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Procedía Engineering 81 (2014) 2171-2176

Procedía Engineering

www.elsevier.com/locate/procedia

11th International Conference on Technology of Plasticity, ICTP 2014, 19-24 October 2014,

Nagoya Congress Center, Nagoya, Japan

Modelling of wrinkling in NC bending of thin-walled tubes with large diameters under multi-die constraints using hybrid method

Nan Liu, He Yang*, Heng Li, M. Zhan, Zhijun Tao, Xiao Hu

State Key Laboratory of Solidification Processing, School of Materials Science and Engineering, Northwestern Polytechnical University, P. O.

Box542, Xi'an 710072, P. R China

Abstract

Numerical control thin-walled tube bending is an advanced technology for manufacturing precision bent tube parts in aerospace, aviation and automobiles, etc. With an increase in the diameters of thin-walled Al-alloy tubes and a decrease in bending radii, accurate prediction of wrinkling instability under multi-die constraints is still one challenge and a focused issue, especially for the thin-walled Al-alloy tubes in aviation. Solely using pure analytical solution method, energy-based analytical solution method, pure explicit algorithm or implicit algorithm cannot accurately predict the wrinkling instability. In this study, the analytical bifurcation solution, analytical-numerical energy-based model and eigenvalue buckling analysis are conducted to generate different kinds of buckling modes, respectively, and a series of imperfections are defined in the shapes of these buckling modes. Second, by assigning the geometrical imperfection into the perfect mesh of the tube, a series of hybrid explicit FE models for numerical control rotary-draw-bending is established. The numerical perturbation analyses are carried out with different bifurcation modes-shapes. Thus, based on the minimum energy principle, the shape of imperfection (bifurcation path) and the magnitude of the imperfection (scaling factors) are chosen, which is corresponded to the lowest level of energy consumption. Third, a set of bending experiments is conducted on Al-alloy tube to verify the predictive capability. The predicted results are in good agreement with the experimental results.

© 2014 PublishedbyElsevier Ltd.Thisisan open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibility of the Department of Materials Science and Engineering, Nagoya University Keywords: Thin-walled tube bending; Implicit; Explicit; Instability; Imperfection; Hybrid method

* Corresponding author. Tel.: 86-029-88495632; fax: 86-029-88495632. E-mail address: yanghe@nwpu.edu.cn(He Yang). liheng@nwpu.edu.cn(Heng Li).

1877-7058 © 2014 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibility of the Department of Materials Science and Engineering, Nagoya University doi: 10.1016/j .proeng.2014.10.304

1. Introduction

As one type of lightweight components for 'blood transfusion' or structural weight-bearing, bent tubular thin-walled parts have been increasingly used in applications requiring diverse geometrical specifications and different quality tolerances for various industries, such as the aerospace, automobile, shipbuilding, energy and health care ones(Li et al., 2014). Wrinkling is one of the major defects in thin-walled parts metal forming processes along with tearing, springback and other geometric and surface defects. Wrinkling may be a serious obstacle to implementing the forming process and assembling the parts, and may also play a significant role in the wear of the tool (Kim et al., 2000). It is well-known that most thin-walled parts forming processes all involve complicated boundary conditions, such as in-plane roll-bending of strip, Numerical control rotary-draw-bending of thin-walled tube, thin-walled part spinning, etc (Li et al., 2009). The complicated boundary conditions are defined as: complex loading paths and history, complicated contact conditions caused by multi-die constraints and complicated friction and clearance between workpiece and dies. While it is known that the boundary conditions play vital roles in restraining wrinkling in current forming processes. Under multi-die constraints as shown in Fig. 1(a), the tube bending is a typical complicated boundary conditions process. In the bending process, unequal tension and compression stress inevitably occurs at the extrados and intrados of the bend tube. Halvorsen and Aukrust (2006) pointed that the probability of buckling is related to the magnitude of the compressive stresses, they confirmed that the magnitude of the compressive stresses distribution influenced whether buckling occurs or not. The result of compressive stress is that buckling occurs and wrinkles are formed (shown in Fig. 1(b)). Especially, with an increase in diameters of tube and a decrease in Rd and tube thickness, the thin-walled structure cannot ever support compressive stress because of their small bending rigidity, and the wrinkling instabilities are more and more severe and easier to occur, i

Fig. 1. Large diameters thin-walled tube numerical control bending: (a) with multi-die constraints; (b) photographs of wrinkled tube (150mm (D) x1.5mm (t) x1.75Dmm).

Up to now, mass and great effort has been devoted to studying the wrinkling from both experimental and theoretical. However, an accurate prediction of wrinkling instability is still one challenging and focused issue, especially for large diameter thin-walled tube with complex boundary constrains effects (Li et al., 2009). To sum up, solely using analytical solution method, energy-based analytical solution method, pure explicit FEM or implicit FEM cannot accurately predict wrinkling instabilities with multi-die constraints (Liu et al., 2014). In the present study, taking a large diameter thin-walled Al-alloy tube as research objective, a robust perturbation method is proposed to accurately predict wrinkling instability in tube bending with multi-die constraints. The method is sufficiently robust to find a bifurcation point after applying a small displacement at each node of the tube surface (imperfection). On account of the principle of minimum energy consumption, the shape of imperfection (bifurcation path) and the magnitude of the imperfection (scaling factors) are optimized and determined. One bending case study is carried out. The prediction results which considering geometrical imperfection and not considering geometrical imperfection (perfect mesh) are both compared with experimental data.

2. Implementation of hybrid method for wrinkling instability in NC bending

Three kinds of hybrid methods are proposed to accurately predict wrinkling instabilities in rotary-draw-bending of thin-walled tube. First, considering the characteristics of the distributions of tension and compression stress in the tube, the actual boundary conditions of the rotary-draw-bending of thin-walled tube is thus simplified to obtain wrinkling bifurcation solution. By using analytical bifurcation solution (Corona et al., 2006), analytical-numerical energy-based model and eigenvalue buckling analysis, three different buckling modes of tube are estimated, respectively. Second, after computing the buckling mode-shapes, three different selected buckling modes are introduced into the structure as a geometrical imperfection. On account of the principle of minimum energy consumption, the shape of imperfection (bifurcation path) and the magnitude of the imperfection (scaling factors) are obtained. Third, based on the above strategy, the appropriate imperfections are seeded into the explicit 3D-FE models for the purpose of perturbing the bending tube.

A new 3D elastic-plastic FEM model assigned the geometrical imperfection is established to predict the wrinkling instability using the FE platform ABAQUS (shown in Fig. 2(a)). Taking the tube with 150 mm (D)x1.5 mm(t)x1.75D mm as the research object. Experiment of rotary-draw-bending under the multi-die constrained is shown in Fig. 2(b). The three buckling modes are used in the seeding the imperfection, with the same imperfection amplitude which discussed in next. Four separate input files, which are the same as the primary input file except for the use of the *IMPERFECTION option, are provided, with the shape of perfect of the geometry, half-circle imperfection generated by eigenvalue buckling analysis, full-circle imperfection generated by analytical bifurcation analysis and full-circle imperfection X2 generated by energy-based wrinkling prediction model, respectively. The strategy of application of the hybrid method for wrinkling prediction is presented in Fig. 3. The detailed strategy of application is articulated in Fig. 3.

Full-circle imperfection XI

Fig. 2. The 3D elastic-plastic FEM model for rotary-draw-bending of thin-walled tube: (a) assigning three different imperfections into the 3D elastic-plastic FE model. (b)The W21YPC-159 bender machine and dies used in experiments.

3. Results and discussion

Recoverable elastic strain energy Ee and the plastic dissipation energy Ep which dissipated through plastic processes are the two components of the deformation energy in a bending process. Therefore, using the sum of Ee and Ep, an indicator of deformation energy is proposed as follow: Esum= Ee+ Ep. An investigation concerning evolution of elastic strain energy Ee and plastic dissipation energy Ep by time is carried out as shown in Fig. 4(a) and (b). It can be seen from the Fig. 4 that the order of the energy consumption of the four kinds of imperfection is: perfect of the geometry gets the largest, half-circle imperfection comes second, full-circle imperfection comes third and full-circle imperfection %2 is the lowest one. This is due to the imperfection of analytical bifurcation analysis considers the uniform ovality in the tube pure bending, while the rotary-draw-bending has no obvious section flattening in practice (the support of mandrel and core balls). For this reason, as an imperfection for wrinkling simulation the one corresponding to the lowest level of energy consumption would be the best one. According to the minimum energy principle, the full-circle imperfection X2 is chosen and employed in the following studies. The influence of the imperfection amplitude b on the calculated responses is illustrated in

Perform eigenvalue buckling analysis

Simplified model for wrinkling instability

Define Mesh, Boundary, Loads and Material according to the specific workpiece

Increase the external load proportionally A,times

Solve the determinant of stiffness matrix [K]

Perform analytical bifurcation analysis Simplified model for wrinkling instability

Define Boundary, Loads and Material according to the specific workpiece

w = R cos — ^ cn cos nO,

> = R cos — ^ dn sin nO,,

X n-0 ^x N

u = R cos — ^ en cos nO

Perform energy-based wrinkling prediction analysis

Simplified model for wrinkling instability

Define Boundary, Loads and Material according to the specific workpiece

W < T instable state

Find the critical mcr = tfKTK

Write the modes in the global system as nodal data

A mathematical model is established to describe the buckling mode

w = - R

a + b sin

cos j ,0 < x < L, NX = L

Use the *IMPERFECTION to read the buckling modes

Selection of the shape of imperfection (bifurcation path) Update the shape of imperfection

Shapel: perfect geometry Shape2: Half-circle imperfection Shape3: Full-circle imperfection A, 1 Shape4: Full-circle imperfection A, 2

No b = 0.01

b = 0.0075

b = 0.005

b = 0.0025

Decrease the magnitude of imperfection

b = 0.01 b = 0.0075 b = 0.005 b = 0.0025

Increase the magnitude of imperfection

Fig. 3. The strategy of application of hybrid method for wrinkling prediction in tube bending.

80000 -,

60000 -

r^ 40000 -

c u 20000 -

-20000 -

W -40000 -

Perfect geometry —&— Half-circle imperfectio —o—Full-circle imperfection XI —V— Full-circle imperfection X2

PQ 40000 -

PÛ -20000-

60000005000000

Cu 4000000

£ 1000000 1 0 g -1000000 ^ -2000000 -3000000

■ Perfect geometry,

- Half-circle imperfection -Full-circle imperfection ¡U

- Full-circle imperfection k 2

8000000 H

7000000

a 6000000 PJ

.- 5000000

.2 4000000 a 3000000

.O 1000000

is 0 ^ -1000000 -2000000

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Fig.4. Selection of imperfection on wrinkling in perturbation analysis: (a) and (b) are the recoverable elastic strain energy and plastic dissipation energy for different imperfection shapes; (c) and (d) are recoverable elastic strain energy and plastic dissipation energy for different imperfection amplitude.

The imperfection 12 Amplitude b=0.005

Fig. 5. Simulation results of the perturbation method: (a) with a perfect geometry, (b) with the imperfection 12 and amplitude b=0.005 and (c) photograph of the corresponding wrinkled tube.

Full-circle imperfection Perfect geometry V Experimental results

X axis direction

-a—| Full-circle imperlectio: -o— Perfect geometry ^ Experimental results

0 5 10 15 20 25 30 35 40 45 50 55

Angles from certain section to initial bending section(°)

Fig. 6. Comparison of simulation results obtained by full-circle imperfection, perfect geometry and experiment result: (a) wrinkling profiles; (b) wall thickening degree.

Fig. 4(c) and (d). Based on experimental measurements the value b=0.005 used in the rest of the study was selected as the most representative for the tubes tested. The results presented and discussed from now on are all based on this imperfection (full-circle imperfection X2 and amplitude b=0.005).

Fig.5 shows the wrinkling profiles obtained by full-circle imperfection X2 and amplitude b=0.005, perfect geometry and the experiment result, respectively. In order to show the difference in detail, the deviations of the wrinkling height of the nodes along the intrados of the tube are shown in Fig. 6(a). It is found obviously that a perfect geometry cannot induce the instability of the wrinkling, and the hybrid method gets the best results compared with the experimental ones. This phenomenon is also accordant with the wall thickening degree in Fig. 6(b). So this study proves the conclusion that the pure explicit algorithm with perfect geometry is unreliable and inaccurate for wrinkling prediction in the tube bending, and the full-circle imperfection generated by analytical-numerical energy-based model does better than the perfect geometry model on the wrinkling detection, which is realized by perturbing the mesh of the structure with tiny imperfection.

4. Conclusions

A hybrid method is proposed to accurately predict wrinkling instability in the rotary draw bending of thin-walled Al-alloy tube. Taking a large diameter thin-walled Al-alloy tube (diameter D=150 mm, thickness t=1.5 mm and bending radius 1.175D) as research objective, the effectiveness of the proposed procedure is validated by comparing the tube bending experimental results. The main conclusions are as follows:

Duo to multi-die constraints, the winkling in the NC bending process are usually can't be simulated by an implicit solver. Therefore, based on the analytical bifurcation solution, analytical-numerical energy-based model and eigenvalue buckling analysis, three buckling modes of tube are generated and embedded into the explicit FE model for NC bending. On account of the minimum energy principle, the shape of imperfection (bifurcation path) and the magnitude of the imperfection (scaling factors) are chosen. As a result, the optimal appropriate imperfection shape (full-circle imperfection X2) generated by energy-based wrinkling prediction model and the magnitude of the imperfection (amplitude=0.005) are obtained.

Using the above optimal appropriate imperfection, a series of 3D elastic-plastic FE models assigned the initial imperfection are established. Meanwhile, the bending experiment is carried out, which is used to verify the predictive capability of the hybrid method. The hybrid method is more accurate and robust in plastic wrinkling prediction than the pure explicit finite element method (with prefect mesh), especially for the issues with complex multi-die constraints. It also provides instructive understanding of the plastic wrinkling in thin-walled part forming.

Acknowledgements

The authors would like to thank the National Natural Science Foundation of China (No. 50905144 and 51275415), the Program for New Century Excellent Talents in University, National Basic Research Program of China (No. 2010CB731701) and the 111 Project (B08040).

References

Li, H., Yang, H., Zhang, Z.Y., Li, G.J., Liu, N., Welo, T., 2014. Multiple instability-constrained tube bending limits. J Mater Process Tech214,445-455.

Kim, J., Yang, D., Yoon, J., Barlat, F., 2000. The effect of plastic anisotropy on compressive instability in sheet metal forming. Int J Plasticity16,649-676.

Li, H., Yang, H., Zhan, M., 2009. A study on plastic wrinkling in thin-walled tube bending via an energy-based wrinkling prediction model.

Modelling Simul. Mater. Sci. Eng,1-33. Halvorsen, F., Aukrust, T., 2006. Studies of the mechanisms for buckling and waving in aluminum extrusion by use of a Lagrangian FEM

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J Mater Process Tech214,1173-1189. Corona, E., Lee, L.H., Kyriakides, S., 2006. Yield anisotropy effects on buckling of circular tubes under bending. Int J Solids Struct43,7099-7118.