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Forming limit diagrams of tubular
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Forming limit diagrams of tubular materials by bulge tests
YeongMaw Hwang
∗
, YiKai Lin, HanChieh Chuang
Department of Mechanical and ElectroMechanical Engineering, National Sun YatSen University, Kaohsiung 804, Taiwan
a r t i c l e i n f o
Article history:
Received 6 August 2008Received in revised form 27 January 2009Accepted 31 January 2009Available online xxx
Keywords:
Tube hydroformingForming limit diagramPlastic instability criteriaBulge testsTensile tests
a b s t r a c t
Thisstudyusesbulgeteststoestablishtheforminglimitdiagram(FLD)oftubularmaterialAA6011.Aselfdesigned bulge forming apparatus of ﬁxed bulge length and a hydraulic test machine with axial feedingare used to carry out the bulge tests. Loading paths corresponding to the strain paths with a constantstrain ratio at the pole of the bulging tube are determined by FE simulations linked with a selfcompiledsubroutine and are used to control the internal pressure and axial feeding punch of the test machine.After bulge tests, the major and minor strains of the grids beside the bursting line on the tube surfaceare measured to construct the forming limit diagram of the tubes. Furthermore, Swift’s diffused neckingcriterion and Hill’s localized necking criterion associated with Hill’s nonquadratic yield function areadopted to derive the critical principal strains at the onset of plastic instability. The critical major andminor strains are plotted to construct the forming limit curve (FLC). The effects of the exponent in theHill’s nonquadratic yield function and the normal anisotropy of the material on the yield locus andFLC are discussed. Tensile tests are used to determine the anisotropic values in different directions withrespect to the tube axis and the
K
and
n
values of the ﬂow stress of the tubular material. The analyticalFLCs using the
n
values obtained by tensile tests and bulge tests are compared with the forming limitsfrom the forming limit experiments.© 2009 Elsevier B.V. All rights reserved.
1. Introduction
Due to increasing demands for lightweight parts, hydroformingprocesses have been widely used to manufacture parts in variousﬁelds, such as automobile, aircraft, aerospace, and ship buildingindustries (Dohmann and Hartl, 1996).Concerning studies of tube
and pipe hydroforming processes,Ahmetoglu et al. (2000)havecarriedoutaseriesofsimulationsandexperimentsontubeformability tests.Dohmann and Hartl (1996)have also investigated tube
hydroforming processes, including the manufacturing of axisymmetric parts and Tshaped parts by expansion and feeding.Asnaﬁand Skogsgardh (2000)proposed a mathematical model to predicttheformingpressureandtheassociatedfeedingdistanceneededtohydroformacirculartubeintoaTshapeproductwithoutwrinklingand bursting.During tube hydroforming, several forming parameters, including the loading path, material properties, die design, and frictionat the tube–die interface, signiﬁcantly inﬂuence the results. Forexample,AhmedandHashmi(1997)proposedatheoreticalmethod
to estimate the forming parameters required for hydraulic bulgeforming of tubular components; in particular, they studied the factors of internal pressure, axial load and clamping load.Sokolowski
∗
Corresponding author. Tel.: +886 7 5252000x4233; fax: +886 7 5254299.
Email address:
ymhwang@mail.nsysu.edu.tw(Y.M. Hwang).
et al. (2000)proposed a tooling and experimental apparatus todetermine the material properties of tubes.Vollertsen and Plancak(2002)proposedaprincipleforthemeasurementofthecoefﬁcientoffrictionintheformingzone.Leietal.(2002)usedtherigidplastic
ﬁniteelementmethodcombinedwithaductilefracturecriteriontoevaluate the forming limit of hydroforming processes. The presentauthors (Hwang and Lin, 2006)proposed a mathematical model
considering the forming tube as an ellipsoidal surface for the purposeofanalyzingtheformingpressureandmaximumbulgeheight.The properties of tubular materials were additionally evaluated byhydraulicbulgetestscombinedwiththeaboveproposedanalyticalmodel (Hwang and Lin, 2007).The forming limit diagram (FLD) of tubular materials ought tobe established, because it directly inﬂuences the formability of thehydraulic forming processes. A few studies concerning the loadingpaths or the forming limit of tubes and sheets have been reported.For example,Tirosh et al. (1996)explored an optimized loading
pathformaximizingthebulgestrainbetweenneckingandbucklingexperimentallywithaluminumA5052tubes.Zhaoetal.(1996)dis
cussedanalyticallyandexperimentallytheeffectsofthestrainratesensitivity of the sheet material on the FLD in sheet metal forming based on the M–K model and Graf–Hosford anisotropic yieldfunction.TheyfoundthatFLDswithdifferentprestrainsaresignificantly inﬂuenced by the straining paths. However, the convertedforming limit stress diagrams (FLSD) appear not to be stronglyinﬂuenced by the straining paths.Xing and Makinouchi (2001)
09240136/$ – see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.jmatprotec.2009.01.026
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Nomenclature
A
crosssectional area
F
d
feeding distance of the hydraulic cylinder punch
F
yield function
g
plastic potential function
K
strength coefﬁcient of the material
L
material length
m
exponent of the yield function
n
strainhardening exponent of the material
P
1
,
P
2
forming loads in principal directions
P
i
internal pressure
r
normal anisotropy of the material
R
0
initial tube outer radius
t
0
initial thickness of the tube
W
p
plastic work
Z
d
subtangent of the stress–strain curve for diffusednecking
Z
l
subtangent of the stress–strain curve for localizednecking
Greek symbols
1
,
2
,
3
principal stresses
ε
1
,
ε
2
,
ε
3
principal strains¯
,
¯
ε
effective stress and effective strain
˛
principal stress ratio (
2
/
1
)
principal strain ratio (
ε
2
/
ε
1
)¯
ε
cd
critical effective strain for diffused necking¯
ε
cl
critical effective strain for localized necking
ε
1
c
,
ε
2
c
critical major and minor principal strains for forming limitinvestigated the differences in forming limits of tubes under internal pressure, independent axial load or torque based on Yamada’splastic instability criteria and Hill’s quadratic yield function. Theabove theory coupled with an inhouse ﬁnite element code ITAS3dwas used to control the material ﬂow and to prevent the ﬁnal failure modes from occurring.Nefussi and Combescure (2002)used
Swift’scriteriaforsheetsandtubesandtookintoaccountthebucklinginducedbyaxialloadinginordertopredictplasticinstabilityfortubehydroforming.TheyconcludedthatthetwoSwift’scriteriaareapplicable to predict necking and that a special attention has to bepaid to plastic buckling, because the critical strains correspondingto buckling are much smaller than the critical strains predicted bytheneckingcriteria.However,experimentsarerequiredtovalidatetheir theoretical results.Yoshida and Kuwabara (2007)discussed
the FLD of steel tubes subjected to a combined axial load andinternal pressure. They proposed a FLSD, and concluded that theforming limit stress of the steel tube is not fully pathindependentand that the path dependence of forming limit stress is stronglyaffected by the strain hardening behavior of the material for givenloading paths.Korkolis and Kyriakides (2008)investigated the per
formanceofHosfordandKaraﬁllisBoycenonquadraticanisotropicyield functions in predicting the response and bursting of tubesloadedundercombinedinternalpressureandaxialload.Theyconcluded that the predicted structural responses are generally, butnot universally, in good agreement with the experimental results,while the predicted strains at the onset of rupture are somewhatlarger than the values measured. So far, a consistent conclusion forforming limit theorems of tubular materials has not been establishedandtheforminglimitdiagramforAA6011tubeshasnotbeenfound.Inthispaper,hydraulicformingmachinesaredeveloped.Experiments of bulge tests with and without axial feeding are carriedout. Loading paths, which correspond to the strain paths with constant strain ratios at the pole of the forming tube, are determinedby “LSDYNA” software linked with a selfcompiled subroutineand are used to control the internal pressure and axial feeding in the forming limit experiments. Swift’s diffused neckingcriterion and Hill’s localized necking criterion are also used topredict the forming limit curves of the tubes. The experimentally obtained forming limits are compared with analyticallyobtained FLCs using different
n
values by tensile tests and bulgetests.
2. Formulation of plastic instability criteria
Swift’s diffused necking criterion (Swift, 1952)for thin sheets
andHill’slocalizedneckingcriterion(Hill,1952)associatedwiththe
Hill’snonquadraticyieldfunction(Hill,1979)areusedtoconstruct
theFLCforthebiaxialtensilestrainzoneandtensile–compressivestrain zone, respectively. Throughout the analysis of plastic instability, the following are assumed:(1) The elastic deformation of the material is neglected;(2) The stress state of the tubes is planar; and(3) The principal stress ratio at the pole of the forming tube isconstant during the bulge tests.The subtangents of the stress–strain curve,
Z
d
, and
Z
l
, as diffused necking and localized necking occur, respectively, are givenas below,d¯
d¯
ε
=
¯
Z
d
,
(1)d¯
d¯
ε
=
¯
Z ,
(2)
Z
d
=
1
(
∂g/∂
1
)
+
2
(
∂g/∂
2
)
1
∂g/∂
1
2
+
2
∂g/∂
2
2
d
g
d¯
,
(3)
Z
l
=
d
g/
d¯
(
∂g/∂
1
)
+
(
∂g/∂
2
)(4)where ¯
and¯
ε
are the effective stress and effective strain, respectively.
g
is the plastic potential function. The physical meaning of subtangents
Z
d
and
Z
l
is shown inFig. 1.It is clear that
Z
increasesas the strain at necking increases. For the detailed derivation of
Z
d
and
Z
l
, please refer to appendixes A1 and A2.For consideration of the effects of normal anisotropy of thematerial,theHill’snonquadraticyieldfunction(Hill,1979)isused
to derive the critical strains for diffused necking and localizednecking. At ﬁrst, let the plastic potential function equal the Hill’s
Fig.1.
Schematic ﬁgure of the subtangent of a stress strain curve as necking occurs.
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3
nonquadratic yield function with a plane stress state:
g
=
¯
m
=
12(1
+
r
)
(1
+
2
r
)

1
−
2

m
+
1
+
2

m
(5)where
m
is the exponent of the yield function and
r
the normalanisotropy of the material. Then, substituting Eq.(5)into Eqs.(3)
and(4),thesubtangentsfordiffusedneckingandlocalizednecking,respectively, are expressed as
Z
d
=
[2(1
+
r
)]
1
/m
⎧⎨⎩
(1
+
2
r
)(1
−
˛
)
1
−
˛
m
−
1
+
(1
+
˛
)
1
+
˛
m
−
1
(1
+
˛
)
(1
+
2
r
)
2
1
−
˛
2
m
−
2
+
1
+
˛
2
m
−
2
+
2(1
+
2
r
)(1
−
˛
)
1
−
˛
2
m
−
1
⎫⎬⎭
×
(1
+
2
r
)
1
−
˛
m
+
1
+
˛
m
(
m
−
1)
/m
(6)
Z
=
[2(1
+
r
)]
1
/m
(1
+
2
r
)
1
−
˛
m
+
1
+
˛
m
(
m
−
1)
/m
2
1
+
˛
m
−
1
(7)where
˛
is the principal stress ratio (
˛
=
2
/
1
).Let the effective stress of the material be expressed by a powerlaw of its equivalent strain:¯
=
K
¯
ε
n
,
(8)where
K
and
n
are the strength coefﬁcient and strainhardeningexponent,respectively,ofthematerial.SubstitutingEq.(8)intoEqs.
(1) and (2),the critical effective strains for diffused necking andlocalized necking can be obtained respectively as¯
ε
cd
=
nZ
d
,
¯
ε
cl
=
nZ
l
.
(9)From the ﬂow rule (Chen and Han, 1995),d
ε
1
=
∂g ∂
1
d
=
m
2(1
+
r
)
(1
+
2
r
)

1
−
2

m
−
1
+
1
+
2

m
−
1
d
(10)d
ε
2
=
∂g ∂
2
d
=
m
2(1
+
r
)
−
(1
+
2
r
)

1
−
2

m
−
1
+
1
+
2

m
−
1
d
(11)whered
isapositivescalarfactorofproportionality.Theprincipalstrain increment ratio can be obtained as
=
d
ε
2
d
ε
1
=−
(1
+
2
r
)
1
−
˛
m
−
1
+
1
+
˛
m
−
1
(1
+
2
r
)
1
−
˛
m
−
1
+
1
+
˛
m
−
1
(12)After arrangement of the above equation, the stress ratio can beobtained as
˛
=
(1
+
2
r
)
1
+
1
/
(
m
−
1)
−
1
−
1
/
(
m
−
1)
(1
+
2
r
)
1
+
1
/
(
m
−
1)
+
1
−
1
/
(
m
−
1)
(13)Fromtheplasticworkincrementd
W
p
=
ij
d
ε
ij
=
¯
d¯
ε
,itfollowsthatd¯
ε
=
1
d
ε
1
+
2
d
ε
2
¯
=
[2(1
+
r
)]
1
/m
((1
+
˛
)
/
(1
−
˛
))(d
ε
1
+
d
ε
2
)
+
(d
ε
1
−
d
ε
2
)2
(1
+
2
r
)
+
1
+
˛
1
−
˛
m
1
/m
(14)After combining with Eqs.(10) and (11),the major principal strain
increment can be expressed as a function of the effective strainincrement as below.d
ε
1
=
2[2(1
+
r
)]
1
/m
×
d¯
ε
1
+
m/
(
m
−
1)
+
(1
/
1
+
2
r
)
1
/
(
m
−
1)
1
−
m/
(
m
−
1)
(
m
−
1)
/m
(15)During the forming process the stress ratio
˛
is assumed to beconstant; accordingly, the strain increment ratio,
, equal to thestrain ratio is a constant. The forming limit for the major principalstrain
ε
1
c
can be obtained by integration on both sides of Eq.(15),as given below:
ε
1
c
=
2[2(1
+
r
)]
1
/m
×
¯
ε
1
+
m/
(
m
−
1)
+
(1
/
1
+
2
r
)
1
/
(
m
−
1)
1
−
m/
(
m
−
1)
(
m
−
1)
/m
=
2[2(1
+
r
)]
1
/m
×
nZ
1
+
m/
(
m
−
1)
+
(1
/
1
+
2
r
)
1
/
(
m
−
1)
1
−
m/
(
m
−
1)
(
m
−
1)
/m
(16)where
Z
is equal to
Z
d
and
Z
l
, as given in Eqs.(6) and (7),for dif
fusedneckingandlocalizednecking,respectively.Thecriticalminorprincipal strain can be obtained from
ε
2
c
=
ε
1
c
. A ﬂow chart fordetermining the forming limit strains is shown inFig. 2.At ﬁrst,
the exponent of the yield function,
m
, the strainhardening exponent,
n
,andthenormalanisotropy,
r
,ofthematerialareinput.Afterthe strain ratio
is given, the stress ratio can be calculated by Eq.(13).If
>0,diffusedneckingcriterionisused.Otherwise,localizednecking criterion is used. The critical major strains correspondingto different strain ratios can be obtained by Eq.(16).Finally, the
forming limit curve can be constructed using the obtained criticalstrain pairs (
ε
2
c
,
ε
1
c
) for 1>
>
−
0.5.
3. Analytical results and discussion
Fig. 3(a) and (b) shows the effects of the exponent of the yieldfunction,
m
, on the yield locus and the forming limit curve, respectively, using Hill’s nonquadratic yield function with
r
=0.5 and
n
=0.3. The region for stress ratios (
˛
=
2
/
1
) from 0.5 to 0 in theyield locus ﬁgure corresponds to that for strain ratios (
=
ε
2
/
ε
1
)from 0 to
−
0.5 in FLC ﬁgure. Combining Eqs.(7) and (13),the crit
ical strain from Eq.(16)turns out to be independent of “
m
” value,whereas using Eqs.(6) and (13),Eq.(16)is not independent of “
m
”,as shown inFig. 3(b).
Fig.4(a)and(b)showstheeffectsofthenormalanisotropyofthematerial,
r
, on the yield locus and the forming limit curve, respectively, using Hill’s nonquadratic yield function with
m
=2.0 and
n
=0.3. FromFig. 4(a), it is known that a larger
r
value makes thematerial more difﬁcult to yield. Accordingly, a larger
r
value canraise the forming limit curve in the tensile–tensile strain regionas shown inFig. 4(b). In the tensile–compressive strain region, the
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Fig. 2.
Flow chart for determining the critical major and minor principal strains.
forminglimitcurvesarenotinﬂuencedbythe
r
value.Itseemsthatthe forming limit curves in the tensile–compressive strain regionusing Hill’s localized necking criterion are not inﬂuenced by the
m
and
r
values in the Hill’s nonquadratic yield function.Fig. 5shows the effects of the strainhardening exponent of the tube material,
n
, on the forming limit curves using Hill’s nonquadratic yield function with
m
=1.4 and
r
=0.5. It is apparent thattheforminglimitcurvesareinﬂuencedsigniﬁcantlybythe
n
value.A material with a larger
n
value undergoes larger plastic deformation before necking occurs, accordingly a larger
n
value raises theforming limit curves.
4. Determination of ﬂow stresses and anisotropic values of tubular materials
4.1. Tensile tests of tubes
Tensiletestsareconductedtoobtainthestress–straincurveandthe anisotropic values of AA6011 tubes. Specimens in the longitudinal (or axial) direction of the tube for the tensile test are cutdirectly from the tube with an ASTM standard dimension. Specimens in the circumferential and 45
◦
directions to the tube axis forthe tensile test are cut from a ﬂattened tube. The ﬂattened tubeswere annealed before the tensile tests to eliminate the residualstress resulted from the bending operation. The annealing condition is exactly the same as that used in the heat treatment of thetubes for bulge tests to get almost the same material properties forthe specimens and tubes. The tensile test was conducted under aconstant strain rate of 2
×
10
−
3
s
−
1
at the room temperature usingan INSTRON universal testing machine. After the tensile tests, therecordedtensileforcesandspecimen’selongationswereconvertedinto true stresses and true strains, respectively. Three specimensweretestedforeachtestconditiontochecktherepeatabilityoftheresults. Quite good agreement was found.Anisotropy is caused by preferred orientations or textures of grains due to manufacturing processes. The anisotropic
r
values
Fig. 3.
Effects of the
m
value on the yield locus and forming limit curve with Hill’snonquadratic yield function.
in different directions are used to denote the extent of anisotropyof the materials. The normal strains in the width and thicknessdirections,
ε
w
and
ε
t
, during the tensile test in the axial, 45
◦
, andcircumferential directions of AA6011 tubes are shown inFig. 6.Anisotropic
r
valuesaretheratioofthestraininthewidthdirectionto that in the thickness direction. Thus,
r
values can be obtained bycalculating the slope of a straight line that best ﬁts the strain datainFig. 6.FromFig. 6,it is known that the anisotropic values in the
axial and circumferential directions are
r
0
=0.466 and
r
90
=0.497.Theanisotropicvaluein45
◦
tothetubeaxisbytensiletestsis0.666.Thenormalanisotropy
r
valueusedintheHill’snonquadraticyieldfunction can be obtained as
r
=(
r
0
+2
r
45
+
r
90
)/4=0.574.Fig.7showstheeffectivestress–effectivestraincurvesconsidering isotropic and anisotropic effects by the tensile tests in the axialandcircumferentialdirections.Ifisotropyofthematerialisconsidered, the effective stress ¯
and effective strain¯
ε
, are equivalent totheaxialstressandaxialstraininthetensiletest,
0
and
ε
0
,respectively.However,iftheanisotropiceffectofthematerialistakenintoaccount, the effective stress ¯
and effective strain¯
ε
are no longer