- Email: [email protected]
Analytical approach to bursting in tube hydroforming using diffuse plastic instability
ARTICLE IN PRESS
International Journal of Mechanical Sciences 46 (2004) 1535–1547 www.elsevier.com/locate/ijmecsci
Analytical approach to bursting in tube hydroforming using diffuse plastic instability Jeong Kima, Sang-Woo Kima, Woo-Jin Songa, Beom-Soo Kangb, a
Department of Aerospace Engineering, Pusan National University, 30 Changjeon dong, Kumjeong Ku, Busan 609-735, Republic of Korea b ERC/NSDM, Department of Aerospace Engineering, Pusan National University, 30 Changjeon dong, Kumjeong Ku, Busan 609-735, Republic of Korea Received 5 February 2003; received in revised form 12 August 2004; accepted 5 September 2004 Available online 27 October 2004
Abstract Analytical studies on onset of bursting failure in tube hydroforming under combined internal pressure and independent axial feeding are carried out. Bursting is irrecoverable phenomenon due to local instability under excessive tensile stress. In this paper, in order to predict the bursting failure diffuse plastic instability based on the Hill’s quadratic plastic potential is introduced. The incremental theory of plasticity for anisotropic material is adopted and then the hydroforming limit and bursting failure diagram with respect to axial feeding and hydraulic pressure are presented. The influences of the plastic anisotropy on plastic instability, the limit stress and the bursting pressure are also investigated. Finally, the stress-based hydroforming limit diagram obtained from the above approach is verified with experimental results. r 2004 Elsevier Ltd. All rights reserved. Keywords: Hydroforming; Plastic instability; Diffuse necking; Plastic anisotropy; Stress-based FLD
1. Introduction Since bursting in hydroforming processes is a consequence of necking, which is a condition of local instability under excessive tensile stresses, prediction of necking initiation is an important Corresponding author. Tel.: +82 51 510 2310; fax: +82 51 513 3760.
E-mail address: [email protected] (B.-S. Kang). 0020-7403/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2004.09.001
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
1536
issue before designing the details of processes [1]. The well-known diffuse necking criterion for a close-end tube can be deduced from Hill’s uniqueness principle, as in Yamada and Aoki [2]. This criterion was applied to the hydroforming process by Xing and Makinouchi [3]. Tirosh et al. [4] and Xia [5] presented an analytical simplified prediction of bursting as well as for buckling instability at the same time. Nefussi and Combescure [6] discussed Swift’s two criteria for sheet forming and for tube hydroforming where necking and buckling are considered simultaneously. In addition, Asnafi [7] constructed analytical model of tube hydroforming in point of yield limit and plastic deformation and employed local necking criterion [8] for prediction of fracture. However, even though a few previous works take into account plastic anisotropy of tube material, the influences on the bursting failure are not investigated. Also the hydroforming limit obtained from their approaches is not verified with experiment result at all. In this paper, the stress-based hydroforming limit and bursting failure diagram with respect to axial feeding and hydraulic pressure are presented and confirmed by a series of tube bulging tests. It is found in the literature that path-independent stress-based FLD (FLSD) can be used well for all forming processes without concern for path effects [9,10]. In addition, it is reported that the pathdependent nature of the strain-based FLD causes the formability prediction method to become ineffective in the analysis of complex forming process, especially hydroforming [11]. Finally, the influences of the plastic anisotropy on plastic instability, limit stress and bursting pressure are investigated. Consequently, it is shown that this approach will provide a feasible method to satisfy the increasing practical demands for evaluating the formability in hydroforming processes.
2. Theory 2.1. Incremental theory of plasticity for anisotropic material In order to predict the onset of necking condition, the Swift’s criterion for diffuse plastic instability is used based on the Hill’s general theory for the uniqueness to the boundary value problem. Consider a thin-walled, close-end tube with original thickness to and radius ro under internal hydraulic pressure p and axial load P, which are applied independently. The tube is assumed to be thin enough for the plane stress hypothesis to be valid and to maintain the orthogonal anisotropic character along the axial and the hoop directions. From the constitutive equation the ratios of the plastic strain increment are defined by dij ¼
qf ðs; ¯ YÞ dl qsij
(1)
in which l is the plastic multiplier, s¯ is the effective stress, Y is the yield stress in simple tension and f is the plastic potential identified as the scalar function that defines the elastic limit surface and is described by Hill’s quadratic yield criterion [12]: 2f ðsij Þ ¼ F ðsy sz Þ2 þ Gðsz sx Þ2 þ Hðsx sy Þ2 þ 2Lt2yz þ 2Mt2zx þ 2Nt2xy 2 ¼ ðF þ G þ HÞs¯ 2 ; 3 where F, G, H, L, M, N are the anisotropy parameters.
ð2Þ
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
1537
Under plane-stress condition, the effective stress in terms of the principal axial and hoop stress is defined as s¯ 2 ¼
3 ½ðG þ HÞs21 þ ðF þ HÞs22 2Hs1 s2 ; 2ðF þ G þ HÞ
(3)
where s1 and s2 are the principal axial and hoop stress, respectively. The plastic strain increments are given by considering the normality condition, incompressibility condition and the equivalent work definition of effective strain increment d¯ 3 d¯ ½s1 ðG þ HÞ Hs2 ; 2ðF þ G þ HÞ s¯ 3 d¯ d2 ¼ ½s2 ðF þ HÞ Hs1 ; 2ðF þ G þ HÞ s¯ d3 ¼ ðd1 þ d2 Þ;
d1 ¼
ð4Þ
where d1 ; d2 and d3 are the plastic strain incremental components along the principal axial, hoop and thickness directions, respectively. 2.2. Diffuse plastic instability of tube Consider a long tube enough for the stresses assumed to be uniformly distributed. From the equilibrium equations, P þ pr2 p pr and s2 ¼ ; (5) 2prt t where P is negative when the axial force is compressive, r and t are the current values of the tube’s radius and wall thickness (t5r). Let a ¼ s1 =s2 and, if it is assumed that instability occurs when the following simultaneous constraints are satisfied: ds1 ¼ s2 d2 þ s1 d1 and ds2 ¼ s2 ð2d2 þ d1 Þ [13], and the cross section of the tube at end remains circular during deformation, the variation of the principal stress components in terms of the principal hoop stress can be derived as s1 ¼
3 d¯ ½ðG þ HÞa2 2Ha þ F þ Hs22 ; 2ðF þ G þ HÞ s¯ 3 d¯ ds2 ¼ ½ðG HÞa þ 2F þ Hs22 : 2ðF þ G þ HÞ s¯
ds1 ¼
Based on Eqs. (3) and (6), the variation of the effective stress is deduced as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 a½ðG þ HÞa H2 þ 2ðF þ H aHÞðF þ aGÞ ds¯ ¼ sd¯ : ¯ 2ðF þ G þ HÞ ½ðG þ HÞa2 2Ha þ F þ H3=2
ð6Þ
(7)
Thus the plastic instability condition in terms of sub-tangent Z for tube hydroforming is obtained as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ds¯ 3 a½ðG þ HÞa H2 þ 2ðF þ H aHÞðF þ aGÞ C ¼ : ¼ p (8) Z s¯ d¯ 2ðF þ G þ HÞ O ½ðG þ HÞa2 2Ha þ F þ H3=2
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
1538
More frequently, the anisotropy of material in Eq. (8) is represented by the quantities r0, r90 known as strain ratios, where 0 and 90 refer to the angle that makes with the rolling direction. In this paper, these parameters are assumed to have the following relationships F ¼ r0 ;
G ¼ r90 ;
H ¼ r0 r90 :
(9)
Fig. 1 shows that the plastic instability according to r-values: it is reduced with the relatively decreasing the plastic anisotropy parameters r0 ð¼ y =t Þ-value but it is increased with the relatively decreasing the plastic anisotropy parameters r90 ð¼ z =t Þ-value. 2.3. Predicted limit strains and bursting pressure To obtain forming limit curves in terms of stresses in hydroforming, it is necessary to firstly introduce the principal critical strains c1 ; c2 along the axial and hoop direction, respectively. With the work-hardening law s¯ ¼ K ¯n ; Eq. (8) can be written as 1 1 ds¯ n C ¼ ¼ ¼ : (10) Z s¯ d¯ ¯ O Let b ¼ d1 =d2 and assuming the proportional loading, thus the equivalent strain can be described as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 2 ðH þ F Þ21 þ ðG þ HÞ22 þ 2H1 2 ¼ Y2 ; (11) ðF þ G þ HÞ ¯ ¼ 3 FG þ FH þ GH where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 2 ðH þ FÞb2 þ 2Hb þ G þ H Y¼ ðF þ G þ HÞ : 3 FG þ FH þ GH
2.0
2.0 r0=0.5 r0=1.0 r0=1.5
1.5 Plastic Instability 1/Z
Plastic Instability 1/Z
1.5 1.0 0.5 0.0 -0.5
r90=0.5 r90=1.0 r90=1.5
1.0 0.5 0.0 -0.5
-1.0 -2.0 (a)
(12)
-1.5
-1.0 -0.5 0.0 Stress Ratio α
0.5
-1.0 -2.0
1.0 (b)
-1.5
-1.0 -0.5 0.0 Stress Ratio α
0.5
1.0
Fig. 1. Plastic instability with different plastic anisotropy parameters: (a) influence of r0 ð¼ y =t Þ; (b) influence of r90 ð¼ z =t Þ:
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
1539
According to Eqs. (10) and (11), the limit strains based on the plasticity instability yield as c1 ¼ bc2 ;
(13)
On c2 ¼ YC :
Also the bursting pressure pc is determined from Eq. (5) as tc c s; (14) rc 2 where tc and rc are current tube thickness and radius at the onset of necking respectively and can be derived as the following: pc ¼
c
c
tc ¼ to eð1 þ2 Þ ; c
rc ¼ ro e2 :
ð15Þ
From Eq. (4) the limit stress sc1 and sc2 ; when the bursting failure is occurred, can be obtained as 2ðFþGþHÞ s¯ c c sc1 ¼ 3ðFGþGHþHF Þ ¯ ½ðF þ HÞ1 þ H2 ; 2ðFþGþHÞ s¯ c c sc2 ¼ 3ðFGþGHþHF Þ ¯ ½ðG þ HÞ2 þ H1 :
(16)
3. Tube hydroforming limit The forming limit is conventionally described as a curve plotting major strain versus minor strain. However, the strain-based FLD approach to evaluate formability has been shown to be valid only for the cases of proportional loading, where the ratio between the principal stresses remains constant throughout the forming process. In the case of tube hydroforming process carried out in this study, the ratio between the principal stresses does not remain constant throughout the forming process under the loading conditions as shown in Fig. 3. Accordingly, the experimental result was compared with analytical stress-based forming limit which is more general forming limit criterion, especially hydroforming, and also analytical bursting pressure curve versus axial feed displacement is presented as a bursting failure diagram. 3.1. Experimental verification In order to evaluate the forming limit in hydroforming process and verify the stress-based approach proposed in this paper, a series of tube bulging tests are executed. The tube material is assumed to be obeyed the stress–strain relationship s¯ ¼ 1400¯0:17 (MPa) and normal anisotropy with anisotropy value R(=r0=r90)=2.14. This test uses internal hydraulic pressure to bulge a tube that is supported between a lower and an upper die. The lower part of the tube is fixed in movement, while the other is free to be able to move in the downward direction. This condition can provide axial feeding during the test. During the hydroforming operation, only a limited amount of material can be fed into the die cavity. Thus as applied internal
ARTICLE IN PRESS 1540
J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
pressure increases, the thickness of material within the die cavity will be reduced rapidly and necking occurs consequently. On the other hand, the compressive load applied at the ends of the material increases amount of material inflow into the die cavity so that it prevents abrupt thinning of tube wall and can delay onset of the necking. The dimensions and configurations of the die and final bulged part are shown in Fig. 2. In order to observe the busting failure, high internal pressures under relatively low axial feeding as shown in Fig. 3 are applied. The internal pressures and the axial feeding displacements as the input loading are controlled by the PC-based controller of the experimental set-up for specified process sequence for a series of the bulging tests. In addition, the process parameters can be stored and visualized on the PC. As a result of excessive pressurizing during bulging process, bursting occurs at the middle of the tube wall as illustrated in Fig. 4. Under the five load cases carried out in this φ50.8
Y
Z
Die
20.0 20.0 final product
235.0
workpiece 100.0
1.4 φ80.0
Fig. 2. Dimensions and configurations of die and final bulged part [unit:mm].
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547 160
1541
150 Case A
Internal Pressure Axial Feeding Displacement
Case B
80
Case C Case D Case E
90
60
Axial feeding [mm]
Internal pressure [MPa]
120 120
40 30
0 0
3
6
9
12
15
0 18
Time [sec]
Fig. 3. Input loading paths with five different cases.
study, the bursting failure had been occurred before the outer surface of the bulged tube reached inner wall of the die. Thus it is reasonable that the experimental conditions can be considered to be in free forming process. The limit strains directly obtained from experiments are transformed to limit stresses using the method suggested by Stoughton [10] and are used to compare with analytical FLSD. The detailed procedure can be found in the reference. Fig. 5 shows forming limit stress diagram, plotted as black dots, experimentally obtained from bulge tests. According to Eq. (16), the analytically predicted principal stress limits based on diffuse plastic instability are shown in Fig. 6. As depicted in the figure, the hydroforming limit predicted from the stress-based approach is somewhat below experimental forming limit curve. 3.2. Influence of material properties on formability Fig. 7 shows that the hydroforming limits with respect to plastic anisotropy and strainhardening exponent of the material: (a) the formability is reduced with the relatively increasing r0 value but (b) it is increased with the relatively increasing the r90 -value and also (c) it is increased with the relatively decreasing the n-value in the range over s1=200 MPa that axial feed displacement is within 50 mm. And (d) the allowable forming zone is reduced with the relatively decreasing K-value. It can be explained that (a) large r0 ð¼ y =t Þ-value makes it easy to deform in the hoop direction compared with the axial direction, thus the major limit stress is decreased. As similar with the influence of r0 ; (b) large r90 ð¼ z =t Þ-value makes it easy to deform in the axial direction at this time, thus the major limit stress is increased. It is also noted that (c) the materials with large strain hardening coefficient n-value have the large limit strain from Eq. (14), however its corresponding stress value at that limit strain is smaller than that at the limit strain for
ARTICLE IN PRESS 1542
J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
Fig. 4. Experimental bursting failure obtained from bulge tests under different loading paths: (a) Case A; (b) Case B; (c) Case C; (d) Case D; (e) Case E.
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
1543
1800 1600
Major Stress σ2 (MPa)
1400 1200 1000 800 600 400 200 0 -800
-600
-400
-200
0
200
400
600
800
Minor Stress σ1 (MPa)
Fig. 5. Hydrofoming stress limit diagram obtained experimentally.
1800
Analytical results (R=2.14) Analytical results (R=1.0) Experimental results
1600 Major Stress σ2 (MPa)
1400 1200 1000 800 600 400 200 0 -800
-600
-400
-200
0
200
400
600
800
Minor Stress σ1 (MPa)
Fig. 6. Hydroforming limit predicted from diffuse necking criterion.
materials with relatively lower n-value. It is easily demonstrated from the effective stress–strain curve obeying the work-hardening law s¯ ¼ K ¯n on which effective stress decreases with the increasing n-value in the region of smaller strain than 1.0. By reason of this effective stress–strain
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
1544
1600
2000
1800
r0=1.5
Major stress σ2 (MPa)
Major stress σ2 (MPa)
r90=0.5 r90=1.0 r90=1.5
r0=0.5 r0=1.0
1600
1400
1200
1400
1200
1000
1000
800
800 0
200
400
800
Minor stress σ1 (MPa)
(a)
-1400 -1200 -1000 -800 -600 -400 -200
n=0.13 n=0.15 n=0.17 n=0.19
1600
1400
1200
200
400
600
K=1300 K=1400 K=1500
1800
1600
1400
1200
1000
1000 -800
(c)
0
Minor stress σ1 (MPa)
(b)
1800
Major stress σ2 (MPa)
600
Major stress σ2 (MPa)
-1200 -1000 -800 -600 -400 -200
-600 -400 -200
0
200
400
Minor stress σ1 (MPa)
600
800
1000
-800
(d)
-600 -400 -200
0
200
400
600
800
1000
Minor stress σ1 (MPa)
Fig. 7. Effects of material properties on limit stresses: (a) influence of r0 ð¼ y =t Þ-value; (b) influence of r90 ð¼ z =t Þvalue; (c) influence of strain-hardening exponent n-value; (d) influence of strength coefficient K-value.
curve characteristic, limit stress calculated from Eq. (16) is decreased with increasing n-value by implying that formability of the material is decreased. (d) The effect of strength coefficient K-value on limit stress also can be explained from simple stress–strain curve characteristic in the same manner. In order to get an optimal loading path, under which there is no bursting failure during any hydroforming stage, the critical internal hydraulic pressure according to axial feeding are calculated from Eq. (15) and depicted in Fig. 8. Any loading combination of internal pressure and axial feeding below the curve indicates no bursting failure expected. Fig. 9 illustrates effects of material properties on the bursting pressure: (a) the bursting pressure is increased with decreasing the n-value, (b) it is increased with increasing strength coefficient K-value, (c) the forming window, which refers to the safe region below the bursting failure curve, is increased with the relatively decreasing r0 -value but (d) that is reduced with the relatively decreasing the
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
1545
Fig. 8. Analytically predicted bursting pressures.
r90 -value. All results for influences on bursting pressure give the similar trend with FLSD as shown in Fig. 7.
4. Conclusions By contrast with buckling and wrinkling, bursting is an irrecoverable failure in tube hydroforming. In order to evaluate the forming limit of hydroforming process in view of bursting failure, analytical approach under combined internal pressure and axial loading is presented in this study. Using the Swift’s plastic instability criterion occurrence of diffuse necking of tube is estimated. Also the influences of the plastic anisotropy on plastic instability, limit stress and bursting pressure are investigated. These findings are summarized to (i) the plastic instability is reduced with relatively decreasing the plastic anisotropy parameters r0 -value while increased with decreasing r90 -value, (ii) the limit stresses are reduced with relatively increasing r0 -value but it is increased with increasing the r90 -value and (iii) the predicted bursting pressure is increased with relatively decreasing r0 -value while reduced with decreasing r90 -value. Finally, the comparison with the experimental results has shown that the hydroforming limit and bursting pressures are successfully predicted using the analytical stress-based approach. Therefore, it is expected that these analytical method developed here can be used as an effective tool for evaluating the formability in wide range of practical tube hydroforming processes.
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
1546
60
60 n=0.13 n=0.15 n=0.17 n=0.19
50
K=1300 K=1400 K=1500
55 Internal pressure [MPa]
Internal pressure [MPa]
55
45
40
50 45
40
35
35
30
30 0
5
10
(a)
15
20
25
30
0
5
(b)
Axial feeding [mm]
10
15
20
30
60
60 r0=0.5
r90=0.5
r0=1.0
55
r90=1.0
55
r90=1.5
Internal pressure [MPa]
r0=1.5
Internal pressure [MPa]
25
Axial feeding [mm]
50 45
40
50 45
40
35
35
30
30 0
5
10
(c)
15
20
Axial feeding [mm]
25
30
0
(d)
5
10
15
20
25
30
Axial feeding [mm]
Fig. 9. Effects of material properties on bursting pressure: (a) influence of strain-hardening exponent n-value; (b) influence of strength coefficient K-value; (c) influence of anisotropy parameter r0 -value; (d) influence of anisotropy parameter r90 -value.
Acknowledgements This work has been completed by the support of 2003 Pusan National University Research Grant, and the first author wishes to thank for this support. Also the last author would like to acknowledge the support of Brain Busan 21 Project.
References [1] Kim J, Kim YW, Kang BS, Hwang SM. Finite element analysis for bursting failure prediction in bulge forming of a seamed tube. Finite Elements in Analysis and Design 2004;40:953–66.
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
1547
[2] Yamada Y, Aoki I. On the tensile plastic instability in axi-symmetric deformation of sheet metals. Journal of the Japan Society for Technology of Plasticity 1966;67:393–406 [in Japanese]. [3] Xing HL, Makinouchi A. Numerical analysis and design for tubular hydroforming. International Journal of Mechanical Sciences 2001;43:1009–26. [4] Tirosh J, Neuberger A, Shirizly A. On tube expansion by internal fluid pressure with additional compressive stress. International Journal of Mechanical Sciences 1996;38:839–51. [5] Xia ZC. Failure analysis of tubular hydroforming. Journal of Engineering Materials Technology 2001;123:423–9. [6] Nefussi G, Combescure A. Coupled buckling and plastic instability for tube hydroforming. International Journal of Mechanical Sciences 2002;44:899–914. [7] Asnafi N. Analytical modelling of tube hydroforming. Thin-Walled Structures 1999;34:295–330. [8] Semiatin SL, Jonas JJ. Formability and workability of metals: plastic instability and flow localization. American Society of Metals 1984: 204–6. [9] Arrieux R, Bedrin C, Boivin M. Determination of an intrinsic forming limit stress diagram for isotropic metal sheets. Proceedings of the 12th Biennial Congress IDDRG, 1982. p. 61–71. [10] Stoughton TB. A general forming limit criterion for sheet metal forming. International Journal of Mechanical Sciences 1999;42:1–27. [11] Stoughton TB, Zhu X. Review of theoretical models of the strain-based FLD and their relevance to the stressbased FLD. International Journal of Plasticity 2004;20:1463–86. [12] Hill R. The mathematical theory of plasticity. New York: Oxford University Press; 1983. [13] Swift HW. Plastic instability under plane stress. Journal of Mechanics and Physics of Solids 1952;1:1–18.
International Journal of Mechanical Sciences 46 (2004) 1535–1547 www.elsevier.com/locate/ijmecsci
Analytical approach to bursting in tube hydroforming using diffuse plastic instability Jeong Kima, Sang-Woo Kima, Woo-Jin Songa, Beom-Soo Kangb, a
Department of Aerospace Engineering, Pusan National University, 30 Changjeon dong, Kumjeong Ku, Busan 609-735, Republic of Korea b ERC/NSDM, Department of Aerospace Engineering, Pusan National University, 30 Changjeon dong, Kumjeong Ku, Busan 609-735, Republic of Korea Received 5 February 2003; received in revised form 12 August 2004; accepted 5 September 2004 Available online 27 October 2004
Abstract Analytical studies on onset of bursting failure in tube hydroforming under combined internal pressure and independent axial feeding are carried out. Bursting is irrecoverable phenomenon due to local instability under excessive tensile stress. In this paper, in order to predict the bursting failure diffuse plastic instability based on the Hill’s quadratic plastic potential is introduced. The incremental theory of plasticity for anisotropic material is adopted and then the hydroforming limit and bursting failure diagram with respect to axial feeding and hydraulic pressure are presented. The influences of the plastic anisotropy on plastic instability, the limit stress and the bursting pressure are also investigated. Finally, the stress-based hydroforming limit diagram obtained from the above approach is verified with experimental results. r 2004 Elsevier Ltd. All rights reserved. Keywords: Hydroforming; Plastic instability; Diffuse necking; Plastic anisotropy; Stress-based FLD
1. Introduction Since bursting in hydroforming processes is a consequence of necking, which is a condition of local instability under excessive tensile stresses, prediction of necking initiation is an important Corresponding author. Tel.: +82 51 510 2310; fax: +82 51 513 3760.
E-mail address: [email protected] (B.-S. Kang). 0020-7403/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2004.09.001
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
1536
issue before designing the details of processes [1]. The well-known diffuse necking criterion for a close-end tube can be deduced from Hill’s uniqueness principle, as in Yamada and Aoki [2]. This criterion was applied to the hydroforming process by Xing and Makinouchi [3]. Tirosh et al. [4] and Xia [5] presented an analytical simplified prediction of bursting as well as for buckling instability at the same time. Nefussi and Combescure [6] discussed Swift’s two criteria for sheet forming and for tube hydroforming where necking and buckling are considered simultaneously. In addition, Asnafi [7] constructed analytical model of tube hydroforming in point of yield limit and plastic deformation and employed local necking criterion [8] for prediction of fracture. However, even though a few previous works take into account plastic anisotropy of tube material, the influences on the bursting failure are not investigated. Also the hydroforming limit obtained from their approaches is not verified with experiment result at all. In this paper, the stress-based hydroforming limit and bursting failure diagram with respect to axial feeding and hydraulic pressure are presented and confirmed by a series of tube bulging tests. It is found in the literature that path-independent stress-based FLD (FLSD) can be used well for all forming processes without concern for path effects [9,10]. In addition, it is reported that the pathdependent nature of the strain-based FLD causes the formability prediction method to become ineffective in the analysis of complex forming process, especially hydroforming [11]. Finally, the influences of the plastic anisotropy on plastic instability, limit stress and bursting pressure are investigated. Consequently, it is shown that this approach will provide a feasible method to satisfy the increasing practical demands for evaluating the formability in hydroforming processes.
2. Theory 2.1. Incremental theory of plasticity for anisotropic material In order to predict the onset of necking condition, the Swift’s criterion for diffuse plastic instability is used based on the Hill’s general theory for the uniqueness to the boundary value problem. Consider a thin-walled, close-end tube with original thickness to and radius ro under internal hydraulic pressure p and axial load P, which are applied independently. The tube is assumed to be thin enough for the plane stress hypothesis to be valid and to maintain the orthogonal anisotropic character along the axial and the hoop directions. From the constitutive equation the ratios of the plastic strain increment are defined by dij ¼
qf ðs; ¯ YÞ dl qsij
(1)
in which l is the plastic multiplier, s¯ is the effective stress, Y is the yield stress in simple tension and f is the plastic potential identified as the scalar function that defines the elastic limit surface and is described by Hill’s quadratic yield criterion [12]: 2f ðsij Þ ¼ F ðsy sz Þ2 þ Gðsz sx Þ2 þ Hðsx sy Þ2 þ 2Lt2yz þ 2Mt2zx þ 2Nt2xy 2 ¼ ðF þ G þ HÞs¯ 2 ; 3 where F, G, H, L, M, N are the anisotropy parameters.
ð2Þ
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
1537
Under plane-stress condition, the effective stress in terms of the principal axial and hoop stress is defined as s¯ 2 ¼
3 ½ðG þ HÞs21 þ ðF þ HÞs22 2Hs1 s2 ; 2ðF þ G þ HÞ
(3)
where s1 and s2 are the principal axial and hoop stress, respectively. The plastic strain increments are given by considering the normality condition, incompressibility condition and the equivalent work definition of effective strain increment d¯ 3 d¯ ½s1 ðG þ HÞ Hs2 ; 2ðF þ G þ HÞ s¯ 3 d¯ d2 ¼ ½s2 ðF þ HÞ Hs1 ; 2ðF þ G þ HÞ s¯ d3 ¼ ðd1 þ d2 Þ;
d1 ¼
ð4Þ
where d1 ; d2 and d3 are the plastic strain incremental components along the principal axial, hoop and thickness directions, respectively. 2.2. Diffuse plastic instability of tube Consider a long tube enough for the stresses assumed to be uniformly distributed. From the equilibrium equations, P þ pr2 p pr and s2 ¼ ; (5) 2prt t where P is negative when the axial force is compressive, r and t are the current values of the tube’s radius and wall thickness (t5r). Let a ¼ s1 =s2 and, if it is assumed that instability occurs when the following simultaneous constraints are satisfied: ds1 ¼ s2 d2 þ s1 d1 and ds2 ¼ s2 ð2d2 þ d1 Þ [13], and the cross section of the tube at end remains circular during deformation, the variation of the principal stress components in terms of the principal hoop stress can be derived as s1 ¼
3 d¯ ½ðG þ HÞa2 2Ha þ F þ Hs22 ; 2ðF þ G þ HÞ s¯ 3 d¯ ds2 ¼ ½ðG HÞa þ 2F þ Hs22 : 2ðF þ G þ HÞ s¯
ds1 ¼
Based on Eqs. (3) and (6), the variation of the effective stress is deduced as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 a½ðG þ HÞa H2 þ 2ðF þ H aHÞðF þ aGÞ ds¯ ¼ sd¯ : ¯ 2ðF þ G þ HÞ ½ðG þ HÞa2 2Ha þ F þ H3=2
ð6Þ
(7)
Thus the plastic instability condition in terms of sub-tangent Z for tube hydroforming is obtained as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ds¯ 3 a½ðG þ HÞa H2 þ 2ðF þ H aHÞðF þ aGÞ C ¼ : ¼ p (8) Z s¯ d¯ 2ðF þ G þ HÞ O ½ðG þ HÞa2 2Ha þ F þ H3=2
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
1538
More frequently, the anisotropy of material in Eq. (8) is represented by the quantities r0, r90 known as strain ratios, where 0 and 90 refer to the angle that makes with the rolling direction. In this paper, these parameters are assumed to have the following relationships F ¼ r0 ;
G ¼ r90 ;
H ¼ r0 r90 :
(9)
Fig. 1 shows that the plastic instability according to r-values: it is reduced with the relatively decreasing the plastic anisotropy parameters r0 ð¼ y =t Þ-value but it is increased with the relatively decreasing the plastic anisotropy parameters r90 ð¼ z =t Þ-value. 2.3. Predicted limit strains and bursting pressure To obtain forming limit curves in terms of stresses in hydroforming, it is necessary to firstly introduce the principal critical strains c1 ; c2 along the axial and hoop direction, respectively. With the work-hardening law s¯ ¼ K ¯n ; Eq. (8) can be written as 1 1 ds¯ n C ¼ ¼ ¼ : (10) Z s¯ d¯ ¯ O Let b ¼ d1 =d2 and assuming the proportional loading, thus the equivalent strain can be described as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 2 ðH þ F Þ21 þ ðG þ HÞ22 þ 2H1 2 ¼ Y2 ; (11) ðF þ G þ HÞ ¯ ¼ 3 FG þ FH þ GH where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 2 ðH þ FÞb2 þ 2Hb þ G þ H Y¼ ðF þ G þ HÞ : 3 FG þ FH þ GH
2.0
2.0 r0=0.5 r0=1.0 r0=1.5
1.5 Plastic Instability 1/Z
Plastic Instability 1/Z
1.5 1.0 0.5 0.0 -0.5
r90=0.5 r90=1.0 r90=1.5
1.0 0.5 0.0 -0.5
-1.0 -2.0 (a)
(12)
-1.5
-1.0 -0.5 0.0 Stress Ratio α
0.5
-1.0 -2.0
1.0 (b)
-1.5
-1.0 -0.5 0.0 Stress Ratio α
0.5
1.0
Fig. 1. Plastic instability with different plastic anisotropy parameters: (a) influence of r0 ð¼ y =t Þ; (b) influence of r90 ð¼ z =t Þ:
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
1539
According to Eqs. (10) and (11), the limit strains based on the plasticity instability yield as c1 ¼ bc2 ;
(13)
On c2 ¼ YC :
Also the bursting pressure pc is determined from Eq. (5) as tc c s; (14) rc 2 where tc and rc are current tube thickness and radius at the onset of necking respectively and can be derived as the following: pc ¼
c
c
tc ¼ to eð1 þ2 Þ ; c
rc ¼ ro e2 :
ð15Þ
From Eq. (4) the limit stress sc1 and sc2 ; when the bursting failure is occurred, can be obtained as 2ðFþGþHÞ s¯ c c sc1 ¼ 3ðFGþGHþHF Þ ¯ ½ðF þ HÞ1 þ H2 ; 2ðFþGþHÞ s¯ c c sc2 ¼ 3ðFGþGHþHF Þ ¯ ½ðG þ HÞ2 þ H1 :
(16)
3. Tube hydroforming limit The forming limit is conventionally described as a curve plotting major strain versus minor strain. However, the strain-based FLD approach to evaluate formability has been shown to be valid only for the cases of proportional loading, where the ratio between the principal stresses remains constant throughout the forming process. In the case of tube hydroforming process carried out in this study, the ratio between the principal stresses does not remain constant throughout the forming process under the loading conditions as shown in Fig. 3. Accordingly, the experimental result was compared with analytical stress-based forming limit which is more general forming limit criterion, especially hydroforming, and also analytical bursting pressure curve versus axial feed displacement is presented as a bursting failure diagram. 3.1. Experimental verification In order to evaluate the forming limit in hydroforming process and verify the stress-based approach proposed in this paper, a series of tube bulging tests are executed. The tube material is assumed to be obeyed the stress–strain relationship s¯ ¼ 1400¯0:17 (MPa) and normal anisotropy with anisotropy value R(=r0=r90)=2.14. This test uses internal hydraulic pressure to bulge a tube that is supported between a lower and an upper die. The lower part of the tube is fixed in movement, while the other is free to be able to move in the downward direction. This condition can provide axial feeding during the test. During the hydroforming operation, only a limited amount of material can be fed into the die cavity. Thus as applied internal
ARTICLE IN PRESS 1540
J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
pressure increases, the thickness of material within the die cavity will be reduced rapidly and necking occurs consequently. On the other hand, the compressive load applied at the ends of the material increases amount of material inflow into the die cavity so that it prevents abrupt thinning of tube wall and can delay onset of the necking. The dimensions and configurations of the die and final bulged part are shown in Fig. 2. In order to observe the busting failure, high internal pressures under relatively low axial feeding as shown in Fig. 3 are applied. The internal pressures and the axial feeding displacements as the input loading are controlled by the PC-based controller of the experimental set-up for specified process sequence for a series of the bulging tests. In addition, the process parameters can be stored and visualized on the PC. As a result of excessive pressurizing during bulging process, bursting occurs at the middle of the tube wall as illustrated in Fig. 4. Under the five load cases carried out in this φ50.8
Y
Z
Die
20.0 20.0 final product
235.0
workpiece 100.0
1.4 φ80.0
Fig. 2. Dimensions and configurations of die and final bulged part [unit:mm].
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547 160
1541
150 Case A
Internal Pressure Axial Feeding Displacement
Case B
80
Case C Case D Case E
90
60
Axial feeding [mm]
Internal pressure [MPa]
120 120
40 30
0 0
3
6
9
12
15
0 18
Time [sec]
Fig. 3. Input loading paths with five different cases.
study, the bursting failure had been occurred before the outer surface of the bulged tube reached inner wall of the die. Thus it is reasonable that the experimental conditions can be considered to be in free forming process. The limit strains directly obtained from experiments are transformed to limit stresses using the method suggested by Stoughton [10] and are used to compare with analytical FLSD. The detailed procedure can be found in the reference. Fig. 5 shows forming limit stress diagram, plotted as black dots, experimentally obtained from bulge tests. According to Eq. (16), the analytically predicted principal stress limits based on diffuse plastic instability are shown in Fig. 6. As depicted in the figure, the hydroforming limit predicted from the stress-based approach is somewhat below experimental forming limit curve. 3.2. Influence of material properties on formability Fig. 7 shows that the hydroforming limits with respect to plastic anisotropy and strainhardening exponent of the material: (a) the formability is reduced with the relatively increasing r0 value but (b) it is increased with the relatively increasing the r90 -value and also (c) it is increased with the relatively decreasing the n-value in the range over s1=200 MPa that axial feed displacement is within 50 mm. And (d) the allowable forming zone is reduced with the relatively decreasing K-value. It can be explained that (a) large r0 ð¼ y =t Þ-value makes it easy to deform in the hoop direction compared with the axial direction, thus the major limit stress is decreased. As similar with the influence of r0 ; (b) large r90 ð¼ z =t Þ-value makes it easy to deform in the axial direction at this time, thus the major limit stress is increased. It is also noted that (c) the materials with large strain hardening coefficient n-value have the large limit strain from Eq. (14), however its corresponding stress value at that limit strain is smaller than that at the limit strain for
ARTICLE IN PRESS 1542
J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
Fig. 4. Experimental bursting failure obtained from bulge tests under different loading paths: (a) Case A; (b) Case B; (c) Case C; (d) Case D; (e) Case E.
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
1543
1800 1600
Major Stress σ2 (MPa)
1400 1200 1000 800 600 400 200 0 -800
-600
-400
-200
0
200
400
600
800
Minor Stress σ1 (MPa)
Fig. 5. Hydrofoming stress limit diagram obtained experimentally.
1800
Analytical results (R=2.14) Analytical results (R=1.0) Experimental results
1600 Major Stress σ2 (MPa)
1400 1200 1000 800 600 400 200 0 -800
-600
-400
-200
0
200
400
600
800
Minor Stress σ1 (MPa)
Fig. 6. Hydroforming limit predicted from diffuse necking criterion.
materials with relatively lower n-value. It is easily demonstrated from the effective stress–strain curve obeying the work-hardening law s¯ ¼ K ¯n on which effective stress decreases with the increasing n-value in the region of smaller strain than 1.0. By reason of this effective stress–strain
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
1544
1600
2000
1800
r0=1.5
Major stress σ2 (MPa)
Major stress σ2 (MPa)
r90=0.5 r90=1.0 r90=1.5
r0=0.5 r0=1.0
1600
1400
1200
1400
1200
1000
1000
800
800 0
200
400
800
Minor stress σ1 (MPa)
(a)
-1400 -1200 -1000 -800 -600 -400 -200
n=0.13 n=0.15 n=0.17 n=0.19
1600
1400
1200
200
400
600
K=1300 K=1400 K=1500
1800
1600
1400
1200
1000
1000 -800
(c)
0
Minor stress σ1 (MPa)
(b)
1800
Major stress σ2 (MPa)
600
Major stress σ2 (MPa)
-1200 -1000 -800 -600 -400 -200
-600 -400 -200
0
200
400
Minor stress σ1 (MPa)
600
800
1000
-800
(d)
-600 -400 -200
0
200
400
600
800
1000
Minor stress σ1 (MPa)
Fig. 7. Effects of material properties on limit stresses: (a) influence of r0 ð¼ y =t Þ-value; (b) influence of r90 ð¼ z =t Þvalue; (c) influence of strain-hardening exponent n-value; (d) influence of strength coefficient K-value.
curve characteristic, limit stress calculated from Eq. (16) is decreased with increasing n-value by implying that formability of the material is decreased. (d) The effect of strength coefficient K-value on limit stress also can be explained from simple stress–strain curve characteristic in the same manner. In order to get an optimal loading path, under which there is no bursting failure during any hydroforming stage, the critical internal hydraulic pressure according to axial feeding are calculated from Eq. (15) and depicted in Fig. 8. Any loading combination of internal pressure and axial feeding below the curve indicates no bursting failure expected. Fig. 9 illustrates effects of material properties on the bursting pressure: (a) the bursting pressure is increased with decreasing the n-value, (b) it is increased with increasing strength coefficient K-value, (c) the forming window, which refers to the safe region below the bursting failure curve, is increased with the relatively decreasing r0 -value but (d) that is reduced with the relatively decreasing the
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
1545
Fig. 8. Analytically predicted bursting pressures.
r90 -value. All results for influences on bursting pressure give the similar trend with FLSD as shown in Fig. 7.
4. Conclusions By contrast with buckling and wrinkling, bursting is an irrecoverable failure in tube hydroforming. In order to evaluate the forming limit of hydroforming process in view of bursting failure, analytical approach under combined internal pressure and axial loading is presented in this study. Using the Swift’s plastic instability criterion occurrence of diffuse necking of tube is estimated. Also the influences of the plastic anisotropy on plastic instability, limit stress and bursting pressure are investigated. These findings are summarized to (i) the plastic instability is reduced with relatively decreasing the plastic anisotropy parameters r0 -value while increased with decreasing r90 -value, (ii) the limit stresses are reduced with relatively increasing r0 -value but it is increased with increasing the r90 -value and (iii) the predicted bursting pressure is increased with relatively decreasing r0 -value while reduced with decreasing r90 -value. Finally, the comparison with the experimental results has shown that the hydroforming limit and bursting pressures are successfully predicted using the analytical stress-based approach. Therefore, it is expected that these analytical method developed here can be used as an effective tool for evaluating the formability in wide range of practical tube hydroforming processes.
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
1546
60
60 n=0.13 n=0.15 n=0.17 n=0.19
50
K=1300 K=1400 K=1500
55 Internal pressure [MPa]
Internal pressure [MPa]
55
45
40
50 45
40
35
35
30
30 0
5
10
(a)
15
20
25
30
0
5
(b)
Axial feeding [mm]
10
15
20
30
60
60 r0=0.5
r90=0.5
r0=1.0
55
r90=1.0
55
r90=1.5
Internal pressure [MPa]
r0=1.5
Internal pressure [MPa]
25
Axial feeding [mm]
50 45
40
50 45
40
35
35
30
30 0
5
10
(c)
15
20
Axial feeding [mm]
25
30
0
(d)
5
10
15
20
25
30
Axial feeding [mm]
Fig. 9. Effects of material properties on bursting pressure: (a) influence of strain-hardening exponent n-value; (b) influence of strength coefficient K-value; (c) influence of anisotropy parameter r0 -value; (d) influence of anisotropy parameter r90 -value.
Acknowledgements This work has been completed by the support of 2003 Pusan National University Research Grant, and the first author wishes to thank for this support. Also the last author would like to acknowledge the support of Brain Busan 21 Project.
References [1] Kim J, Kim YW, Kang BS, Hwang SM. Finite element analysis for bursting failure prediction in bulge forming of a seamed tube. Finite Elements in Analysis and Design 2004;40:953–66.
ARTICLE IN PRESS J. Kim et al. / International Journal of Mechanical Sciences 46 (2004) 1535–1547
1547
[2] Yamada Y, Aoki I. On the tensile plastic instability in axi-symmetric deformation of sheet metals. Journal of the Japan Society for Technology of Plasticity 1966;67:393–406 [in Japanese]. [3] Xing HL, Makinouchi A. Numerical analysis and design for tubular hydroforming. International Journal of Mechanical Sciences 2001;43:1009–26. [4] Tirosh J, Neuberger A, Shirizly A. On tube expansion by internal fluid pressure with additional compressive stress. International Journal of Mechanical Sciences 1996;38:839–51. [5] Xia ZC. Failure analysis of tubular hydroforming. Journal of Engineering Materials Technology 2001;123:423–9. [6] Nefussi G, Combescure A. Coupled buckling and plastic instability for tube hydroforming. International Journal of Mechanical Sciences 2002;44:899–914. [7] Asnafi N. Analytical modelling of tube hydroforming. Thin-Walled Structures 1999;34:295–330. [8] Semiatin SL, Jonas JJ. Formability and workability of metals: plastic instability and flow localization. American Society of Metals 1984: 204–6. [9] Arrieux R, Bedrin C, Boivin M. Determination of an intrinsic forming limit stress diagram for isotropic metal sheets. Proceedings of the 12th Biennial Congress IDDRG, 1982. p. 61–71. [10] Stoughton TB. A general forming limit criterion for sheet metal forming. International Journal of Mechanical Sciences 1999;42:1–27. [11] Stoughton TB, Zhu X. Review of theoretical models of the strain-based FLD and their relevance to the stressbased FLD. International Journal of Plasticity 2004;20:1463–86. [12] Hill R. The mathematical theory of plasticity. New York: Oxford University Press; 1983. [13] Swift HW. Plastic instability under plane stress. Journal of Mechanics and Physics of Solids 1952;1:1–18.