Hydroforming of anisotropic aluminum tubes: Part II analysis

Hydroforming of anisotropic aluminum tubes: Part II analysis

International Journal of Mechanical Sciences 53 (2011) 83–90 Contents lists available at ScienceDirect International Journal of Mechanical Sciences ...

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International Journal of Mechanical Sciences 53 (2011) 83–90

Contents lists available at ScienceDirect

International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

Hydroforming of anisotropic aluminum tubes: Part II analysis Yannis P. Korkolis 1, Stelios Kyriakides n Research Center for Mechanics of Solids, Structures & Materials WRW 110, C0600, The University of Texas at Austin, WRW 110, Austin, TX 78712, USA

a r t i c l e in f o

abstract

Article history: Received 20 August 2010 Received in revised form 23 November 2010 Accepted 27 November 2010 Available online 4 December 2010

Part I presented an experimental investigation of hydroforming of Al-6260-T4 tubes and a simple twodimensional model of the process. Relatively long, extruded circular tubes were formed against a square die with rounded corners, with simultaneous application of axial feeding. Localized wall thinning was reported to occur at mid-span which, accentuated by friction, led to burst. Part II presents fully 3D models of the process that include friction as well as more advanced constitutive models shown in previous studies to be essential for simulation of burst in free hydroforming of aluminum alloy tubes. The models are used to simulate several of the experiments of Part I, emphasizing the prediction of all aspects of the forming process, including wall thinning and its localization that lead to rupture. A shell element model is shown to capture the majority of the structural features of the process very successfully. However, even with the implementation of advanced constitutive models, it fails to reproduce correctly the localization of wall thinning. It is demonstrated that switching to solid elements coupled to non-quadratic yield functions results in accurate predictions of all aspects of the problem, including the onset of rupture. Apparently, slow growing depressions that develop at the interface between the flattened part of the cross section that is in contact with the die and the rounded part that is not, have a complex three dimensional stress state requiring accurate modeling offered by solid elements. Furthermore, the evolution of these depressions is only reproduced with accuracy when in addition non-quadratic yield functions are adopted. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Hydroforming Aluminum tubes Manufacturing Modeling Anisotropy

1. Introduction In the companion paper (Part I) we reported results from hydroforming experiments on Al-6024-T4 seamless tubes conducted in a custom facility. The experiments involved relatively long (L¼32 in. (813 mm)) circular tubes (D ¼2.358 in. (59.89 mm) and t ¼0.080 in (2.0 mm)) formed into a square shape (2.4 in. (61 mm) sides, 0.5 in. (12.7 mm) radius corners and 24 in. (610 mm) long). The tubes were formed against external dies by internal pressure. Simultaneously, the material was axially fed into the dies in order to counter the induced wall thinning and thus avoid or delay burst. The pressure–axial feed loading path was selected using a 2D numerical model so as to avoid the possible limiting states of wrinkling, buckling and burst. Despite careful lubrication of the tubes, friction affected the forming. Indeed, the effect of friction was aggravated by the length of the tubes formed. Thus, at mid-span, the tubes experienced less than half of the axial compressive strain prescribed at the ends. Inadequate compression in this area led to failures by burst, which

n

Corresponding author. E-mail address: [email protected] (S. Kyriakides). 1 Presently at Department of Mechanical Engineering, University of New Hampshire, USA. 0020-7403/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2010.11.004

initiated from local thickness depressions that develop where a flattened side of the cross section meets a curved one. Obviously, a 2D numerical model, while used in designing the experiments, cannot capture friction-induced axial variations in the cross sectional deformation of the formed tubes. Thus, here we present fully 3D finite element models of the process. Both shell and solid element models are considered. In addition it will be shown that, as described in Refs. [24–26],I reproducing the evolution of the localized wall thinning and the onset of burst for the Al alloy used require the adoption of a non-quadratic yield function that is also capable of incorporating material anisotropy. The paper presents strengths and weaknesses of the shell and solid 3D models, each coupled with the advanced constitutive models developed previously, in providing accurate predictions of the onset of rupture. 2. Shell element model (3D-Sh) 2.1. Model set-up We start by developing a 3D shell element model of our hydroforming process, as this presently constitutes the standard I

Refers to references and figures that appear in Part I.

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Y.P. Korkolis, S. Kyriakides / International Journal of Mechanical Sciences 53 (2011) 83–90

practice for sheet metal forming processes (similar models have been developed for hydroforming in Refs. [1–3] among others). A schematic of the shell element model developed in ABAQUS/ Standard is shown in Fig. 1. Taking into account the symmetries of the problem (see Figs. 3I and 4I), only one-eighth of the actual tube is considered. The model is meshed with linear, reduced integration shell elements (S4R), capable of course for large deformations. Convergence studies [22]I showed that 40 elements around the quadrant, 77 along the length and Simpson’s rule with 7 integration points through the thickness are adequate. Finer meshes result in ill-shaped elements because of the relative thinness of the shell. At the same time, more accurate integration rules or more integration points through the thickness did not yield different results. The mid-surface of the tube was also assumed to be the reference surface for contact between the tube and the die (default option). For the problem considered, choosing alternative reference surfaces did not affect the results. The hard contact option available in ABAQUS was adopted. The model was loaded by prescribing the pressure in the mid-surface of the shell and the axial displacement at the tube end (see Fig. 1). The non-quadratic constitutive models calibrated and evaluated in [24,25]I are incorporated in the shell FE model, along with the same true stress–logarithmic strain material response. They include models based on the isotropic yield function of Hosford [4] and on the anisotropic one of Barlat et al. ([5,6]; this model includes deformation-induced anisotropy as described in [25]I). Recall that these models were shown to perform adequately for both proportional and non-proportional biaxial stress paths [26]I that are anticipated in hydroforming processes. They are implemented in ABAQUS through user material subroutines (UMATs) developed in our referenced work.

2.2. Shell element model results

(For alternative experimental methods for establishing m see Refs. [7–9] among others.) A more detailed view of the final thickness distribution of the tube in Exp. HY5 is shown in Fig. 4. The contours clearly show the

Fig. 2. Deformed configuration for experiment HY5 using shell elements with m ¼ 0.2 and the H(8) plasticity model (P ¼4000 psi (276 bar); surrounding die has been removed for clarity; model has been reflected about y ¼0 and y ¼ p/2 planes). Notice the variation of shape along the length and compare with the experimental image in Fig. 7aI.

-

x

(%)

25

3D-Sh Model

20

0 0.05

15

In the numerical simulations the models were loaded under the exact pressure–axial feed loading path of each experiment. Fig. 2 shows the calculated deformed configuration for HY5 (for better visualization the model has been reflected about the y ¼0 and p/2 planes). The axial non-uniformity of the shape of the tube caused by friction (see Figs. 7aI and 8I) is reproduced by the numerical model very well using the Coulomb friction coefficient (m) of 0.2. The optimum value of m was determined in separate calculations that produced results such as the ones shown in Fig. 3 for HY8. Comparing the final axial strain along the length of the specimen for different values of m with the corresponding measured profile, m ¼0.2 is seen to provide the best correlation between experiment and analysis. This value is thus adopted throughout this work.

µ

0.1

Exp.

0.15

10

0.2 5 Al-6260-T4

0 0

0.1

HY8 0.2

0.3

0.4

x/L

0.5

Fig. 3. Predicted axial strain along the formed tube for different coefficients of friction and comparison to experiment (vM plasticity).

Tube End

L/2 Transition Zone

Tube End Support Die Entrance

Main Die

y R x Tube Mid-Span

z to

Fig. 1. Undeformed mesh of 77  40 shell elements (L  y), showing 1/8th of the tube (surrounding die has been removed for clarity).

Fig. 4. Predicted thickness reduction contours for the shell element model (m ¼0.2) for experiment HY5, using the H(8) plasticity model. Color gray represents deformed zones that are thicker than to (compare to Fig. 10I).

Y.P. Korkolis, S. Kyriakides / International Journal of Mechanical Sciences 53 (2011) 83–90

significant variation in wall thickness along the length of the tube with the mid-span area being the thinnest. As in the experiments and in the results from the 2D model of Part I, localized wall thinning is predicted to occur at the locations where the flat sides meet the curved ones (see also Ref. [36]I). The areas of the figure that are colored in gray become thicker than the undeformed tube (0.080 in (2.032 mm)) because of the friction encountered in the die. The actual measurements of thickness for this tube (HY5) shown in Fig. 10I confirm the trend of the numerical results, including the wall thickening at the feeding end of the tube. We next compare results obtained using the 3D shell element model and the various constitutive models considered. Thus Fig. 5 shows a comparison of the measured axial force–displacement response for HY8 and predictions from the model using three different yield functions: von Mises (vM), Hosford with exponent 8 (H(8)) and the Yld2000-2D as calibrated in Refs. [24,26]I. Switching to a non-quadratic yield function lowers somewhat the predicted response, but the effect is quite modest. Similarly insensitive to the constitutive model is the prediction of the axial compressive strain variation along the tube [22]I. The final shape of the mid-surface at mid-span was found to be dependent, but to a limited extent, on the constitutive model, with the higher exponents leading to more pronounced deformation of the curved part of the cross section [22]I. These findings are to be expected since both the overall shape as well as the force–displacement response constitute more global aspects of the structural response of the tube. Hence their sensitivity to the constitutive details of the material behavior is limited, with even a quadratic model providing very good agreement between experiments and analysis. What is more interesting in the context of failure prediction is the prediction of wall thinning at the tube mid-span. Fig. 6 compares the final wall thicknesses yielded by the various constitutive models to the experimental results from HY5. The effect of the yield function exponent on wall thinning is examined in Fig. 6a using the isotropic Hosford yield function. It can be seen that increasing the exponent intensifies the depth of the wall depressions, at the expense, however, of the quality of the prediction of wall thinning in the rest of the circumference. It is worth noting that there would be no way of deciding on the appropriate exponent to capture the localization if experimental measurements are not available. Fig. 6b compares results from the three main models considered, two isotropic (vM, H(8)) and one anisotropic (Yld2000-2D). vM and Yld2000-2D yield approximately the same results. H(8) produces more wall thinning and somewhat more pronounced

70

(kips)

to

1.00

HY5 Exp.

vM

0.98 0.96 0.94

H(8)

H(16)

0.92 0.90 0

t( ) to

μ=0.2 3D-Sh Model

Al-6260-T4 30

60

1.00

90

HY5 Exp.

vM

0.98 0.96

H(8)

0.94

Yld2000-2D

0.92

μ=0.2 3D-Sh Model

Al-6260-T4

0.90 0

30

60

90

Fig. 6. (a) Effect of the exponent of H yield function on the prediction of wall thinning at mid-span for HY5. (b) Effect of the yield function employed on the prediction of wall thinning at mid-span for HY5.

localization, approaching the depression values of the experiment. This apparent insensitivity of the depth of the depressions to the constitutive model employed is in drastic contrast with the results of simulation of the free inflation experiments [24–26]I where the constitutive model had a profound effect on the prediction of the onset of rupture. It should be noted that this trend persisted in the simulations of the other hydroforming experiments. In summary then, the 3D shell element model yields reliable predictions of the final (cross sectional) shape of the part, including the friction-induced axial variations and the end-feed loads required. However, this model could not correctly predict the evolution of localization that results in rupture, irrespective of which constitutive model that was adopted. It will now be shown that this deficiency is alleviated when the structure is modeled with solid elements.

300

3D-Sh Model μ = 0.2

Fspec. 60

t( )

85

250

Exp.

Fspec. (kN)

50

3. Solid element model (3D-Sol) 3.1. Model set-up

200

40

vM H(8) Yld2000-2D

30 20 10

HY8 Al-6260-T4

0 0

4

8

12

150 100 50 0

16 / L (%)

Fig. 5. Comparison of measured and predicted axial force vs. axial feed response for HY8 using different plasticity models (shell element model, m ¼ 0.2).

The same problem domain is now modeled with 8-node solid elements with full integration (C3D8), in order to investigate whether the more accurate representations of the normal stresses and of the through thickness deformation improve the prediction of the evolution of localization and the onset of rupture (see also Refs. [10,11] among others). The one-eighth of the actual tube considered is meshed with 45 elements in the axial direction, 60 around the circumference and 3 through the thickness (adequacy confirmed through detailed convergence studies in Ref. [22]I), which results in six integration points through the thickness. Several constitutive frameworks will once again be employed, including in this case, the 3D version of the anisotropic Yld2000-2D of Barlat et al. [12] designated as Yld2004-3D that is outlined in

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Y.P. Korkolis, S. Kyriakides / International Journal of Mechanical Sciences 53 (2011) 83–90

Appendix A together with its calibration procedure; its implementation is based on a UMAT subroutine provided by Yoon [13] (see also Ref. [14]). As with the shell element model, a hard contact between the tube and the die is adopted. The model is then loaded by prescribing the pressure in the internal surface of the tube and the axial displacement at its end, replicating exactly the loading paths traced in each experiment. Depending on the constitutive model, the computational time required by the solid element model ranged between 2 and 4 times of the time taken by the shell element model.

3.2. Solid element model results Fig. 7 shows the deformed configuration of the tube (model reflected about the y ¼0 and p/2 planes as before). The effect of friction in the nonuniform final shape is quite obvious, as was the case with the shell element model in Figs. 2 and 3. The equivalent plastic strain contours are shown in Fig. 8a for the isotropic H(8) model. It can be seen that there is significant deformation at the feeding end of the tube which, as reported earlier, is due to wall thickening resulting from the friction. At mid-span, the model predicts localized wall thinning at the expected locations. An interesting feature is that in any one quadrant, one of the thickness wells tends to deepen at a faster rate than the other (see later discussion). Contours of the contact pressure between the tube and the die are depicted in Fig. 8b. The boundary of contact is outlined very effectively by the highest pressure contour. Next we compare results based on the three constitutive models considered. All three yield similar predictions of the axial load-feed responses, the distribution of the axial compressive strain and the shape of the formed tube cross section at mid-span. These responses approach the experimental results closely and will not be shown here (see Ref. [22]I). As reported earlier for the shell element simulations, these more overall structural aspects of the problem are relatively insensitive to the constitutive model used. Where the models produce distinct differences however is in the induced wall thinning and the resulting localization. This point is illustrated in Fig. 9a and b, which shows comparisons of predicted and measured wall thinning at mid-span from two experiments HY6 and HY2, respectively, at P¼4450 psi (307 bar) and 3500 psi (241 bar)). It can be seen that the two non-quadratic yield functions promote the deepening of the two depressions at about 251 and 651

Fig. 7. Deformed configuration for experiment HY6 using solid elements (H(8), m ¼ 0.2, P ¼4450 psi (307 bar); surrounding die has been removed for clarity). Notice the variation of shape along the length and compare with the experiment in Fig. 7aI.

Fig. 8. Prediction of (a) equivalent plastic strain and (b) contact pressure between the formed tube and the die for experiment HY6 (P¼ 4450 psi (307 bar), solid element model, H(8), m ¼0.2). Note the large deformation at the feeding end of the tube. The contours in (b) outline the boundaries of contact between the tube and the die (compare with the experiment at Fig. 7aI).

much more than the quadratic (vM) plasticity model. Indeed, the predictions from H(8) and Yld2004-3D are both close to the experimental results. It is interesting to also observe that at some stage, one of the depressions starts growing faster than the other, as is natural in localization problems. This is also in concert with the experiments where typically one of the eight thickness depressions dominated and led to rupture whenever it occurred. Furthermore, the two sets of results show that by depending on the specifics of each loading path, incorporation of anisotropy may or may not have a noticeable effect on the wall thinning predictions, at least for the present material (compare Fig. 9a for HY6 with Fig. 9b for HY2, noting the different vertical scales). Overall, the specifics of the description of anisotropy appear to be much less crucial in these tube hydroforming simulations than they were for the free inflation experiments studied earlier. However, unlike in the shell element predictions, here the depth of the calculated thickness wells approaches the experimental measurements more realistically (compare Figs. 6 and 9). Several simulations were performed with the solid element model to evaluate the effect of the exponent of the yield function on wall thinning. Fig. 10 shows the predicted wall thinning plots at mid-span for HY6 (at P¼4450 psi (307 bar)) using different exponents in the isotropic Hosford yield function (H(k)). Notice that despite the thickness being approximately the same in the part of the cross section in contact with the die, increasing the exponent leads to

Y.P. Korkolis, S. Kyriakides / International Journal of Mechanical Sciences 53 (2011) 83–90

t( ) to

1.00

1

t( )

0.95

to

0.96 0.92

Exp. vM

0.88

H(8) Yld2004-3D

0.84 0.80 0

30

60

0.85 0.8 0.75

to

HY6 Al-6260-T4

0.7 0

30

B

µ = 0.2 Yld2004-3D 3D-Sol. Model 60

90

90 1

t( ) t( )

1.5 2.25 3.0 P 3.6 (ksi) 4.1 4.6 5.1 5.3

A

0.9

µ = 0.2 3D-Sol. Model

HY6 Al-6260-T4

87

0.98

t o 0.95 Exp.

0.96

vM

0.9

C

0.85

0.94 H(8)

0.92

0.8

Yld2004-3D

0.75

0.90 0.88 0

0.7

µ = 0.2 3D-Sol. Model

HY2 Al-6260-T4 30

60

0

90

1

t Fig. 9. Predictions of wall thinning for (a) HY6 at P¼ 4450 psi (307 bar) and (b) HY2 at P ¼3500 psi (241 bar) using solid elements with m ¼0.2 and different plasticity models included is the experiment.

HY6 Al-6260-T4

1.5 2.25 3.0 P (ksi) 3.6 4.1 4.6 5.1 5.36

30

µ = 0.2 Yld2000-2D 3D-Sh Model 60

90

Sh. Model C

µ = 0.2

min

C

t

o

Sol. Model

0.9

B

1 t( ) to

vM

0.96

6

0.92

8 H(k) 16

0.88

32

0.76

HY6 Al-6260-T4

0.7 0

0.84 0.8

A

0.8

μ = 0.2 3D-Sol. Model

HY6 Al-6260-T4 0

30

60

90

Fig. 10. Effect of exponent of H(k) yield function on the prediction of wall thinning at mid-span, using solid elements (HY6, P ¼4450 psi (307 bar)).

much more intense localization of wall thinning (vM showed no signs of localization perhaps because of inadequate growth of the depressions). This observation is in accord with the recent observations that non-quadratic yield functions promote earlier localization than the classical J2 plasticity (e.g., see Ref. [24]I, [15,16]). The evolution of wall thinning with the monotonically increasing pressure is shown in Fig. 11a. While the wall thickness is initially uniform, friction promotes the development of two wells in the circumferential distribution of the thickness as the deformation progresses and the tube–die contact is established (here at about 925 psi (64 bar)). This deformation pattern is stable up to some pressure (approximately 4400 psi (300 bar)), but at higher values the deformation localizes in one of the two wells and grows precipitously, potentially leading to the rupture. Note that a failure

1

2

3

4

5 6 P (ksi)

Fig. 11. Thickness profiles at different pressure levels across the quadrant analyzed for (a) solid elements and (b) shell elements, for HY6. (c) Wall thinning at the two thickness depressions as a function of pressure, for solid and shell element models.

criterion (e.g., see Refs. [17,18]) is not included in the numerical model, hence the simulation is not interrupted at this point. Simultaneously, the wall thinning at the second site is arrested and this thickness well stops growing. The localization in wall thinning as it exists at the highest pressure of the calculation is also illustrated in the through thickness strain contour plot in Fig. 12a. This ‘‘bifurcated’’ behavior is better illustrated in Fig. 11c, where the thickness reduction in the two wells is plotted against pressure. Interestingly, the critical pressure (approximately 4400 psi (303 bar)) is very close to the pressure at which the HY6 tube ruptured in the experiment (see Table 1I). The same simulation was repeated with the shell element model using the Yld2000-2D constitutive model and a similar set of thickness profiles at mid-span are plotted in Fig. 11b. The difference between the shell and solid element results is striking. As for the solid model, depressions in the shell simulations develop early in the forming history but their growth with pressure is

88

Y.P. Korkolis, S. Kyriakides / International Journal of Mechanical Sciences 53 (2011) 83–90

applied (corresponds to about 15% of the equivalent stress at these locations; not shown). By contrast, the shell model has zero normal stress. In addition, there are some differences in the way shear is represented at the point where the wall lifts off the die. These differences change the stress state and the triaxiality at the thickness depression sites leading to the observed divergent results. Before closing this section we observe that although shell element modeling of sheet metal forming processes remains the rule, in recent times increasingly more authors point to the need for a switch to solid elements in predictions of necking in sheet metal forming processes (e.g., see Refs. [15,19–21]). This need appears to be more critical when the sheet is in contact with a stiff surface like a die or a tool, as it results in local triaxial stress states that can only be captured by 3D representations, as indeed was the case in the present hydroforming problem.

4. Discussion and conclusions

Fig. 12. Solid element model predicted (a) through thickness true strain contours and (b) normal thickness stress contours for HY6 at P¼ 5300 psi (366 bar).

anemic in comparison to the solid model results. Even at the very high pressure of 5360 psi (369.7 bar), the two depressions remain very small and have the same wall thickness (note that this specimen ruptured at about 4400 psi—303 bar). In other words, some aspect of the plane stress idealization of the shell formulation does not promote localization. In the absence of localization, as the pressure increases the shell continues to conform to the die shape filling out the rounded corners. In the present results this is confirmed by the inward movement of the depressions and by the more significant reduction in the wall thickness of the central section of the contours. The thickness of the two wells, depicted by ‘‘C’’ in Fig. 11b, is plotted against pressure in Fig. 11c together with the corresponding results from the solid element model. The plot demonstrates slower thinning of the wall thickness in the depressions and the lack of localization (most of the change corresponds to the overall wall thinning). As mentioned earlier, the drastic difference between the shell and solid element results with respect to capturing the evolution and growth of localization in this problem must be related to 3D effects not properly represented by the simplifying assumptions of plane stress. For example, Fig. 12b shows contours of the through thickness normal stress from the solid model. As expected, the straightened parts of the tube exhibit a nearly uniform stress through the thickness that corresponds to the internal pressure

Part II of the two sister papers deals with the challenges associated with the modeling of the hydroforming experiments presented in Part I. In Part I it was shown that 2D models of our relatively long tubes, although useful as an initial design tool, are inadequate primarily because they do not capture the axial variation in the cross sectional shape and deformation induced by friction. Since insufficient axial compression and the associated wall thinning are responsible for the burst failures that terminated the forming, capturing these effects through fully 3D models became essential. To this end two 3D families of models were developed, the first based on conventional shell elements and the second based on solid elements. A second component of the work is the introduction of non-quadratic isotropic and anisotropic constitutive models to the problem, shown earlier to be essential for the prediction of rupture in free hydroforming of Al-6260-T4 and other Al alloy tubes ([14–26]I, [16]). By adopting the Coulomb friction with a coefficient of 0.2, both families of 3D models were able to reproduce accurately the overall structural effects such as the variation in shape and axial strain and circumferential wall thickness along the length, as well as the force–displacement response of the loading history. Both models also predicted correctly the location of thickness depressions that develop and gradually grow at points on the tube cross section where the straightened sectors that are in contact with the die meet the curved parts that are not. The shell models, however, failed to reproduce correctly the evolution and localization of these thickness depressions. Since these narrow zones of localized deformation govern the onset of rupture, the main limit state encountered in our setup, this deficiency is debilitating for this hydroforming operation. Success in this respect was achieved by switching to solid elements in the 3D simulations coupled with the incorporation of non-quadratic yield functions. Clearly, gradually growing thickness depressions that develop in the presence of pressure on the inner surface and contact with a stiff surface on the outer are associated with locally fully threedimensional stress and deformation states. The same can of course be said about the necking process itself, which here is also influenced by the tube–die contact and by the resulting friction. Such 3D effects can only be dealt with the completeness offered by solid elements. Furthermore, the adoption of a non-quadratic yield function with exponent 8 was shown to be equally important for the correct prediction of the onset of rupture. Inclusion of the anisotropy in such models produced additional improvements for some loading paths. A similar need for the use of solid elements coupled with nonquadratic yield functions for the simulation of slowly growing wall

Y.P. Korkolis, S. Kyriakides / International Journal of Mechanical Sciences 53 (2011) 83–90

depressions was recently reported in Ref. [15]. The localizations develop during lateral crushing of finite length Al-6061-T6 shells radially constrained at the ends by rounded solid end-plugs. They appear early in the crushing history, grow gradually and evolve into necks that lead to rupture. Here again contact with the solid plugs coupled with the necking produce a complex triaxial state of stress. The use of non-quadratic yield function was shown to be indispensable in reproducing the evolution of the necks that develop. Finally it is also interesting to contrast these conclusions with those from the numerical simulation of free hydroforming experiments on similar tubes loaded to rupture under various biaxial stress paths (pressure–axial load). Because of the live pressure, in all cases rupture was sudden and catastrophic and was precipitated by a structural limit load instability. The tube was in a state of nearly plane stress up to the onset of rupture and consequently it could be simulated adequately using shell elements. In other words, it was not necessary to track the evolution of the threedimensional localized wall thinning. However, inclusion of anisotropy in the same non-quadratic models used here played a crucial role in the prediction of the strain paths and the onset of rupture. In contrast, hydroforming inside a die led to a much more stable deformation, where multiple necks form around the circumference and slowly grow with the loading. Clearly, the biaxial radial and corner stress paths followed in the free hydroforming place different demands on the constitutive models than in the tube hydroforming considered here.

Acknowledgements The authors acknowledge with thanks the financial support for 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 and Dr. Jeong-Whan Yoon, formerly of Alcoa and currently at Swinburne University of Technology in Australia, for graciously providing his subroutine for the Yld20043D model used in solid element calculations.

Appendix A. Yld2004-3D-18p Anisotropic Yield Function The Yld2004-3D (18 parameters) constitutive model suggested by Barlat et al. [12] follows the lines of their earlier Yld2000-2D function [5] but affords a complete three-dimensional description of anisotropy. Two linear transformations are used here as well, operating on the deviatoric stress tensor and offering a total of eighteen parameters for the description of anisotropy (compares with the eight parameters of Yld2000-2D). The starting point is again the non-quadratic isotropic yield function of Hosford [4], which in terms of the principal values of the stress deviator is written as k

k

k

9s1 s2 9 þ9s2 s3 9 þ 9s3 s1 9 ¼ 2sko

Su ¼ Cus ¼ CuT r ¼ Lur 0

00

0

and 00

00

as follows: 0

0

0

0

0

0

0

u c44

0

0

u c55

0

0

3

7 0 7 7 0 7 7 7 0 7 7 0 7 5 u c66

ðA:3aÞ

and 2

0 6 00 6 c21 6 00 6 c11 6 C 00 ¼ 6 6 0 6 6 0 4 0

00 c12

00 c13

0

0

0

00 c23

0

0

00 c32

0 0

0 00 c44

0 0

0

3

0

0

0

0

00 c55

0

0

0

0

7 0 7 7 0 7 7 7 0 7 7 0 7 5

ðA:3bÞ

00 c66

The principal values,ðS1u ,S2u ,S3u Þ and ðS100 ,S200 ,S300 Þ, of the linearly transformed stress tensors S0 and S00 , respectively, are evaluated analytically using Cardan’s method. The solutions as well as the first and second derivatives of the yield function with respect to the stress components, which are required for the flow rule and the consistent tangent modulus, are given in Refs. [12,14]. The Yld2004-3D yield function is then written as k

k

k

k

k

f ¼ 9S1u S100 9 þ 9S1u S200 9 þ9S1u S300 9 þ 9S2u S100 9 þ 9S2u S200 9 k

k

k

k

þ 9S2u S300 9 þ 9S3u S100 9 þ9S3u S200 9 þ 9S3u S300 9 ¼ 4sko

ðA:4Þ

In concert with the rest of our work on Al-6260-T4, the exponent k is again assigned the value of 8, typical for FCC alloys [22]. In contrast to working with sheets, where the yield stresses and r-values from uniaxial tests along different orientations are available, we performed a series of biaxial experiments on Al-6260-T4 [24]I, from the same batch as the hydroformed tubes, loaded along radial paths in the engineering stress space (axial–hoop). The resulting points at equal levels of plastic work of 2 ksi (13.8 MPa) are shown in Fig. A.1 as blue dots (note that the material exhibits significant work hardening, see Fig. 3a in [24]I). An advantage of using such an approach is that there are sufficient in-plane data points for the optimization algorithm to easily converge to a realistic shape for the yield locus. The model is then calibrated by fitting the twelve parameters ciju and cij00 (i,j ¼1,3) to the stress work contour in Fig. A.1. These parameters are associated with normal stresses. The remaining six parameters, ciju and cij00 (i¼4,5,6,

σθ

1.2

σo

Exps. (WP= 2 ksi) 0.8

0.4

Yld2004-3D

ðA:1Þ

Anisotropy is now introduced by two linear transformations, which are used to construct the tensors S0 and S00 from the actual stress tensor r as follows: 00

18 anisotropy parameters 2 0 c12 u c13 u 6 u 0 c23 u 6 c21 6 6 c31 u c u 0 32 6 Cu ¼ 6 0 0 6 0 6 6 0 0 0 4 0 0 0

89

00

00

S ¼ C s ¼ C Tr ¼ L r

ðA:2Þ

where C , C , T, L and L are appropriate transformation matrices that allow introduction of anisotropy. T is the standard linear transformation of r to its deviator s, while the C0 and C00 contain the

0 -0.2 -0.2

Al-6260-T4 0

0.4

0.8

σx / σo

1.2

Fig. A.1. Experimental data representing the 2 ksi (13.8 MPa) work contour and the Yld2004-3D yield function.

90

Y.P. Korkolis, S. Kyriakides / International Journal of Mechanical Sciences 53 (2011) 83–90

Table A.1 Anisotropy parameters for Yld2004-3D (k¼8, Wp ¼2 ksi (13.8 MPa)). c12 u

c13 u

c21 u

c23 u

c31 u

c32 u

c44 u

c55 u

c66 u

1.02

1.21

1.14

0.91

0.64

0.73

1.0

1.0

1.0

00 c12

00 c13

00 c21

00 c23

00 c31

00 c32

00 c44

00 c55

00 c66

1.01

0.85

0.82

1.0

1.03

0.98

1.0

1.0

1.0

no sum), which are associated with shear stresses are set equal to 1 as no data was available ([24]I reported that shear anisotropy did not affect the onset of rupture in their free hydroforming work). The extraction of the 12 parameters was performed using a MATLAB program that iteratively minimizes the error between an initial guess for f and the constants and the data points forming the yield surface (unconstrained nonlinear optimization). The set of eighteen parameters thus arrived at is listed in Table A.1 and the resulting plane stress work contour is shown in Fig. A.1. Clearly the agreement between the predicted and the experimental contour is excellent, except perhaps for the uniaxial loading path, which is far from the stress states encountered in tube hydroforming. References [1] Abrantes JP, de Lima CEC, Batalha GF. Numerical simulation of an aluminum alloy tube hydroforming. J Mater Process Technol 2006;179:67–73. [2] Guan Y, Pourboghrat F, Barlat F. Finite element modeling of tube hydroforming of polycrystalline aluminum alloy extrusions. Int J Plast 2006;22:2366–93. [3] Jansson M, Nilsson L, Simonsson K. On strain localization in tube hydroforming of aluminium extrusions. J Mater Process Technol 2008;195:3–14. [4] Hosford WF. A generalized isotropic yield criterion. ASME J Appl Mech 1972;39:607–9. [5] Barlat F, Brem JC, Yoon JW, Chung K, Dick RE, Lege DJ, et al. Plane stress yield function for aluminum alloy sheets-Part 1: theory. Int J Plast 2003;19:1297–319.

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