Path-dependent failure of inflated aluminum tubes

International Journal of Plasticity 25 (2009) 2059–2080

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Path-dependent failure of inflated aluminum tubes Yannis P. Korkolis, Stelios Kyriakides * Research Center for Mechanics of Solids, Structures & Materials, The University of Texas at Austin, ASE/EM, 210 E 24th, WRW 110, C0600, Austin, TX 78712, USA

a r t i c l e

i n f o

Article history: Received 30 June 2008 Received in final revised form 29 December 2008 Available online 23 January 2009

Keywords: Aluminum Tube hydroforming Burst Path-dependent failure

a b s t r a c t Our recent investigation on the formability of Al alloy tubes under combined internal pressure and axial load is expanded by examining the effect of the loading path traced. A set of Al-6260-T4 tubes were loaded along orthogonal stress paths to failure and the results are compared to those of the corresponding radial paths. It is confirmed that failure strains are path-dependent, but also is demonstrated that failure stresses become path-dependent if the prestrain is significant. The experiments are simulated using the previously developed finite element models and the calibration of the Yld2000-2D [Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.-H., Chu, E., 2003. Plane stress yield function for aluminum alloy sheets-part I: theory. Int. J. Plasticity 19, 1297–1319] anisotropic yield function shown in [Korkolis, Y.P., Kyriakides, S., 2008b. Inflation and burst of anisotropic aluminum tubes. Part II: an advanced yield function including deformation-induced anisotropy. Int. J. Plasticity 24, 1625–1637] to yield accurate predictions of rupture for nine radial paths. The models are shown to reproduce the path dependence of the failure stresses and strains quite well. A group of additional radial and corner paths are subsequently examined numerically to enrich the existing data on path-dependence of failure. It is again shown that the amount of plastic prestraining in either of the two directions influences the difference of the failure stresses and strains between the radial and the corner stress paths. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction We recently reported results from a series of experiments, conducted in support of a larger project on tube hydroforming, that involved inflation of relatively thin-walled Al-6260-T4 tubes under * Corresponding author. Tel.: +1 5124714167; fax: +1 5124715500. E-mail address: [email protected] (S. Kyriakides). 0749-6419/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2008.12.016

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combined internal pressure and axial load (Korkolis and Kyriakides – K-K – 2008a). The tubes were loaded to failure along radial paths in the engineering stress space. The specimens developed localized axisymmetric bulging associated with a limit load, that lead to localized failure. In axial tension dominant loading paths, the tubes failed by circumferential rupture whereas for hoop stress dominant paths the rupture was along a tube generator (see also Kuwabara et al., 2005; Davies et al., 2000). The experiments were then simulated with finite element models (FE) in which three different anisotropic yield functions were employed, using isotropic hardening plasticity: Hosford (1979), Karafillis and Boyce (1993), and Barlat et al. (2003). The added flexibility provided by the eight anisotropy parameters of the last model, coupled with the incorporation of some deformation-induced anisotropy observed in the experiments were found to produce the most accurate predictions of the strain paths traced in the experiments (K-K, 2008b). On the other hand, optimal prediction of the strains at rupture for the whole set of experiments required additional small amendments to the anisotropy parameters. The most successful scheme was based on a hybrid procedure that also included adjustments of the yield function variables based on the structural performance of the FE models. A significant body of work dealing with strain-based forming limit diagrams (FLDs) has shown experimentally that such failure limits are path-dependent and consequently are only applicable to forming operations with loading paths similar to the ones used to construct them. For example, the path dependence of FLDs was demonstrated for a wide range of materials and loading paths that typically involved prestraining, unloading and reloading in a variety of failure strain combinations by Muschenborn and Sonne (1975), Kleemola and Pelkkikangas (1977), Lloyd and Sang (1979), McCandless and Bahrani (1979), Wagoner and Laukonis (1983) and others. In more recent investigations, Graf and Hosford (1993, 1994), Hosford and Caddell (1993), and Korhonen and Manninen (2008), among others, confirmed and expanded these earlier observations. In an effort to develop a more general failure framework for metal forming, it has been postulated that despite the failure strains being path-dependent, the associated failure stresses are not (or at least much less so; e.g., Gronostajski (1984), Arrieux (1995) and more recently Stoughton (2000), Wu et al. (2005), and others). Some experimental support for this proposal appears in Yoshida et al. (2005) although later work by the same group (Yoshida and Kuwabara, 2006, 2007) points to limitations of this concept. The present work presents results from a number of new biaxial experiments involving axial loading and internal pressure of tubes of the same Al alloy as that used in K-K (2008a) but now using non-proportional stress paths. The aim is to establish the extent of path dependence of limit states in strain space and simultaneously test the extent of validity of path independence of the corresponding limit stresses. This is accomplished by prescribing rather extreme non-proportional corner paths is the rx–rh plane. The experiments are subsequently simulated numerically using the FE models developed in K-K (2008a) along with the specially calibrated anisotropic yield function of Barlat et al. (2003) which successfully predicted the onset of failure for the radial paths in K-K (2008b). One of the objectives is to determine if the thus calibrated constitutive models are also capable of predicting with accuracy the onset of failure under the more demanding non-proportional stress paths.

2. Experimental 2.1. Experimental setup and procedure The experimental program involved testing seamless Al-6260-T4 tubes under combined axial tension and internal pressure. The experiments were conducted in the custom biaxial testing facility shown schematically in Fig. 1. The facility consists of a 50 kip (222 kN) servo-hydraulic testing machine that can operate in conjunction with a 10,000 psi (690 bar) pressurizing unit with an independent closed-loop control system (inside the dashed boundary). By connecting the two systems through feedback, the axial and pressure modes of loading can be related, in order to generate various paths in the engineering stress plane rx–rh (x and h are the axial and circumferential directions, respectively).

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Fig. 1. Schematic drawing of the axial load – internal pressure testing facility used to test tubes under various loading paths.

C D

A

•

x

E

R

0

x B x

Fig. 2. Definition of the radial and corner loading paths prescribed in the engineering stress space.

For a radial path, such as OA in Fig. 2, the two stresses are related through

rx ¼ arh ; a ¼ const:

ð1Þ

If F is the force measured by the load cell and P is the internal pressure, the axial and circumferential stresses are then

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rx ¼

F PR þ 2pRt 2t

and

rh ¼

PR t

ð2Þ

where R and t are, respectively, the initial mid-surface radius and wall thickness of the tube. Thus, for example, if the load cell reads zero force then the specimen is reacting the pressure end-load and consequently develops an axial stress of PR/2t. Using (1) and (2) one can find a relationship between the pressure transducer signal and the command signal for the axial load. Results for three radial paths will be presented in which a = {0.75, 0.9, 1.25}. Here the pressurization was run under volume control and the axial machine under load control with the pressure as command signal. For x ? h corner paths, such as OBC in Fig. 2, the aim was to load the specimen to an axial stress that corresponds to the value attained at failure for a particular radial path, hold this value and increase the hoop stress until the specimen ruptured. For path OB the axial machine was run in displacement control and on reaching B was switched to load control. The subsequent pressurization was conducted under volume control with the pressure being used as the command signal for the axial loading. In this way, as the pressure increased, the machine force was reduced in accordance with Eq. (2) in order to keep the axial stress in the specimen constant (BC). For h ? x corner paths, such as ODE in Fig. 2, the two systems were coupled during the pressurization phase of the path (OD) so that the axial stress remained zero despite the end-cap loading due to the pressure. The hoop stress was increased to a level that matched the value at failure for the corresponding radial path. The two systems were then decoupled and pressurization was switched to pressure control. With the pressure kept constant in this manner, the tube was loaded axially to failure under displacement control. In all tests performed typical strain rates were of the order of 104 s1. The tubes tested had a diameter of approximately 2.36 in (60 mm) and wall thickness of 0.080 in (2 mm). The specimen total length varied between 12.625 and 13.5 in (321–343 mm). The test setup includes two solid steel grips that mount onto the testing machine. The specimen is sealed with solid end-plugs as shown in Fig. 1. The assembly is mounted into the grips using Ringfeder axisymmetric locking devices leaving a test section that ranged in length between 7.625 in and 8.5 in (194– 216 mm). The average circumferential strain at the mid-span of the test section was measured using a chain circumferential extensometer, which was modified to have a strain range of about 17%. The axial strain was measured with a 1-in. gage length extensometer with a range of 30%. In addition, two pairs of axial and circumferential strain gages, placed diametrically opposite to each other in the neighborhood of the mid-span, were used to measure the strain in the early parts of the tests. Local strain measurements in the zone of failure were obtained using a square mesh with a 0.25 in (6.4 mm) spacing that was lightly scribed in the central part of the tube. The tubes had a small amount of wall eccentricity that was sufficient to cause rupture to occur systematically on the thinner side of the tube wall. Therefore, a grid was placed on only one-half of the circumference. In addition, several thickness measurements were performed in the area of failure using an ultrasonic thickness gage. The pressure, axial force, axial displacement, and the extensometer and strain gage signals were recorded in a computer operated data acquisition system using the LabVIEW software. 2.2. Experimental results for radial paths The tubes used in the tests are from the same batch that the specimens used in Korkolis and Kyriakides (2008a) work originated from and consequently the basic material properties are the same. Results from three of the original radial path experiments will form the basis of the comparison with the new corner paths. These tests had proportionality constants of a = {0.75, 0.9, 1.25} while the geometric characteristics of the specimens are given in Table 1a (henceforth called RXX). The prescribed stress paths in the rx–rh plane plotted in Fig. 3 are seen to be linear and the corresponding strain paths are seen in Fig. 4 to be also nearly linear. For completeness, we also include the true stress (sx–sh) and logarithmic strain (ex–eh) versions of the same results in Appendix A (Figs. A1 and A2). As was reported in K-K (2008a), both the stress and strain paths exhibit small nonlinearities and end at higher values. Fig. 5 shows plots of the circumferential and axial stress–strain responses. All three cases developed a limit load instability that is marked on the responses with a caret (^). For R0.75 and R0.9 a pressure maximum developed, associated with a mild axisymmetric bulge at the specimen mid-span. In both

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Table 1a Geometric characteristics of the tubes tested under radial and corner paths. Exp No.

D (in.)

t (in.)

2L (in.)

No (%)

A17-2 A19-1 A20-2 A20-4 A20-5 A21-1 A21-3

2.359 2.356 2.359 2.360 2.359 2.359 2.359

0.0800 0.0800 0.0800 0.0796 0.0797 0.0795 0.0800

13.38 13.00 12.63 13.50 12.75 13.50 13.50

0.69 0.63 0.75 0.82 0.75 0.70 0.88

1 in. = 25.4 mm.

of these cases the volume-controlled pressurization coupled with the ‘‘stiff” testing system used, enabled tracking of the response past the limit load instability. The mild bulge suddenly precipitated wall thinning along a generator resulting in rupture of the type shown in Fig. 6a. The rupture occurs dynamically at a location not exactly known apriori and consequently the associated localized deformation could not be captured (the wall thinning failure shown in Fig. 8 of Korkolis and Kyriakides, 2008a is the result of such a dynamic burst). The stress at the pressure maximum (rhmax) and at the onset of rupture (rhf) are marked on the stress paths in Fig. 3 (and the true counterparts in Fig. A1) with symbols (d) and (N), respectively, and numerical values are given in Table 1b. The average strains at the pressure maximum (ehL, exL) and those at failure (ehf, exf) are marked with the same symbols on the strain paths in Fig. 4 (logarithmic counterparts in Fig. A2) and their values are listed in Table 1b. Included in the same table are strain values measured locally, adjacent to the failure zone, using the grid, and verified with an ultrasonic thickness gage. Thus for example, for R0.9 the local values were ehf|l = 18.3% and exf|l = 11% whereas ehf = 7.5% and exf = 7.9%, illustrating the local nature of the failure. In the case of R1.25, an axial force maximum developed instead that can be seen in Fig. 5b. The specimen developed a mild bulge at mid-span and failed by localized thinning in the circumferential direction as illustrated in Fig. 7 (similar failure modes were reported in Davies et al. (2000) and

x

0

50

100

150

(MPa)

200

250

40 Al-6260-T4

250

LL F

(ksi)

(MPa)

30

200 x 1.25 R0.75

150 20 R0.9

100

R1.25

10 50

x 0.75

0.9

1.25

0

0 0

10

20

30 x

(ksi)

40

Fig. 3. Corner paths prescribed in the experiments, and the corresponding radial paths from K-K (2008a). Marked are the limit (d) and the failure (N) stresses.

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8 LL F

R0.9 R0.75

(%) 4

1.25

x 0.75

R1.25

0 0.9 x

1.25 Al-6260-T4

-4 -4

0

4

8

12

16

20 x

(%)

Fig. 4. Strain paths traced in the corner experiments, and in the corresponding radial paths from K-K (2008a). Marked are the average strains at the limit load (d) and at failure (N).

a

40 Al-6260-T4

240

(ksi)

(MPa)

30

R0.9

R0.75

180 R1.25

20

120

10

0

b

x

=

60

x

0

2

4

6

40

0

8

(%)

10

Al-6260-T4

240

x

x

(ksi)

(MPa)

R1.25

30

R0.9

180

R0.75

20

120

10

x

=

x

0

0

5

10

15

0 25

20 x

60

(%)

Fig. 5. Stress–strain responses recorded in the radial path experiments (from K-K, 2008a). (a) rh–eh and (b) rx–ex.

Yoshida et al. (2005)). This type of wall thinning can be seen in Fig. 12 of Korkolis and Kyriakides (2008a) where the localization extends over approximately one tube diameter. The stresses and strains at the limit load and at failure are marked in Figs. 3 and 4 and the numerical values are listed in Table 1b.

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Fig. 6. Comparison of failed test specimens for R0.75 (LHS) and x ? h 0.75 (RHS). Notice the remarkably different hoop strain.

2.3. Experimental results for corner paths Three x ? h corner path tests were performed. Since one of the objectives of the new tests is to compare the stresses and strains at failure with those of the radial paths, in each case the maximum axial stress was selected to correspond to the axial stress at failure for one of the three corner paths. The prescribed stress paths are shown in red color in Fig. 3 where they are labeled by the proportionality number of the radial path they correspond to (e.g., 0.9). The induced strain paths are compared to the radial ones in Fig. 4. The corresponding true stress and logarithmic strain trajectories are included in Figs. A1 and A2 where they are seen to exhibit some small nonlinearities. The hoop and axial stress– strain responses appear in Fig. 8. The first part of each axial response in Fig. 8a comes from pure uniaxial loading so the three trajectories are seen to match quite well. During this leg (OB in Fig. 2) the induced eh is purely due to the Poisson effect and is negative. During the second leg of the corner paths (BC in Fig. 2) the axial stress is constant and this is responsible for the horizontal parts of the rx–ex responses in Fig. 8a. The corresponding rh–eh responses exhibit initial linear trajectories that end in tight knees followed by limited hardening. Each response develops a pressure maximum that is marked in Fig. 8b, and soon after each is attained the specimen bursts along a generator at relatively

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Fig. 7. Comparison of failed test specimens for R1.25 (LHS) and for the corresponding corner path x ? h 1.25 (RHS). Of interest is the failure mode change, and the very limited hoop strain that developed despite the axial rupture mode in the tube on the right.

small values of hoop strain. The stresses and average strains at the pressure maximum and at failure are marked once more with symbols (d) and (N) in Figs. 3 and 4 (and the true counterparts in Figs. A1 and A2). For example, for the 1.25 case failure occurred at an average value of eh = 0.25% in other words the hoop strain never reached a positive value. A photograph of this specimen is shown in Fig. 7 where its diameter is seen not to have changed very much (compare the ends that are gripped with the test section) and furthermore bulging in the failure zone is very limited if any at all. The average failure strains are {ex, eh} = {9.27%–0.25%}, which compare with {19.4%, 3.3%} for the radial path. This striking difference is also apparent in the strain trajectories in Fig. 4 (and in Fig. A2). Interestingly, the mode of rupture is also different in this case as for the R path the specimen failed by circumferential

Y.P. Korkolis, S. Kyriakides / International Journal of Plasticity 25 (2009) 2059–2080

a

40

Al-6260-T4

240

x

(ksi)

2067

x

1.25

30

(MPa) 180

0.9 x

0.75

20

120

10

60 x

0

0

2

4

6

0 10

8 (%) x

b

40

1.25

0.9

240

0.75

(ksi)

(MPa) 180

30

20

120 x

10

60 Al-6260-T4

0

-4

-2

0

2

4

(%)

0 6

Fig. 8. (a) Axial and (b) circumferential stress–strain responses recorded in the three x ? h corner path experiments. (d) in (a) corresponds to the limit pressure marked in (b).

localization and in the x ? h path it burst along a generator as shown in Fig. 7. One more difference that is illustrated in Fig. 3 is that the hoop stresses at both the limit pressure and failure are significantly higher than the value for the R path ({39.33, 38.33} ksi for x ? h vs. {26.95, 25.63} ksi for R). A similar observation can be made from the corresponding true stress trajectories in Fig. A1. The results for the other two x ? h paths are similar. For cases 0.9 and 0.75 the initial axial straining is smaller and this allows them to reach higher hoop strains at failure. However, once again as evidenced in Fig. 4, the failure strains are much lower than the values achieved in the corresponding R paths. Both cases failed by bursting along a generator, which is the same as the failure modes of the R paths (see comparison of 0.75 specimens in Fig. 6). The hoop stresses at the limit pressures and at failure are again significantly higher than the corresponding values for the R paths (see Figs. 3 and A1 and Table 1b). In addition one h?x path test was also performed that corresponds to R1.25 following the loading procedure described in the previous section (see Fig. 3). The first leg (OD in Fig. 2) represents in essence uniaxial loading in the hoop direction, which results in expansion of the tube diameter and axial contraction (see Fig. 4). During the axial loading leg (DE in Fig. 2), the presence of the constant pressure causes simultaneous growth of eh as well as ex as illustrated in Fig. 4. Although the generation of the stress and strain results is a welcome addition to the data base, the specimen failed at the grips, perhaps prematurely. Consequently comparison of failure stresses and strains to those of the radial path is not possible. We still see however that the axial stress reached before failure exceeded the value at failure of the corresponding R path. It was not possible to perform h?x experiments corresponding to R0.75 and R0.9 because the hoop stress at point D (see Fig. 2) is greater than the failure stress for pure circumferential loading (see Fig. 5, K-K, 2008a).

2068

Exp No.

R

x?h

h?x

a

rh jrx max

rxmax

rhf

rxf

ehL

exL

ehf

exf

ehfIl

exfIl

(ksi)

(ksi)

(ksi)

(ksi)

(%)

(%)

(%)

(%)

(%)

(%)

34.03 (38.82) 36.77 (43.58) 33.88 (41.45) 38.44 (41.20) –

25.50 (28.04) 24.56 (28.80) 30.49 (35.27) 29.78 (31.80) –

–

–

–

–

–

–

–

–

39.33 (34.30) n/a

31.89 (34.30) n/a

26.95 (32.39) –

33.61 (39.38) –

n/a

n/a

33.31 (38.28) 35.77 (42.92) 33.31 (41.47) 37.33 (40.42) 25.63 (32.669) 38.33 (41.71) n/a

25.00 (27.62) 24.56 (28.94) 30.02 (35.25) 29.78 (32.02) 31.98 (39.21) 31.89 (34.67) n/a

5.86 (5.64) 2.79 (2.75) 6.73 (6.49) 0.58 (0.60) 2.19 (2.18) -0.79 (-0.76) n/a

1.80 (1.78) 3.49 (3.45) 7.39 (7.14) 6.61 (6.39) 15.12 (14.06) 8.98 (8.60) n/a

6.08 (5.92) 3.08 (3.09) 7.54 (7.23) 0.88 (0.90) 3.26 (3.25) -0.25 (-0.25) n/a

1.80 (1.78) 3.49 (3.45) 7.85 (7.60) 6.61 (6.39) 19.38 (17.73) 9.27 (8.87) n/a

21.6 (19.6) 11.5 (10.9) 18.3 (16.8) 5.3 (5.2) 3.3 (3.2) 2.4 (2.4) n/a

1.8 (1.8) 4.6 (4.5) 11.0 (10.44) 6.7 (6.5) 29.5 (25.9) 10.0 (9.5) n/a

(ksi)

A19-1

0.75

–

–

A20-4

–

0.75

–

A20-2

0.9

–

–

A20-5

–

0.9

–

A17-2

1.25

–

–

A21-1

–

1.25

–

A21-3

–

–

1.25

1 ksi = 6.897 MPa.

rx jrh max

rhmax

(ksi)

Mode of failure A A A A C A n/a

Y.P. Korkolis, S. Kyriakides / International Journal of Plasticity 25 (2009) 2059–2080

Table 1b Summary of radial and corner path test results. Included are the engineering stresses (true in parentheses) and strains (logarithmic in parentheses) at the limit instability and at failure, and the mode of failure (A: axial and C: Circumferential rupture as per Fig. 7).

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x

0

100

(MPa)

200

50

40 (ksi)

3000 Wp 2000 (psi) 1500 1000 750 500 300 150 40

Yld2000-2D-III

300

(MPa)

30

200

20 100 10

Al-6260-T4

0 -10

0

10

0 20

30 x

40 (ksi)

Fig. 9. Loci of experimental points representing various levels of constant plastic work and corresponding contours based on the Yld2000-2D yield function (from K-K, 2008b).

3. Analysis We will use the same finite element models developed in K-K (2008a) to simulate the corner path experiments and to conduct calculations for additional cases. The models were generated in the nonlinear code ABAQUS using shell elements with reduced integration (S4R). Two models were developed one for rupture along a generator and the other for rupture around the circumference. A small initial imperfection was introduced in each model in the form of a groove of reduced wall thickness in order to trigger the instability. Internal pressure was applied using hydrostatic elements (F3D3 and F3D4) and Riks’ path-following scheme was employed in order to allow for the anticipated limit load instabilities to develop. More details about the FE models can be found in Section 4.1 of K-K (2008a). 3.1. Constitutive modeling A user-defined subroutine was developed for the constitutive model that uses isotropic hardening and is based on the non-quadratic, anisotropic yield function put forward by Barlat et al. (2003) in the manner outlined in K-K (2008b). Barlat et al. start from the isotropic yield function of Hosford (1972), which can be written in terms of the principal stress deviators as

js1  s2 jk þ j2s1 þ s2 jk þ js1 þ 2s2 jk ¼ 2rko :

ð3Þ 0

Anisotropy is introduced by two linear transformations, one applied to the first term (/ ) and the other to the second and third terms (/00 ):

 k  k  k / ¼ /0 þ /00 ¼ S01  S02  þ 2S001 þ S002  þ S001 þ 2S002  ¼ 2rko : ðS01 ; S02 Þ

ðS001 ; S002 Þ

ð4Þ 0

Here and are the principal values of the linearly transformed stress tensors S and S00 , respectively. These tensors are obtained from the stress deviator s and the stress tensor r by:

S0 ¼ C0 s ¼ C0 Tr ¼ L0 r and S00 ¼ C00 s ¼ C00 Tr ¼ L00 r 0

00

00

ð5Þ

where C , C ,T, L’and L are appropriate transformation tensors that allow introduction of the anisotropy (Barlat et al., 2003). Thus, for our 2-D stress state

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a

40

Al-6260-T4

Rupture

240

x

(ksi)

x

30

(MPa)

Analysis

180 20

x

Exp.

1.25

120

10

60 x

0

0 0

2

4

6

8

10 x

b

12 (%)

40 Exp.

Rupture

Analysis

1

(ksi) 30

240

(MPa)

Onset of Localized Wall Thinning

20

180 x

1.25

120

10

60 Al-6260-T4

0

-4

-2

0

2

4

6

0 8

(%) Fig. 10. Comparison of measured and calculated stress–strain responses for the x ? h 1.25 path experiment. (a) Axial and (b) circumferential results. (d) in (a) corresponds to the limit pressure marked in (b).

8 0 9 2 0 L11 > = < Sx > 6 0 S0y ¼ 4 L21 > ; : 0 > 0 Sxy

L012 L022 0

9 8 00 9 2 00 38 L11 > > = = < rx > < Sx > 00 7 6 00 S and ¼ 4 L21 0 5 ry y > > > ; : : 00 > rxy ; L066 0 Sxy

L0012

0

L0022 0

9 38 > = < rx > 7 0 5 ry > > : rxy ; L0066 0

ð6aÞ

0

where L and L00 are related to parameters ai (i = 1, 8) as follows:

8 0 9 2 8 00 9 3 2=3 0 0 L11 > L11 > > > > > > > 9 8 > > > > > > 7 6 0 > 00 > > > > > 1=3 0 0 a L L > > > > > 7 6 1 12 > = 1 < 12 = 6 < = < 7 00 7 a2 ¼ 0 1=3 0 and L021 ¼ 6 L 21 7 6 > 6 > > 00 > 9 > > > ; : > 7> 0 > > > > > 0 2=3 0 a > > > > L22 > 4 L22 > 5 7 > > > > > ; : 0 > : 00 ; 0 0 1 L66 L66

3 8 9 > > > a3 > > > 7 > 6 > a4 > > > 6 1 4 4 4 0 7 > = < 7 6 6 4 4 4 1 0 7 a 7 > 5> 6 7 > > 6 a > 2 2 0 5 > > > 4 2 8 > > > 6> ; : a8 0 0 0 0 9 2

2

2

8

2 0

ð6bÞ The exponent k is assigned the value of 8 as is typical for aluminum alloys. Further, since it has been demonstrated earlier (K-K, 2008a) that shear anisotropy does not influence the predictions, it will be neglected here too (i.e., a7 = a8 = 1). The remaining anisotropy parameters ai (i = 1, 6) adopted are taken from the hybrid calibration scheme outlined in Korkolis and Kyriakides (2008b). They were arrived at by trying to closely match the work contours generated from the nine radial stress paths in K-K (2008a) while simultaneously trying to optimize the structural performance of the FE models in predicting the rupture. The parameters arrived at evolve with plastic work as shown in Table 3 of K-K (2008b). The resulting work contours are compared to the experimental data in Fig. 9 ((sx, sh) are true stresses).

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Fig. 11. Calculated deformed configuration from the x ? h 1.25 path just after the onset of localization (marked with  in Fig. 10(b)).

x 0

50

100

150

(MPa)

200

250

40

Al-6260-T4 250 (ksi)

(MPa) x 1.25

30

200

150 20 0.9 x

0.5

100

0.75 1.75

10

Exp.

50

1.25

LL F

0

0 0

10

20

30

40 x

(ksi)

Fig. 12. Engineering stress paths prescribed in numerical calculations. Marked are the stresses at the limit load and at failure (experimental blue, predicted red and green). (For interpretation of color mentioned in this figure the reader is referred to the web version of the article.)

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Table 2 Summary of radial and corner path results from numerical simulation. Included are the stresses and strains at the limit load instability and at failure, and the mode of failure (A: axial and C: circumferential rupture). Ra

x?h

h?x

rhmax (ksi)

0.5 – 0.75 – 0.9 – 1.25 – – 1.75 – –

– 0.5 – 0.75 – 0.9 – 1.25 – – 1.75 –

– – – – – – – – 1.25 – – 1.75

34.12 35.65 33.84 36.55 33.15 37.21 – 38.20 – – 37.66 –

rx jrh max

rh jrx max

rxmax

rhf

rhx

ehL

exL

ehf

exf

(ksi)

(ksi)

(ksi)

(ksi)

(%)

(%)

(%)

(%)

Mode of failure

17.08 16.56 25.42 25.51 29.86 29.67 – 31.85 – – 32.11 –

– – – – – – 26.04 – 25.34 18.72 – 18.49

– – – – – – 32.50 – 33.27 32.56 – 33.89

33.57 34.76 33.57 36.12 33.00 36.55 25.21 38.20 25.34 18.05 37.10 18.49

16.81 16.56 25.28 25.51 29.72 29.67 31.53 31.88 31.61 31.44 32.11 31.89

8.55 8.63 7.91 4.77 7.91 2.76 1.11 1.47 3.39 0.10 0.55 0.26

0.14 -0.18 1.89 3.41 6.78 7.49 13.44 10.00 4.62 8.87 10.97 8.63

9.03 9.20 8.72 4.93 8.83 3.08 1.23 2.36 3.51 0.10 1.91 0.26

0.34 0.34 2.05 3.37 7.15 7.54 14.35 10.36 4.93 9.19 11.70 8.91

A A A A A A C A C C A C

(ksi)

1 ksi = 6.897 MPa.

3.2. Numerical results The FE models are now used to simulate the experiments and their performance is evaluated by direct comparisons of the two sets of results. Additional runs are subsequently performed in order to span more broadly the biaxial stress and strain spaces examined and help extract more general trends. 3.2.1. x ? h paths We will discuss the main characteristics of the x ? h path simulations by comparing experimental and predicted results from the 1.25 test. Fig. 10 shows comparisons of the measured and calculated eh is averaged over the circumference). axial and circumferential stresses and strains at mid-span ( The prescribed stress path is included in Fig. 12 and the induced strain paths are compared to the experimental ones in Fig. 13. Fig. 10a demonstrates that during the first branch of this loading path the axial stress replicates the experimental one as expected for uniaxial loading. Fig. 13 shows that the induced hoop strain follows a slightly different trajectory than the measured one, apparently

Exp.

Al-6260-T4

0.5

Anal.

8 (%) 0.75 1.25

4

0.9 1.25 1.75

0

-4 0

4

8

12 x

(%)

Fig. 13. Comparison of calculated and measured engineering strain paths from various corner paths. Marked are the average strains at the limit load and at failure.

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because of some deficiency in the anisotropic yield function adopted in the neighborhood of the rx axis. (Observe that as shown in Fig. 9, this area of the yield function was not probed sufficiently in our original radial path tests used to calibrate the Yld2000 function. This however, should not have a significant impact on the discussion that follows.) Because of this small difference, the calculated rh–eh response in Fig. 10b starts at a slightly smaller value of eh but follows a similar trajectory to that of the experimental one. Indeed, the position of yielding is reproduced well as is the limited hardening that follows. The tube experiences some mild circumferential expansion but a limit load instability develops at a strain of about 1.5%, a value that is somewhat larger than the corresponding experimental one (see also Fig. 13). Beyond this point, the tube starts of develop some mild non-axisymmetric bulging that precipitates localized wall thinning in the axial groove. This is responsible for the sharp drop in pressure that is seen to occur at a strain of about 2.4%. As in our preceding work on the subject, we identify this point with the onset of rupture. The increasing circumferential strain in the descending part of the response, drawn in dashed line, is driven by the localizing deformation in the grooved zone illustrated in the deformed configuration shown in Fig. 11 (corresponds to the point marked on this part of the response with a symbol ‘‘”). In other words, the axial rupture observed in the corresponding experiment is reproduced by the model. The stresses and strains at the limit load and at rupture are listed in Table 2 and are marked in Figs. 12 and 13 with red bullets (d, N). It is important to note that the limit and rupture stresses are both close to the experimental ones (see Table 2 and blue bullets - -d, N- - in Fig. 12). In other words, the analysis reproduces the significant path-dependence of the failure stresses observed in the experiment. The other two x ? h path experiments (0.75 and 0.9) were simulated in a similar fashion. The prescribed stress paths are included in Fig. 12, the induced strain paths in Fig. 13 while the rx–ex and rh– eh trajectories appear in Fig. 14a and b, respectively. The general trends of the predictions follow those of the experimental results. Both cases ruptured along a generator, which is in agreement with the experiments. The delay in yielding during the pressurization phase of the response observed in the experiments is reproduced by the simulations. So is also the relatively low rate of hardening of the inelastic part of this branch. Consequently, the stresses at the limit loads and at the points of rupture are in close agreement with the measured values (see Tables 1b and 2 and Fig 12). Furthermore, both the experimental and numerical results indicate that re-yielding in the circumferential direction is delayed until a stress level develops that approximately corresponds to the maximum value reached during the axial stressing phase of the path. This ‘‘delayed” re-yielding is at least partly responsible for the much higher failure stresses of the x ? h paths compared to the corresponding radial ones. Incidentally, we observe that since our calculations are based on pure isotropic hardening plasticity, the level of agreement between the measured and calculated stress–strain trajectories achieved in these simulations would indicate that this particular Al alloy must experience combined isotropic and kinematic hardening with the former dominating. This is said realizing that any deformation-induced shape changes of the yield surface, such as those purported to take place in the direction of loading, that may affect other performance criteria of the model (e.g., the onset of instability), are obviously not accounted for by this constitutive model. The calculated strain paths are seen in Fig. 13 to follow the experimental ones quite well. However, as was the case for the 1.25 experiment, both the onset of the limit loads and the points of rupture are delayed somewhat compared to the experiments. It is reasonable to assume that this disagreement may be related to local changes in the yield surface shape not captured by our isotropic hardening plasticity model. This is suggested despite the deformation-induced adjustment of the anisotropy and the additional corrections to the yield functions performed in K-K (2008b), as such effects are most probably path dependent. In summary then, the performance of the x ? h path simulations and of the constitutive model as calibrated can be declared quite satisfactory as they support both qualitatively and quantitatively the experimental observations that for this type of corner paths the strains as well as the stresses of the limit states are strongly path dependent. For completeness two additional pairs of cases were run numerically and will be discussed: R1.75 and R0.5 along with the corresponding x ? h paths. The results from the x ? h cases are included in Figs. 12–14. Both sets of results fall in line with the other three cases. Thus, for 1.75 the axial strain induced in the first leg of the path is larger than for 1.25 as is the corresponding contraction of the

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a

40

Al-6260-T4

1.25

240

x

x

1.75

(ksi) 30

(MPa) 180

0.9 0.75

20

120 0.5

10

60 x

0

0

2

4

6

8

10

0 14

12 (%) x

b (ksi)

40 240 0.5

(MPa)

30 0.75

180

0.9 1.75 1.25

20

120

10

60 Al-6260-T4

0

-4

-2

0

2

4

6

8

(%)

0 10

Fig. 14. Stress–strain responses from numerical simulations of various x ? h corner paths. (a) Axial and (b) circumferential results. The symbol (d) in (a) corresponds to the limit pressure marked in (b).

circumference (Fig. 13). A similar delay in yielding observed for the other cases during the pressurization phase of the path is seen to take place again (Fig. 14). The specimen develops a limit load instability close to 1% average eh and ruptures along a generator. The stress and strain trajectories are compared to those of the corresponding radial path in Fig. 15. Once again a very significant difference between the limit and failure stresses from the two paths is observed in Fig. 15. Interestingly, Fig. 16 shows that R1.75 induced a nearly pure axial straining that lead to circumferential rupture at (exf, ehf) of (9.19%, 0) (see Table 2). These values compare with (exf, ehf) of (11.7%, 1.91%) for the x ? h path. In other words, in the radial path rupture occurs at smaller strains reversing the trend seen in the other cases (0.75, 0.9 and 1.25). The case of x ? h 0.5 develops no plastic strain during the axial phase of the path. Furthermore, this path engages the yield surface in the zone of relatively small curvature (see Fig. 9) as does the corresponding R path. (We note that experimental results for R0.5 presented in K-K (2008a) are in good agreement with the present simulation). Consequently, during the pressurization phase an essentially purely hoop strain trajectory is traced that is nearly congruent to that of the R path (Fig. 16). The values of the limit and rupture strains from the two cases are also almost coincident which indicates that the strain path plays a decisive role in influencing rupture. The corresponding stress values from the two paths (Fig. 15 and Table 2) differ slightly but the difference is by far the smallest amongst the five cases compared in this set. 3.2.2. h ? x paths The h ? x 1.25 experiment was also simulated numerically and the induced stress and strain paths are included in Figs. 12, 13, 15 and 16. In addition, the axial and circumferential stress–strain results

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are compared to the corresponding measurements in Fig. 17. The calculated stress–strain paths are very close to those measured. The end of the experimental strain path in Fig. 13 represents the point at which the specimen failed prematurely at the grips. In the simulation, rupture was circumferential and occurred at strains that are slightly lower than those of the end of the experimental trajectory. From this we conclude that in the experiment the tube must have been very close to rupture before the test was terminated by end-failure. A deformed configuration illustrating circumferential localization at mid-span of the specimen is shown in Fig. 18. It corresponds to the point marked on the descending part of the calculated response in Fig. 17b with the symbol ‘‘”. The stresses at rupture in the simulation are seen in Figs. 12 and 15 to be in good agreement to those of the experiment also. Included in Figs. 15 and 16 are the stress and strain paths for the corresponding R path. The critical stresses of the h ? x path are close to those of the R path but the strains differ considerably (as indeed was also the case in the experimental results in Figs. 3 and 4). One additional pair of cases was run numerically comparing R1.75 and the corresponding h ? x path. The calculated stress and strain paths are compared in Figs. 15 and 16 while the axial and circumferential stress–strain responses for the h ? x run are included in Fig. 17. This particular radial path results in almost no plastic hoop strain and consequently the strain path is along the ex axis or, in other words, plastic deformation is essentially limited to the axial direction. The axial response develops a limit load at a strain of 8.87% and ruptures circumferentially at a strain of 9.19%. These values are of course significantly smaller than the failure strains for the 1.75 x ? h path included in Figs. 15 and 16 as are the failure stresses. Interestingly, the h ? x path induces a strain trajectory that is very similar to that of the R path. Apparently, despite the different stresses the model develops very limited plastic circumferential deformation during the pressurization phase. During the rx leg of the path the material is initially elastic and plastic deformation, when it occurs, is limited to the axial strain. The axial response develops a limit load at exL = 8.63% and ruptures circumferentially at a strain of 8.91%. Both of these values are very close to those of the R1.75 path while the maximum axial stress achieved is slightly higher than that of the R path. This supports once more the premise that the strain path is a major factor in deciding rupture. Surprisingly, the two rupture strains are also very close to the corresponding eh

x

0

50

100

150

(MPa)

200

250

40

Al-6260-T4 250 (ksi)

(MPa) 30

x 1.25

200

x 1.75

150

R0.5

20

x

1.75 100

R1.25

10 x x

1.25

50

0.5

R1.75

0

0 0

10

20

30

40 x

(ksi)

Fig. 15. Engineering stress paths prescribed in numerical calculations. Marked are the stresses at the limit load and at failure.

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Al-6260-T4

8

x

R0.5

0.5

LL F

(%)

4

x 1.25

1.25

x x 1.75

0

R1.25

R1.75 x

-4

0

4

8

1.75

12 x

(%)

16

Fig. 16. Strain paths corresponding to stress paths in Fig. 15.

a

40 Al-6260-T4

240 (ksi)

(MPa)

30

Failure at grip

1.25 Exp.

180 1.25 Analysis

20

x

120

1.75 Analysis

10

60 x

0

0

2

4

6

8

0

10

12 (%)

b

40

Failure at grip

x

(ksi)

1.25 Exp.

1.25 Analysis

240 x

30

Onset of Localized Wall Thinning

(MPa) 180

1

20

1.75 Analysis

120 x

10

60 Al-6260-T4

0

-2

0

2

4

6 x

8 (%)

0

Fig. 17. Stress–strain responses from numerical simulations of various x ? h corner paths. (a) Axial and (b) circumferential results. The symbol (d) in (a) corresponds to the limit pressure marked in (b).

values of the R0.5 and x ? h 0.5 paths, which ruptured along a generator. In other words, despite the difference in the mode of rupture, the critical strains in the two pairs of calculations are similar.

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Fig. 18. Calculated deformed configuration from the h ? x 1.25 path showing localization around the circumference at the specimen mid-span (marked with  in Fig. 17(b)).

Finally, it is also worth noting the significant difference between the critical stress levels of the 1.25 and 1.75 h ? x and x ? h paths observed in Fig. 15. As mentioned above, the critical stresses of the former paths are close to those of the R paths while the ones for the x ? h paths are significantly higher. One contributor to this difference may be the fact that the stress levels of the initial rh legs of the h ? x paths are significantly lower than the axial stresses achieved during the rx legs of the x ? h paths. Consequently, the induced rh stresses did not cause significant expansion of the yield surfaces whereas, as observed earlier, the rx legs did. Accordingly, during the axial stressing legs of the h ? x paths plastic deformation recommences earlier and apparently this also leads to failure at lower stresses levels. 4. Discussion and conclusions In an effort to establish forming limits for tube hydroforming, the path dependence of failure of Al6260-T4 tubes loaded under combined internal pressure and axial tension has been examined using experiment and analysis. In particular, the stress and strain paths and the values at the onset of failure in radial stress paths and in two types of corner stress paths are compared. The first corner path involved uniaxial loading in the axial direction to a stress level that corresponds to the failure stress of a radial path, followed by pressurization to failure while keeping the axial stress constant. The second corner path involved an initial pure hoop stress loading, again to a stress level that corresponds to the failure hoop stress under radial stressing, followed by axial tension keeping the hoop stress constant. We note that in such experiments two instabilities typically manifest: first a limit load

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instability develops either in pressure or in load that in several of the experiments was associated with the formation of a mild axisymmetric bulge at mid-span. The way the loading was controlled enabled us to follow the deformation beyond the limit load instability. Following some axisymmetric deformation, the tubes ruptured catastrophically either along a generator or circumferentially. Failure occurs suddenly and dynamically at a location that is not exactly known apriori. Consequently the recorded strains and stresses stop at the onset of failure. As expected, the strain paths induced by the corner paths are significantly different from those of the corresponding radial paths and so are the failure strains (this holds for both the engineering and the logarithmic versions of the trajectories). This of course confirms the well-established path dependence of FLDs that has originated mainly from work on sheet metal. In addition, in one case in which in the radial path test the tube ruptured circumferentially, the corresponding corner path ruptured along a generator. Interestingly, the corner path stresses at the onset of failure were also found to differ from those of the radial paths to various degrees (for both the engineering and true stresses). Realizing that the stresses reported here represent the onset of failure, the results raise at least a question on the notion that FLSDs are always path independent. Since this aspect of the problem was not the main focus of the present investigation this issue deserves a more thorough study. This difference is at least partly caused by the fact that the corner paths chosen introduced significant prestraining to the specimens, something that previously was not sufficiently investigated. This tended to expand the yield surface, which in turn delayed re-yielding during the second legs of the corner paths (see also similar results in Fig. 6b in Wu et al., 2005 and Fig. 10 in Yoshida et al., 2005). The burst experiments were subsequently simulated numerically using FE models in which rupture was induced by small groove imperfections. A custom isotropic hardening plasticity model based on the Barlat et al. (2003, Yld2000-2D) anisotropic yield function was implemented using the calibration found previously (K-K, 2008b) to result in quite accurate predictions of failure for nine radial path tests on the same Al alloy tubes. The simulations reproduced quite well the stress and strain paths of the corner tests too. The predicted strains at failure were somewhat larger than the measured values while the failure stresses were very close to the experimental ones. Thus, the simulations confirmed the strong path dependence of both the failure strains and stresses exhibited in the experiments. Simultaneously it was demonstrated that for this Al alloy the radial path calibration of the constitutive model was also adequate for non-proportional loadings. (It is worth noting that for this alloy isotropic hardening dominated the evolution of subsequent yield surfaces – e.g., see Fig. 8b.) Having confirmed the fidelity of the FE and constitutive models, they were subsequently used to perform additional calculations comparing corner and radial paths in order to develop a broader picture of the failure trends. The results confirmed that the amount of plastic prestraining in either of the two directions plays an important role in the observed difference of the failure strains and stresses between radial and corner stress paths. Thus for example, in the case of paths with limited plastic prestraining the strains and stresses at failure were very close for the corner and radial paths. By contrast, when the prestraining is large the opposite is true. In these cases the numerical results also showed a switch in the failure mode between radial and corner paths. Finally, the simulations also confirmed that plane strain inflation and biaxial loading that results in pure axial straining (no hoop strain) result in the smallest failure strains. Indeed, the strains at failure in these two paths were found to be very similar despite the distinctly different modes of rupture exhibited by the two cases (axial in the first and circumferential in the second).

Acknowledgements The authors acknowledge with thanks financial support of this work received from the National Science Foundation through Grant DMI-0140599 and supplementary funding provided by G.M. with Robin Stevenson as coordinator. Special thanks to Alcoa and Edmund Chu for providing initial seed funding for the project and the tubes analyzed and tested. Discussions with Frederic Barlat were instrumental in the adoption of the Yld2000-2D yield function in this body of work.

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Appendix A. The work performed was motivated by tube hydroforming where the active variables are the internal pressure and the axial force. Consequently, these were also the prescribed variables in both the radial and corner path tests performed (see K-K, 2008a). In order to enable direct comparison with these experiments the same variables were prescribed in the numerical simulations. Thus, the results and the conclusions regarding the performance of the constitutive framework adopted are tailored to hydroforming applications. At the same time, the reported dependence of the onset of failure on the loading paths has implications in other forming processes where failure values are usually reported in terms of true stresses. For this reason the experimental loading histories and the induced strain trajectories were converted to true stresses (sx, sh) and logarithmic strains (ex, eh) and are reported in Figs. A1 and A2, respectively. These are evaluated from the force, pressure and extensometer

x

0

50

100

150

200

(MPa) 250

300 300

Al-6260-T4

LL F

40

250 (MPa)

(ksi)

R0.75

30

200

x 1.25

150

R0.9

20

R1.25

100 10 50

x 0.75

0.9

1.25

0

0 0

10

20

30

40 x

(ksi)

Fig. A1. Corner paths prescribed in the experiments, and the corresponding radial paths from K-K (2008a) in terms of true stresses. Marked are the limit (d) and the failure (N) stresses.

8 e

LL F

R0.9 R0.75

(%) 4

1.25

x 0.75

R1.25

0 0.9 x

1.25 Al-6260-T4

-4 -4

0

4

8

12

16 ex (%)

20

Fig. A2. Strain paths traced in the corner experiments, and in the corresponding radial paths from K-K (2008a) in terms of logarithmic strains. Marked are the average strains at the limit load (d) and at failure (N).

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measurements described in Section 2.1. Marked on the trajectories are the points corresponding to the limit load instabilities and those at the onset of failure. References Arrieux, R., 1995. Determination and use of the forming limit stress diagrams in sheet metal forming. J. Mater. Process. Tech. 53, 47–56. Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.-H., Chu, E., 2003. Plane stress yield function for aluminum alloy sheets-part I: theory. Int. J. Plasticity 19, 1297–1319. Davies, R., Grant, G., Herling, D., Smith, M., Evert, B., Nykerk, S., Shoup, J., 2000. Formability investigation of aluminum extrusions under hydroforming conditions. SAE Paper 2000-01-2675. Graf, A., Hosford, W.F., 1993. Effect of changing strain paths on forming limit diagrams of Al 2008-T4. Metall. Trans. 24A, 2503– 2512. Graf, A., Hosford, W.F., 1994. The influence of strain-path changes on forming limit diagrams of Al 6111 T4. Int. J. Mech. Sci. 36, 897–910. Gronostajski, J., 1984. Sheet metal forming-limits for complex strain paths. J. Mech. Working Tech. 10, 349–362. Hosford, W.F., 1972. A generalized isotropic yield criterion. ASME J. Appl. Mech. 309, 607–609. Hosford, W.F., 1979. On yield loci of anisotropic cubic metals. In: Proceedings of the 7th North American Metalworking Research Conference, Society of Manufacturing Engineers, Dearborn, MI, pp. 191–196. Hosford, W.F., Caddell, R.M., 1993. Metal Forming: Mechanics and Metallurgy, second ed. Prentice Hall, Englewood Cliffs, NJ. Karafillis, A.P., Boyce, M.C., 1993. A general anisotropic yield criterion using bounds and a transformation weighting tensor. J. Mech. Phys. Solids 41, 1859–1886. Kleemola, H.J., Pelkkikangas, M.T., 1977. Effect of predeformation and strain path on the forming limits of steel copper and brass. Sheet Metal Ind., 591–599. Korhonen, A.S., Manninen, T., 2008. Forming and fracture limits of austenitic stainless steel sheets. Mater. Sci. Eng. A488, 157– 166. Korkolis, Y.P., Kyriakides, S., 2008a. Inflation and burst of anisotropic aluminum tubes for hydroforming applications. Int. J. Plasticity 24, 509–543. Korkolis, Y.P., Kyriakides, S., 2008b. Inflation and burst of anisotropic aluminum tubes. Part II: an advanced yield function including deformation-induced anisotropy. Int. J. Plasticity 24, 1625–1637. Kuwabara, T., Yoshida, K., Narihara, K., Takahashi, S., 2005. Anisotropic plastic deformation of extruded aluminum alloy tube under axial forces and internal pressure. Int. J. Plasticity 21, 101–117. Lloyd, D.J., Sang, H., 1979. The influence of strain path on subsequent mechanical properties – orthogonal tensile paths. Metall. Trans. 10A, 1767–1772. McCandless, A.J., Bahrani, A.S., 1979. Strain paths, limit strains and the forming limit diagram. In: Proceedings of the 7th North American Metalworking Research Conference, Society of Manufacturing Engineers, Dearborn, MI, pp. 184–190. Muschenborn, W., Sonne, H.-M., 1975. Influence of the strain path on the forming limits of sheet metal. Arch. Eisenhuttenwes. 9, 597–602 (in German). Stoughton, T.B., 2000. A general forming limit criterion for sheet metal forming. Int. J. Mech. Sci. 42, 1–27. Wagoner, R.H., Laukonis, J.V., 1983. Plastic behavior of aluminum-killed steel following plane-strain deformation. Metall. Trans. 14A, 1487–1495. Wu, P.D., Graf, A., MacEwan, S.R., Lloyd, D.J., Jain, M., Neale, K.W., 2005. On forming limit stress diagram analysis. Int. J. Solids Struct. 42, 2225–2241. Yoshida, K., Kuwabara, T., Narihara, K., Takahashi, S., 2005. Experimental verification of the path-independence of forming limit stresses. Int. J. Forming Proc. 8, 283–298. Yoshida, K., Kuwabara, T., 2006. Experimental verification of path-dependence of forming limit stress for a steel tube. In: Proceedings on Plasticity ’06, Halifax, Canada, pp. 106–108. Yoshida, K., Kuwabara, T., 2007. Effect of strain hardening behavior on forming limit stresses of steel tube subjected to nonproportional loading paths. Int. J. Plasticity 23, 1260–1284.