Prediction of Necking in Tubular Hydroforming Using an Extended Stress-Based Forming Limit Curve

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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

aded 12 Oct 2007 to 129.97.69.217. Redistribution subject to ASME

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.
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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


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

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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


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.

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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.



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.

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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



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
JANUARY 2007, Vol. 129 / 41


dflcTaptlwmett

Tdttdocniob

tddcil

mplicit time integration.

4

Downlo

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

2 / Vol. 129, JANUARY 2007


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.


oitt

aiat

wfpfsvaepsdl

tecdts

Fe5laTw

J

Downlo

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.



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

JANUARY 2007, Vol. 129 / 43


fahwsta

am

nrptaeiiswpactsnmspmwcA

mo

cm

stfvewuTsuiiwfts

tdfintAi

4

Downlo

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.

4 / Vol. 129, JANUARY 2007


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


lcs

pdcwTct

tpmcsd

ctv

, re

J

Downlo

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-



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-

JANUARY 2007, Vol. 129 / 45


tttptvlmcSXaid

7

tdwbkgbsoucnbsb

FtwdFlmXu

4

Downlo

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


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.


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.

eferences1 Keeler, S. P., and Backofen, W. A., 1963, Plastic Instability and Fracture in

Sheets Stretched Over Rigid Punches, ASM Trans. Q., 56, pp. 2548.2 Goodwin, G. M., 1968, Application of Strain Analysis to Sheet Metal Form-

ing in the Press Shop, SAE Paper, No. 680093.3 Ghosh, A. K., and Laukonis, J. V., 1976, The Influence of Strain Path

Changes on the Formability of Sheet Steel, 9th Biennial Congress of TheInternational Deep Drawing Research Group, Sheet Metal Forming and En-ergy Conservation, ASM, Metals Park, OH.

4 Graf, A., and Hosford, W., 1993, Effect of Changing Strain Paths on FormingLimit Diagrams of Al 2008-T4, Metall. Trans. A, 24A, pp. 25032512.

5 Stoughton, T. B., 1999, A General Forming Limit Criterion for Sheet MetalForming, Int. J. Mech. Sci., 42, pp. 127.

6 Stoughton, T. B., 2001, Stress-Based Forming Limits in Sheet-Metal Form-ing, ASME J. Eng. Mater. Technol., 1234, pp. 417422.

7 Embury, J. D., and LeRoy, G. H., 1977, Failure Maps Applied to MetalDeformation Processes, Advances in Research on the Strength and Fractureof Materials, Pergamon Press, London, pp. 1542.

8 Ko, M., and Altan, T., 2001, An Overall Review of the Tube HydroformingTHF Technology, J. Mater. Process. Technol., 1083, pp. 384393.

9 Gotoh, M., Chung, C., and Iwata, N., 1995, Effect of Out-of-Plane Stress onthe Forming Limit Strain of Sheet Metals, JSME Int. J., Ser. A, 381, pp.123132.

10 Smith, L. M., Averill, R. C., Lucas, J. P., Stoughton, T. B., and Matin, P. H.,2003, Influence of Transverse Normal Stress on Sheet Metal Formability,Int. J. Plast., 1910, p. 1567-1583.

11 Saanouni, K., Nesnas, K., and Hammi, Y., 2000, Damage Modeling in MetalForming Processes, Int. J. Damage Mech., 9, pp. 196240.

12 Cherouat, A., Saanouni, K., and Hammi, Y., 2002, Numerical Improvement



of Thin Tubes Hydroforming With Respect to Ductile Damage, Int. J. Mech.Sci., 4412, pp. 24272446.

13 Baradari, G. J., 2006, Damage in Hydroforming of Pre-Bent Aluminum AlloyTubes, Ph.D. thesis, Department of Mechanical Engineering, University ofWaterloo.

14 Gurson, A. L., 1977, Continuum Theory and Ductile Rupture by Void Nucle-ation and Growth: Part I Yield Criteria and Flow Rules for Porous DuctileMedia, ASME J. Eng. Mater. Technol., 991, pp. 215.

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Prediction of Necking in Tubular Hydroforming Using an Extended Stress-Based Forming Limit Curve

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