to the prestrain imposed during the linear portion of the
loadpath. An alternative approach to describe the formability limit
wasadvanced by Stoughton 5 who developed a stress-based
form-ability limit curve-FLC. By assuming a stress-strain
responseand an appropriate yield function, he transformed the FLC
intotrue principal stress 2 ,1 space. The resulting limit curve
instress space is somewhat sensitive to the assumed
stress-straincurve and the assumed yield function. However, the
most note-worthy demonstration in Stoughtons paper 5 is that when
theFLCs of the prestrained sheet are transformed into stress
spaceafter accounting for the prestrains, the resulting forming
limitcurve is nearly coincident with the stress-based limit curve
for theas-received sheet. In other words, to within the limits of
experi-mental uncertainty and within the framework of the
constitutiveassumptions, there exists a single curve in principal
stress spacethat represents the formability limit of the sheet. It
has been ar-gued that the various FLCs transformed into nearly
coincidentalcurves in stress space because of the insensitivity of
the stress-strain relation at large strains. Stoughton 6 addressed
this byshowing that when the FLC was shifted by an order of
magni-tude, the FLC shifted by 5 MPa. In passing, it should be
notedthat a similar procedure to transform FLCs to stress space
wasproposed by Embury and LeRoy 7. However, they did not
dem-onstrate that the stress-based limit curve for the as-received
andthe prestrained sheet were nearly the same.
Since the FLC is measured through plane stress loading, the
FLC represents the formability limit for plane stress load paths.
007 by ASME Transactions of the ASME
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he FLC appears to be particularly attractive to predict neckingn
sheet metal with or without prestrains and non-proportionaload
paths under plane stress conditions. This paper considers
thepplication of stress-based forming limit approaches to the case
ofeam-welded tubes, which are fabricated from sheet, and wherehe
fabrication process stores plastic strains in the part. In
thetress-based framework, the FLC of the tube is obtained
byransforming FLC of the as-received sheet with the
stress-strainurve into principal stress space.
Necking and/or failure in the hydroforming of tubes can occurt
locations where in addition to the in-plane stresses a
through-hickness compressive component acts. For a review of
hydro-orming technology see Ko and Altan 8. During hydroforming,hen
the tube is initially expanding due to the internal pressure,
he stress state in the thin-walled tube is indeed
approximatelylane stress. However, after the tube wall contacts the
die, thelane stress approximation is no longer valid. For example,
whenstraight tube of circular cross section is being formed into
a
quare cross section, when the expanding tube comes into
contactith the die, a through-thickness compressive stress due to
the
nternal pressure acts in addition to the in-plane stresses.
Neckingnder these conditions, usually originates at the location
where theube wall becomes a tangent to the die wall.
The through-thickness compressive component of stress re-uires
an additional consideration, which is a potential increase inhe
formability and its influence on the FLC and the FLC. Go-oh et al.
9 have presented an analytical expression that predictsn increase
in the plane strain forming limit in strain space due tohe presence
of through-thickness compressive stresses. Smitht al. 10, arguing
that the model of Gotoh and co-workers cannote used for rate
sensitive materials, developed an alternative ex-ression that
predicts an increase in formability due to a compres-ive 3.
However, they assume that the formability curve in stresspace is
not affected by a compressive 3, where 3 acts in
thehrough-thickness direction. This assumption is adopted in
theurrent work.
The foregoing are continuum approaches to predicting the onsetf
necking. That is, when the variables such as stress and straineach
critical levels, a neck is predicted. Alternatively,
damage-echanics based approaches have also been applied to
metal
orming 11. For hydroforming, in particular, the article by
Cher-uat et al. 12, uses a fully coupled damage mechanics
model,hich is used in finite element computations, to predict
failureuring hydroforming. In this approach, damage is treated as
acalar variable, and used to model failure. The dissertation
byaradari 13, from our group, uses the Gurson-Tvergaard-eedleman
14,15 constitutive model to model failure of alumi-um alloy tube
during hydroforming. In this work, it is assumedhat void growth,
nucleation and coalescence are responsible forailure. In this
approach, the model predictions of the void volumeraction at
failure compare favorably to metallographic observa-ions. Though
the results of these efforts using damage mechanicsased approaches
are promising, they involve either fairly com-lex modeling schemes
or an involved calibration effort to obtainaterial damage
parameters.
Table 1 Experimental details hydroformingOutside corner fill.
ICF inside corner fill. Dtube length.
Experiment Material Geo
Straight tube SCF EN-AW 5018 L 4Pre-bent tube OCF EN-AW 5018
R/Pre-bent tube ICF EN-AW 5018 R/Pre-bent tube SCF DP600 RPre-bent
tube OCF DP600 R/
The main objective of the current work is to develop a con-
ournal of Engineering Materials and Technology
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tinuum mechanics based approach to predict necking under
three-dimensional stress states and apply the method to predict
neckingduring tubular hydroforming. To this end, an extended
stress-based FLC XSFLC is proposed. With suitable
assumptions,Stoughtons FLC is transformed into the XSFLC; the
XSFLCcan predict necking under a three-dimensional state of stress.
TheXSFLC is applied to two alloys: EN-AW 5018 aluminum alloyand
dual phase DP600 steel. Finite element simulations using
theexplicit dynamic finite element code LS-DYNA, in conjunctionwith
the XSFLC, are used to predict necking during hydroformingof
straight and pre-bent tubes. For the case of pre-bent tube
hy-droforming, additional assumptions required to use the XSFLCare
presented. The computations are used to estimate the
internalhydroforming pressure at the onset of necking and these
valuesare compared with the experimental results. The paper
concludeswith an appraisal of the stress-based method.
2 Hydroforming ExperimentsThis section presents a brief
description of the hydroforming of
straight and pre-bent tubes and discusses the conditions
underwhich a neck initiates. Two materials are considered in this
work.The first is EN-AW 5018, an Al-Mg-Mn alloy, and the second is
adual-phase steel, DP600. Table 1 summarizes details of the
hydro-forming experiments using these materials, including the tube
ma-terial, geometry, and hydroforming lubricant. The table also
pre-sents values for the tube-die friction coefficients that
weredetermined from twist-compression tests. At least, two
hydro-forming trials were performed for each case. Figure 1 shows
sec-tional views of the tubes in the hydroforming dies, and the
diecross sections. The EN-AW 5018 aluminum tubes had a
wallthickness of 2 mm, whereas the DP600 wall thickness was1.85 mm.
All tubes had an initial outside diameter of 76.2 mm.The EN-AW 5018
tubes were annealed after tubing, and theDP600 tubes were not.
CF Square cross-section, corner fill. OCFe diameter. R
center-line bend radius. L
tryHydroforming
lubricantTube-die friction
coefficient
mm AL070 0.05.5 Hydrodraw 711 0.15.5 Hydrodraw 711 0.152
Hydrodraw 625 0.03.5 Hydrodraw 625 0.03
Fig. 1 Cross section of dies used in the hydroforming
. Stub
me
53D 2D 2/DD 2
experiments
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In the straight tube corner fill experiments, EN-AW 5018
alu-inum alloy tubes of 453 mm length were constrained in a die
of
quare cross section with rounded corners radius 3.17 mm. Aolid
film lubricant, ALO70, was applied to the tubes. Pressurizeduid was
admitted into the tube through the end plugs. The endlugs were
loaded with sufficient force to provide a seal for theressurized
fluid, and there was no end feeding applied to the tuben these
experiments. The fluid pressure and the forces on the endeals were
recorded, and the internal pressure was increased untilhe tubes
burst. Subsequently, experiments were interrupted torevent bursting
and tubes with an incipient neck were recovered.ore details about
the experiments can be found in the article byholipour et al. 16.
The values of the corner fill expansion and
he internal pressure at the onset of the neck are presented in
Table.
EN-AW 5018 tubes were pre-bent and then hydroformed in thenside
corner fill and outside corner fill dies Fig. 1. To carry
outydroforming of bent tubes, the tubes were bent in an
instru-ented rotary draw bender. Mandrels were used to prevent
oval-
zation, and bend torque and forces on the tooling were
monitoreduring the bending process. After bending, the tubes
wererimmed to size for hydroforming. Bend geometries of theN-AW
5018 tubes are presented in Table 1. As in the case oftraight tube
hydroforming, the process variables, which includednternal
pressure, end plug forces, and expansion of the tube were
onitored. The burst pressure and the pressure to cause
incipientecks were determined for bent tube hydroforming. Necking
pres-ures and the final cross-section geometry at necking at 45
degn the bend angle are presented in Tables 1 and 2,
respectively.
ore details of the bending and hydroforming can be found in
therticles by Dwyer et al. 17 and Oliveira et al. 18.
DP600 tubes were pre-bent and hydroformed in the die with aquare
cross section Fig. 1. The methods employed were similaro the
bending of EN-AW 5018 tubes. Tables 1 and 2 presentetails of the
DP600 experiments, and more details about similarxperiments can be
found in article 19.
Failure of the tubes during hydroforming showed a marked
fea-ure. Though the failure location depended on the tube
geometrynd material, failure was found to occur at locations where
thexpanding tube came into contact with the die. Figure 2
illustrateshe conditions at which a neck originates in the tube. As
the tube
Table 2 Comparison of experimental and comgeometry
NeckingMP
Experiment Material Experiment
Straight tube SCF EN-AW 5018 31.8Prebent tube OCF EN-AW 5018
20.8Prebent tube ICF EN-AW 5018 21.6Prebent tube SCF DP600
44.5*Prebent tube OCF DP600 96.5
aRo-radius of tube before hydroforming. See Fig. 1 for
defininecking pressure was not measured in this case; the
reported
ig. 2 Schematic of conditions under which a neck originates
n tubular hydroforming
8 / Vol. 129, JANUARY 2007
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expands and fills the die, a greater portion of the tube comes
intocontact with the die wall. A segment of the tube is shown in
thefigure. One portion of the segment is in contact with the die
wall,which sets up a through-thickness compressive component
ofstress, and the outer surface of this portion is consequently
sub-jected to a tube-die friction force that retards the motion of
thisportion. The compressive component acts in addition to the
tensilehoop stress and, possibly, an axial component of stress if
endfeeding is applied. The portion of the segment not in contact
withthe die is undergoing free expansion and is only acted on by
atensile hoop stress and the internal fluid pressure. This portion
isapproximately under plane stress loading and plastic
deformationis higher in this portion. Thinning and expansion in the
planestress portion can be reduced by material flow from the
three-dimensional portion. However, when the magnitude of the
frictionforce on the three-dimensional portion is large enough,
which oc-curs when a sufficient portion of the tube is in contact
with the diewall, material forming into the plane stress region is
restricted.Under these conditions, a neck and eventual failure
originates atthe interface of the three dimensional portion and the
plane stressportion. Thus, the critical conditions for neck
formation are char-acterized by a three-dimensional stress state
and the friction forcedue to the tube-die interaction. The
computational modeling, de-scribed in a later section, highlights
these two conditions. Sincethe FLC and FLC describe the formability
limit for plane stressloading, a formability limit curve in
three-dimensions is devel-oped from the FLC under the assumption
that stress state at theonset of necking under in-plane plane
stress loading are equiva-lent to those at the onset of necking
under three-dimensionalloading.
3 Extended Stress-Based Forming Limit Curve (XS-FLC)
The FLCs adopted for the EN-AW 5018 aluminum alloy andthe DP600
steel are shown in Fig. 3a. No conventional strain-based FLC was
available for the aluminum; instead, the left sideof the curve was
obtained from free-expansion experiments car-ried out on EN-AW 5018
tubes 20, whereas the right side of thecurve is approximated by
that of AA5754 aluminum alloy sheet.These two alloys are similar in
composition, but differ somewhatin the magnesium content and
stress-strain curves, EN-AW 5018having higher strength and a higher
magnesium content 3.5%versus 3.0%. The plane strain intercept in
this approximate FLCis equal to that obtained from free expansion
experiments 20.Since the tubes were annealed, the plastic strains
produced duringtubing need not be accounted for. For the DP600
steel, the ap-proximation due to Keeler-Brazier is used. This
involves assum-ing the shape of the FLC as the one given by Keeler
and Brazier21, and determining the plane strain intercept from the
harden-ing exponent, n, and sheet thickness, t. A value of n equal
to 0.115was obtained from the tensile stress-strain curve.
Figure 3 also shows the stress-strain response of the two
alloys.
tational necking pressures and cross-section
sure Cross-section Geometrya
mmputation Experiment Computation
32.2 2Ro+s 91.6 91.522.4 2Ro+s 79 80.227.8 2Ro+s 78.5 82.748.5 -
- -105.3 2Ro+s 86.8 87.4
of s. This cross section is at 45 in the bend angle. * Thee is
95% of the burst pressure.
pu
presaCom
tionvalu
These were determined by using tensile test samples obtained
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rom the tubes and conform to the ASTM E8M standards. Notehat the
EN-AW 5018 and DP600 tensile-test samples fail at
20% elongation and that the rest of each curve is an
extrapola-ion. These curves, which are relationships between the
longitudi-al stress and longitudinal strain, are equivalent to a
functionalelationship between the equivalent stress, eq defined
below,nd effective plastic strain, p =2/3ijij. Here, the
Einsteinummation convention is used.
Two assumptions are introduced to transform the FLC into
theSFLC. They are as follows:ASSUMPTION 1. The alloys are assumed
to be described by the J2
ow theory with isotropic hardening. Hardening is described byhe
functional relationship y =yp, assuming proportionalityf the load
path.
ASSUMPTION 2. The stress invariants, namely the equivalenttress
and mean stress, that characterize the formability limit un-er
plane stress loading are representative of the formability
limitnder three-dimensional stress states.
Using equations developed by Stoughton 5 and Assumption 1,he FLC
is transformed into the FLC and these curves arehown in Fig. 3c.
The FLC for DP600 lies above that of theluminum alloy since the
dual phase steel has higher strength ands more formable than the
aluminum alloy. From Assumption 2,he equivalent stress and mean
stress in the neck are describedhrough the invariants, equivalent
stress, eq, and mean stress,
hyd, given by
eq = 12 + 22 12 and hyd =1 + 2
31
here the i is the principal stress the FLC. The variables eqnd
hyd are the variables that constitute the XSFLC.
The XSFLCs for the two alloys are shown in Fig. 3d. Meantress is
assumed to be positive in tension. The left edge of theSFLC
corresponds to the left edge of the FLC and is the form-
bility limit under a uniaxial stress load path. The right edge
cor-esponds to the formability limit under biaxial loading, and the
dip
Fig. 3 a Strain-based FLCs for EN-Ab stress strain curves for
the alloyobtained assuming isotropic hardeninmean stress is assumed
to be positiv
n the XSFLC corresponds to the plane strain limit.
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To apply the XSFLC to predict the onset of necking in sheetmetal
forming, the evolution of eq and hyd in the part beingformed are
required. When the load path described by these vari-ables, at any
location in the part, intersects the XSFLC, the onsetof necking is
predicted. Since the XSFLC is described in terms ofstress
invariants, it can be used to predict the onset of necking inparts
being formed under plane stress loading as well as partsformed
under three-dimensional loading. Here, its utility in theprediction
of necking in tubular hydroforming is investigated.
4 Computational Details
4.1 Constitutive Model. The experiments described abovewere
modelled using the explicit dynamic finite element code LS-DYNA.
The computations were geared toward using the XSFLCto predict the
onset of necking in tubular hydroforming. To thisend, a
user-subroutine was programmed to model the tube mate-rial. The
tube material was modeled as an elastic-plastic materialusing J2
flow theory with isotropic hardening. In the context ofsmall
elastic strains, the increment of the strain tensor, d, is givenby
the additive decomposition of the elastic and plastic straintensors
as d=de+dp. Bold-faced letters are used to denote ten-sors and
subscripts e and p denote elastic and plastic
components,respectively. The plastic strain is computed through an
associatedflow rule as
dp = dp
where is the yield function, dp the increment of the
effectiveplastic strain, and the stress tensor. The yield function
isgiven by
= 3J2 yp =32sijsij yp = 0where yp is the hardening curve
obtained through the uni-axial tensile test and s is the deviatoric
stress.
5018 aluminum alloy and DP600 steel,c stress-based FLCs for the
alloysnd d XSFLCs for the two alloys. Thetension.
Ws,g, ae in
Algorithmic implementation of the stress update for the
mate-
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ial model utilized the radial return scheme of Wilkins 22. Arief
description of this scheme is provided in the Appendix; ahorough
treatment can be found in the book by Belytschko et al.23. The
XSFLC serves as an input to the subroutine. In additiono
implementing the stress update for the element, the subroutinelso
tracks whether the load path described by the variables eqnd hyd
has crossed the XSFLC. A formability variable for aiven hyd is
defined as follows:
=eq
XSFLC2
here eq is the equivalent stress, and XSFLC is the
equivalenttress from the XSFLC, for the given hyd. This variable
deter-ines the proximity of the load path hyd ,eq to the XSFLC.hen
is unity the load path has intersected the XSFLC. An
lternative definition of as a binary variable =0, safe, and 1,
failed, is sometimes useful to highlight the potential failure
ocation in the mesh. Contour plots of can be used to
determineocations in the mesh where the element load paths have
crossedhe XSFLC.
4.2 Material Parameters. For the EN-AW 5018 aluminumlloy, the
shear modulus and bulk modulus K were taken to be6 GPa and 68 GPa,
respectively. For the DP600 steel, 80 GPa and K=164 GPa. The
uniaxial tensile stress strainurves presented in Fig. 3b were
provided in tabular form to theonstitutive subroutine. In the
standard ASTM uniaxial tensile testor tube samples, the total
strain at uniform elongation is less thanbout 20%, after which the
test results are not usable due to thenset of necking instability.
In the straight tube hydroformingroblem studied in this work, the
effective plastic strain at thenset of necking is about 20%; thus,
the flow stress relationshipbtained from the ASTM uniaxial tensile
test can be used toodel the flow stress of the tube.The pre-bent
tubes, however, present an additional challenge.
ince the effective plastic strain at the end of bending can
reach30%, an extrapolation of the uniaxial stress strain curve
is
eeded to model the tube material. Such an extrapolation
willntroduce uncertainities into the flow stress versus effective
plastictrain curve. It is worthwhile to note that the increment of
plastictrain during the hydroforming of pre-bent tubes, for the
materialsonsidered in this work, is about 12%.
To circumvent such an extrapolation, Koc et al. 24 proposed
aydraulic bulge test to evaluate the flow stress of tubular
materialsnder biaxial strain conditions that approximate
hydroforming.hey presented flow stress-strain data for tubes of
SS304 stainlessteel, 6260-T4 aluminum alloy, and 1008 low carbon
steel. How-ver, Ko and Altan 8 point out that the stress strain
curvesbtained from the bulge test agree with those obtained from
theSTM uniaxial test for materials such as copper, brass, and
alu-inum, with the exception of titanium.On the other hand, Levy et
al. 25 used hydraulic burst tests on
KDQ and HSLA steel tubes and showed that the tensile proper-ies
measured from uni-axial ASTM tests on tube samples coulde used to
adequately model the tube flow stress. Indeed, they alsoroposed a
way to use the uniaxial stress strain curve obtainedrom the sheet,
after accounting for the tube-making strains, toredict the flow
stress of the tube.
Thus, there is no consensus in the literature as to how to
char-cterize the flow stress of tubular material. Papers by
Gholipourt al. 16, Dwyer et al. 17, and Oliveira et al. 18
presentesults of bending computations wherein the tubes were
modeledy using an extrapolation of the ASTM tensile test curves.
Theomputational results were in excellent agreement with the
strainseffective strain during bending is 30% and loads
measureduring the bending process. Thus, extrapolations of the
uniaxialtress-strain curves obtained from the ASTM tensile tests
weredopted in the current work due the absence of hydraulic
bulge
ata.
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4.3 Finite Element Mesh Details. All simulations were car-ried
out with eight-noded solids elements with reduced integra-tion. For
the straight-tube hydroforming, due to symmetry, one-eighth of the
tube was modeled with five elements through thethickness and a
total of 45,000 elements. To contrast these com-putations with
those in which the through-thickness compressivestress is zero,
shell element computations were also carried outusing the shell
element developed by Belytschko et al. 26 withseven
through-thickness integration points. The loading conditionsfor
bending and hydroforming that were measured in the experi-ments
were used as inputs to the simulations. The bend tooling,the
mandrels, wiper die, and bend die, were modeled as rigidbodies. A
similar approach was used for the modeling of the hy-droforming
dies and tooling. The computational time and theloading history
were adjusted so that the computations were per-formed in a
reasonable amount of time and also minimized inertialeffects.
Further details of such bending and hydroforming compu-tations can
be found in the article by Bardelcik and Worswick27.
For bent-tube hydroforming, one-half of the tube and toolingwere
meshed in the bending and hydroforming computations. Inthese
models, the tube was modeled using eight-noded solid ele-ments with
five elements through the thickness and a total of21,600 elements.
These computations were performed in a se-quence of steps: bending,
springback, die-close, and hydroform-ing. At the end of the bending
computations, springback calcula-tions were performed. The level of
hardening described by p inthe elements at the end of bending and
the elemental stresses atthe end of springback were transferred
with the mesh to the hy-droforming computation.
4.4 Tube-Die Friction. Friction in tube hydroforming is acomplex
phenomenon depending on the interacting materials, lu-bricant, the
interfacial pressure, and sliding distance. A variety oftechniques
exist to characterize the coefficient of friction betweenthe tube
and die. In tubular hydroforming, three distinct frictionzones,
namely, the guiding, transition and expansion zone havebeen
identified 28. However, in computational practice, a con-stant
value of the coefficient of friction COF is often adopted inthe
contact algorithm to model tube-die friction for all the zones.
Ko 29 used a combined experimental-numerical procedureand
reported COF values of 0.080.09 and 0.120.14 for two wetlubes. In
addition, values of 0.040.05 for a dry lube and 0.190.22 for a
paste lube were reported 29 specifies the composi-tions of these
lubricants. Ngaile et al. 30 used a limiting domeheight test to
estimate an overall COF in the expansion zone forfour lubricants
given as Lube A 0.125, Lube B 0.15, Lube C0.15, and Lube D 0.075.
In addition, they used the pear-shapedtest to estimate the COF in
the expansion zone for the same lu-bricants as 0.074, 0.22, 0.20,
and 0.10. Vollertsen and Plancak31 have used a tube-upsetting test
to estimate COF to be be-tween 0.01 and 0.1, for three different
lubricants.
The methods described above and the estimates of COF
thesemethods yield show that there is not a clear consensus as to
howthis quantity can be characterized. In the current work, the
twistcompression test 32 was used to estimate a COF between thetube
and the die; however, it is recognized that this value may notbe
valid for all regions of the hydroforming die. The values
fortube-die friction presented in Table 1 are considered
reasonablewhen compared with the estimates from other methods.
Thesevalues were used in the penalty function-based contact
algorithms33 available in LS-DYNA to model tube-die friction.
5 Computational Results: Application of XSFLC toTubular
Hydroforming
The application of the XSFLC to predict the onset of
neckingduring tubular hydroforming is presented here. A straight
tubecorner fill expansion operation is examined first; this case
repre-
sents a relatively monotonic strain path. Next, the XSFLC is ap-
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lied to the hydroforming of pre-bent tubes for which the
strainath changes between the pre-bend and hydroforming
operations.
5.1 Application of XSFLC to Straight Tubeydroforming. Figure 4
shows results of the simulations of hy-
roforming of an EN-AW 5018 straight tube. The contour plotsre
the results from solid element calculations and plot
formabilityariable as defined in Eq. 2. A value of unity indicates
therossing of the XSFLC by the load path hyd ,eq. Note thathese
computations were carried out by modeling one-eighth ofhe tube. The
contour plots were reflected about the xz plane. Theocations
designated as 3D in Fig. 4b are under a three-imensional state of
stress, and the locations designated as dieontact remain in contact
with the die from the start of hydroform-ng. The portions of the
tube not in contact with the die are underapproximately plane
stress loading and are designated as the freexpansion region.
In Fig. 4a, when the internal pressure is 32.2 MPa, the
ini-ially circular tube has expanded and the variable indicates
thatany elements are at or above 95% of their formability in
stress
pace. At this state, many elements located on the inside
surface
ig. 4 Contour plots of straight tube hydroforming EN-AW018
aluminum using solid elements. Formability variable Eq. 2 indicates
whether the load path hyd ,eq in the ele-ent has crossed the XSFLC.
a Shows one-quarter of the
ube when the internal pressure is 32.2 MPa. Several elementsn
the inside of the tube have crossed the XSFLC. b Close upf the tube
center where the locations under a three-imensional state of stress
are labeled as 3D. At these loca-
ions, the load paths in all of the elements through the
thick-ess of the tube have crossed the XSFLC =1. The
regionsesignated as free expansion are under plane stress
loadingapproximately. The regions labeled die contact stay in
con-act with the die during the entire process.
f the tube have crossed the XSFLC =1. In particular, =1 at
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the locations on the inside of the tube designated as 3D. In
Fig.4b, when the internal pressure is 34.1 MPa, at the locations
la-beled as 3D, the load paths in all of the elements through
thethickness of the tube have crossed the XSFLC. This is the
onlylocation in the mesh where this feature is observed;
consequently,these are the locations where the tube will eventually
fail. Thus,necking is predicted to originate at the locations
labeled as 3Dwhen the internal pressure is 32.2 MPa. It must be
emphasizedthat the mesh is not designed to model an actual neck.
The equiva-lent stress and mean stress in the load path are
compared with thevalues from the XSFLC curve. The XSFLC, in the
present ex-ample, can be used to predict the internal pressure 32.2
MPa atwhich a neck will start to form in the tube. Table 2 compares
thenecking pressure and cross-section geometry obtained through
thecomputation with those measured in the experiment and it can
beseen that they are in excellent agreement. Furthermore
qualitativepredictions of failure location can be obtained from the
contourplot in Fig. 4b; the location designated as 3D is where the
tubewill burst. Since this is the only location where the load
paths inall of the elements through the thickness of the tube cross
theXSFLC.
Figures 5a and 5b present load paths from solid
elements,obtained through the explicit time integration scheme,
located inthe 3D, free expansion, and die contact regions. These
paths arenonsmooth, which is a characteristic of the explicit
dynamic timeintegration scheme used in the finite element
computation see theAppendix. Load paths from elements in the free
expansion anddie contact regions do not cross the XSFLC. The load
path fromthe element at location 3D, on the other hand, does cross
theXSFLC. Note that the path changes slope while crossing the
XS-FLC; this is the point at which the mesh comes into contact
withthe die. Until this point, the stresses are predominantly
tensile.Once the tube contacts the die the through-thickness
pressure actsas a compressive component of stress and serves to
reduce themean stress; consequently, the curve acquires a negative
slope.
Figures 5c and 5d present load paths from solid
elements,obtained through the implicit time integration scheme,
located atthe 3D, the free expansion and the die contact regions.
In thiscase, the built-in implicit solver within LS-DYNA was
used.These results show that there are negligible difference
between theresults obtained using explicit integration and those
obtained us-ing implicit integration. The implicit results are
smoother thanthose obtained using explicit integration, since both
the constitu-tive update and the time integration use implicit
integration. Inaddition, in this computation, the onset of necking
is predicted atan internal pressure of 32 MPa, which is identical
to the valuepredicted using explicit time integration.
To investigate the effect of neglecting the
through-thicknesscomponent of compressive stress, a computation
wherein the tubewas meshed with plane stress shell elements was
carried out. Thiscomputation shows a rather different result when
compared withthe one using the solid elements. Results of the
computation usingplane stress shell elements can be seen in Fig. 6,
where load paths,hyd ,eq, are plotted with respect to the XSFLC.
This compu-tation is similar to the solid element one with regard
to loadingand boundary conditions. The only difference is in the
type ofelements that constitute the mesh of the tube. It must be
noted thatthe shell elements in this mesh do not admit a
three-dimensionalstate of stress, and in this mesh, the locations
labeled 3D are notunder a three-dimensional state of stress; the
notation has beenretained to compare load paths from equivalent
locations in theshell element mesh with those from the solid
element computa-tion. It can be seen in Fig. 6 that the load path
from element in thefree expansion region crosses the XSFLC,
whereas, the load pathfrom the element in the 3D region does not.
Since these shells areplane stress elements, load paths in
principal stress space 2 ,1can be plotted with respect to the FLC
Fig. 7. Again, the loadpath from an element in the free expansion
region crosses the
FLC, whereas the load path from the element in the 3D region
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oes not. That is, the shell element computations indicate
thatailure will occur in the free expansion region. The feature of
theoad path acquiring a negative slope when the mesh comes
intoontact with the die is also not seen in the shell element
results.hese results highlight the effect of the through-thickness
stressnd show that incorrect predictions can be obtained if this
com-onent of stress is neglected. Furthermore, the internal
pressure athe onset of necking was predicted to be 27 MPa, which is
muchower than the experimentally observed value of 32 MPa. It
isorthwhile to note that the FLC, in conjunction with shell
ele-ents, can be used to predict necking in tubes undergoing
free
xpansion 34. In the case of the free expansion of tubes,
sincehere is no die, the plane stress approximation is reasonably
validhrough the entire process, and shell elements can be
utilized.
5.2 Application of XSFLC to Hydroforming of Pre-bentubes. The
simulations in which the XSFLC was applied to pre-ict the onset of
necking during the hydroforming of pre-bentubes indicated that
additional assumptions were required to ob-ain realistic
predictions. For example, in pre-bent tube OCF hy-roforming, the
method predicts the onset of necking at the insidef the bend as
well as at the outside of the bend. This prediction isounter to the
experimental observation in which the tube failed ateither of these
locations. The reason for the incorrect predictions that the XSFLC,
as presented, does not predict the formabilityf the material when
the direction and state of loading changeetween the bending and
hydroforming operations.
During bending primary loading, the material at the outside ofhe
bend is plastically deformed under tension along the longitu-inal
axis of the tube. In the inside of the bend, though the
plasticeformation is also along the longitudinal axis, the plastic
strain isompressive. During hydroforming secondary loading, the
load-ng is approximately perpendicular to the direction of the
primary
Fig. 5 Load paths from solid elemenforming computations EN-AW
5018 ausing explicit time integration. Load pload path crosses the
XSFLC and whilwhen the mesh comes into contact wiin the die contact
and at the free expathe XSFLC. The load paths are fromlayer. b , d
Results obtained using i
ts computations of straight tube hydro-luminum. These results
were obtainedath from an element at location 3D. Thise doing so the
path changes slope pointth the die. c Load paths from elementsnsion
regions. These paths do not crosselements that are located in the
middle
oading and is tensile on the inside of the bend as well as on
the
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Fig. 6 Load paths from Beytschko-Lin-Tsay plane stress
shellelement computations of straight tube hydroforming EN-AW5018
aluminum alloy. a Load path hyd ,eq from element atlocation 3D
plotted with the XSFLC. b Load path hyd ,eqfrom an element in the
free expansion region plotted with theXSFLC. The load path in the
free expansion region crosses the
XSFLC, whereas, the path in the 3D region does not.
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utside. The load path change between bending and hydroformingn
the inside of the bend is designated as compression-tension, andhe
path change in the outside of the bend is designated
asension-tension.
It will be argued below that the XSFLC, when strictly applieds
presented in previous sections, cannot account for the formabil-ty
limit during secondary loading. However, with some
additionalssumptions the XSFLC can be applied successfully. Before
in-roducing these assumptions, it is useful to define a variable,
*, as
* =0
t
sgnhydp dt 3
here p is the rate of effective plastic strain, and sgn is the
signunction. It can be seen that if the material yields under a
com-ressive state of stress * will be negative since hyd is
negativeor compressive stress states. For yielding under tensile
stresstates, * will be positive. From a computational standpoint,
thisariable identifies whether or not the plastic strain in the
elementccumulated due to yielding in compression or tension. It
must bemphasized that this is a nonphysical variable since
effectivelastic strain cannot be negative and is only used to track
thetress state that is the source of the plastic deformation,
especiallyuring bending. Depending on the sign of *, the
tension-tensionoad path can be distinguished from the
compression-tension path.
5.2.1 Tension-Tension. The constitutive response of materialhat
has been subjected to tensile prestrains, of the
magnitudesncountered in the outside of the bend, and then unloaded
can belassified into two types. Zandrahimi et al. 35 present a
succinctiscussion of the two. Figure 8a presents a schematic that
showshe two types of response, and these are compared with the
re-
ig. 7 Load paths from Beytschko-Lin-Tsay plane stress
shelllement computations of straight tube hydroforming EN-AW018
aluminum alloy. a Load path 2 ,1 from element at
ocation 3D plotted with the FLC. b Load path 2 ,1 fromn element
in the free expansion region plotted with the FLC.he load path in
the free expansion region crosses the XSFLC,hereas, the path in the
3D region does not.
ponse to monotonic single loading of the as-received
material.
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The secondary deformation is along an axis different from that
ofthe primary loading, as is the case during bending followed
byhydroforming. Upon secondary loading, in the Type 1 response,the
material displays a reduced yield stress but a higher
hardeningrate. For a Type 2 response, on the other hand, the
material showsan increased yield stress but a lower hardening rate.
The Graf andHosford 5 stress-strain data for as-received and
prestrainedsheets of Al 2008-T4 display Type 1 response. By way of
contrast,Lloyd and Sang 36 present data for an aluminum
AA3003-0alloy that displays Type 2 response. Other representative
examplesof Type 1 and Type 2 responses for steels and other alloys
can befound in the articles by Laukonis and Ghosh 37 and
Zandrahimiet al. 35. A marked feature of both these responses is
that if thesecondary deformation imposes sufficient plastic strain,
the re-sponse of the material tends to that obtained under single
loading.However, during the secondary loading, there exists a
transientregime when the plastic strains accrued during the
secondaryloading are small, and when isotropic hardening fails to
ad-equately model the constitutive response. Consequently, the
form-ability during the secondary loading will not be correctly
modeledby isotropic hardening. In addition, anisotropic strength
effectsmay be playing a role since there is a change in loading
betweenbending and hydroforming. Therefore, under Assumption 1,
theXSFLC will predict a premature neck during hydroforming in
theregion of the bend that has tensile prestrains. The introduction
ofkinematic hardening and anisotropy of strength will be one way
toobtain a better formability estimate during the secondary
loading.
It is assumed that during secondary loading the transient
regimeis in operation and causes the material to have a higher
formabil-ity than predicted by the XSFLC. For Type 1 response,
since thematerial displays a higher hardening rate, the formability
is higher
Fig. 8 Schematic illustrating Assumption 3. a Material re-sponse
under monotonic single loading solid line. Dashedlines illustrate
Type 1 and Type 2 responses. These curves areshifted by the
prestrain values. b Effect of Assumption 3.Load paths oa and ob
have been drawn as straight lines for thepurpose of illustration,
in reality they are not.
than that derived using the monotonic stress-strain curve
obtained
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orm single loading. Type 2 response, on the other hand, will
havehigher yield strength and consequently a higher formability.
Thisigher formability will operate during the transient regime
andill only hold when the increment of plastic strain imposed by
the
econdary deformation process is small. The following assump-ion
is made to account for the higher formability due to Type 1nd Type
2 material responses.
ASSUMPTION 3. The formability of a material element that hasn
tensile effective tensile pre-strain, p, is assumed to beaxyp
,XSFLC.Note that Assumption 3 is equivalent to assuming that onset
of
ecking, during secondary loading, is not possible until the
mate-ial yields during the secondary loading. Yoshida et al. 38
haveresented data for an aluminum alloy that supports this
assump-ion. This assumption is required because isotropic
hardening, asrgued above, underpredicts the formability and a
kinematic hard-ning model may better account for the increased
formability dur-ng the transient regime. The incorporation of
kinematic hardeningn the stress-based FLC framework will be
addressed later. As-umption 3 is a catch-all and augments the XSFLC
irrespective ofhether Type 1 or Type 2 response is in operation.
Figure 8bresents a scenario where the assumption plays a role. Path
oa ispproximately uniaxial, as is the case during bending, and when
itrosses o the material yields and starts to harden. Load path oa
iserminated at a, the end of bending, and the hardening corre-ponds
to an equivalent stress of oa. Note that the path termi-ates before
the material necks or intersects the XSFLC, and theaterial is
unloaded. The material is then loaded along the plane-
train path ob. This path roughly corresponds to the
hydroformingrocess. This load path first intersects the XSFLC,
however, theaterial element will not neck under Assumption 3, but
neckshen it intersects its yield surface oa. At the end of the
bending
omputation, if *0, and depending on the level of
hardening,ssumption 3 is invoked during the hydroforming
computation.
5.2.2 Compression-Tension. The formability of material ele-ents
that underwent yielding due to compression, as in the inside
f the bend, is augmented according to the following
assumption.ASSUMPTION 4. The formability of a material element that
has a
ompressive effective plastic pre-strain, p, is assumed to
beaxy2p ,XSFLC.At the beginning of the hydroforming computation,
this as-
umption is applied to elements in which *0. Unlike Assump-ion 3,
there can be no appeal to experimental evidence as supportor
Assumption 4. However, microstructural arguments can be in-oked.
For example, if the necking under tensile loading is gov-rned by
growth and nucleation of voids, compressive prestrainill suppress
these mechanisms and delay the onset of neckingnder a secondary
tensile loading. A consideration of the Gurson-vergaard-Needleman
14,15 framework for void growth willupport this claim. A material
element that underwent yieldingnder compression during primary
deformation will have stored int an effective plastic strain of
magnitude p1 that will be equal to*. When the secondary tensile
deformation process results in anncrement of effective plastic
strain of p1, the magnitude of *ill be zero and the total effective
plastic strain will be 2p1. The
actor 2 arises from the assumption, that the increment p1
underensile loading will annihilate the effects the compressive
pre-train of p1.The factor 2 is, at this stage, purely
arbitrary.
Assumptions 3 and 4 do not affect the results of the straightube
computations. These assumptions were added to the user-efined
subroutine and applied to the problem of bent-tube hydro-orming. At
the end of the bending and springback computations,n addition to
the effective plastic strain and the stress compo-ents, the
elemental values of * are transferred with the mesh tohe
hydroforming calculation. Depending on the sign of *,
eitherssumption 3 or 4 is invoked at the beginning of the
hydroform-
ng computation.
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5.2.3 Results for Pre-Bent Hydroformed Tubes. Figure 9 pre-sents
the load paths in bending and hydroforming from the com-putations
for the pre-bent OCF EN-AW 5018 aluminum alloy. Thecolumns
correspond to element locations in the outside and insideof the
bend, and the failure location in the tube. In Fig. 9, the rowsa,
b, and c are plots in principal strain, principal stress
andinvariant space. The FLCs in each space are also shown.
Thesucceeding discussion treats each plot as an element of a
nine-element matrix. The strain paths are continuous; however,
thestress paths are not. Partly due to the complexity of the
formingprocesses under consideration and partly due to the nature
of theexplicit time integration, the stress-based paths are complex
andragged. In the interest of clarity, salient inflection points
have beenchosen and connected by straight lines. All hydroforming
loadpaths are terminated when the internal pressure has reached
thecondition for the predicted onset of the neck. For the plots
instrain and stress space, since these are results of solid
elementcomputations, the third component has been ignored. First,
con-sider the plots in strain space. The largest amount of plastic
strainis accumulated during the bending process; whereas, the
incre-ment of plastic strain is small prior to failure during
hydroform-ing. This relative magnitude of plastic strain emphasizes
the majorimpact of the pre-bend on the formability during
hydroforming.These plots suggest that the strain paths are linear
during bendingand hydroforming. However, note that the slope of the
path inbending is different from that during hydroforming. As
pointedout in the Introduction, the path during hydroforming has to
becompared with the FLC corresponding to the level of
prestrainimposed during bending to predict the onset of necking
duringhydroforming. As plotted in Fig. 9, the FLC of the
as-receivedsheet provides no useful information.
The FLCs plots in Fig. 9 provide no useful information
either.For example, in Fig. 9 Outside-b the bend path is
roughlyuniaxial along the major principal stress axis and the path
actuallycrosses the FLC, indicating necking during bending, which
is anincorrect prediction. The hydroforming path in this figure
suggestsuniaxial loading from the springback state S and intersects
theFLC, indicating failure. Again, this is not where the
pre-bentOCF aluminum alloy tube fails. Figure 8 Inside-b, on the
otherhand, indicates that the loading is completely compressive, as
ex-pected, and failure in the inside of the bend is indicated
incor-rectly, when the hydroforming path intersects the FLC. Figure
9Fail-b, also incorrectly predicts that necking will occur
duringbending.
A more complex picture is revealed by the row of plots
ininvariant space the XSFLC plots. The bending paths show com-plex
changes in mean stress, as when the mesh comes into contactwith the
mandrel labeled M. At this point, a through-thicknesscompressive
component is imposed on the tube, which decreasesthe mean stress
that is predominantly tensile. This leads to achange in slope of
the load path Fig. 9c. Consider the loadpaths in the outside of the
bend, Fig. 9 Outside-c. After twoslope changes due to contact with
the mandrel, the highest hard-ening level equivalent stress of 324
MPa is attained duringbending. Subsequently, during hydroforming,
the load path under-goes a slope change when the tube undergoes a
draw-in due to themotion of the end seals and proceeds to intersect
the XSFLC.However, the formability has been augmented in accordance
withAssumption 3 and is shown by the dashed horizontal line.
Sincethe load path during hydroforming does not intersect this
upperlimit, a neck does not originate in this location.
The role of Assumption 4 can be seen in the load path from
theelement in the inside of the bend Fig. 9 Inside-c. The
bendingpath results in a hardening of 319 MPa, which corresponds
toan effective plastic strain of 24%. In accordance with
Assumption4, the formability limit is set to an equivalent stress
that corre-sponds to a plastic strain of 48%, which corresponds to
an equiva-
lent stress of 359 MPa and is indicated by the dashed horizontal
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ine in the figure. Though the load path during
hydroformingrosses the XSFLC, it does not intersect the dashed line
and con-equently, a neck does not originate in the element.
In the element at the failure location, Fig. 9 Fail-c, the
bendath results in a hardening level of 316 MPa as indicated by
theashed horizontal line. During hydroforming, the load pathrosses
this line and then undergoes a slope change due to contactith the
die and finally intersects the XSFLC leading to failure.his feature
of a slope change in the load path when the meshomes into contact
with the die was also pointed out in the straightube
computation.
The preceding discussion and the accompanying plots highlighthe
complexity of the forming operations under discussion. Twooints
deserve to be emphasized. First, plots in strain space
areisleading. If the third component of stress is neglected, then
the
omplexity of the forming process is hidden. Second, it can be
beeen that both Assumptions 3 and 4 introduce a stress path
depen-ence into the stress-based framework for describing
formability.
Table 2 presents quantitative results from the
hydroformingomputations. The necking pressures and final cross
section ob-ained in the experiments are compared with the
experimental
Fig. 9 Strain paths and stress paths for bending analloy tube.
Plots in each column correspond to elemenrespectively see mesh
outline plot at bottom left. PloThe linear paths shown are
simplifications to thosetreated as an element in a matrix. Note
that the equivaldo not start at zero. Also note that the augmented
fdashed horizontal line in plots Outside-c and Inside-c
alues. In the context of the bent tube OCF experiments,
Assump-
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tion 4 plays a role and prevents the load paths in the elements
inthe inside of the bend from crossing the XSFLC, and it is
As-sumption 3 that controls the final failure locations. For all of
thebent tube OCF cases, the predicted necking pressures are only10%
higher than the experimentally obtained value and the finalcross
section obtained in the computations are within 1% of
theexperimentally measured final cross section. Therefore, this
as-sumption appears to be reasonable. For the bent tube ICF
predic-tions, Assumption 4 plays a key role, since the failure
location isin the region that yielded under compression during
bending. Thepredicted necking pressure is 28% higher than the
experimentalvalue in the computation for the EN-AW 5018 bent tube
ICF.Although Assumption 4 played a satisfactory role in the bent
tubeOCF case, it results in an over-estimate of the formability of
theinside of the tube in the bent tube ICF. This suggests that
thefactor of two used in Assumption 4 is too high. When more data
isavailable, a better estimate of this factor may be possible.
6 Prediction of Failure Location Using the XSFLCThe proposed
method also provides qualitative predictions of
ydroforming of pre-bent OCF EN-AW 5018 aluminumcations at the
inside, outside, and the failure location,
rows correspond to strain, stress and XSFLC space.puted. For the
purpose of discussion, each plot isstress axes in the XSFLC plots
Outside-c and Fail-cability, as per Assumptions 3 and 4, is shown
as aspectively.
d ht lots incomentorm
the final failure location during tubular hydroforming. The
loca-
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ion in the mesh where all of the elements through the thickness
ofhe mesh cross the XSFLC is identified as the location where
theube will burst. Contour plots of from the hydroforming
com-utations are shown in Fig. 10 along with the photographs ofubes
tested to failure. In these contours, is treated as a binaryariable
for clarity, and it can be seen that the predicted failureocation
are in excellent agreement with those seen in the experi-
ents. Note that for the two DP600 tubes, the computations
indi-ate additional failure locations as shown by the gray
arrows.everal approximations have been made to implement theSFLCthe
stress-strain curves, the FLC, and Assumptions 3
nd 4. These are all potential sources of the incorrect
prediction. Its not possible, though, to pinpoint the cause of the
incorrect pre-iction at this stage of the research.
SummaryThis paper presents an extended stress-based FLC that
allows
he prediction of the onset of necking in sheet metal under
three-imensional states of stress. A conventional plane stress
FLCas converted to a plane stress FLC, using the method proposedy
Stoughton 5, which was transformed into the XSFLC. Theey assumption
behind the XSFLC is that the triaxial stress statesenerated in the
neck under plane stress loading can be describedy the invariants
hyd and eq and can be used to describe thetress states in necks
that originate under three-dimensional statesf stress. Necking was
shown to occur in tubular hydroformingnder three-dimensional stress
states. Consequently, solid elementomputations were used in
conjunction with the XSFLC to predictecking pressures and tube
expansion. From the good agreementetween the computational
predictions and the experiments for thetraight tube hydroforming
and some of the bent tube cases, it can
ig. 10 Comparison of contour plots predicting failure loca-ions
left and photographs right from the experiments. Thehite arrows
indicate the failure location. The gray arrows in-icate additional
locations in the DP600 tubes where the XS-LC approach indicates
failure. Though the variable =1 in the
ocation indicated by the black arrow DP600 SCF, all the ele-ents
throught the thickness of the mesh have not crossed theSFLC. The
binary definition of the formability variable, , issed to plot
these contours.
e concluded that Assumption 2 is valid. Inasmuch as the
quali-
6 / Vol. 129, JANUARY 2007
aded 12 Oct 2007 to 129.97.69.217. Redistribution subject to
ASME
tative and quantitative predictions of the approach advanced
inour paper provide support for this assumption, independent
vali-dation of this assumption is warranted. A forthcoming
publicationwill present an approach that can be used to validate
this assump-tion. Furthermore, computations carried out using plane
stressshell elements, in which the through-thickness stress is
zero, wereshown to lead to erroneous predictions.
Isotropic hardening Assumption 1 appears to be sufficient
todescribe the material response to predict the onset of
neckingusing the XSFLC. However, as already mentioned, in the case
ofbent-tube hydroforming, the isotropic hardening assumption wasnot
sufficient to obtain reasonable predictions of the
experimentalresults. It was argued that because of the presence of
prestrains atthe onset of hydroforming and a change in the loading
directionbetween bending and hydroforming, the isotropic hardening
as-sumption under predicted the formability during the
secondarydeformation due to hydroforming. Assumptions 3 and 4 were
in-troduced to circumvent complications that would arise with
thephysically more realistic kinematic hardening. However, these
as-sumptions introduce a stress path dependence in the
XSFLCframework. The predictions for the case of bent-tube OCF
hydro-forming EN-AW 5016 and DP600 tubes are not in as
goodagreement with the experimental results as in the case of
straighttube hydroforming. A potential improvement to the approach
pre-sented in this work is to incorporate kinematic hardening
andanisotropy of strength. Though it is not clear as to how
kinematichardening can be incorporated into the stress-based
formabilityassessments. Consequently, the approach adopted in this
work hasbeen to make the simplest set of assumptions to develop a
toolwith predictive capability.
AcknowledgmentThe authors wish to acknowledge the following
agencies and
companies for their support: Auto21 Network Centres of
Excel-lence, the Ontario R&D Challenge fund, General Motors
ofCanada, Dofasco, Stelco, Nova Tube, DA Stuart, Eagle
PrecisionTechnologies, Natural Sciences and Engineering Research
Coun-cil of Canada, and the Canada Research Chair Directorate.
AppendixIn a finite element code that uses an explicit time
integration
scheme, for stability, the time step is governed by the
Courant-Friedrichs-Lewy criterion 23, which is given by
t minleCe
where le is the characteristic length of element e, and Ce is
thewave speed in the element. At the end of the computation for
timetn, for each element, a time step is computed and the least
valueof these time steps is used as the time step for the entire
domain.
The stress update algorithm is identical to the the built-in
ma-terial model designated as MAT 24 in LS-DYNA 33. The
user-developed subroutine that was used for the computations, in
ad-dition to carrying out the constitutive update, checks whether
theload path described by the equivalent stress and mean stress
hascrossed the XSFLC. At the start of the time step tn+1, the rate
ofthe increment of the total strain tensor, n+1 is known. The
taskof the constitutive subroutine is to update the stress tensor,
n,from time tn to the tensor n+1 at tn+1. The stress tensor n
has been corrected for the rotation of the element between tn
andtn+1 before entry into the subroutine 33. The Lam constants and
are used for the elastic update. Index notation for tensorsand the
summation convention is used below.
1. Compute the trial elastic stresses through the
co-rotational
Jaumann rate.
Transactions of the ASME
license or copyright, see
http://www.asme.org/terms/Terms_Use.cfm
-
t
itts
R
J
Downlo
ijT = ij
n + 2ij + kkt
where T denotes trial state.2. Compute the trial deviators, mean
stress, and equivalent
stress.
sijT = ij
T 1
3kk
T ij, pT =
1
3kk
T and eqT =3
2sij
TsijT
3. Compute T and check for yielding.
T = eqT yp
n
if 0 then material is elastic go to step 6, other wisecompute
plastic strain increment.
4. Compute plastic strain increment.
dp =eq
T ypn
3 + dydp
nwhere dy /dpn is the slope of the hardening curve.
5. Scale the deviators back to the yield surface.
sijn+1 = sij
T ypn
eqT
6. Update stresses, mean stress, and effective plastic
strain.
ijn+1 = sij
n+1 + pT
pn+1 = pT
and
pn+1 = p
n + dp7. Check whether the stress state pn+1 ,eq
n+1 has crossed theXSFLC curve.
Note that the stress state at the end of the update, n+1, is
onhe yield surface whose radius in deviatoric space is yp
n. Thats, the yield surface corresponding to the hardening at
tn, and nothe yield surface at tn+1. This feature and inertial
effects in theime explicit integration could lead to some
non-smoothness in thetress computations.
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