forming limit diagrams of tubular materials by bulge tests

Journal of Materials Processing Technology 209 (2009) 50245034

Contents lists available at ScienceDirect

Journal of Materials Processing Technology

journa l homepage: www.e lsev ier .com/

Formin b

Yeong-MDepartment of hsiung

a r t i c l

Article history:Received 6 AuReceived in reAccepted 31 Ja

Keywords:Tube hydroforForming limitPlastic instabilBulge testsTensile tests

the fxed. Loadtube ae intr strag limcrite

al strt thend thed ton va

ensilfrom the forming limit experiments.

2009 Elsevier B.V. All rights reserved.

1. Introdu

Due to inprocesses helds, suchindustries (and pipe hcarried outbility tests.hydroformimetric partand Skogsgthe forminghydroformand burstin

During ting the loadat the tubeexample, Ahto estimateforming of ttors of inter

CorresponE-mail add

0924-0136/$ doi:10.1016/j.jction

creasing demands for lightweight parts, hydroformingave been widely used to manufacture parts in variousas automobile, aircraft, aerospace, and ship buildingDohmann and Hartl, 1996). Concerning studies of tubeydroforming processes, Ahmetoglu et al. (2000) havea series of simulations and experiments on tube forma-Dohmann and Hartl (1996) have also investigated tubeng processes, including the manufacturing of axisym-s and T-shaped parts by expansion and feeding. Asnaardh (2000) proposed a mathematical model to predictpressure and the associated feeding distance needed to

a circular tube into a T-shape productwithoutwrinklingg.ube hydroforming, several forming parameters, includ-ing path, material properties, die design, and frictiondie interface, signicantly inuence the results. FormedandHashmi (1997)proposeda theoreticalmethodthe forming parameters required for hydraulic bulgeubular components; in particular, they studied the fac-nal pressure, axial load and clamping load. Sokolowski

ding author. Tel.: +886 7 5252000x4233; fax: +886 7 5254299.ress: [email protected] (Y.-M. Hwang).

et al. (2000) proposed a tooling and experimental apparatus todetermine the material properties of tubes. Vollertsen and Plancak(2002) proposed a principle for themeasurement of the coefcientof friction in the forming zone. Lei et al. (2002) used the rigid-plasticnite elementmethod combinedwith a ductile fracture criterion toevaluate the forming limit of hydroforming processes. The presentauthors (Hwang and Lin, 2006) proposed a mathematical modelconsidering the forming tube as an ellipsoidal surface for the pur-pose of analyzing the forming pressure andmaximumbulge height.The properties of tubular materials were additionally evaluated byhydraulic bulge tests combinedwith the above-proposed analyticalmodel (Hwang and Lin, 2007).

The forming limit diagram (FLD) of tubular materials ought tobe established, because it directly inuences 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 loadingpath formaximizing the bulge strain betweennecking andbucklingexperimentally with aluminumA5052 tubes. Zhao et al. (1996) dis-cussed analytically and experimentally the effects of the strain ratesensitivity of the sheet material on the FLD in sheet metal form-ing based on the MK model and GrafHosford anisotropic yieldfunction. They found that FLDswith different pre-strains are signif-icantly inuenced by the straining paths. However, the convertedforming limit stress diagrams (FLSD) appear not to be stronglyinuenced by the straining paths. Xing and Makinouchi (2001)

see front matter 2009 Elsevier B.V. All rights reserved.matprotec.2009.01.026g limit diagrams of tubular materials by

aw Hwang , Yi-Kai Lin, Han-Chieh ChuangMechanical and Electro-Mechanical Engineering, National Sun Yat-Sen University, Kao

e i n f o

gust 2008vised form 27 January 2009nuary 2009

mingdiagramity criteria

a b s t r a c t

This study uses bulge tests to establishdesigned bulge forming apparatus ofare used to carry out the bulge testsstrain ratio at the pole of the bulgingsubroutine and are used to control thAfter bulge tests, the major and minoare measured to construct the formincriterion and Hills localized neckingadopted to derive the critical principminor strains are plotted to construcHills non-quadratic yield function aFLC are discussed. Tensile tests are usrespect to the tube axis and the K andFLCs using the n values obtained by tlocate / jmatprotec

ulge tests

804, Taiwan

orming limit diagram (FLD) of tubularmaterial AA6011. A self-bulge length and a hydraulic test machine with axial feedinging paths corresponding to the strain paths with a constantre determined by FE simulations linked with a self-compiledernal pressure and axial feeding punch of the test machine.ins of the grids beside the bursting line on the tube surface

it diagram of the tubes. Furthermore, Swifts diffused neckingrion associated with Hills non-quadratic yield function areains at the onset of plastic instability. The critical major andforming limit curve (FLC). The effects of the exponent in thee normal anisotropy of the material on the yield locus anddetermine the anisotropic values in different directions withlues of the ow stress of the tubular material. The analyticale tests and bulge tests are compared with the forming limits

Y.-M. Hwang et al. / Journal of Materials Processing Technology 209 (2009) 50245034 5025

Nomenclature

A cross-sectional areaFdFgKLmnP1, P2PirR0t0WpZd

Zl

Greek sym1, 2, 1, 2, 3, cdcl1c,2c

investigatednal pressurplastic instaabove theorwas used toure modesSwifts criteling inducedtube hydrofapplicable tpaid to plasto bucklingthe neckingtheir theorethe FLD ofinternal preforming limand that thaffected byloading patformanceofyield functiloaded undcluded thatnot universwhile the plarger thanforming limlished and tfound.

In this paiments of b

out. Loading paths, which correspond to the strain paths with con-stant strain ratios at the pole of the forming tube, are determinedby LS-DYNA software linked with a self-compiled subroutineand are used to control the internal pressure and axial feed-

the forming limit experiments. Swifts diffused neckingn and Hills localized necking criterion are also used to

t the forming limit curves of the tubes. The experimen-btained forming limits are compared with analyticallyed FLCs using different n values by tensile tests and bulge

mulation of plastic instability criteria

fts diffused necking criterion (Swift, 1952) for thin sheetslls localizednecking criterion (Hill, 1952) associatedwith theon-quadratic yield function (Hill, 1979) are used to constructfor the bi-axial tensile strain zone and tensilecompressive

zone, respectively. Throughout the analysis of plastic insta-he following are assumed:

e elase stree pristan

subeckiw,

Zd,

Z,

1feeding distance of the hydraulic cylinder punchyield functionplastic potential functionstrength coefcient of the materialmaterial lengthexponent of the yield functionstrain-hardening exponent of the materialforming loads in principal directionsinternal pressurenormal anisotropy of the materialinitial tube outer radiusinitial thickness of the tubeplastic worksubtangent of the stressstrain curve for diffusedneckingsubtangent of the stressstrain curve for localizednecking

bols3 principal stressesprincipal strainseffective stress and effective strainprincipal stress ratio (2/1)principal strain ratio (2/1)critical effective strain for diffused neckingcritical effective strain for localized neckingcritical major and minor principal strains for form-ing limit

the differences in forming limits of tubes under inter-e, independent axial load or torque based on Yamadas

ing incriteriopredictally oobtaintests.

2. For

SwiandHiHills nthe FLCstrainbility, t

(1) Th(2) Th(3) Th

con

Thefused nas belo

dd

=

dd

=[bility criteria and Hills quadratic yield function. They coupled with an in-house nite element code ITAS3dcontrol the material ow and to prevent the nal fail-from occurring. Nefussi and Combescure (2002) usedria for sheets and tubes and took into account the buck-byaxial loading inorder topredictplastic instability for

orming. They concluded that the two Swifts criteria areo predict necking and that a special attention has to betic buckling, because the critical strains correspondingare much smaller than the critical strains predicted bycriteria. However, experiments are required to validatetical results. Yoshida and Kuwabara (2007) discussedsteel tubes subjected to a combined axial load andssure. They proposed a FLSD, and concluded that theit stress of the steel tube is not fully path-independente path dependence of forming limit stress is stronglythe strain hardening behavior of the material for givenhs. Korkolis and Kyriakides (2008) investigated the per-HosfordandKarallis-Boycenon-quadratic anisotropicons in predicting the response and bursting of tubeser combined internal pressure and axial load. They con-the predicted structural responses are generally, but

ally, in good agreement with the experimental results,redicted strains at the onset of rupture are somewhatthe values measured. So far, a consistent conclusion forit theorems of tubular materials has not been estab-he forming limit diagram for AA6011 tubes has not been

per, hydraulic formingmachines are developed. Exper-ulge tests with and without axial feeding are carried

Zd =1(

Zl = (g/where antively. g is tsubtangentas the strainand Zl, plea

For consmaterial, thto derive tnecking. At

Fig. 1. Schematic deformation of the material is neglected;ss state of the tubes is planar; andncipal stress ratio at the pole of the forming tube ist during the bulge tests.

tangents of the stressstrain curve, Zd, and Zl, as dif-ng and localized necking occur, respectively, are given

(1)

(2)

(g/1) + 2(g/2)g/1

)2 + 2(g/2)2]

dgd

, (3)

dg/d

1) + (g/2)(4)

d are the effective stress and effective strain, respec-he plastic potential function. The physical meaning ofs Zd and Zl is shown in Fig. 1. It is clear that Z increasesat necking increases. For the detailed derivation of Zd

se refer to appendixes A1 and A2.ideration of the effects of normal anisotropy of thee Hills non-quadratic yield function (Hill, 1979) is usedhe critical strains for diffused necking and localizedrst, let the plastic potential function equal the Hills

tic gure of the subtangent of a stress strain curve as necking occurs.

5026 Y.-M. Hwang et al. / Journal of Materials Processing Technology 209 (2009) 50245034

non-quadratic yield function with a plane stress state:

g = m = 12(1 + r)

[(1 + 2r)|1 2|m + |1 + 2|m

](5)

where m isanisotropyand (4), therespectively

Zd = [2(1 +m1

1

[(1

Z =[2(1 +

where is tLet the e

law of its eq

= Kn,where K anexponent, r(1) and (2)localized ne

cd = nZd,From the

d1 =g

1d

d2 =g

2d

where d isstrain incre

= d2d1

=

After arrobtained as

=[(1 + 2[(1 + 2

From thethat

d = 1 d

= [2(1 +

After combincrement

increment as below.

d1 =2

[2(1 + r)]1/m

[

ing tnt; aratio1c cn be

[2(1

[1[2(1

[1Z iseckial stinin

pone, andain r >0g crierentg limpairs

lytic

3(a)n, musinTheocusto ain fas usn in

4(a)al, r,usinFromal mhe fon inthe exponent of the yield function and r the normalof the material. Then, substituting Eq. (5) into Eqs. (3)subtangents for diffused necking and localized necking,, are expressed as

r)]1/m

(1 + 2r)(1 )

1 m1 + (1 + )1 + (1 + )

[(1 + 2r)2

1 2m2 + 1 + 2m2]+ 2(1 + 2r)(+ 2r)

1 m + 1 + m](m1)/m

r)]1/m[(1 + 2r)

1 m + 1 + m](m1)/m21 + m1 (7)

he principal stress ratio (=2/1).ffective stress of the material be expressed by a poweruivalent strain:

(8)

d n are the strength coefcient and strain-hardeningespectively, of thematerial. Substituting Eq. (8) into Eqs., the critical effective strains for diffused necking andcking can be obtained respectively as

cl = nZl. (9)ow rule (Chen and Han, 1995),

= m2(1 + r)

[(1 + 2r)|1 2|m1 + |1 + 2|m1

]d

(10)

= m2(1 + r)

[(1 + 2r)|1 2|m1 + |1 + 2|m1

]d

(11)

a positive scalar factor of proportionality. The principalment ratio can be obtained as

(1 + 2r)1 m1 + 1 + m1

(1 + 2r)1 m1 + 1 + m1 (12)

angement of the above equation, the stress ratio can be

r)1 + ]1/(m1) 1 1/(m1)

r)1 + ]1/(m1) + 1 1/(m1) (13)

plastic work increment dWp = ij dij = d, it follows

1 + 2 d2

r)]1/m((1 + )/(1 ))(d1 + d2) + (d1 d2)

2[(1 + 2r) +

1+1

m]1/m (14)ining with Eqs. (10) and (11), the major principal straincan be expressed as a function of the effective strain

Durconstastrainstrainas give

1c =

=

wherefusednprincipdetermthe exnent, nthe str(13). Ifneckinto diffforminstrain

3. Ana

Fig.functiotively,n=0.3.yield lfrom 0ical strwhereas show

Fig.materitively,n=0.3.materiraise tas show)1 2m1

(6)

d

1 + m/(m1) + (1/1 + 2r)1/(m1)1 m/(m1)](m1)/m

(15)

he forming process the stress ratio is assumed to beccordingly, the strain increment ratio, , equal to theis a constant. The forming limit for the major principalan be obtained by integration on both sides of Eq. (15),low:

2

+ r)]1/m

+ m/(m1) + (1/1 + 2r)1/(m1)1 m/(m1)](m1)/m

2

+ r)]1/m

nZ

+ m/(m1) + (1/1 + 2r)1/(m1)1 m/(m1)](m1)/m

(16)

equal to Zd and Zl, as given in Eqs. (6) and (7), for dif-ngand localizednecking, respectively. The criticalminorrain can be obtained from 2c = 1c. A ow chart forg the forming limit strains is shown in Fig. 2. At rst,nt of the yield function, m, the strain-hardening expo-the normal anisotropy, r, of thematerial are input. Afteratio is given, the stress ratio can be calculated by Eq., diffused necking criterion is used. Otherwise, localizedterion is used. The critical major strains correspondingstrain ratios can be obtained by Eq. (16). Finally, theit curve can be constructed using the obtained critical(2c, 1c) for 1 > >0.5.

al results and discussion

and (b) shows the effects of the exponent of the yield, on the yield locus and the forming limit curve, respec-g Hills non-quadratic yield function with r=0.5 andregion for stress ratios (=2/1) from 0.5 to 0 in thegure corresponds to that for strain ratios ( = 2/1)0.5 in FLC gure. Combining Eqs. (7) and (13), the crit-rom Eq. (16) turns out to be independent of m value,ing Eqs. (6) and (13), Eq. (16) is not independent of m,Fig. 3(b).

and (b) shows the effects of the normal anisotropy of theon the yield locus and the forming limit curve, respec-g Hills non-quadratic yield function with m=2.0 and

Fig. 4(a), it is known that a larger r value makes theore difcult to yield. Accordingly, a larger r value canrming limit curve in the tensiletensile strain regionFig. 4(b). In the tensilecompressive strain region, the


Fig. 2. Flow ch

forming limthe formingusing Hillsand r values

Fig. 5 shthe tube mquadratic ythe formingA material wtion beforeforming lim

4. Determitubular ma

4.1. Tensile

Tensile tthe anisotrotudinal (ordirectly fromens in thethe tensilewere anneastress resultion is exactubes for buthe specimeconstant stran INSTRONrecorded teinto true stwere testedresults. Qui

Anisotrograins dueart for determining the critical major and minor principal strains.

it curves are not inuenced by the r value. It seems thatlimit curves in the tensilecompressive strain region

localized necking criterion are not inuenced by the min the Hills non-quadratic yield function.ows the effects of the strain-hardening exponent ofaterial, n, on the forming limit curves using Hills non-ield function with m=1.4 and r=0.5. It is apparent thatlimit curves are inuenced signicantly by the n value.ith a larger n value undergoes larger plastic deforma-

necking occurs, accordingly a larger n value raises the

it curves.

nation of ow stresses and anisotropic values ofterials

tests of tubes

ests are conducted to obtain the stressstrain curve andpic values of AA6011 tubes. Specimens in the longi-axial) direction of the tube for the tensile test are cutm the tube with an ASTM standard dimension. Speci-circumferential and 45 directions to the tube axis for

test are cut from a attened tube. The attened tubesled before the tensile tests to eliminate the residualted from the bending operation. The annealing condi-tly the same as that used in the heat treatment of thelge tests to get almost the same material properties forns and tubes. The tensile test was conducted under aain rate of 2103 s1 at the room temperature usinguniversal testing machine. After the tensile tests, the

nsile forces and specimens elongationswere convertedresses and true strains, respectively. Three specimensfor each test condition to check the repeatability of thete good agreement was found.py is caused by preferred orientations or textures ofto manufacturing processes. The anisotropic r values

Fig. 3. Effectsnon-quadratic

in differentof the matedirections,circumferenAnisotropicto that in thcalculatingin Fig. 6. Froaxial and cThe anisotrThe normalfunction ca

Fig. 7 shoing isotropiand circumered, the efthe axial strtively. Howaccount, thof the m value on the yield locus and forming limit curve with Hillsyield function.

directions are used to denote the extent of anisotropyrials. The normal strains in the width and thickness

w and t, during the tensile test in the axial, 45, andtial directions of AA6011 tubes are shown in Fig. 6.r values are the ratio of the strain in thewidth directione thickness direction. Thus, r values can be obtained bythe slope of a straight line that best ts the strain datam Fig. 6, it is known that the anisotropic values in the

ircumferential directions are r0 = 0.466 and r90 =0.497.opic value in 45 to the tube axis by tensile tests is 0.666.anisotropy r value used in theHills non-quadratic yieldn be obtained as r= (r0 +2r45 + r90)/4 =0.574.ws the effective stresseffective strain curves consider-c and anisotropic effects by the tensile tests in the axialferential directions. If isotropy of thematerial is consid-fective stress and effective strain , are equivalent toess and axial strain in the tensile test, 0 and 0, respec-ever, if the anisotropic effect of thematerial is taken intoe effective stress and effective strain are no longer


Fig. 4. Effectsnon-quadratic

equal to 0strain can b(Hwang and

=

32

((

=

23

((

where 0 atests, respe curveals. If the ostrain, = Kexponent nK and n vatively, and tClearly, theanisotropiccal.

ffects of the n value on the forming limit curve with Hills non-quadraticction.Fig. 5. Eyield funof the r value on the yield locus and forming limit curve with Hillsyield function.

and 0, respectively. The effective stress and effectivee obtained in terms of r0, r90, 0 and 0 as given belowLin, 2007).

1 + (1/r0)1/r90) + 1 + (1/r0)

)1/20 (17)

1/r90) + 1 + (1/r0)1 + (1/r0)

)1/20 (18)

nd 0 are the axial stress and axial strain in the tensilectively. Eqs. (17) and (18) are adopted when plotting thewith consideration of the anisotropic effect of materi-w stress is represented by a power law of its equivalentn, the strength coefcient K and the strain-hardeningcan be obtained using the least-squares method. Thelues for isotropy are 304.9MPa and 0.308, respec-hose for anisotropy are 287MPa and 0.308, respectively.n values for the ow stresses considering isotropic andeffects from the tensile tests are approximately identi-

4.2. Bulge t

The yielstate are usThus, Koc eferent methof low carbtests are coA self-desighydraulic pduct the buset, inwhictogether bynormally fring dies arebolts. In thconducted

Figests of tubes

d locus and the effective stress under a biaxial stressually different from those under a uniaxial stress state.t al. (2001), and Strano and Altan (2004) proposed dif-ods by bulge tests to determine the ow stress curveson steel and stainless steel tubes. In this section, bulgenducted to determine the ow stress of AA6011 tubes.ned experimental apparatus composed of a tool set, aower system, and a pressure intensier is used to con-lge tests. Fig. 8 shows the schematic diagram of the toolh the upper and lower plates and the container are xedlarge bolts on each plate. These bolts carry the loadsom the hydraulic pressure. The upper and lower x-used to hold the upper and lower die inserts by the

is way, bulge tests for different tube diameters can beeasily by simply changing the die inserts with different. 6. Strain relationships of AA6011 tubes by tensile tests.


Fig. 7. Effective stresseffective strain curves considering isotropy and anisotropy.

entry radii. In the bulge test, a urethane ring is used to clamp eachend of the tube. A pressure transducer connected to a digital displayis used to measure the forming pressure. The bulge height duringthe bulge test can be read from a dial-gauge through a transmis-sion rod, as shown in Fig. 8. After bulging to a certain height level,the tube is taken out of the tool set and the wall thickness at thepole of the tube is measured by a dial-gauge combined with a self-designedmechanism. A tailoredmechanism is required because anordinary dial-gauge cannot reach the central part of a long tube.Furthermore, it is difcult tomeasure the thickness at a curved sur-face with anthe bulgedand thickne

Fig. 8. Schemaxial feeding.

Fig. 9. S

respectivelyare 15mmof the bulgepressure anaboveexpetive strain fmodel. Thebulge testsstrain-hardobtained us

tively

erim

ore blon

ealeesuseterallyulgeg as smentordinary dial-gauge. A micrometer is used to measurediameter of the tube at the pole. The initial diameterss of the aluminum tubes are 51.91mm and 1.86mm,

respec

5. Exp

Bef200mmare annthe tuba diamchemic

A bfeedinexperiatic diagram of the experimental apparatus for bulge tests without

limit diagrawith axial fstrain pathtensile and

5.1. Hydrau

A hydrodesigned bthe left-sidconsists of ting the toolsource of thbased contrpressure upcient for hsteel tubes.the toolingHwang et atressstrain curves of AA6011 with bulge tests and tensile tests.

. The die entry radius and bulge length for the bulge testand 60mm, respectively. During bulge tests, ve levelsheight are scheduled and the corresponding forming

d the wall thickness at the pole are recorded. Once therimental data areobtained, theeffective stress andeffec-or each level can be determined using a self-developedstressstrain values of the tubular material AA6011 byis shown in Fig. 9. The strength coefcient K and theening exponent n in the ow stress equation = Kning the least-squares method are 254.9MPa and 0.265,(Hwang et al., 2007a).

ents for construction of forming limit diagrams

ulge tests, the tubes of AA6011 aluminum alloy areg, 1.86mm thick, and 51.91mm in outer diameter. Theyd at 410 C with a holding time of 2h. Additionally, fored for the forming limit experiments, circular gridswithof 5mm and an internal spacing of 1mm are electro-etched on the tube surface before the experiments.test apparatus with a xed bulge length without axialhown in Fig. 8 is used to implement the forming limits to obtain the strain path on right side of the forming

m and a newly developed hydroforming test machineeeding is used to conduct the experiments to obtain theon the left side of the forming limit diagram, in whichcompressive strains occur.

lic test machine

forming test machine with axial feeding is newlyy the present authors to conduct the experiments fore data in the forming limit diagram. This test machinehree main parts: a platform or foundation for support-ing; a hydraulic power system for providing thepressuree internal pressure and the feeding punches; and a PC-ol system. The test machine can operate with internalto 70MPa and axial force up to 24 tons; this limit is suf-ydraulically forming aluminum, copper and low-carbonFig. 10 shows a schematic diagram of the platform andset. For the details of this test machine, please refer tol. (2007b).


st machine for bulge tests with axial feeding.

5.2. Determ

During tpole of theloading pataxial feedinpath is detecode DYNAsoftware isincrement.

During tplastic matThe loadingenced by tthe anisotrthe formingAccordinglying is adoptobjects areis imposedand an inteshell elemeious lubrica(Hwang anthe guidingfore, a coninterface besimulations

A ow chratio of =tions are inxed pressurunning theferential anof the bulgi0.10.01,Otherwise,until the rean internalenough to m

Four dif =0.1 to in Fig. 12. Tto be used tin the formit is clear th

creases with increasing the absolute value of the strain ratio

rming limit experiments

C-based control system is used to control the forming pres-nd the left and right axial feeding distances of the testFig. 10. Schematic diagram of the platform and tooling set of the te

ination of loading paths by FE-simulations

he forming limit experiments, the strain ratio at theforming tube has to be kept as a constant value. Theh or the relationship between the internal pressure andg distance that can generate a constant strain ratiormined using an explicit and dynamic nite element3D linkedwith a self-compiled subroutine, because thiseasy to link a subroutine to control the axial feeding

he simulations, the die is assumed to be rigid. An elasto-erial model with strain hardening for the tube is used.path or the internal pressure ismore signicantly inu-he strain hardening (n value) of the ow stress thanopy (r value) of the material, just as the tendency inlimit diagrams shown in Figs. 5 and 4(b), respectively., von-Mises yield function considering strain harden-ed in DYNA 3D. Due to symmetry, only one half of theadopted for the simulations. An axial feeding velocityon the nodes of the elements at the end of the tubernal pressure is imposed on the center of the four-nodents for the boundary conditions. Friction tests using var-nts have been conducted by one of the present authorsd Huang, 2005). The friction coefcients obtained atzone during hydroforming are about 0.030.09. There-stant friction coefcient of 0.05 is assumed at thetween the tube and the die during the nite element.art for determining an adequate loading path for strain0.1 is shown in Fig. 11. At rst, the initial forming condi-

sure in.

5.3. Fo

A Psure aput. Then, an internal pressure increment is set. For there, a feeding distance and an increment are set. Aftersimulation with DYNA 3D, the strains in the circum-

d meridian directions and the strain ratio at the poleng tube can be obtained. If the strain ratio is betweenthe programgoes through to the next internal pressure.the program adjusts the feeding distance incrementquirement is satised. The calculation is iterated untilpressure of 15MPa is reached, which pressure is largeake busting occur.

ferent strain paths with accompanying strain ratios of0.4 obtained using the ow chart in Fig. 11 are shown

he corresponding loading paths to the four strain pathso control the internal pressure and the feeding distanceing limit experiments are shown in Fig. 13. From Fig. 13,at the slope of the feeding distance to the internal pres-

Fig. 11. Flow =0.1.chart for determining an adequate loading path for strain ratio of


Fig. 12. Strain pathswith different strain ratios for construction of the forming limitdiagram.

machine according to the loading paths shown in Fig. 13. The actualresponses of the forming pressures and axial feeding distances dur-ing the forming limit experiments (bulge tests with axial feeding)for differenand (b), resloadingpathtively.Hollofor the leftrespectivelymoved withnal pressureprescribedimplies tha

Fig. 13. Loadinwith constantt strain ratios, =0.1 and 0.2, are shown in Fig. 14(a)pectively. Solid symbols () and () are the prescribeds for the feedingdistanceand formingpressure, respec-wsymbols, (), () and (), are theactual loadingpathsand right axial feeding distances and forming pressure,. It is clear that the left and right axial feeding punchesthe same speed. Generally speaking, the actual inter-and the axial feeding distances faithfully followed the

loading paths. The moment when the pressure dropst bursting of the tube occurs. The bursting pressureg paths for the test machine corresponding to different strain pathsstrain ratios.

Fig. 14. Loadin

decreases sstrain ratio.of the tube.ness at thevolume inspressure.

The resufor differenis known tthe bulgedthe increasdimensionaand minormeasured mconstruct thsection.g paths in the forming limit experiments: (a) =0.1, (b) =0.2.

lightly with the increase of the absolute value of theIn Fig. 14(b), there is a small drop right prior to burstingIt is probably because of an abrupt change of the thick-thinnest part, which results in an abrupt change of theide the tube and leads to an abrupt drop of the internal

lts of the products after bulge tests with axial feedingt strain ratios, =0.1 to 0.4, are shown in Fig. 15. Ithat cracks or bursting lines occur around the pole oftubes and the maximum bulge height increases withe of the absolute value of the strain ratio . A three-l image processing system is used tomeasure themajorstrains of the deformed grids. After the bulge tests, theajor and minor strains on the tube surface are used toe forming limit diagramtobediscussed in the following


Fig. 1

6. Construlimit curve

In Sectioand Hills inthe criticaldeterminin5.2, the loastant strainow chart floading pattine is showstrain paththe ow ching limit exFig. 13 to instant straincurves obtastrains obtabelow.

From thand forminas shown itube for diffstrainpathsDYNA 3D acthe deformconstant stpath for nostatic nitetion. Duringstrainharde() represewhere the bFromthegferent strainpaths and iare acceptaat the earlyThat impliedouble thethe strain ra

=0.5axial and eqow stressFLC of AA6

Exper

strained neto c

d neto c, forthusisotrThe dhe tempard neromthat

formhe n

clus

actuxial feeding followed faithfully the prescribed ones that cor-d to the strain paths with a constant strain ratio at the poleforming tube. A forming limit diagram from forming limitments was successfully constructed using an experimentaltus with xed bulge length and a test machine with axial5. Results of the formed product for different strain paths.

ction of forming limit diagrams and formings

n 2 of this paper, an analytical model combining Swiftstability criteria is proposed, which is used to predictstrains in the forming limit diagram. A ow chart forg the forming limit strains is shown in Fig. 2. In Sectionding paths that can generate strain paths with a con-ratio at the pole of the forming tube are determined. Aor determining adequate axial feeding distances in theh using DYNA 3D linked with a self-compiled subrou-n in Fig. 11. The loading paths corresponding to the

s with different constant strain ratios obtained usingart in Fig. 11 are shown in Fig. 13. In Section 5.3, form-periments are carried out using the loading paths insure the plastic deformation of the tube with a con-ratio at its pole. In this section, the forming limit

ined from the analytical models and the forming limitined from the forming limit experiments are compared

e experimental data and the analytical results, the FLDg limit curves (FLC) of AA6011 tubes are constructedn Fig. 16. The strain paths at the pole of the formingerent strain ratios are also shown in the gure. The fourindicatedby =0.1 to0.4 in Fig. 16 are obtained fromcording to the ow chart shown in Fig. 11. Apparently,ation at the pole of the forming tube is nearly under arain ratio during the bulge forming process. The strainfeeding is obtained by DEFORM 2D, an implicit andelement code suitable for simulations of large deforma-

Fig. 16.curves.

majorlocalizis useddiffuseis usedtion, m1979),mal an0.574.from tthe codiffusevalue fis clearto theusing t

7. Con

Thewith aresponof theexperiapparasimulations, a rigid-plasticmaterialmodel consideringningobtained frombulge tests is used. Symbols () andnt the major and minor principal strains of the meshursting line passes by and passes through, respectively.ure, it is known that theactually obtained strains fordif-ratios are close to the corresponding prescribed strain

t could be said that the simulation results by DYNA 3Dble. For bulge testswithout axial feeding, the strain pathstage of the bulge tests is close to the plane strain state.s that the stress in the hoop direction is approximatelystress in the axial direction. As the bulge test proceeds,tio of the strain path at the pole increases slightly.and 1 shown in Fig. 16 represent strain paths with uni-ual biaxial stress states, respectively. The n value of theobtained from the tensile tests is used to construct the011 tubes. From Eq. (16), it is known that the critical

feeding.AnSwifts diffuHills non-qvalue for nexperimenttubes is recpredict thediffused anHills non-q

Acknowled

The authScience Cou2212-E110-acknowledgimental forming limit diagram of tubes and predicted forming limit

is proportional to the n value. The subtangent for thecking criterion, given in Eq. (7), combined with Eq. (16)onstruct the left part of the FLC, whereas that for thecking criterion, given in Eq. (6), together with Eq. (16)onstruct the right part. The exponent of the yield func-aluminum alloys is usually between 1.38 and 1.47 (Hill,,m=1.4 is adopted during constructing the FLC. The nor-opy of thematerial, r, obtained from the tensile testes isashed and solid lines represent the FLC using n valuesnsile tests (n=0.308) and bulge tests (n=0.265). Fromison, it is known that the theoretical FLC from Swiftscking and Hills localized necking criteria using the nthe tensile tests overestimates the real forming limits. Itthe FLC using the n value from the bulge tests is closer

ing limits from the forming limit experiments than thatvalue from the tensile tests.

ions

al loading paths implemented by the hydraulicmachinealytical forming limit curveswerealsoconstructedusingsed andHills localized necking criteria associatedwithuadratic yield function. The predicted FLC using theobtained by bulge tests is in better agreement with theal forming limits. Therefore, the bulge test of AA6011ommended to obtain the ow stress which is used toFLCs of tubular materials effectively adopting Swiftsd Hills localized necking criteria associated with theuadratic yield function.

gements

ors would like to extend their thanks to the Nationalncil of the Republic of China under Grant no. NSC 93-002. The advice and nancial support of NSC are greatlyed.


Appendix A. A1: Swifts diffused necking criterion

The plastic potential function g(1, 2, 3) describes therelationshipamong the threeprincipal stressesduringplasticdefor-mation. The yield function F(Wp) can be expressed in terms of theplastic work Wp. The associated ow rule (Chen and Han, 1995) ispostulated, which means that the yield function F and the plasticpotential function g coincide:

g = F(Wp).

The plasstate can be

dg = g1

d

where 1 anstate. The y

dF = F (WpDue to equa

g

1d1 +

Diffused ne(Swift, 1952

dP1 = d(1AdP2 = d(2A

From volubecomes

d1 = 1 d

From the pl

d1 =g

1d

where d isEqs. (A6) an

1

(g

1

)2The plas

the effectivto the effec

dg ()d

= dF

Since dWp =

F (Wp) = 1Substitu

dd

= Zd

,

Zd =[

1(

1(g

where antively. Zd isdiffused ne

Appendix B. A2: Hills localized necking criterion

Hills localized necking criterion is usually used to construct theFLC in the tensilecompressive strain zone. At the onset of localizednecking, ththe followin

d11

= d22

olum

1 (d

utin

d2)

utin

Z,

g/

Zl is

nces

M., He formglu, March, a

., Skorolled.F.,Hanan, ppn, F., Hs. J. M1952.ing in979.of theY.M., HB: J. EY.M., Lnisot1928Y.M.,ring anol. 1Y.M., Le test.Y.M.,coun

, Y.P., KydrofAue-uoform772.Kim,esses. MecG., Cooformski, T.nd manol. 9., Alt

bular6..W., 19

., Neusure(A1)

tic potential function increment, dg, for a plane stressexpressed as

1 +g

2d2 (A2)

d 2 are the two principal stresses of the biaxial stressield function increment, dF, can be expressed as

)dWp = F (Wp)(1 d1 + 2 d2) (A3)

lity of dg and dF, it follows that

g

2d2 = F (Wp)(1 d1 + 2 d2) (A4)

cking occurs when the loading reaches the maximum), i.e.,

1) = 1 dA1 + A1 d1 = 0,

2) = 2 dA + A2 d2 = 0 (A5)

me constancy (dA/A) = (dL/L) = d, then, Eq. (A5)

1, d2 = 2 d2 (A6)

astic ow rule (Chen and Han, 1995),

, d2 =g

2d (A7)

a positive scalar factor of proportionality. Substitutingd (A7) into Eq. (A4) gives

+ 2(

g

2

)2= F (Wp)

(1

g

1+ 2

g

2

)(A8)

tic potential function can be expressed as a function ofe stress, i.e., g () = F(Wp). Differentiating with respecttive stress yields

(Wp)d

= dF(Wp)dWp

dWpd

= F (Wp)dWpd

(A9)

d, Eq. (A9) becomes

dgd

dd

. (A10)

ting Eq. (A10) into Eq. (A8) gives

(A11)

g/1) + 2(g/2)/1)

2 + 2(g/2)2]

dgd

, (A12)

d are the effective stress and effective strain, respec-the so-called subtangent of the stressstrain curve ascking occurs.

From v

d1 =

Substit

(d1 +

Substit

dd

=

Zl = (where

Refere

Ahmed,bulg

Ahmetorese231.

Asna, Ncont

Chen,WTaiw

Dohmanpart

Hill, R.,neck

Hill, R., 1ings

Hwang,Eng.

Hwang,ing a1921

Hwang,sideTech

Hwang,bulg

Hwang,with

Korkolisfor h

Koc, M.,hydr761

Lei, L.P.,procInt. J

Nefussi,hydr

Sokolowity aTech

Strano, Mof tu929

Swift, H18.

Tirosh, Jpres851.ere is no normal strain increment along a direction andg equation must hold (Hill, 1952):

= d3 (A13)

e constancy,

1 + d2) , d2 = 2 (d1 + d2) (A14)

g the above equations into Eq. (A4),(1

g

1+ 2

g

2

)= F (Wp)(1 d1 + 2 d2) (A15)

g Eqs. (A7) and (A10) into the above equation,

(A16)

dg/d

1) + (g/2)(A17)

the subtangent for localized necking.

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Theoretical plasticity of textured aggregates. Mathematical Proceed-Cambridge Philosophical Society, p. 17.uang, L.S., 2005. Friction tests in tube hydroforming. Proc. Inst. Mech.ng. Manufact. 219, 587594.in, Y.K., 2006. Analysis of tube bulge forming in an open die consider-ropic effects of the tubular material. Int. J. Mach. Tools Manufact. 46,.

Lin, Y.K., 2007. Evaluation of ow stresses of tubular materials con-nisotropic effects by hydraulic bulge tests. Tran. ASME, J. Eng. Mater.29, 414421.in, Y.K., Altan, T., 2007a. Evaluation of tubularmaterials by a hydraulicInt. J. Mach. Tools Manufact. 47, 343351.

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Forming limit diagrams of tubular materials by bulge testsIntroductionFormulation of plastic instability criteriaAnalytical results and discussionDetermination of flow stresses and anisotropic values of tubular materialsTensile tests of tubesBulge tests of tubes

Experiments for construction of forming limit diagramsHydraulic test machineDetermination of loading paths by FE-simulationsForming limit experiments

Construction of forming limit diagrams and forming limit curvesConclusionsAcknowledgementsA1: Swifts diffused necking criterionA2: Hills localized necking criterionReferences

forming limit diagrams of tubular materials by bulge tests

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