Inflation and burst of anisotropic aluminum tubes for hydroforming applications

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International Journal of Plasticity 24 (2008) 509–543 www.elsevier.com/locate/ijplas

Inflation and burst of anisotropic aluminum tubes for hydroforming applications Yannis P. Korkolis, Stelios Kyriakides * Research Center for Mechanics of Solids, Structures & Materials, WRW 110, C0600, The University of Texas at Austin, WRW 110, Austin, TX 78712, United States Received 23 February 2007; received in final revised form 6 July 2007 Available online 2 August 2007

Abstract Burst due to internal pressure and axial load is a common mode of failure in tube hydroforming. Comparison between numerical simulations and hydroforming experiments on aluminum tubes have indicated that localized wall thinning and burst can be very sensitive to the constitutive description employed for the material. This observation motivated the present combined experimental/analytical effort. The investigation assesses the performance of different anisotropic yield functions in predicting the response and burst of tubes loaded under combined internal pressure and axial load. The experiments involved Al-6260-T4 tubes loaded under combined internal pressure and axial tension or compression. Several radial paths in the axial-circumferential stress space were prescribed until failure occurred. In all cases deformation was initially uniform until a limit load instability was attained. For axial tension dominant loading paths, the tube eventually bulged axisymetrically, and localization resulted in circumferential rupture. For hoop stress dominated paths, the axisymmetric bulge evolved into a non-axisymmetric one leading to localized thinning and rupture along a tube generator. The results were used to calibrate the non-quadratic anisotropic yield functions of Hosford and Karafillis–Boyce. These were then employed in FE simulations of the experiments. The predicted structural responses are generally, but not universally, in good agreement with the experimental results, while the predicted strains at the onset of rupture are somewhat larger than the values measured. It is demonstrated that discrepancies between experimental and predicted results may be caused by inadequate representation of the anisotropy by the constitutive models adopted, and by the omission of deformation-induced anisotropy. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Aluminum; Tube hydroforming; Burst; Anisotropy

*

Corresponding author. Tel.: +1 512 471 4167; fax: +1 512 471 5500. E-mail address: [email protected] (S. Kyriakides).

0749-6419/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2007.07.010

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1. Introduction Tube hydroforming is a process in which a relatively thin-walled circular tube is inflated by internal pressure and is forced to conform to the shape of a stiff die surrounding it. Inflation causes wall thinning, and consequently failure by bursting is a major limiting factor of the process. Burst can be delayed by compressing the tube as it is inflated (axial feed) (Chu and Xu, 2004a). Axial compression is itself limited by buckling of the part either in an overall manner or by local wrinkling. Thus, establishing the working envelope in the pressure–axial compression plane is a major design requirement for the manufacture of parts by this process (Chu and Xu, 2004b). The use of aluminum alloys in the place of steel components in automotive applications has seen significant increase during the last 10 years. In this regard, hydroforming of aluminum tubes is a very desirable manufacturing alternative to sheet metal forming. The present work is part of a larger project that aims to understand the limitations and develop the necessary mechanical tools that enable the use of modeling in the design of aluminum tube hydroforming. These objectives are challenged by the more complex constitutive behavior of aluminum alloys and by the smaller ductility they exhibit by comparison to steel. The present work is concerned with burst failure as a limiting factor in hydroforming. Comparison between numerical simulations and hydroforming experiments performed on aluminum tubes has shown that localized wall thinning and burst are very sensitive to the constitutive description employed for the material, and are further complicated by contact with the die. The study involves a combined experimental/numerical effort that aims to assess the performance of different constitutive models in predicting burst in the simpler setting that excludes the die. Existing knowledge of constitutive modeling of aluminum has been mainly generated from the analysis of sheets produced by rolling. Since seamless aluminum tubes are typically extruded and/or drawn, they can have significantly different crystallographic structure to that of sheets (Rollett and Wright, 1998; Guan et al., 2006). It is thus imperative that constitutive models be calibrated and evaluated vis-a`-vis experimental results derived from tubes similar to ones used in hydroforming. It is also known that the cylindrical geometry of hydroformed tubes can affect the onset of localization and possibly the failure strains (Stout and Hecker, 1983). Furthermore, forming limits are known to be influenced by the loading paths followed (Graf and Hosford, 1993; Stoughton, 2000). All of these issues make it necessary that failure strains for tube hydroforming be developed from tubes tested under stress and deformation paths that are close to those seen during hydroforming. The present experimental program involves a series of tests on Al-6260-T4 tubes loaded under combined internal pressure and axial tension or compression. Several radial paths in the axial–circumferential stress space are prescribed until failure occurs [experiments of this type have been reported in Davies et al. (2000), Yoshida et al. (2005) and Kuwabara et al. (2005)]. The results are then used to first evaluate the suitability of a class of nonquadratic yield functions as constitutive models for reproducing the material response under the biaxial loading histories considered. The best performing constitutive models are then adopted in finite element models that have the capacity to simulate the onset of localization that leads to rupture. The models are used to simulate the experiments and establish the onset of failure. The results of the experiments and the analyses will be used to draw conclusions about strengths and weaknesses of the models adopted.

Feedback Signal Load Cell Pressure Transducer

Span

Signal Generator

Upper Grip Span Set Point

Span Set Point

Command Signal Test Specimen Circumferential Extensometer Solid Insert

Servo Control

Press. Control

Strain Gages Axial Extensometer

Volume Control Locking Assembly Servovalve Seals

Spacer

Pressurizing Fluid

Lower Grip Servovalve

Pressure Intensifier

LVDT

Actuator

Hydraulic Power

511

Fig. 1. Experimental set-up used to load tubes under combined internal pressure and axial load.

Volume Feedback

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Amplifiers

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2. Experimental 2.1. Experimental setup and procedure The experimental program involved testing seamless Al-6260-T4 tubes under combined axial tension/compression 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. The pressurization unit has an independent closed-loop control system shown schematically inside the dashed boundary in the figure. By connecting the two systems through feedback, tests can be performed under purely displacement or load control. In the present tests, the tubes were pressurized under volume control. The axial actuator was run under load control and through feedback it was made to follow the induced pressure at a prescribed ratio producing a radial trajectory in the engineering stress plane rx–rh (x and h are, respectively, the axial and circumferential directions). 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 in. 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

Fig. 2. Tube ductile rupture from experiment A20-1 (a = 0.5).

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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 as shown in Fig. 2. The thickness was also measured at the same locations using an ultrasonic thickness gage. The tubes had a small amount of wall eccentricity, which 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 as shown in the figure. In addition, several thickness measurements were performed in the area of failure using an ultrasonic thickness gage as described in the next section. A solid metal insert was placed inside the cavity of the specimen to reduce its size and consequently the energy stored in the system. The cavity was filled and pressurized with a white mineral oil using the unit shown schematically in Fig. 1. It consists of the 10,000 psi (690 bar) pressure intensifier mentioned above. The intensifier operates on standard 3000 psi (200 bar) hydraulic power, has its own independent closed-loop control system, and was run under volume control. The pressure was monitored via a 10,000 psi (690 bar) pressure transducer whose output was amplified so that at full scale it had the standard output of 10 V. The testing machine was run in load-control using the pressure signal as the command signal. The tubes were loaded along a radial stress path such that rx ¼ arh ;

a ¼ const:

ð1Þ

Let the force measured by the load cell of the machine be F, and the internal pressure be P. The axial and circumferential stresses are then 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. ((sx, sh) that appear in several of the figures that follow are the corresponding true stresses.) Thus, for example, if the load cell reads zero force it means that the specimen is reacting the pressure end-load and consequently develops an axial stress of PR/2t (equivalent to stress state in a pressurized tube that is closed with end-caps). Using (1) and (2), one can find a relationship between the pressure transducer signal (10 V  10 ksi) and the command signal for the axial load (±10 V  ±50 kips). Experiments were performed for 0.2 6 a 6 1.25. The volume-controlled pressurization of all tests was conducted at the same rate, in other words oil was supplied to the cavity at a constant volume rate. In view of this, the strain rate varied to some extent in the two directions and from test to test. Typical strain rates were of the order of 104 s1. 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 The diameter and wall thickness of each test specimen were measured prior to each test and the average values of the measurements are listed under D and t, respectively, in Table 1.

514

Experiment number

D (in.)

t (in.)

2L (in.)

No (%)

a

rh max (ksi)

rx jrh max ðksiÞ

rh jrx max ðksiÞ

rx max (ksi)

ehL (%)

exL (%)

ehf (%)

exf (%)

ehfjl (%)

exfjl (%)

A17-4 A17-3 A19-3 A17-1 A20-1 A19-1 A20-2 A19-2 A17-2 A18-3a

2.359 2.359 2.358 2.359 2.358 2.356 2.359 2.358 2.359 –

0.0800 0.0800 0.0800 0.0801 0.0800 0.0800 0.0800 0.0800 0.0800 –

13.38 13.50 13.38 13.38 12.63 13.00 12.63 13.25 13.38 –

0.69 0.63 0.69 0.63 0.56 0.63 0.75 0.69 0.69 –

0.2 0.1 0 0.25 0.5 0.75 0.9 1.0 1.25 –

31.21 32.07 32.87 33.58 33.68 34.03 33.88 – – –

6.19 3.02 0.185 8.58 16.87 25.50 30.49 – – –

– – – – – – – 32.27 26.95 0

– – – – – – – 32.40 33.61 32.55

18.1 18.0 11.8 9.3 7.0 5.8 6.7 5.0 2.2 –

11.9 11.0 5.7 2.2 0.2 1.8 7.4 10.6 15.1 19.5

22.1 19.1 12.6 9.7 7.4 6.1 7.5 9.8 3.3 –

14.4 11.0 5.7 2.2 0.2 1.8 7.9 12.8 19.4 24.7

44.0 35.3 29.7 25.5 19.4 21.6 18.3 11.9 3.3 –

14.4 11.0 5.7 3.6 0.0 1.8 11.0 27.7 29.5 –

1 in. = 25.4 mm, 1 ksi = 6.897 Mpa. a Uniaxial test on an axial strip of the material.

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Table 1 Summary of biaxial burst test results. Included are the geometric characteristics of the tubes and the stresses and strains at the limit load instability and at failure

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The tubes exhibited a small amount of wall eccentricity that is quantified by the following variable: tmax  tmin No ¼ : ð3Þ tmax þ tmin Values of No resulting from the measurements appear in Table 1; they range from 0.56% to 0.75%. The stress–strain response of the material was measured from uniaxial tension tests on strips cut from the axial direction. Fig. 3a shows one of the measured responses (rx–ex) while others were very similar. The maximum stress occurred at a strain of 19.5%, following which the strip developed a diffuse inclined neck. The stress–strain response in the circumferential direction (rh–eh) was also measured in a test in which a tube was pressurized so that the axial stress was maintained at zero level (i.e., F = PpR2; Kyriakides and Yeh, 1988). Initially, the specimen inflated so that the test section remained uniform. However, eventually it developed an axisymmetric bulge followed by localized thinning along a generator that led to ductile rupture. The bulge and the rupture were both limited to a section about one tube diameter long (see

a

240 σθ- εθ

σ 30 (ksi)

σ

200

σx - εx

(MPa) 160

20 120 80 10 40 Al-6260-T4

0

0

5

10

15

20

30

b

200

Shear

σ

0

25 ε (%)

Hoop

e

σe

160 (MPa)

(ksi) 20

120

Axial

80 10 40 Al-6260-T4

0

0

0.2

0.4

0.6

0.8

0

1 εep (%)

Fig. 3. (a) Axial and circumferential stress–strain responses of Al-6260-T4 tubes tested to failure. (b) Axial, circumferential and shear equivalent plastic stress–strain responses.

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photograph for a = 0 in. Fig. 4). This response, included in Fig. 3a, is somewhat higher than the axial one indicating that the tube exhibits some yielding anisotropy. A more striking difference is the fact that this response developed a maximum at a strain of only 11.8%. Although the tube was initially in a state of uniaxial stress, the formation of the bulge and the onset of wall thinning generate a more complex state of stress in the neighborhood of the localization, which precipitate the earlier development of the load maximum (see also Stout and Hecker, 1983). This issue will be further pursued in what follows. A third test involved torsion of one of the tubes. The resultant shear stress–strain response is compared to the two uniaxial responses in Fig. 3b (the von Mises equivalent stress and equivalent plastic strain are adopted for this comparison). The response in shear is clearly stronger indicating a more complex anisotropy. Nine biaxial experiments were conducted for values 0.2 6 a 6 1.25. The resultant engineering stress (rx–rh) responses are shown in Fig. 5 and, as expected, they follow the prescribed radial paths. The results were also used to plot the true stress trajectories (sx–sh) shown in Fig. 6. These paths are initially radial but become slightly curved at higher stress and deformation values [similar to results generated by Hecker and Stout that appear in Hill et al. (1994)]. Results from two representative experiments corresponding to a = 0.75 and 0.2 shown in Figs. 7–9 will be used to describe some of the measurements made, as well as some of the observed trends. In the case of a = 0.75, shown in Fig. 7, both the axial and circumferential stresses increase monotonically until a pressure maximum develops. A mild axisymmetric bulge occurs at the specimen mid-span, which subsequently evolves into a non-axisymmetric one. This precipitates wall thinning along a generator that results in rupture. The failure of this specimen is seen in Fig. 4 to extend over a length of approximately one tube diameter (this is a function of the energy stored in the test system as a

Fig. 4. A set of failed test specimens tested at different biaxiality ratios: from left to right a = 0.2, 0, 0.75, 1.0.

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517

σ (MPa) x

0

σ

100

200

50

Al-6260-T4

θ

300

α

(ksi) 0

θ

(MPa)

40 -0.2 -0.1

σ

0.75

0.5

0.25

0.9 1.0

30

200

1.25

20 100 10

LL Rupt.

0

0 -10

0

10

20

30

σ (ksi)

40

x

Fig. 5. Radial engineering stress paths prescribed. Marked are the limit and failure stresses.

τ (MPa) x

0

100

200

50

Al-6260-T4

τ

θ

(ksi)

300

τ

θ

(MPa)

40

30

200

20 100 10

LL Rupt.

0 -10

0 0

10

20

30

40

τ (ksi) x

Fig. 6. True stress paths traced. Marked are the limit and failure stresses.

whole). The stress at the pressure maximum (rhmax) is 34.03 ksi (235.7 MPa) and the corresponding strain (ehL) is 5.8% (see Table 1). The corresponding axial stress and strain are,

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a

σθ

40

Al-6260-T4

250 σθ (MPa) 200

(ksi) 30

150

20

100 σθ

10

50

σx = 0.75σθ σx

Exp. A19-1

0

0

1

2

3

4

5

0

6 ε (%)

7

θ

b

σx

40

Al-6260-T4

250 σx (MPa)

(ksi) 30

200 150

20

100 σθ

10

50

σ = 0.75σ

θ

x

σx

Exp. A19-1

0 0

1

2

3

4

5

εx (%)

0 6

7

Fig. 7. (a) Circumferential and (b) axial stress–strain responses recorded in experiment with a = 0.75.

14

Δt t 12 (%)

Exp. A19-1 α = 0.75

Al-6260-T4

10 B

8 6 A

C

4 C B A

2

Rupture

0 -180

-120

-60

0

60

120

o

180

θ

Fig. 8. Thickness reduction profiles at three locations around the circumference of a burst tube, that illustrate localization of wall thinning.

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a

519

240 Exp. A17-4

σθ

30

200

(ksi)

σθ (MPa)

160 20 120 σθ σ = -0.2σ x

10

80

θ

σx

40 Al-6260-T4

0

0

5

10

15

20

ε (%)

0 25

θ

b

0

σx 0 (ksi) -5 -10 -15 -25

σx

(MPa) -40 -80

α = -0.2 Al-6260-T4

-20

Exp. A17-4

-15

-10

-5

ε (%)

0

x

Fig. 9. (a) Circumferential and (b) axial stress–strain responses recorded in experiment with a = 0.2.

respectively, 25.50 ksi (175.9 MPa) and 1.8%. Failure occurred at a decreasing pressure so the maximum strain recorded by the circumferential extensometer (ehf) was somewhat higher, 6.1%. Following the response past the limit load is possible because of the volume-controlled pressurization scheme adopted. Included in Table 1 are strain values measured locally, adjacent to the failure zone, using the grid and an ultrasonic thickness gage. For this experiment the local values were ehfjl = 18.3% and exfjl = 11.0%. These values are significantly larger than the average strains at failure illustrating the local nature of the failure. Fig. 8 shows thickness contours around the circumference taken at the completion of the test at the three axial locations indicated in the inset. The contours illustrate that wall thinning is localized both axially and circumferentially. Results for a = 0.2 are shown in Fig. 9. In this case, the tube was compressed as it was pressurized. The circumferential response occurs at a lower stress reaching a maximum of 31.21 ksi (215.2 MPa). Compression has however the beneficial effect of delaying burst as the circumferential strain at the pressure maximum is now 22.1% and the corresponding axial strain is 14.4%. In this case, the capacity of the circumferential extensometer was exceeded and thus the last part of the rh–eh response was constructed from the grid measurements (drawn in dashed line). The specimen, included in Fig. 4, developed a bulge and failed by localized wall thinning along a generator of the cylinder. The circumferential and axial stress–strain responses recorded in all nine experiments are shown in Fig. 10a and b, respectively. All tubes developed limit load instabilities. This once more indicates that a purely stress-controlled loading is not an optimal option if capturing the failure is a goal of the tests. In the case of the seven cases with

520

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a

40

σθ

0.9

0.75

0.5

α 0.25

0

-0.1

(ksi) 30

240 σθ (MPa) 200

-0.2

1

160 1.25

20

120 80

10 40 Al-6260-T4

0

0

5

10

15

20

0

25 ε (%) θ

b

240 Al-6260-T4

σx 30

0.9

(ksi)

1.25

1

0.75

20

α

200 σx (MPa) 160 120

0.5

80

0.25

10

40 0

0

-0.1

-40

-0.2

-10 -15

-10

-5

0

5

10

15

20 ε (%)

25

x

Fig. 10. (a) Circumferential and (b) axial stress–strain responses recorded in nine experiments.

0.2 6 a 6 0.9, the tubes developed some bulging followed by localized thinning along a generator of the cylinder that resulted in ductile rupture (see three cases shown in Fig. 4). The bulging is illustrated in Fig. 11 for the case with a = 0.1 while the ductile rupture is illustrated in Fig. 2. For two cases, corresponding to a = 1.0 and 1.25, the specimens developed a mild bulge at mid-span and failed by localized thinning in the circumferential direction as illustrated in Fig. 4 for a = 0.2 (similar failure modes were reported in Davies et al. (2000) and Yoshida et al. (2005)). Fig. 12 shows thickness profiles along the lengths of these two tubes taken after the tests. The profiles demonstrate that the localization extended over approximately a length of one tube diameter at mid-span. The stress–strain responses in Fig. 10 demonstrate that as a decreases from 1.0 to 0.2 the circumferential stress sustained is reduced but more importantly the circumferential strain at the limit load increases. This is illustrated in Fig. 13 where plots of the strain paths recorded appear. Fig. 13a shows the engineering (ex–eh) strain results and Fig. 13b the corresponding logarithmic ones (ex–eh). Both sets of strain paths remain nearly radial up to the limit load instability depicted in these plots with the symbol . Values of the stresses and strains at the limit load are listed in Table 1. Because of space con-

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Fig. 11. Bulging of specimen A17-1 (a = 0.1).

35

Δt t (%)

Al-6260-T4

Exp. A17-2 and A19-2

30

α=1

25

α = 1.25

20 15 10 5 0 -5

-4

-3

-2

-1

0

1

2

3

4

5

x/R Fig. 12. Axial thickness reduction profiles from two experiments that exhibited circumferential rupture. Both illustrate localized wall thinning.

straints, the axial extensometer was mounted just above the chain of the circumferential one, in other words above the axial centerline of the specimen. For this reason, in some

522

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a Al-6260-T4

40

ε

θ

(%) 30

20

10

0

-10

-20

-10

0

10

20

30

40

ε (%) x

b e

θ

Al-6260-T4

40

(%) 30

20

10

0

-10

-20

-10

0

10

20

30

40

e (%) x

Fig. 13. Strain paths traced in the experiments. (a) Engineering strains and (b) logarithmic strains (d, average strains at limit load; N, average strains at rupture; and s, local strains in zone of rupture).

cases where failure involved bulging followed by localization around the mid-span, the axial extensometer did not record accurately the maximum axial strain causing the abrupt nonlinearity seen in a couple of the strain paths past the points corresponding to the limit load. The maximum average strains recoded by the extensometers are listed in Table 1 and are marked in the figure with the symbol N. Finally, the local strains in the zones of failure

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523

τx (MPa) 0

100

200

p

τθ

40

(ksi) 30

Al-6260-T4

W 2000 (psi) 1500 1000 750 500 300 150 80 40 18 4.5

300

τθ (MPa) 200

20 100 10

0

0 -10

0

10

20

30

τx (ksi) Fig. 14. Loci of experimental points representing various levels of constant plastic work.

were measured from the grid and local thickness measurements using the ultrasonic thickness gage. These strains are listed under (exjl, ehjl) in Table 1 and are marked with an open circle symbol () in Fig. 13. They follow a similar trend as the average strains at failure but have much higher values. The results illustrate that the onset of instability, as indicated by attainment of a load maximum, is influenced by the cylindrical geometry of the specimens. Consequently, they occur at relatively small strains as observed by Stout and Hecker (1983). By contrast, the local strains in the proximity of the failure zone are significantly higher. The magnitude and trend of the failure strains is similar to those reported by Davies et al. (2000) for the same material (Al-6260-T4). It has been long realized that deformation can induce anisotropy to materials. To evaluate this effect for our material we use the experimental results to construct contours of constant plastic work in the spirit of Hill (Hill, 1991; Hill and Hutchinson, 1992; Hill et al., 1994). Each contour consists of 10 points of equal plastic work plotted in the (sx–sh) plane in Fig. 14 (the points are joined by straight construction lines). The innermost contour is essentially equivalent to the initial yield surface. Subsequent ones are not related to yield surfaces except through the fact that the local normal is in the direction of the plastic strain increment. In that sense, for each level of plastic work the contour can be considered as the envelope of the current yield surfaces. It is apparent that the shapes of successive contours gradually change with the sides becoming flatter and the rounded zone around equibiaxial tension changing curvature. Since the plastic strain increment is normal to these contours such distortions can be expected to affect the deformation. Consequently, this indicates that some deformation-induced anisotropy due to microstructural evolution has taken place.

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3. Constitutive modeling We intend to model the inelastic behavior of our material through an associated flow rule that is based on an anisotropic yield function. Isotropic hardening will be adopted. Furthermore, based on our experience (Miller and Kyriakides, 2003; Kyriakides et al., 2004) as well that of others, aluminum alloys require the adoption of non-quadratic yield functions (Hosford, 1972, 1979; Hill, 1979, 1990; Barlat and Lian, 1989; Barlat et al., 1991, 1997, 2003; Hopperstad et al., 1998; Banabic, 2000; Bron and Besson, 2004). In the radial path experiments conducted, the tubes deform homogeneously over a significant part of their loading histories, with bulging and localization that precipitate rupture only occurring in the latter parts of the histories. Thus, in the early parts of each test, a principal state of stress develops (rx,rh,rr). In these regimes, we adopt Hosford’s principal stress anisotropic yield function [1979] as it is relatively easy to calibrate and manipulate. By contrast, in the numerical simulation of the onset of rupture, the state of stress is no longer axisymmetric and thus the general stress state anisotropic yield function of Karafillis and Boyce (1993) is more appropriate. 3.1. Principal stress yield function Hosford’s principal stress anisotropic yield criterion can be written as follows:        1 1 1 1 1 1 1 k k k 1 þ k  k jrh  rx j þ 1 þ k  k jrx  rr j þ k þ k  1 jrr  rh j ¼ rko 2 Sh Sr Sr Sh Sr Sh ð4Þ

1.2

σθ

σo

0.8

Exps. Strain Offset = 0.05% WP= 9.2 psi

0.4

{Sr, Sθ} = {1.01, 1.04} k=8 0

Al-6260-T4 -0.4 -0.4

0

0.4

0.8

1.2

σx / σo Fig. 15. Experimental data representing the initial yield surface according to two different definitions of yielding and Hosford’s anisotropic yield criterion.

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where S h ¼ rroho , S r ¼ rroro and ro, roh, ror are the yield stresses in the axial, circumferential and radial directions, respectively. Prompted by previous works on aluminum alloys, the exponent k is assigned the value of 8 (Logan and Hosford, 1980). Sh = 1.04 is found to bring together the early parts of the two uniaxial stress–strain responses shown in Fig. 3b. The value of Sr was chosen for optimal matching of all of the biaxial responses. The resultant initial yield surface in the (rx–rh) plane is shown in Fig. 15. It has a sharper curvature along the equibiaxial tension/compression direction and flatter sides than the von Mises yield criterion. Included in the figure are two sets of experimental points. The first set corresponds to a strain offset of 0.05% and the second to a plastic work of 9.2 psi (63 kPa). The two sets of points are seen to agree quite well with each other indicating the near equivalence of the two methods of establishing yielding. The data are also in quite good agreement with Hosford’s yield function as calibrated. The true stress version of the yield function was used to generate an associative flow rule. The flow rule was subsequently used to calculate the equivalent true stress–equivalent logarithmic plastic strain ðse –depe Þ response for the early part of each of the nine biaxial experiments performed. The results are shown in Fig. 16. The yield function bundles the responses around the uniaxial one to a reasonable degree. The extent of spreading between the responses is an indication of the limitations of this yield function. The flow rule was also used to calculate contours of equal plastic work for the values for which the experimental contours in Fig. 14 were generated. Several calculated and experimental contours are compared in Fig. 17. The calculated work contours do not match perfectly the experimental points which is another indication that the yield function adopted, coupled with the assumption of isotropic hardening, are only approximations that do not represent the material behavior perfectly. 3.2. Karafillis–Boyce yield function [1993] The isotropic version of this yield criterion is a combination of two distinct yield surfaces. The first is Hosford’s isotropic yield criterion [1972] which in terms of the principal deviatoric stresses, si (i = 1, 3), is given by: 0

τ 40

-0.1

300

-0.2

0.25

e

τ 250 e (MPa)

0.75 0.5

(ksi) 30

0.9

1

1.25 Uniaxial

200 150

20

100 10 50

k=8 {S , S } = {1.01,1.04} r

θ

Al-6260-T4

0 0

2

4

6

8

10

12

14 e

p e

0 16 (%)

Fig. 16. Experimental data presented in the form of equivalent true stress vs. the work conjugate equivalent logarithmic plastic strain based on Hosford’s yield function.

526

Y.P. Korkolis, S. Kyriakides / International Journal of Plasticity 24 (2008) 509–543 τx (MPa) 0

τθ

40

(ksi)

30

100

2000 p 1500 W 1000 (psi) 750 500 300 150 40

200

Sr=1.01, Sθ=1.04

300

τθ (MPa) 200

20 100 10

Al-6260-T4

0 -10

0

10

0 20

30

τx (ksi)

Fig. 17. Loci of experimental points representing various levels of constant plastic work and corresponding contours based on Hosford’s yield function.

i 1h 2k 2k 2k ðs1  s2 Þ þ ðs2  s3 Þ þ ðs3  s1 Þ ¼ r2k o 2

ð5Þ

where ro is the yield stress in a uniaxial test. Yield surfaces bounded by the von Mises (k = 1) and Tresca (k = 1) surfaces can be generated by selecting the exponent to be an integer k 2 [1, 1). The second yield criterion, expressed again in terms of si, is written as  32k  2k 2k s1 þ s2k ¼ r2k 2 þ s3 o 2k 2 þ2

ð6Þ

with k once more having the same range. Eq. (6) represents surfaces that lie between the von Mises and the outer bound for isotropic convex surfaces (Mendelson, 1968). Karafillis and Boyce (K–B) associated f1 and f2 with the LHSs of (5) and (6) and combined them to form the following convex yield function: f ¼ ½ð1  cÞf1 þ cf2 

1=2k

;

c 2 ½0; 1:

ð7Þ

Anisotropy was introduced by evaluating the isotropic plasticity equivalent deviatoric stress tensor s, related to the stress tensor r acting on the anisotropic material point through a linear operator as follows: s ¼ Lr:

ð8Þ

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527

Adopting the symmetries for L appropriate for orthotropy, and limiting attention to plane stress states of interest to our tube problems, (8) can be written as

a

1.2

σθ

σo

0.8

Exps. Strain Offset = 0.05% WP= 9.2 psi

0.4

{Γ , α1, α2} = {0.68, 0.94, 0.97} 2k = 8, c = 0 0

Al-6260-T4 -0.4 -0.4

0

0.4

0.8

1.2

σx / σ

o

τx (MPa)

b 0

τθ

40

(ksi) 30

100

200

{Γ , α1, α2} = {0.68, 0.94, 0.97} 2k = 8, c=0

2000 p 1500 W (psi) 1000 750 500 300 150 40

300

τθ (MPa) 200

20 100 10

0 -10

Al-6260-T4 0

10

0 20

30

τx (ksi) Fig. 18. (a) Experimental data representing the initial yield surface according to two different measures and the K–B anisotropic yield criterion. (b) Loci of experimental points representing various levels of constant plastic work and corresponding contours based on the K–B yield function.

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8 9 sx > > > =

2

1 6 b h 1 ¼ C6 4b > sr > 2 > > : ; 0 srh

b1 a1 b3 0

b2 b3 a2 0

9 38 0 > rx > > > = < 07 7 rh > 0 > 0 5> > ; : d rrh

ð9Þ

where 1 1 1 b1 ¼ ða2  a1  1Þ; b2 ¼ ða1  a2  1Þ; and b3 ¼ ð1  a1  a2 Þ: 2 2 2 Thus, the anisotropy is defined by constants {C, a1, a2, d}. The principal deviatoric stress components si (i = 1, 3) are evaluated from sij in (9) in the usual manner and are substituted in (7) to complete the anisotropic yield function. The parameters {C, a1, a2} can be determined as discussed in Appendix 2 of K–B. The value of d then becomes  1=2k ro 32k d¼ ð1  cÞð22k1 þ 1Þ þ c 2k1 : ð10Þ Crxyo 2 þ1 Two versions of this model are considered. In the first version c = 0, which reduces f in (7) to Hosford’s yield function. In this case, 2k in (5) was selected to be 8, in other words the same exponent as in the principal stress version of the anisotropic Hosford function. The procedure for determining the constants {C, a1, a2} described in K–B was suited for sheet metal. In the present case, it was found more convenient to follow a different calibration approach. The constants were selected for optimal agreement of the initial yield surface with the experimental data, as well as for best performance in bringing together the equivalent stress–equivalent plastic strain versions of the experimental responses (similar to Fig. 16). The parameters chosen are {C, a1, a2} = {0.68, 0.94, 0.97}. The resulting 1.2

σθ

σo

Exps. 0.8

Strain Offset = 0.05% WP= 9.2 psi

{Γ , α1, α2} = {0.6665, 0.96, 0.99}

0.4

2k = 24, c=0.8

0

Al-6260-T4 -0.4 -0.4

0

0.4

0.8

1.2

σx / σ

o

Fig. 19. Experimental data representing the initial yield surface according to two different measures and the K–Bc anisotropic yield criterion.

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529

initial yield surface is compared to the experimental data that correspond to the two different definitions of yielding discussed earlier, in Fig. 18a. The shape is seen to be in reasonable agreement with the data. The performance of the model with regards to the equivalent true stress–equivalent plastic strain responses is comparable to the corresponding results from Hosford’s model shown in Fig. 16 and will not be repeated here. Constant plastic work contours were generated with this yield function for the values that correspond to the experimental ones in Fig. 14. The results are compared to the experimental ones in Fig. 18b. The calculated contours underpredict the experimental values to some degree, most prominently for values of a between 0.5 and 1.0. We expect this degree of disagreement to have an impact in the numerical simulations of the burst experiments that follow. The performance of the full version of the K–B yield function was also investigated. The model was calibrated following a similar procedure. In this case, the presence of f2 dictates that a much higher exponent be employed, 2k = 24 (in accord with K–B findings). The parameters arrived at are {c, C, a1, a2} = {0.8, 0.6665, 0.96, 0.99}. The corresponding initial yield surface is shown in Fig. 19. In general the shape is very close to the one in Fig. 18 with c = 0. The performance in pulling together the measured equivalent true stress–equivalent plastic strain responses is once more of similar quality as that of the Hosford yield function shown in Fig. 16. 4. Numerical simulation of inflation tests 4.1. Finite element models The inflation experiments were simulated numerically using finite element models developed in the nonlinear code ABAQUS/Standard. The models are used in conjunction with the constitutive equations described in the previous section to reproduce the inflation experiments, including the onset of localization and burst. Because of the two distinctly different failure modes observed in the experiments, two different FE models have been developed. Model I is tailored to simulate rupture along a generator that was observed in seven of the experiments. In this case, symmetry allows consideration of one quarter of the tube as shown in Fig. 20 with the origin of the coordinate system used being at the mid-span of the tube (i.e., the planes x = 0 and h = [0, p] are planes of symmetry). In each simulation the model is assigned the length (2L), mid-surface radius (R) and wall thickness (t) corresponding to the values given in Table 1. The wall thickness is assumed to be uniform but a local imperfection in the form of an axial groove of reduced thickness (tg) is added as shown in Fig. 20 (shaded band). The domain is discretized using linear, fournode shell elements with reduced integration (S4R). These elements allow for large strains and large rotations both required for burst instabilities. A nearly isotropic mesh is used with a refinement in the zone surrounding the groove. The main mesh has 45 elements along the length and 45 around the half circumference. The zone around the groove (L1  s1) has double the mesh density in both directions and the groove itself is two elements wide and 25 elements long (Lg = R and sg ffi t arrived at from convergence studies). The wall thinning imperfection is defined by g¼

t  tg P 0: t

ð11Þ

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L

Lg

L1 sg

s1

x θ

R t

Fig. 20. Model geometry and FE mesh for cases that rupture in the axial direction.

The value of g = 0.05 is used in all cases analyzed. The instabilities are imperfection sensitive. Although different imperfection amplitudes might yield limit strains that are closer to the experimental results, this value was selected for best model performance for the whole set of experiments. Model II is customized to model circumferential rupture that was observed in two experiments with high stress ratio (a = 1.0 and 1.25). Again symmetries allow consideration of just one quarter of the tube. In this case, the model is discretized with 61 elements

L

wg Imperfection x GL1

R

θ t

GL2 Fig. 21. Model geometry and FE mesh for cases that rupture in circumferential direction.

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531

Table 2 Anisotropy variables for two K–B models Model

c

2k

C

a1

a2

d

K–B K–Bc

0 0.8

8 24

0.680 0.6665

0.94 0.96

0.97 0.99

1.249 1.287

along the halflength and 60 around the half-circumference. A groove of width wg ffi t is placed along the circumference at the plane of symmetry at x = 0 as shown in Fig. 21. The groove is two elements wide and has a wall thickness tg(h) with a wall thinning imperfection of g = 0.05. In order to allow for the possibility of non-axisymmetric localization to develop, tg was given a small perturbation described by   n  tg ðhÞ ¼ tg 1 þ cos h ; ð12Þ tg where n=tg is a wall eccentricity variable that was assigned the value of 0.0105. The boundary at x = L is fixed radially and constrained to remain plane. Each model is loaded by prescribing incrementally both the internal pressure and the axial load. In order to allow for the anticipated limit load to develop, Riks’ path-following procedure is employed. The internal cavity of the tube was meshed with ABAQUS’ hydrostatic elements F3D3 and F3D4 (removed from the figures for clarity), which allowed the pressure to be applied. The external axial force is prescribed at a reference node at (x, r, h) = (L, 0, 0). All nodes in the plane x = L (i.e., those of the tube as well as those of the cavity) were constrained to follow the reference node. Because of the kinematic coupling of the nodes, the axial load is shared between the specimen and pressurized cavity, thus corresponding to the load measured by the load cell in our experimental setup. In this way, the proportional loading histories prescribed in our experiments are exactly reproduced. Thus, for example, when the external axial force is set to remain zero throughout the simulation, the response of the model matches that of a hydrostatic pressure loading test (closed ends tube inflation). Similarly, when the external load/pressure ratio is set at pR2, a purely lateral pressure loading is reproduced (zero axial stress). The FE models use any one of the three constitutive models described in the previous section, through user defined material subroutines (UMATs). In addition to the anisotropy parameters reported earlier (see Table 2), shear anisotropy recorded experimentally was also included in the two K–B models. For each case the appropriate values of d evaluated using (10) are listed in Table 2. 4.2. Numerical results We will discuss the main characteristics of the numerical responses through two representative examples. Fig. 22a shows the calculated pressure-average circumferential strain response ðP  eh Þ for the loading case a = 0.1. Fig. 22b shows the corresponding axial stress–strain response (rx–ex). The results were established using the K–B constitutive model with the parameters given in Table 2. In an attempt to present results in a manner similar to the experimental ones, the circumferential deformation measure ðeh Þ adopted is the change in circumference/initial circumference at x = 0. By contrast, the axial strain (ex)

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Fig. 22. Results from numerical simulation of experiment for a = 0.1. (a) P –eh and (b) rx–ex responses using two plasticity models and the corresponding experimental ones. (c) Model deformed configuration just after the onset of localization (K–B). (d) Calculated wall thinning at two locations vs. eh (K–B).

is based on the change in length/original length of a 0.5-in. gage length in the imperfection at (x, h) = (0, p). Included in the two figures are the corresponding experimental responses based on the average strain measured by the chain extensometer at the specimen mid-span and the strain measured by the axial extensometer. The predicted P –eh response is seen to track the experimental one well, over-predicting the pressure slightly. The corresponding rx–ex response overpredicts the experimental one to a larger extent although the level of axial stress that builds up for this loading path is relatively small compared to rh (see Fig. 5). The tube deforms essentially axisymmetrically, shortening while expanding circumferentially. The stiffness of the model is gradually reduced reaching a maximum pressure at eh ¼ 17:85%. This value compares well with the corresponding experimental strain of 18.0%. The descending part of the response is followed using Riks’ path following technique. With increasing pressure the deformation starts to localize in the neighborhood of the thickness imperfection where non-axisymmetric bulging starts to develop. At approximately eh ¼ 19:0%, the wall in the local imperfection starts to undergo precipitous thinning while simultaneously the pressure drops sharply. Fig. 22c shows a view of the

Y.P. Korkolis, S. Kyriakides / International Journal of Plasticity 24 (2008) 509–543

533

c

d

80

Al-6260-T4 α = -0.1

Δt t (%) 60

A (x = 0)

40 εθ (P ) max

20 B (x = 0.6L)

0 0

5

10

15

20

ε (%)

25

θ

Fig. 22 (continued)

deformed mesh at the point marked on the two responses with solid bullets (r), with the color contours representing the current wall thickness (compare with corresponding experimental configuration in Fig. 11). The localized bulging is evident, as is the significant stretching and thinning of the elements spanning the imperfection. The localization of wall thinning is also illustrated in Fig. 22d where the reduction in wall thickness (Dt/t) at two locations is plotted against the mean circumferential strain at mid-span. Point A is located in the imperfection at (x, h) = (0, p) whereas point B is located outside the imperfection at (x, h) = (0.6L, p). Initially, the thickness changes at the same rate at the two points. Around a strain of 10% the two trajectories start to diverge with the rate of thinning accelerating at point A and decelerating at B. As the strain corresponding to the pressure maximum in Fig. 22a is approached, the rate of thinning at A accelerates even further, while at B it stops growing. The kink on the wall thinning response at A corresponds to the sharp downward trend in pressure in Fig. 22a at about

534

Y.P. Korkolis, S. Kyriakides / International Journal of Plasticity 24 (2008) 509–543

19% strain. The model does not include a rupture criterion and as a result the thickness at the crown point of the bulge is down to about 80% of its original value in this configuration. The material cannot sustain such large strains, and consequently rupture can be expected to take place in the early parts of the descending response, as indeed occurred in the corresponding experimental response. The average strain at failure in the experiment was eh = 19.1% which compares well with eh ¼ 19:0% of the calculation at the point the precipitous localization commenced. A similar calculation was performed using the constitutive model K–Bc, with the parameters listed in Table 2. The calculated P –eh and rx–ex responses are included in Fig. 22a and b (dashed lines). The responses are essentially identical to the ones based on the K–B model, with the localization now occurring at the somewhat larger strain of approximately eh ¼ 20%. All other aspects of the simulation are essentially identical to those described above. As stated earlier, the problems considered in this study remain essentially axisymmetric until the pressure maximum is reached, when non-axisymmetric localized bulging that precipitates wall thinning and rupture commences. In other words, in the latter part of the response in-plane shear cannot be precluded. At the same time, a certain amount of shear anisotropy was recorded in the pure shear test performed on a section of our tube (Fig. 3a), which for completeness should be incorporated in the constitutive model. This prompted the adoption of the K–B constitutive framework in our modeling efforts in the first place. The role of this shear anisotropy on the problem at hand was evaluated by running simulations of all experiments using the K–B model with and without the shear anisotropy (shear anisotropy is precluded when d = 1/C). Interestingly, in all cases the two simulations ended up being very close to each other. This is illustrated in Fig. 23, which shows the P –eh responses from the two simulations for the loading case a = 0.1 (K–B1 does not include the shear anisotropy). Having reached the conclusion that the shear anisotropy did not influence the bursting response in any tangible manner, it became apparent that Hosford’s principal stress anisotropic yield fiction could also serve the purposes of this class of problems. Included in Fig. 23 is the P –eh response calculated with this

2.5

Al-6260-T4 α = -0.1

16

K-B

P 2 (ksi)

H

14

P (MPa)

12 K-B1

1.5

10 8

1

6 4

0.5

2 0

0 0

5

10

15

20

ε (%)

25

θ

Fig. 23. Comparison of P –eh responses calculated using two Karafillis–Boyce models and Hosford’s model.

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535

constitutive model (H) using the anisotropy constants listed in the inset of Fig. 15. The response is very close to the two in which the K–B model was implemented, with the

a

35 Al-6260-T4

240

Rupture K-B

1

σ 30 x (ksi)

GL1

1

25 Exp.

Onset of Localized Wall Thinning

GL2

20

σ 200 x (MPa) 160 120

15 80 10

σθ

σ = ασ x

α=1

5

θ

40

σx

0 0

2

4

6

8

10

12

0 14 16 ε (%) x

b

2.5

Al-6260-T4 σ=1

P (ksi) 2

16 P (MPa) 14 Exp.

K-B

12 1.5

10 8

1

6 4

0.5

2 0 0

2

4

6

8

0 10 12 ε (%) α

c

Fig. 24. Results from numerical simulation of experiment for a = 1. (a) rx –ex responses with strain measured at two locations and the corresponding experimental response. (b) P –eh response and the corresponding experimental one. (c) Model deformed configuration just after the onset of localization.

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pressure maximum and the onset of localization being slightly delayed. In view of the good quality of these results in the rest of the paper we will be quoting results from simulations that use both the K–B and the H constitutive models. Fig. 24 shows predictions for the loading case a = 1. In the experiment, rupture was around the circumference (see Fig. 4) and as a result FE model II with a circumferential imperfection at the plane of symmetry (x = 0) was adopted, along with the K–B constitutive framework. Fig. 24a shows the calculated engineering stress-average axial strain response ðrx –ex Þ and Fig. 24b the P –eh response. The corresponding experimental results are included in both plots. Using the numerical results we evaluated ex based on the change in length/original length using two different 0.5 in. gage lengths (GL1 and GL2 shown in Fig. 21). GL1 starts at x = 0 so that it includes the imperfection. In this manner, it is anticipated that it will capture the localized deformation expected to develop in the imperfection band. GL2 spans several elements immediately next to the imperfection and may not sense the localization. In each case, results from three locations, 0°, 90° and 180° from the position of the thinnest wall (h = p, see Eq. (12)), were then averaged to generate ex . The circumferential strain eh was evaluated in the manner described earlier. The two axial responses trace trajectories that are initially very close, following the experimental one quite well. Their small deviation is caused by the small difference in

a

40 σ

θ

0.5

0.9

α

0.25

0

-0.1

250 σ

-0.2

θ

0.75

(ksi) 30

(MPa) 200 150

1

20 1.25

100 10 50 K-B Al-6260-T4

0

0 0

5

10

15

20

25 ε (%) θ

b σx 30 (ksi)

Al-6260-T4 K-B 1.25

0.9 0.75

20

100

0.25

0

50 0

-0.1

-50

-0.2

-10 -15

-10

150

α

0.5

10

1

200 σx (MPa)

-5

0

5

10

15 20 εx (%)

Fig. 25. Results from numerical simulations of nine experiments. (a) rh –eh and (b) rx–ex responses.

Y.P. Korkolis, S. Kyriakides / International Journal of Plasticity 24 (2008) 509–543

537

thickness in part of GL1. The responses attain a stress maximum at somewhat different strain levels, ex ¼ 9:5% for GL2 and 10.5% for GL1. These values compare favorably with the 10.6% value recorded in the experiment by the 1-in. axial extensometer that was adjacent to the rupture zone. The circumferential response differs to some degree from the one recoded in the experiment. Part of this disparity may be caused by a difference between the location of the chain extensometer in the experiment and the location at which eh is evaluated in the analysis (x = 0), but part of it is due to inadequacies of the class of constitutive models adopted. The loading causes simultaneous circumferential expansion and axial stretching. In contrast to the ‘‘bottleneck” deformation of the tube seen in Fig. 22b for a = 0.1 loading, in this case the radial fixity condition at x = L results in the frustum-like deformed shape shown in Fig. 24c (compare these to the test specimens in Fig. 4). On reaching the axial stress maximum, deformation starts to localize in the imperfection band and its neighborhood. This is illustrated by the increasingly larger strain recorded by GL1. At some point, the thinning becomes precipitous while the load is decreasing. The wall thinning is mainly axial but with a circumferential bias towards the initially thinner side. This bias results in loss of the essential axisymmetry (see Fig. 24c, which corresponds to points marked on the responses with a solid bullet and r) that prevailed before the load maximum. Concurrently, GL2 is registering unloading, something that is typical of localization problems. Once again, the model does not include a rupture criterion and as a result it is not possible to pinpoint the failure, but one can consider the strain regimes inside the construction ellipse to be reasonable estimates of rupture strains. Interestingly, this construction includes the point of rupture recorded in the experiment corresponding to a strain of 12.8%.

σ (MPa) x

0

100

200

50

Al-6260-T4 K-B

σ

θ

(ksi)

300

σ

θ

(MPa)

40

30

200

20

100 10

Exp. Ana. LL Rupt.

0 -10

0

10

20

30

0 40

σ (ksi) x

Fig. 26. Engineering stress paths prescribed. Marked are the stresses at the limit load and at rupture.

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ε

θ

Al-6260-T4 K-B

30

(%)

Anal. Exp. -0.2

20

α -0.1 0 0.25 0.5

10

0.75

0.9

1.0

1.25

0 -10

0

10

20

ε (%) x

Fig. 27. Comparison of calculated and measured engineering strain paths for nine loading cases using the K–B yield function.

Simulations were conducted for all nine experiments performed using both the K–B and H yield functions. The complement of the results will be presented in summary form. Figs. 25a and b show, respectively, the circumferential and axial engineering stress–strain responses calculated for each of the nine loading paths. For the seven cases with 0.2 6 a 6 0.9 the average circumferential strain ðeh Þ is based on the change in circumference/initial circumference at the symmetry plane x = 0 and the axial strain (ex) is based on the change in length of a 0.5 in. gage length at (x, h) = (0, p). For the cases with a = 1 and 1.25 the circumferential strain is evaluated in the same manner but the axial strain is based on the average value of the change in length in three 0.5 in. gage lengths located at h = (0, p/2, p) and x = 0. All responses exhibit a limit load instability that leads to localized wall thinning and rupture. The onset of rupture is associated with the sharp downturn in the calculated responses. As was the case in the experiments, in seven cases rupture is axial and in two circumferential. The trends of the responses follow reasonably well those of the experimental results with some interesting differences. Fig. 26 shows a plot of the engineering stress paths traced. Since these were prescribed they are identical to the corresponding experimental ones. Marked on the responses are the positions of the limit loads (, at the end of each trajectory) and the corresponding experimental values (red1 color ). Also marked are the experimental (N) and numerical (red1 color N) points of rupture. For most cases the limit load and rupture stresses are in good agreement, with the largest discrepancy occurring for the three paths with the larger values of a. A different perspective of the performance of the constitutive models can be obtained from Fig. 27, which shows plots of the engineering strain paths followed in the experi1

For interpretation of the references to color in Fig. 26, the reader is referred to the web version of this article.

Y.P. Korkolis, S. Kyriakides / International Journal of Plasticity 24 (2008) 509–543

ε

θ

Al-6260-T4 K-B

539

Rupture

30

Exp.

(%)

Anal.

• •

-0.2

α -0.1

20

0

0.5 0.25 0.75

0.9

1.0

10 1.25

0 -10

0

10

20

ε (%) x

Fig. 28. Comparison of measured and calculated average strains at rupture using the K–B yield function.

ments and in the corresponding simulations. The end of each path corresponds to the load maximum. The calculated paths follow the experimental ones quite well for seven of the paths, but deviate from the experiments for a = 0.9 and 1.0. In other words, the main discrepancy is in the neighborhood of equibiaxial tension. In isotropic hardening plasticity, the high curvature of the yield surface in this regime (see Fig. 18) implies that even a small

ε

θ

Al-6260-T4 H

30

(%)

Anal. Exp. -0.2

20

α -0.1 0 0.25

0.5

10

0.75 0.9 1.0

1.25

0 -10

0

10

20

ε (%) x

Fig. 29. Comparison of calculated and measured engineering strain paths for nine loading cases using the H yield function.

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local deviation between the actual and fitted shapes can result in significant difference in the orientation of the local normal, with corresponding differences in the predicted strains. We believe that such differences are the main contributors for the strain path deviations for a = 0.9 and 1.0. The ends of the trajectories in Fig. 27 correspond to the load maximum that precipitates localization. With two exceptions, the general trend is that the predicted limit loads occur at higher strains than in the experiments. In the case of a = 0.1, the strains at the limit load agree well with the experiment and for a = 1.25 the calculated limit load occurs at strains that are smaller than the experimental values. The calculated average strains at rupture, as defined above, are compared to the ones measured by the extensometers in the experiments in Fig. 28. Here, the agreement is good for a = 0.2 and 0.1. The calculated results overpredict the experimental ones for a = 0, 0.25, 0.5, 0.75 and 0.9, and underpredict the failure strain for a = 1.25. The calculated limit loads and the resultant failure strains are influenced by the shape of the initial yield surface adopted, as it plays a decisive role in the induced strain path. In addition, they are influenced by the evolution of the yield surface, by the shape of the stress–strain response adopted, and by the amplitude of the imperfection introduced in the model. The stress–strain response adopted is the one measured in uniaxial tension tests on axial strips extracted from the tubes used in this study (true stress–logarithmic plastic strain version of the one shown in Fig. 3a). It is a fact that small changes in tangent modulus at higher strains can influence the strain of the calculated limit load instabilities. However, because of the tubular nature of the specimens no other practical option is available for establishing the intrinsic stress–strain response of the material. The imperfection amplitude used was selected through a parametric study as the one that best represents actual thickness variations and other small imperfections unavoidably present in the test setup used. This leaves the yield surface and its evolution as possible contributors to the overprediction of the strains at the limit loads. The effect of a yield surface that better represents the experimental data will be investigated in the future by considering anisotropic yield functions that provide even more flexibility in the choice of shape, such as the one of Barlat et al. (2003). The effect of the evolution of the yield surface can be investigated by making the constants of such a model evolve with history. All experiments were also simulated numerically using Hosfords’s anisotropic yield function in the FE models (material assumed to be isotropic in shear). The main features of the predicted responses are similar to those of the K–B model and will not be shown. The calculated strain paths are compared to the ones measured in Fig. 29. Once again, the trajectories end at the attainment of the limit load instability. The paths are slightly different from those in Fig. 27, in some cases a bit closer to the measured paths and in others somewhat further apart. The strains at the limit loads once more are generally larger than the experimental values. 5. Summary and conclusions The paper presents the results of a combined experimental/numerical effort that aims to assess the performance of different anisotropic yield functions in predicting burst of aluminum tubes used in hydroforming. The experiments involved seamless Al-6260-T4 tubes loaded under combined internal pressure and axial tension or compression. Several radial paths in the axial–circumferential stress space were prescribed until failure occurred. In all

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cases deformation was initially essentially uniform until a limit load instability was approached. For axial tension dominant loading paths, the tube eventually bulged axisymetrically, and localization resulted in circumferential rupture. For hoop stress dominated paths, an initially axisymmetric bulge evolved into a non-axisymmetric one, leading to localized thinning and rupture along a tube generator. The load maximum is influenced by the hardening of the material and by the cylindrical geometry of the test specimens. Thus, for example, the limit load under a purely lateral pressure loading (i.e., a uniaxial hoop stress state until bulging develops) occurred at a strain of 11.8%, which compares with a 19.5% limit strain measured in a uniaxial tension test on an axial strip. This pattern of generally lower limit strains compared to corresponding results from sheet metal, persisted for all experiments. Local strain measurements in the ruptured regions produced estimates of forming limits for the loading histories considered. These strains are significantly larger than the average strains recorded at the onset of rupture. Initial yield surfaces and constant plastic work contours derived from the experimental results indicated the following: (a) The shape of the initial yield surface lies between that of the Tresca and von Mises yield surfaces, as indeed has been observed previously for many aluminum alloys. Thus, a non-quadratic yield criterion is best suited for the present material. (b) The material exhibited some initial anisotropy that appears to be different from that observed in aluminum alloy sheets. (c) The constant plastic work contours indicate that the anisotropy evolved with deformation, to some degree. The response and eventual burst of the nine tubes tested were simulated using appropriate FE models. A class of non-quadratic, anisotropic yield functions along the frameworks of Hosford and Karafillis–Boyce were calibrated to the experimental results and incorporated in the numerical models through an associated, isotropic hardening flow rule. The following conclusions can be drawn from the numerical results. (a) The fittings of the initial yield surface by Hosford’s principal stress anisotropic yield function and two versions of the K–B yield function were close to the experimental data to a comparably acceptable degree. (b) The models do not facilitate evolution of subsequent yield surfaces and as a result it was not possible to capture the evolution of the shape of the plastic work contours observed experimentally. (c) As a consequence of (a) and (b), strain paths calculated with the two models differed from measured ones, in particular in the neighborhood of equibiaxial tension, where the yield surface has the highest curvature. Similar yield functions with more flexibility in the fitting of the initial yield shape may reduce this discrepancy (e.g. Barlat et al., 2003). (d) The stresses corresponding to the limit load instabilities were generally well captured by the simulations. The corresponding strains and those at the onset of rupture were generally overpredicted to some degree. It is expected that at least part of this difference will be alleviated with a better representation of the initial yield surface. By making the anisotropy constants evolve with history, such a model can also capture

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at least part of the evolution of the plastic work contours observed. It would be worthwhile to also assess if such a model will improve the prediction of the strains at the limit loads and at failure. (e) The influence of the sizeable shear anisotropy on burst was found to be small. As a consequence the performance of Hosford’s yield function was comparable to the more complete K–B models. 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 GM 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. We also acknowledge the contributions of Jorge Capeto who was involved in the development of the experimental testing facility used in this study and for the conduct of preliminary tests of the type presented. References Banabic, D., 2000. Anisotropy of sheet metal. In: Banabic, D. (Ed.), Formability of Metallic Materials. Springer, Berlin. Barlat, F., Lian, J., 1989. Plastic behavior and stretchability of sheet metals. Part I. A yield function for orthotropic sheets under plane stress conditions. International Journal of Plasticity 5, 51–66. Barlat, F., Lege, D.J., Brem, J.C., 1991. A six component yield function for anisotropic materials. International Journal of Plasticity 7, 693–712. Barlat, F., Maeda, Y., Chung, K., Yanagawa, M., Brem, J.C., Hayashida, Y., Lege, D.J., Matsui, K., Murtha, S.J., Hattori, S., Becker, R.C., Makosey, S., 1997. Yield function development for aluminum alloy sheets. Journal of the Mechanics and Physics of Solids 45 (11/12), 1727–1763. 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 1: theory. International Journal of Plasticity 19, 1297–1319. Bron, F., Besson, J., 2004. A yield function for anisotropic materials. Application to aluminum alloys. International Journal of Plasticity 20, 063–937. Chu, E., Xu, Y., 2004a. Hydroforming of aluminum extrusion tubes for automotive applications. Part I. Buckling, wrinkling, and bursting analyses of aluminum tubes. International Journal of Mechanical Sciences 46, 263–283. Chu, E., Xu, Y., 2004b. Hydroforming of aluminum extrusion tubes for automotive applications. Part II. Process window diagram. International Journal of Mechanical Sciences 46, 285–297. 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., 1993. Effect of changing strain paths on forming limit diagrams of Al-2008-T4. Metallurgical Transactions A 24A, 2503–2512. Guan, Y., Pourboghrat, F., Barlat, F., 2006. Finite element modeling of tube hydroforming of polycrystalline aluminum alloy extrusions. International Journal of Plasticity 22, 2366–2393. Hill, R., 1979. Theoretical plasticity of textured aggregates. Mathematical Proceedings of the Cambridge Philosophical Society 85, 179–191. Hill, R., 1990. Constitutive modeling of orthotropic plasticity in sheet metals. Journal of the Mechanics and Physics of Solids 38, 405–417. Hill, R., 1991. A theoretical perspective on in-plane forming of sheet metal. Journal of the Mechanics and Physics of Solids 39, 295–307. Hill, R., Hutchinson, J.W., 1992. Differential hardening in sheet metal under biaxial loading: a theoretical framework. ASME Journal of Applied Mechanics 59, 1–9.

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