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Wrinkling analysis for forming limit of tube bending processes
Journal of Materials Processing Technology 152 (2004) 363–369
Wrinkling analysis for forming limit of tube bending processes He Yang∗ , Yan Lin College of Materials Science and Engineering, Northwestern Polytechnical University, P.O. Box 542, Xi’an 710072, PR China Received 13 February 2002; received in revised form 29 December 2003; accepted 25 April 2004
Abstract Thin-walled tube bending processes may produce a wrinkling phenomenon if the process parameters are inappropriate, especially for tubes with large diameter and thin wall thickness. How to predict this phenomenon rapidly and accurately is one of the urgent key problems to be solved for the development of this process at present. In this paper, a wrinkling wave function is proposed and a simplified wrinkling prediction model to predict the minimum bending radius for tubes is established based on thin-shell theory, forming theory, the energy principle and wave function. The minimum bending radius calculated according to the method established is in agreement with data reported in the literature. Furthermore, effects of parameters on the minimum bending radius are also analyzed: (1) the influence of bending angle on the minimum bending radius is negligibly small; (2) the effect of geometrical dimensions and material properties of tubes on the minimum bending radius is significantly large; (3) the minimum bending radius becomes larger with the original radius and strength coefficient of tubes increasing, whereas with the wall thickness and strain hardening exponent decreasing. The results are helpful to the design and optimization of the relevant processes in practice. © 2004 Elsevier B.V. All rights reserved. Keywords: Tube bending; Wrinkling; Forming limit; Energy method; NC
1. Introduction Thin-walled tube parts are playing an important role in automobile, aerospace, oil and other various industries for their satisfactory high strength/weight ratio [1]. The NC precision rotary-draw bending process of thin-walled tube is one of the advanced tube forming processes with high efficiency, high forming precision, low consumption, and good flexibility for bending angle changes. However, the inner side of thin-walled tubes may produce a wrinkling phenomenon if the process parameters are inappropriate especially for tubes with large diameter and thin wall thickness as shown in Fig. 1 because of the compressive stress during the bending process, which will lead to the process failure or even die damage if wrinkling is severe. It is well known that the problem resolution relies heavily on experience and involves repeated trial-and-errors in practice, which spends excessive manpower, raw material, and time in designing and adjusting the process and dies, and moreover makes the production efficiency abate drastically. Even so, sometimes contented outcomes may not be
∗ Corresponding author. Fax: +86 29 8491000. E-mail address: [email protected] (H. Yang).
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.04.410
obtained [2]. As a result, How to predict this phenomenon rapidly and accurately has become one of the key problems urgently to be solved for the development of this advanced NC thin-walled tube bending process at present. Wrinkling research has interested many scholars for a long period. Great efforts have been put on in the prediction of wrinkling initiation during sheet forming processes. The energy method has always been a widely used approach to obtain the critical condition of wrinkling in these processes. However, up to now, literature on the studies of wrinkling in the tube bending process has been scant. Wang and Cao [3] studied the limit of tube bending process by wrinkling analysis. However, it appears that the method proposed by them is not suitable to acquire the minimum bending radius for tubes with large radius/thickness ratio because of the relevant wrinkling wave function proposed, which predicts the critical stress for wrinkling is rather small for these tubes. Therefore, in this paper, a new wrinkling wave function associated with bending angle and wave numbers is established. And furthermore a simplified stress-based model for predicting the process wrinkling is developed with the aim of the exploring the dependence of the forming limit of the process on geometrical parameters and material properties of tubes on the basis of the energy principle and thin-shell theory.
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H. Yang, Y. Lin / Journal of Materials Processing Technology 152 (2004) 363–369
κββ = −
1 ∂2 w w − 2, 2 2 ∂θ r0 r0
καβ = χ = −
Fig. 1. Photograph of a wrinkled tube part.
2. Energy method for predicting thin-shell wrinkling Energy principle is a main and effective approach to establish the wrinkling criteria for thin shell workpieces during forming processes. It believes that the critical moment of wrinkling onset is when the internal energy of the wrinkled shell U equals the work done by the external forces T [4], that is: U=T
(1)
3. Wrinkling prediction model of tube bending process 3.1. Shell model According to the rotary-draw tube process (Fig. 2), an analytical shell model is made up. 0 ∼ ϕ1 is the compressed area along the longitudinal direction of tube, in which wrinkling area is from ϕ0 to ϕ1 . Moreover, it is thought that the shell deformation is only related with the displacement w in the normal direction. Thus, the strains of the shell due to wrinkling are: εαα = εϕ =
w , R
εαβ = γϕθ = 0,
w , εαβ = γϕθ = 0, r0 1 ∂2 w sin θ ∂w w =− 2 2 − − 2, Rr0 ∂θ R ∂ϕ R
εββ = εθ = καα
1 ∂2 w sin θ ∂w + 2 Rr0 ∂ϕ∂θ R ∂ϕ
(2)
where α, β and z are the convective coordinate system on the middle surface of a shell; εij (i, j = α, β) the strain components on the middle surface; κij the curvature change of the shell; R = R0 − r0 cos θ; R0 is the bending radius, r0 the tube radius. And ϕ is the curve coordinate in the tube bending direction which changes from ϕ0 to ϕ1 , while θ is the curve coordinate in the circumferential direction of the tube A–A cross-section (shown in Fig. 2) which changes from 0 to θ 1 that is considered as π/2. According to Yu and Zhang [5], the plasticity deformation theory, assumption of thin-walled shell theory and Eq. (2), the internal energy of the wrinkled shell U includes the energy needed by the membrane deformation and bending deflection. U can be expressed as: U=
t0 /2
S
+
−t0 /2
σij dz
dεij
dS
Mij dκij dS t w2 w2 w2 Es = Rr0 dϕ dθ + 2 + 2ν 2 Rr0 1 − ν 2 R2 r0 2 Er 1 ∂2 w t3 w + + 2 24 1 − ν2 r02 ∂θ 2 r0 2 1 ∂2 w sin θ ∂w w + + + Rr0 ∂θ R2 ∂ϕ2 R2 1 ∂2 w sin θ ∂w 1 ∂2 w w w + 2ν 2 2 + 2 + + 2 Rr0 ∂θ R2 ∂ϕ2 R r0 ∂θ r0 2 sin θ ∂w 1 ∂2 w Rr0 dϕ dθ (3) − 2 + 2(1 − ν) Rr0 ∂ϕ∂θ R ∂ϕ S
Fig. 2. Schematic diagram of the rotary-draw tube bending process.
H. Yang, Y. Lin / Journal of Materials Processing Technology 152 (2004) 363–369
where σ ij (i, j = α, β) is the membrane stress components, Mij the bending moments, S the region of the shell middle √ √ surface where wrinkles occur, Er = 4EEt /( E + Et )2 the reduced modulus, Et = dσ/d¯ ¯ ε the tangent modulus, Es = σ/¯ ¯ ε the secant modulus, ν the Poisson’s ratio which is considered 0.5 in plastic deformation without allowing for the effect of material anisotropic behavior and t0 is the tube wall thickness. 3.2. Introduction of a new wrinkling wave function According to the wrinkling wave function proposed by Wang and Cao [3] and assuming that there are m half waves in the longitudinal direction of tube if wrinkles occur and the boundary conditions: (1) ϕ = 0, ϕ1 − ϕ0 , w = 0; (2) ϕ = 0, ϕ1 − ϕ0 , ∂w/∂ϕ = 0, the displacement in the normal direction w can be characterized with the new following function form: 2πmϕ w = f 1 − cos ϕ1 − ϕ 0 where f can be determined in the following way: if tube wrinkles when the bending radius is R0 , according to the assumption for small deflection the longitudinal displacement of the compressed section of tube lϕ (shown in Fig. 3) is
ϕ1 −ϕ0 ϕ1 −ϕ0 1 dw 2 R 1+ dϕ − R dϕ lϕ = R dϕ 0 0 1 ϕ1 −ϕ0 ∂w 2 ≈ dϕ 2R 0 ∂ϕ and also lϕ = R0 (ϕ1 − ϕ0 ) − R(ϕ1 − ϕ0 ) = r0 cos θ(ϕ1 − ϕ0 ) then
√ r0 R0 (ϕ1 − ϕ0 ) R cos θ f = πm R0
thus
w = w0 √ w0 =
and
r0 R0 (ϕ1 − ϕ0 ) πm
(4)
3.3. Establishment of a wrinkling prediction model From Eqs. (3) and (4), the energy U required for the shell wrinkling can be expressed as: r0 R0 K1 K3 U = 2 m2 + K2 (ϕ1 − ϕ0 ) + 2 (ϕ1 − ϕ0 )3 ϕ1 − ϕ 0 π m (5) when ∂U =0 ∂m we get
mcr = (ϕ1 − ϕ0 ) 4
Umin =
K3 K1
r0 R 0 K1 m2cr + K2 (ϕ1 − ϕ0 ) 2 ϕ1 − ϕ 0 π K3 3 + 2 (ϕ1 − ϕ0 ) mcr
(6)
where t03 π/2 Er 8π4 r0 K1 = cos θ dθ, 24 0 1 − ν2 R2 R0 t 3 π/2 Er 2π2 K2 = 0 − sin θ(r0 sin 2θ − R0 sin θ) 24 0 1 − ν2 R2 R0 2π2 ν − 2r0 cos 2θ − R0 cos θ RR0 r0 (r0 sin 2θ − R0 sin θ)2 − 2RR0 cos θ +
(1 − ν)π2 (r0 sin 2θ − R0 sin θ)2 cos θ R2 R0 r 0
−
4(1 − ν)π2 sin θ (r0 sin 2θ − R0 sin θ) R2 R0
4(1 − ν)π2 r0 2 sin θ cos θ R2 R 0 r0 ν 2 − 4π + cos θ dθ, R0 r0 R2 R0 t03 π/2 Er K3 = 24 0 1 − ν2
sin2 θ 3 (r0 sin 2θ − R0 sin θ)2 × 8R2 R0 r0 cos θ +
Fig. 3. Illustration of the wrinkled bending part of a tube.
2πmϕ R cos θ 1 − cos R0 ϕ1 − ϕ 0
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H. Yang, Y. Lin / Journal of Materials Processing Technology 152 (2004) 363–369
1 + 3 2r0 cos 2θ − R0 cos θ 8r0 R0 cos θ 2 (r0 sin 2θ − R0 sin θ)2 − 2RR0 cos θ 3ν sin θ + (r0 sin 2θ − R0 sin θ) 4RR0 r02 cos θ (r0 sin 2θ − R0 θ)2 × 2r0 cos 2θR0 cos θ − 2RR0 cos θ r0 3 R2 2ν cos θ + + + 2 R2 R0 R 0 r0 R0 r03 3 sin θ ν + + 2 (r0 sin 2θ − R0 sin θ) 2R0 R2 r0 R ν 3 + + 2R0 r03 Rr0 (r0 sin 2θ − R0 sin θ)2 × 2r0 cos 2θ−R0 cos θ− dθ 2RR0 cos θ R2 R t0 π/2 Es 3 r0 cos θ dθ + + 2ν + 2 0 1 − ν 2 2 R0 R 0 r0 R0 3
As shown in Fig. 4, the work T done by the external force can be determined by the following formula mainly associated with σ ϕ :
π/2
T = t0
0
σϕ lϕ r0 dθ
π/2
= t0 0
σϕ r2 (ϕ1 − ϕ0 ) cos θ dθ 0
(7)
where σ ϕ is the compression stress. If the material model of tubes is σ¯ = K (¯ε + ε0 )n , then σ ϕ can be determined by the following equation (its derivation is shown in Appendix A): n 2 1 R σϕ = − √ K √ ln + ε0 R0 3 3 n θ 1 r0 2 R −√ K + ε0 dθ (8) sin θ √ ln R0 3 π/2 R 3 where K is the strength coefficient and n the strain hardening exponent.
3.4. Minimum bending radius From Eqs. (1), (6) and (7), we get R0 K1 K3 2 2 m + K + (ϕ − ϕ ) 2 1 0 cr π2 (ϕ1 − ϕ0 )2 m2cr π/2 2 σϕ r cos θ dθ = t0 0
(9)
0
The minimum bending radius can be obtained by analyzing the above Eqs. (8) and (9).
4. Results and discussion The computations were carried out under various process conditions for various tube materials. Some representative results are given in the following. 4.1. Verification of the wrinkling prediction model The materials used is 304 stainless steel tube with wall thickness being 1.65 mm and its stress–strain characteristics is σ¯ = 1356(¯ε + 0.001)0.549 . Variation of the minimum bending radius as a function of tube radiuses/thickness theoretically predicted using author’s model and its comparison with the experimental data from [6] and with the results from Wang and Cao’s model [3] are shown in Fig. 5 without considering the material anisotropic behavior. It can be seen that the model proposed in this paper is more suitable than Wang and Cao’s model [3] to predict the bending limit for tubes with the ratio of radius to thickness larger than 20. This shows that the new wrinkling wave function proposed and the wrinkling prediction model established are reasonable, considering the assumptions adopted in this paper. By the model proposed in this paper By Wang and Cao's model[3] Experimental from[6]
1000
Minimum bending radius R0 /mm
366
800 600 400
200 0 16
20
24
28
32
36
40
Tube radius/thickness r0/t0
Fig. 4. Illustration of the compression stress distribution in the tube longitudinal direction.
Fig. 5. Minimum bending radius vs. tube radius/thickness theoretically predicted using author’s model and its comparison with the experimental data from [6] and with the results from Wang and Cao’s model for 304 stainless steel tube under t0 = 1.65.
114 112 110 108 106 0.092
367
550
116
minimum bending radius R0 /mm
Minimum bending radius R0 /mm
H. Yang, Y. Lin / Journal of Materials Processing Technology 152 (2004) 363–369
0.094
0.096
0.098
500
304 LF21M
450 400 350 300 250 200 150 100 50 0.50
0.100
0.75
Angle ϕ1-ϕ0 /rad
1.00
1.25
1.50
1.75
2.00
Wall-thickness t0 /mm Fig. 8. Minimum bending radius vs. tube thickness.
Fig. 6. Minimum bending radius vs. bending angle for LF21M aluminum tube.
Minimum bending radius R0 /mm
600
4.2. Effects of geometrical parameters The calculation condition: the material is LF21M with the stress flow model σ¯ = 177¯ε0.21 , and the geometrical dimensions are t0 = 1.0 mm and r0 = 20 mm; where the bending angle ranges from 0.092 to 0.10 rad; the materials are 304 stainless steel and LF21M with the stress flow models σ¯ = 1356(¯ε + 0.001)0.549 and σ¯ = 177¯ε0.21 , respectively; where when t0 = 1.0 mm, r0 ranges from 20 to 50 mm; and when r0 equals 30 mm, t0 ranges from 0.6 to 2.0 mm. The effect of the bending angle is shown in Fig. 6. It can be seen that the bending angle has a rather little influence on the minimum bending radius when the bending angle is small, and the minimum bending radius is almost fixed when the bending angle exceeds a certain value. Thus, the influence of the bending angle on the minimum bending radius can be ignored. The influence of geometrical dimensions of tubes on the minimum bending radius are shown in Figs. 7 and 8. It can be seen that the minimum bending radius increases with the tube radius increasing, while it reduces with wall thickness rising.
500 400 300 200 100 0.0
0.1
0.2
0.3
0.4
0.5
0.6
Strain hardening exponent n Fig. 9. Minimum bending radius vs. strain.
4.3. Effects of material properties The calculation condition: the geometrical dimensions are t0 = 1.0 mm and r0 = 20 mm; where when K = 177 MPa, n varies from 0.05 to 0.6; and when n = 0.21, K ranges from 100 to 1500 MPa. Figs. 9 and 10 illustrate the effect of material properties on the minimum bending radius. It is shown that the minimum
Minimum bending radius R0 /mm
330
minimum bending radius R0 /mm
900 800
304 LF21M
700 600 500 400 300 200 100 0 20
25
30
35
40
45
Tube radius r0 /mm Fig. 7. Minimum bending radius vs. tube radius.
50
315
300
285
270
0
200 400 600 800 1000 1200 1400 1600
Stength coefficient K /MPa Fig. 10. Minimum relative bending radius vs. hardening exponent strength coefficient.
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H. Yang, Y. Lin / Journal of Materials Processing Technology 152 (2004) 363–369
dθ
θ
Infinitesimal element
Y
O1
σθ+dσθ
σr+dσr
X
σr
R
ϕ
σθ
r
dr
O1
50225518), National Natural Science Foundation of China (Nos. 59975076 and 50175092), the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, PRC, and Specialized Research Fund for the Doctoral Program of Higher Education of MOE, PRC (20020699002) for the support given to this research.
Appendix A Y
X
O
dR
R
σr+dσr σϕ
σθ dϕ
ϕ σϕ
R O
Fig. 11. Illustration of a infinitesimal element on the compressive area of tube in the bending process.
bending radius decreases with the strain hardening exponent increasing and with the strength coefficient decreasing.
5. Conclusion By introducing a new wave function, a simplified analytical wrinkling prediction model for thin-walled tube bending is set up, the agreement between theory and experiment having been verified to be satisfactory, especially compared with [3], when the radius/wall thickness ratio of tube is larger than 20. This model can be used to determine the minimum bending radius of given tubes. The influences of geometrical parameters and material properties of tubes are also studied. The results indicate that: (1) the influence of the bending angle on the minimum bending radius is negligibly small; (2) the effect of the geometrical parameters and material properties of tubes on the minimum bending radius is significantly large; (3) the minimum bending radius becomes larger with the original radius and strength coefficient of the tubes increasing, whereas with the wall thickness and strain hardening exponent decreasing. The model proposed in this paper cannot take all factors into account, such as the rotating speed of the bend die, friction and so on, whose influences on the bending limit can be relatively thoroughly researched by FEM simulation.
Acknowledgements The authors would like to thank the National Science Foundation of China for Distinguished Young Scholars (No.
A infinitesimal element (dr × dθ × dϕ) is intercepted from the location (r, θ, ϕ) in the compressive area during the tube bending process (shown in Fig. 11). The X–Y plane is the intersection of the tube and Y coordinate is in the direction of the bending radius. ϕ is a coordinate along the longitudinal direction during the deforming tube. The infinitesimal balance equations in the Y and X directions are, respectively, − [(σr + dσr )(r + dr)(R + dR) dθ dϕ − σr rR dθ dϕ] cos θ + [(σθ + dσθ )(R + dR) dr dϕ − σθ R dr dϕ] sin θ − σϕ r dr dθ dϕ = 0
(A.1)
[(σr + dσr )(r + dr)(R + dR) dθ dϕ − σr rR dθ dϕ] sin θ + [(σθ + dσθ )(R + dR) dr dϕ − σθ R dr dϕ] cos θ = 0 (A.2) From (A.1) and (A.2) 1 (σθ dR + R dσθ ) = σϕ r dθ sin θ
(A.3)
Because of t0 /r0 << 1 for thin-walled tubes, it can be thought of that r ≈ r0 . And then from R = R0 − r cos θ, we can get dR = r0 sin θ dθ. Thus it can be derived from (A.3) that r0 dσθ = sin θ(σϕ − σθ ) dθ (A.4) R If assuming that there is no olivation of the tube intersection during the bending process, then εθ = 0, and σϕ + σr 2
(A.5)
2 σϕ − σr = − √ σ¯ 3
(A.6)
2 ε¯ = √ εϕ 3
(A.7)
σθ =
From (A.5) and (A.6) 1 r dσϕ = − √ dσ¯ + σ¯ sin θ dθ R 3
(A.8)
H. Yang, Y. Lin / Journal of Materials Processing Technology 152 (2004) 363–369
369
According to the material stress flow model σ¯ = K(¯ε + ε0 )n , (A.8) becomes:
References
1 1 r0 dσϕ = − √ d K(¯ε + ε0 )n − √ K sin θ(¯ε + ε0 )n dθ 3 3 R (A.9)
[1] H. Yang, Y. Lin, S. Zhichao, Advanced plastic processing technology and research progress on tube forming, J. Plasticity Eng. 8 (2) (2001) 86–88 (in Chinese). [2] Y. Lin, H. Yang, Thin-walled tube precision bending process and FEM simulation, in: Proceedings of the Fourth International Conference on Frontiers of Design and Manufacturing, vol. 2, Beijing, 2000, pp. 305–308. [3] X. Wang, J. Cao, Wrinkling limit in tube bending, Trans. ASME 123 (2001) 430–435. [4] L. Hongwen, Theory of Plates and Shells, Zhejiang University Press, Hangzhou, 1987 (in Chinese). [5] T.X. Yu, L.C. Zhang, Plastic Bending Theory, Science Press, Beijing, 1992 (in Chinese). [6] W. Tonghai, Tube Forming Technology, Mechanical Industry Press, Beijing, 1998 (in Chinese).
Assuming that the neutral layer of stress, that of strain, and the geometrical center are coincident, then n 1 2 R σϕ = − √ K √ ln + ε0 R0 3 3 n θ 1 r0 2 R −√ K sin θ √ ln + ε0 dθ R0 3 π/2 R 3 (A.10)
Wrinkling analysis for forming limit of tube bending processes He Yang∗ , Yan Lin College of Materials Science and Engineering, Northwestern Polytechnical University, P.O. Box 542, Xi’an 710072, PR China Received 13 February 2002; received in revised form 29 December 2003; accepted 25 April 2004
Abstract Thin-walled tube bending processes may produce a wrinkling phenomenon if the process parameters are inappropriate, especially for tubes with large diameter and thin wall thickness. How to predict this phenomenon rapidly and accurately is one of the urgent key problems to be solved for the development of this process at present. In this paper, a wrinkling wave function is proposed and a simplified wrinkling prediction model to predict the minimum bending radius for tubes is established based on thin-shell theory, forming theory, the energy principle and wave function. The minimum bending radius calculated according to the method established is in agreement with data reported in the literature. Furthermore, effects of parameters on the minimum bending radius are also analyzed: (1) the influence of bending angle on the minimum bending radius is negligibly small; (2) the effect of geometrical dimensions and material properties of tubes on the minimum bending radius is significantly large; (3) the minimum bending radius becomes larger with the original radius and strength coefficient of tubes increasing, whereas with the wall thickness and strain hardening exponent decreasing. The results are helpful to the design and optimization of the relevant processes in practice. © 2004 Elsevier B.V. All rights reserved. Keywords: Tube bending; Wrinkling; Forming limit; Energy method; NC
1. Introduction Thin-walled tube parts are playing an important role in automobile, aerospace, oil and other various industries for their satisfactory high strength/weight ratio [1]. The NC precision rotary-draw bending process of thin-walled tube is one of the advanced tube forming processes with high efficiency, high forming precision, low consumption, and good flexibility for bending angle changes. However, the inner side of thin-walled tubes may produce a wrinkling phenomenon if the process parameters are inappropriate especially for tubes with large diameter and thin wall thickness as shown in Fig. 1 because of the compressive stress during the bending process, which will lead to the process failure or even die damage if wrinkling is severe. It is well known that the problem resolution relies heavily on experience and involves repeated trial-and-errors in practice, which spends excessive manpower, raw material, and time in designing and adjusting the process and dies, and moreover makes the production efficiency abate drastically. Even so, sometimes contented outcomes may not be
∗ Corresponding author. Fax: +86 29 8491000. E-mail address: [email protected] (H. Yang).
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.04.410
obtained [2]. As a result, How to predict this phenomenon rapidly and accurately has become one of the key problems urgently to be solved for the development of this advanced NC thin-walled tube bending process at present. Wrinkling research has interested many scholars for a long period. Great efforts have been put on in the prediction of wrinkling initiation during sheet forming processes. The energy method has always been a widely used approach to obtain the critical condition of wrinkling in these processes. However, up to now, literature on the studies of wrinkling in the tube bending process has been scant. Wang and Cao [3] studied the limit of tube bending process by wrinkling analysis. However, it appears that the method proposed by them is not suitable to acquire the minimum bending radius for tubes with large radius/thickness ratio because of the relevant wrinkling wave function proposed, which predicts the critical stress for wrinkling is rather small for these tubes. Therefore, in this paper, a new wrinkling wave function associated with bending angle and wave numbers is established. And furthermore a simplified stress-based model for predicting the process wrinkling is developed with the aim of the exploring the dependence of the forming limit of the process on geometrical parameters and material properties of tubes on the basis of the energy principle and thin-shell theory.
364
H. Yang, Y. Lin / Journal of Materials Processing Technology 152 (2004) 363–369
κββ = −
1 ∂2 w w − 2, 2 2 ∂θ r0 r0
καβ = χ = −
Fig. 1. Photograph of a wrinkled tube part.
2. Energy method for predicting thin-shell wrinkling Energy principle is a main and effective approach to establish the wrinkling criteria for thin shell workpieces during forming processes. It believes that the critical moment of wrinkling onset is when the internal energy of the wrinkled shell U equals the work done by the external forces T [4], that is: U=T
(1)
3. Wrinkling prediction model of tube bending process 3.1. Shell model According to the rotary-draw tube process (Fig. 2), an analytical shell model is made up. 0 ∼ ϕ1 is the compressed area along the longitudinal direction of tube, in which wrinkling area is from ϕ0 to ϕ1 . Moreover, it is thought that the shell deformation is only related with the displacement w in the normal direction. Thus, the strains of the shell due to wrinkling are: εαα = εϕ =
w , R
εαβ = γϕθ = 0,
w , εαβ = γϕθ = 0, r0 1 ∂2 w sin θ ∂w w =− 2 2 − − 2, Rr0 ∂θ R ∂ϕ R
εββ = εθ = καα
1 ∂2 w sin θ ∂w + 2 Rr0 ∂ϕ∂θ R ∂ϕ
(2)
where α, β and z are the convective coordinate system on the middle surface of a shell; εij (i, j = α, β) the strain components on the middle surface; κij the curvature change of the shell; R = R0 − r0 cos θ; R0 is the bending radius, r0 the tube radius. And ϕ is the curve coordinate in the tube bending direction which changes from ϕ0 to ϕ1 , while θ is the curve coordinate in the circumferential direction of the tube A–A cross-section (shown in Fig. 2) which changes from 0 to θ 1 that is considered as π/2. According to Yu and Zhang [5], the plasticity deformation theory, assumption of thin-walled shell theory and Eq. (2), the internal energy of the wrinkled shell U includes the energy needed by the membrane deformation and bending deflection. U can be expressed as: U=
t0 /2
S
+
−t0 /2
σij dz
dεij
dS
Mij dκij dS t w2 w2 w2 Es = Rr0 dϕ dθ + 2 + 2ν 2 Rr0 1 − ν 2 R2 r0 2 Er 1 ∂2 w t3 w + + 2 24 1 − ν2 r02 ∂θ 2 r0 2 1 ∂2 w sin θ ∂w w + + + Rr0 ∂θ R2 ∂ϕ2 R2 1 ∂2 w sin θ ∂w 1 ∂2 w w w + 2ν 2 2 + 2 + + 2 Rr0 ∂θ R2 ∂ϕ2 R r0 ∂θ r0 2 sin θ ∂w 1 ∂2 w Rr0 dϕ dθ (3) − 2 + 2(1 − ν) Rr0 ∂ϕ∂θ R ∂ϕ S
Fig. 2. Schematic diagram of the rotary-draw tube bending process.
H. Yang, Y. Lin / Journal of Materials Processing Technology 152 (2004) 363–369
where σ ij (i, j = α, β) is the membrane stress components, Mij the bending moments, S the region of the shell middle √ √ surface where wrinkles occur, Er = 4EEt /( E + Et )2 the reduced modulus, Et = dσ/d¯ ¯ ε the tangent modulus, Es = σ/¯ ¯ ε the secant modulus, ν the Poisson’s ratio which is considered 0.5 in plastic deformation without allowing for the effect of material anisotropic behavior and t0 is the tube wall thickness. 3.2. Introduction of a new wrinkling wave function According to the wrinkling wave function proposed by Wang and Cao [3] and assuming that there are m half waves in the longitudinal direction of tube if wrinkles occur and the boundary conditions: (1) ϕ = 0, ϕ1 − ϕ0 , w = 0; (2) ϕ = 0, ϕ1 − ϕ0 , ∂w/∂ϕ = 0, the displacement in the normal direction w can be characterized with the new following function form: 2πmϕ w = f 1 − cos ϕ1 − ϕ 0 where f can be determined in the following way: if tube wrinkles when the bending radius is R0 , according to the assumption for small deflection the longitudinal displacement of the compressed section of tube lϕ (shown in Fig. 3) is
ϕ1 −ϕ0 ϕ1 −ϕ0 1 dw 2 R 1+ dϕ − R dϕ lϕ = R dϕ 0 0 1 ϕ1 −ϕ0 ∂w 2 ≈ dϕ 2R 0 ∂ϕ and also lϕ = R0 (ϕ1 − ϕ0 ) − R(ϕ1 − ϕ0 ) = r0 cos θ(ϕ1 − ϕ0 ) then
√ r0 R0 (ϕ1 − ϕ0 ) R cos θ f = πm R0
thus
w = w0 √ w0 =
and
r0 R0 (ϕ1 − ϕ0 ) πm
(4)
3.3. Establishment of a wrinkling prediction model From Eqs. (3) and (4), the energy U required for the shell wrinkling can be expressed as: r0 R0 K1 K3 U = 2 m2 + K2 (ϕ1 − ϕ0 ) + 2 (ϕ1 − ϕ0 )3 ϕ1 − ϕ 0 π m (5) when ∂U =0 ∂m we get
mcr = (ϕ1 − ϕ0 ) 4
Umin =
K3 K1
r0 R 0 K1 m2cr + K2 (ϕ1 − ϕ0 ) 2 ϕ1 − ϕ 0 π K3 3 + 2 (ϕ1 − ϕ0 ) mcr
(6)
where t03 π/2 Er 8π4 r0 K1 = cos θ dθ, 24 0 1 − ν2 R2 R0 t 3 π/2 Er 2π2 K2 = 0 − sin θ(r0 sin 2θ − R0 sin θ) 24 0 1 − ν2 R2 R0 2π2 ν − 2r0 cos 2θ − R0 cos θ RR0 r0 (r0 sin 2θ − R0 sin θ)2 − 2RR0 cos θ +
(1 − ν)π2 (r0 sin 2θ − R0 sin θ)2 cos θ R2 R0 r 0
−
4(1 − ν)π2 sin θ (r0 sin 2θ − R0 sin θ) R2 R0
4(1 − ν)π2 r0 2 sin θ cos θ R2 R 0 r0 ν 2 − 4π + cos θ dθ, R0 r0 R2 R0 t03 π/2 Er K3 = 24 0 1 − ν2
sin2 θ 3 (r0 sin 2θ − R0 sin θ)2 × 8R2 R0 r0 cos θ +
Fig. 3. Illustration of the wrinkled bending part of a tube.
2πmϕ R cos θ 1 − cos R0 ϕ1 − ϕ 0
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1 + 3 2r0 cos 2θ − R0 cos θ 8r0 R0 cos θ 2 (r0 sin 2θ − R0 sin θ)2 − 2RR0 cos θ 3ν sin θ + (r0 sin 2θ − R0 sin θ) 4RR0 r02 cos θ (r0 sin 2θ − R0 θ)2 × 2r0 cos 2θR0 cos θ − 2RR0 cos θ r0 3 R2 2ν cos θ + + + 2 R2 R0 R 0 r0 R0 r03 3 sin θ ν + + 2 (r0 sin 2θ − R0 sin θ) 2R0 R2 r0 R ν 3 + + 2R0 r03 Rr0 (r0 sin 2θ − R0 sin θ)2 × 2r0 cos 2θ−R0 cos θ− dθ 2RR0 cos θ R2 R t0 π/2 Es 3 r0 cos θ dθ + + 2ν + 2 0 1 − ν 2 2 R0 R 0 r0 R0 3
As shown in Fig. 4, the work T done by the external force can be determined by the following formula mainly associated with σ ϕ :
π/2
T = t0
0
σϕ lϕ r0 dθ
π/2
= t0 0
σϕ r2 (ϕ1 − ϕ0 ) cos θ dθ 0
(7)
where σ ϕ is the compression stress. If the material model of tubes is σ¯ = K (¯ε + ε0 )n , then σ ϕ can be determined by the following equation (its derivation is shown in Appendix A): n 2 1 R σϕ = − √ K √ ln + ε0 R0 3 3 n θ 1 r0 2 R −√ K + ε0 dθ (8) sin θ √ ln R0 3 π/2 R 3 where K is the strength coefficient and n the strain hardening exponent.
3.4. Minimum bending radius From Eqs. (1), (6) and (7), we get R0 K1 K3 2 2 m + K + (ϕ − ϕ ) 2 1 0 cr π2 (ϕ1 − ϕ0 )2 m2cr π/2 2 σϕ r cos θ dθ = t0 0
(9)
0
The minimum bending radius can be obtained by analyzing the above Eqs. (8) and (9).
4. Results and discussion The computations were carried out under various process conditions for various tube materials. Some representative results are given in the following. 4.1. Verification of the wrinkling prediction model The materials used is 304 stainless steel tube with wall thickness being 1.65 mm and its stress–strain characteristics is σ¯ = 1356(¯ε + 0.001)0.549 . Variation of the minimum bending radius as a function of tube radiuses/thickness theoretically predicted using author’s model and its comparison with the experimental data from [6] and with the results from Wang and Cao’s model [3] are shown in Fig. 5 without considering the material anisotropic behavior. It can be seen that the model proposed in this paper is more suitable than Wang and Cao’s model [3] to predict the bending limit for tubes with the ratio of radius to thickness larger than 20. This shows that the new wrinkling wave function proposed and the wrinkling prediction model established are reasonable, considering the assumptions adopted in this paper. By the model proposed in this paper By Wang and Cao's model[3] Experimental from[6]
1000
Minimum bending radius R0 /mm
366
800 600 400
200 0 16
20
24
28
32
36
40
Tube radius/thickness r0/t0
Fig. 4. Illustration of the compression stress distribution in the tube longitudinal direction.
Fig. 5. Minimum bending radius vs. tube radius/thickness theoretically predicted using author’s model and its comparison with the experimental data from [6] and with the results from Wang and Cao’s model for 304 stainless steel tube under t0 = 1.65.
114 112 110 108 106 0.092
367
550
116
minimum bending radius R0 /mm
Minimum bending radius R0 /mm
H. Yang, Y. Lin / Journal of Materials Processing Technology 152 (2004) 363–369
0.094
0.096
0.098
500
304 LF21M
450 400 350 300 250 200 150 100 50 0.50
0.100
0.75
Angle ϕ1-ϕ0 /rad
1.00
1.25
1.50
1.75
2.00
Wall-thickness t0 /mm Fig. 8. Minimum bending radius vs. tube thickness.
Fig. 6. Minimum bending radius vs. bending angle for LF21M aluminum tube.
Minimum bending radius R0 /mm
600
4.2. Effects of geometrical parameters The calculation condition: the material is LF21M with the stress flow model σ¯ = 177¯ε0.21 , and the geometrical dimensions are t0 = 1.0 mm and r0 = 20 mm; where the bending angle ranges from 0.092 to 0.10 rad; the materials are 304 stainless steel and LF21M with the stress flow models σ¯ = 1356(¯ε + 0.001)0.549 and σ¯ = 177¯ε0.21 , respectively; where when t0 = 1.0 mm, r0 ranges from 20 to 50 mm; and when r0 equals 30 mm, t0 ranges from 0.6 to 2.0 mm. The effect of the bending angle is shown in Fig. 6. It can be seen that the bending angle has a rather little influence on the minimum bending radius when the bending angle is small, and the minimum bending radius is almost fixed when the bending angle exceeds a certain value. Thus, the influence of the bending angle on the minimum bending radius can be ignored. The influence of geometrical dimensions of tubes on the minimum bending radius are shown in Figs. 7 and 8. It can be seen that the minimum bending radius increases with the tube radius increasing, while it reduces with wall thickness rising.
500 400 300 200 100 0.0
0.1
0.2
0.3
0.4
0.5
0.6
Strain hardening exponent n Fig. 9. Minimum bending radius vs. strain.
4.3. Effects of material properties The calculation condition: the geometrical dimensions are t0 = 1.0 mm and r0 = 20 mm; where when K = 177 MPa, n varies from 0.05 to 0.6; and when n = 0.21, K ranges from 100 to 1500 MPa. Figs. 9 and 10 illustrate the effect of material properties on the minimum bending radius. It is shown that the minimum
Minimum bending radius R0 /mm
330
minimum bending radius R0 /mm
900 800
304 LF21M
700 600 500 400 300 200 100 0 20
25
30
35
40
45
Tube radius r0 /mm Fig. 7. Minimum bending radius vs. tube radius.
50
315
300
285
270
0
200 400 600 800 1000 1200 1400 1600
Stength coefficient K /MPa Fig. 10. Minimum relative bending radius vs. hardening exponent strength coefficient.
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dθ
θ
Infinitesimal element
Y
O1
σθ+dσθ
σr+dσr
X
σr
R
ϕ
σθ
r
dr
O1
50225518), National Natural Science Foundation of China (Nos. 59975076 and 50175092), the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, PRC, and Specialized Research Fund for the Doctoral Program of Higher Education of MOE, PRC (20020699002) for the support given to this research.
Appendix A Y
X
O
dR
R
σr+dσr σϕ
σθ dϕ
ϕ σϕ
R O
Fig. 11. Illustration of a infinitesimal element on the compressive area of tube in the bending process.
bending radius decreases with the strain hardening exponent increasing and with the strength coefficient decreasing.
5. Conclusion By introducing a new wave function, a simplified analytical wrinkling prediction model for thin-walled tube bending is set up, the agreement between theory and experiment having been verified to be satisfactory, especially compared with [3], when the radius/wall thickness ratio of tube is larger than 20. This model can be used to determine the minimum bending radius of given tubes. The influences of geometrical parameters and material properties of tubes are also studied. The results indicate that: (1) the influence of the bending angle on the minimum bending radius is negligibly small; (2) the effect of the geometrical parameters and material properties of tubes on the minimum bending radius is significantly large; (3) the minimum bending radius becomes larger with the original radius and strength coefficient of the tubes increasing, whereas with the wall thickness and strain hardening exponent decreasing. The model proposed in this paper cannot take all factors into account, such as the rotating speed of the bend die, friction and so on, whose influences on the bending limit can be relatively thoroughly researched by FEM simulation.
Acknowledgements The authors would like to thank the National Science Foundation of China for Distinguished Young Scholars (No.
A infinitesimal element (dr × dθ × dϕ) is intercepted from the location (r, θ, ϕ) in the compressive area during the tube bending process (shown in Fig. 11). The X–Y plane is the intersection of the tube and Y coordinate is in the direction of the bending radius. ϕ is a coordinate along the longitudinal direction during the deforming tube. The infinitesimal balance equations in the Y and X directions are, respectively, − [(σr + dσr )(r + dr)(R + dR) dθ dϕ − σr rR dθ dϕ] cos θ + [(σθ + dσθ )(R + dR) dr dϕ − σθ R dr dϕ] sin θ − σϕ r dr dθ dϕ = 0
(A.1)
[(σr + dσr )(r + dr)(R + dR) dθ dϕ − σr rR dθ dϕ] sin θ + [(σθ + dσθ )(R + dR) dr dϕ − σθ R dr dϕ] cos θ = 0 (A.2) From (A.1) and (A.2) 1 (σθ dR + R dσθ ) = σϕ r dθ sin θ
(A.3)
Because of t0 /r0 << 1 for thin-walled tubes, it can be thought of that r ≈ r0 . And then from R = R0 − r cos θ, we can get dR = r0 sin θ dθ. Thus it can be derived from (A.3) that r0 dσθ = sin θ(σϕ − σθ ) dθ (A.4) R If assuming that there is no olivation of the tube intersection during the bending process, then εθ = 0, and σϕ + σr 2
(A.5)
2 σϕ − σr = − √ σ¯ 3
(A.6)
2 ε¯ = √ εϕ 3
(A.7)
σθ =
From (A.5) and (A.6) 1 r dσϕ = − √ dσ¯ + σ¯ sin θ dθ R 3
(A.8)
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369
According to the material stress flow model σ¯ = K(¯ε + ε0 )n , (A.8) becomes:
References
1 1 r0 dσϕ = − √ d K(¯ε + ε0 )n − √ K sin θ(¯ε + ε0 )n dθ 3 3 R (A.9)
[1] H. Yang, Y. Lin, S. Zhichao, Advanced plastic processing technology and research progress on tube forming, J. Plasticity Eng. 8 (2) (2001) 86–88 (in Chinese). [2] Y. Lin, H. Yang, Thin-walled tube precision bending process and FEM simulation, in: Proceedings of the Fourth International Conference on Frontiers of Design and Manufacturing, vol. 2, Beijing, 2000, pp. 305–308. [3] X. Wang, J. Cao, Wrinkling limit in tube bending, Trans. ASME 123 (2001) 430–435. [4] L. Hongwen, Theory of Plates and Shells, Zhejiang University Press, Hangzhou, 1987 (in Chinese). [5] T.X. Yu, L.C. Zhang, Plastic Bending Theory, Science Press, Beijing, 1992 (in Chinese). [6] W. Tonghai, Tube Forming Technology, Mechanical Industry Press, Beijing, 1998 (in Chinese).
Assuming that the neutral layer of stress, that of strain, and the geometrical center are coincident, then n 1 2 R σϕ = − √ K √ ln + ε0 R0 3 3 n θ 1 r0 2 R −√ K sin θ √ ln + ε0 dθ R0 3 π/2 R 3 (A.10)