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Cross-sectional deformation of H96 brass double-ridged rectangular tube in rotary draw bending process with different yield criteria
CJA 1376 11 September 2019 Chinese Journal of Aeronautics, (2019), xxx(xx): xxx–xxx
No. of Pages 11
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Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
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Cross-sectional deformation of H96 brass doubleridged rectangular tube in rotary draw bending process with different yield criteria
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Yangyang XIA a,b, Yuli LIU a,b,*, Mengmeng LIU a,b
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a
Research & Development Institute of Northwestern Polytechnical University in Shenzhen, Shenzhen 518057, China State Key Laboratory Solidification Process, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an 710072, China b
Received 28 March 2019; revised 19 June 2019; accepted 15 July 2019
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KEYWORDS
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Anisotropy yield criterion; Cross-sectional deformation; H96 double-ridged rectangular tube; H-type rotary draw bending; Inverse method
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Abstract Different yield criterion has great difference in predicting the deformation of tube with different material. In order to improve the prediction accuracy of the cross-sectional deformation of the double-ridged rectangular tube (DRRT) during rotary draw bending (RDB) process, Mises isotropic yield criterion, Hill’48 and Barlat/Lian anisotropic yield criteria commonly used in practical engineering are introduced to simulate RDB of DRRT. The inverse method combining uniaxial tensile test of whole tube and response surface methodology was proposed to identify the parameters of Hill’48 and Barlat/Lian yield criteria of small-sized H96 brass extrusion DRRT as well. Then based on ABAQUS/Explicit platform, the FE models of RDB process of DRRT considering Mises, Hill’48 and Barlat/Lian yield criteria were built. The results show that: The variation trend of cross-sectional deformation ratio is same when using different yield criteria. The cross-sectional deformation ratio by using Mises yield criterion is close to that by using Hill’48 yield criterion. However, there is a quite difference between by using Barlat/Lian yield criterion and by using Mises or Hill’48 yield criteria. The prediction values of cross-sectional height deformation by using three yield criteria all underestimate the experiment ones, and the prediction values of cross-sectional width deformation overestimate the experiment ones. By comparing the simulation results of cross-sectional deformation of the DRRT with different yield criteria and experiment ones, Barlat/Lian yield criterion is found to be suitable for describing the RDB process of DRRT. Ó 2019 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
* Corresponding author at: Research & Development Institute of Northwestern Polytechnical University in Shenzhen, Shenzhen 518057, China. E-mail address: [email protected] (Y. LIU). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier https://doi.org/10.1016/j.cja.2019.08.006 1000-9361 Ó 2019 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: XIA Y et al. Cross-sectional deformation of H96 brass double-ridged rectangular tube in rotary draw bending process with different yield criteria, Chin J Aeronaut (2019), https://doi.org/10.1016/j.cja.2019.08.006
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1. Introduction
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Due to the merits of wider operation bandwidth, lower characteristic impedance and lower cutoff frequency, the bent double-ridged rectangular tube (DRRT) has been widely applied in the fields of the microwave communication, radar and electronic countermeasure.1 Rotary draw bending (RDB) is commonly used to manufacture DRRT bending parts because of its high bending accuracy, stable forming quality and high productivity.2 However, the RDB of DRRT is a high-nonlinearity complex physical process, and defects such as cross-sectional deformation, springback and wrinkling occur easily during RDB process. Moreover, the crosssectional deformation will seriously affect the microwave transmission efficiency, so controlling it becomes a prime priority to improve the precision of DRRT during RDB process. The FE simulation method has been widely used in tube bending research because of its simplicity, high efficiency and accuracy. However, since the tube is generally formed by drawing, extrusion, rolling, etc., which appears to be obvious anisotropy. In order to improve the prediction accuracy of the cross-sectional deformation of DRRT during RDB process, it is very important to select the yield criterion that can accurately describe the anisotropic behavior of the material. Therefore, it is of great significance to study the influence of different yield criteria on prediction accuracy of the cross-sectional deformation of RDB process of DRRT. At present, domestic and foreign scholars have done a lot of research about the influence of yield criterion on the prediction accuracy of cross-sectional deformation of bending tube. Miller and Kyriakides3 used Mises isotropic yield criterion to simulate the bending process of aluminum alloy rectangular tube, the results showed that the prediction value of crosssectional deformation of the tube was seriously underestimated by about 30%. Zhu et al.4 introduced Mises, Hill’48 and Barlat91 yield criteria into the bending process of 304 austenitic stainless steel tube, the parameters of these yield criteria were determined by uniaxial tensile tests at 0°, 45° and 90°. Then by comparing section flatness of tube, it was found that the simulated value obtained by Hill’48 yield criterion is more consistent with the experimental value. Hopperstad et al.5 obtained the parameters of Hill’48 and Barlat/Lian yield criteria by uniaxial tensile tests at 0°, 45° and 90°, and the above yield criteria were introduced into the bending process of tube, then found that the latter is more accurate for predicting the height deformation of the bending tube. From the above research, different yield criteria have great difference in predicting the deformation of different material tubes. So far, no one yield criterion is suitable for forming process of all material tubes. Moreover, the parameters of the anisotropic yield criterion in the above literatures are determined directly by using uniaxial tensile tests with different angles of tube extrusion direction. H96 brass DRRT widely used in microwave transmission is small in size and has many edges, so it is difficult to determine the parameters of the anisotropic yield criterion by using uniaxial tensile tests at 45° and 90° with respect to tube extrusion direction. However, the reverse method is to obtain the material parameters backwardly by using FE method to simulate the deformation process of the specimen, and restric-
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Y. XIA et al. tions on the size and deformation of the specimen are relaxed,6 therefore, the reverse method is widely applied to determine parameters of anisotropic yield criterion. Yan et al.7 determined the parameters of Hill’48 yield criterion of low carbon steel by the inverse method combining plate bending test and orthogonal experiment, and discovered that Hill’48 yield criterion can better describe the deformation behavior of plate during roll bending. Because it is difficult to obtain the specimen that meets the dimensional requirements, Liu et al.8 obtained the strength coefficient and hardening exponent of steel tube by the reverse method and flattening test. And the reliability of the reverse parameters was verified by the tube collision test. Chamekh et al.9 adopted the inverse method combining the cylindrical cup drawing test to obtain the parameters of the steel’s Hill’48 yield criterion. The result showed that Hill’48 yield criterion determined by the reverse method can well describe the earing phenomenon. Material performance parameters in the above studies were obtained based on the inverse method combining the actual forming process, and the reliability of the reverse parameters was verified by experiment. However, a large number of iterative calculations were required in the reverse process, resulting in lower efficiency. In order to improve the efficiency of determining parameters of yield criterion, inverse method combining response surface methodology (RSM) was proposed,10 in which the FE simulations can be executed simultaneously and iterations can be avoided, so the computing time was reduced sharply. Moreover, the accuracy of parameters of yield criterion is largely determined by the suitability of experimental measurements in the inverse method. Two commonly used methods to determine the parameters of anisotropy yield criterion are based on yield stresses and r-values, respectively. Xue and Liao11 determined the parameters of Hill’48 yield criterion by the above two methods of DC05 steel plate, the result showed that Hill’48 yield criterion determined by r-values could describe in-plane principal strain and outline of crossdie deep drawing specimen better. Wang et al.12 found the earing phenomenon of 5754O aluminum alloy can be predicted well by Hill’48 yield criterion based on r-values. The above studies showed that the parameters of yield criterion through r-values was more suitable for describing the workpiece deformation problem. Our research group13–16 has done a lot of research about the cross-sectional deformation of H96 brass DRRT in H-typed bending and E-typed bending by the FE method. The establishment of the FE models are based on Mises isotropic yield13–15 and Hill’48 anisotropy yield criterion,16 while the parameters of Hill’48 yield criterion are determined based on stress measurements. Therefore, in order to confirm the yield criterion which is suitable for predicting the cross-sectional deformation of RDB process of H96 brass DRRT, Mises isotropic yield criterion, Hill’48 and Barlat/Lian anisotropic yield criteria widely used in practical forming are selected and the parameters of Hill’48 and Barlat/Lian anisotropic yield criteria are determined based on deformation measurements in inverse method combining RSM. The prediction accuracy of the three yield criteria on the cross-sectional deformation of RDB process of DRRT was evaluated by comparing simulation results and experiment ones.
Please cite this article in press as: XIA Y et al. Cross-sectional deformation of H96 brass double-ridged rectangular tube in rotary draw bending process with different yield criteria, Chin J Aeronaut (2019), https://doi.org/10.1016/j.cja.2019.08.006
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Cross-sectional deformation of H96 brass double-ridged rectangular tube
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2. Selection of yield criterion and obtainment of the corresponding parameters
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2.1. Selection of yield criterion
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Mises, Hill’48 and Barlat/Lian yield criteria with wide application and high practicability are selected to study the bending process of the DRRT. In bending process, DRRT is usually considered to be in plane stress state due to its wall thickness is very thin. Therefore, the three yield criteria can be described as follows.
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m
gðpÞ ¼
2mr
1 r45 þ @r@fyy r45
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ð7Þ 202 203
2.2. Obtainment of parameters of anisotropy yield criteria
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where rxx and ryy are the normal stress in the rolling and transversal directions, respectively. rxy is the shear stress, r is equivalent stress, and C is a constant related to material properties.
2.2.1. The determination procedure of the parameters of anisotropic yield criterion
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Because the size of H96 DRRT is small, the parameters of the anisotropic yield criterion can’t be determined by the conventional method. So the inverse method combining the actual forming process of the workpiece is proposed. In view of the fact that uniaxial tensile test of the whole tube is simple and the measurement of the deformation process is convenient, the inverse method combining uniaxial tensile test of whole tube and RSM to determine the parameters of Hill’48 and Barlat/Lian yield criteria was proposed. In the reverse method, some responses needed to be determined to quantify deformation degree of DRRT in uniaxial tensile test. The normal anisotropy is usually characterized by the r-value of plastic strain at 15% or 20% of uniaxial tensile test.18 Therefore, the transverse strains of middle position of calibration section DW10, DW20 and DW25 (as shown in Fig. 1) are taken as the responses when the tensile deformation degree of tube are 10%, 20% and 25%, respectively, which can be measured by digital image correlation (DIC). The parameter determination of Hill’48 yield criterion is taken as an example to explain the proposed reverse method in detail. When Hill’48 yield criterion is applied in ABAQUS, the anisotropic parameters F, G, H, and N of Hill’48 yield function is determined by the yield stress ratios R11, R22, R33 and R12 embedded in the software, as shown in Eq. (8)19 in plane stress state: 8 > r 1 1 1 > þ 2 2 ¼ 12 R12 þ R12 R12 > F ¼ 2 2 > r22 r33 r11 22 33 11 > > > > > > > < G ¼ r2 21 þ 21 21 ¼ 12 R12 þ R12 R12 r33 r11 r22 33 11 22 ð8Þ > > > r 1 1 1 1 1 1 1 >H ¼ þ 2 2 ¼ 2 R2 þ R2 R2 > 2 2 > r11 r22 r33 11 22 33 > > > > > : N ¼ 3 s ¼ 32 2 r 2R
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(2) Hill’48 yield criterion Hill’48 yield criterion is expressed in plane stress state in Eq. (2), which can describe the yield behavior of orthotropic materials:
ð2Þ
where the meanings of rxx , ryy , rxy and r are the same with those in Eq. (1). F, G, H, and N are parameters that characterize material anisotropy, which can be expressed by Lankford’s coefficients r0, r45 and r90 as shown in Eq. (3). 8 F ¼ r90 ðrr00 þ1Þ > > > > < G ¼ r01þ1 ð3Þ 0 H ¼ r0rþ1 > > > > : 90 Þð1þ2r45 Þ N ¼ ðr0 þr 2r90 ðr0 þ1Þ where r0, r45 and r90 are the r-values at 0°, 45°, and 90° with respect to rolling direction in uniaxial tensile test, respectively. (3) Barlat/Lian yield criterion
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where r45 is yield stress of 45° to the rolling direction.
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where the value of p has a certain relationship with rh when a and k are known. Many studies show that rh increases with the increase of p when h ¼ 45 , as shown in Eq. (7).
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Mises yield criterion is expressed in plane stress state, as shown in Eq. (1): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ r ¼ r2xx rxx ryy þ r2yy þ 3r2xy ¼ C
2
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@f @rxx
fðrij Þ ¼ ðG þ HÞr2xx þ ðH þ FÞr2yy 2Hrxx ryy þ 2Nr2xy r ¼ 0
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a, c, k and p are material parameters, and a = 2-c, their relationship with Lankford’s coefficient is shown in Eq. (6). 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r0 r90 > 1þr < a ¼ 2 2 1þr 0 90 ð6Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > r 1þr 0 90 :k ¼ r90 1þr0
(1) Mises yield criterion
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Barlat and Lian17 proposed a three-parameter anisotropic yield criterion for solving plane stress state problems in 1989. The expression is shown in Eq. (4): m
fðrij Þ ¼ ajK1 þ K2 jm þ ajK1 K2 jm þ cj2K2 jm 2r ¼ 0 where 8 rxx þkryy > < K1 ¼ 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 rxx kryy > : K2 ¼ þ p2 r2xy 2
ð4Þ
12
ð5Þ
where m is related to the crystal structure, recommended values of the exponent are m = 6 for BCC materials and m = 8 for FCC materials. For H96 brass tube in this paper, m = 8.
12
where rij is the nonzero yield stress of rij , and s ¼ r =3. The yield stress ratios R11, R22, R33 and R12 are defined with respect to the equivalent stress r, and when the equivalent stress r is equal to r11 (the yield stress in the rolling direction), R11 is equal to 1. And R22, R33, and R12 are parameters that need to be determined. Combining the Eqs. (3) and (8), the relation-
Please cite this article in press as: XIA Y et al. Cross-sectional deformation of H96 brass double-ridged rectangular tube in rotary draw bending process with different yield criteria, Chin J Aeronaut (2019), https://doi.org/10.1016/j.cja.2019.08.006
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Fig. 1
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ship between the parameters R22, R33, and R12 and Lankford’s coefficients r0 , r45 and r90 is shown in Eq. (9): qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 1þð1=r0 Þ > > R22 ¼ 1þð1=r90 Þ > > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0Þ ð9Þ R33 ¼ ð1=r1þð1=r 0 Þþð1=r90 Þ > > q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 3½1þð1=r0 Þ :R ¼ 12 ½ð1=r0 Þþð1=r90 Þ½1þ2r45 The flow chart of determining the parameters R22, R33, and R12 of Hill’48 yield criterion of H96 brass DRRT based on inverse method combining RSM is shown in Fig. 2, which consists of the following three main steps. Firstly, the uniaxial tensile of H96 brass DRRT was conducted and three responses DW10, DW20 and DW25 were measured by DIC. Then the parameters (R22, R33, and R12) combination of Hill’48 yield criterion were created by using Box-Behnken design (BBD). Taking these parameters combination as the material data of FE simulations, the FE simulations of uniaxial tensile of H96 brass DRRT were performed and the simulation results of responses were recorded. Based on these simulation results, the mathematical models between each response DW10, DW20 and DW25 and the parameters R22, R33, and R12 were established by using RSM, and its polynomial equation is expressed in Eq. (10): Yk ¼ b0 þ
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Sketch of position of calibration section.
3 X
bm xm þ
3 X
m¼1
m¼1
bmm x2m þ
2 3 X X
bmn xm xn
ð10Þ
m¼1 n¼mþ1
where Yk represents response, that is transverse strain DW10, DW20 and DW25 of the calibration section under different tensile deformation of tube. xm and xn are influence factors, these refers to the parameters R22, R33, and R12. b0, bm, bmm and bmn are the coefficients of intercept, linear, quadratic and interactive terms, respectively. Finally, an objective function describ-
Fig. 2
ing the gap between the simulated results and the experimental values of the response was established as shown in Eq. (11):18 " RSM 2 # T 1 X Yk YEXP k ð11Þ Fobj ¼ uk T k¼1 YEXP k where T is number of responses, in this paper, T = 3. YRSM k RSM represents the response value of DWRSM and DWRSM 10 , DW20 25 calculated from mathematical models shown in Eq. (10). EXP YEXP is response of DWEXP and DWEXP measured by k 10 , DW20 25 DIC. Uk is the weight coefficient representing the relative importance of each term, as shown in Eq. (12): uk ¼ YEXP = k
T X
YEXP k
ð12Þ
k¼1
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2.2.2. The determination of parameters of anisotropic yield criterion
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Before creating the FE models of uniaxial tensile test of DRRT with different combination of three parameters R22, R33 and R12 by using Box-Behnken Design (BBD) as shown in Fig. 2, the reasonable range of three parameters needs to be determined, firstly. According to Eq. (9), the values of the three parameters R22, R33 and R12 can be determined according to r0 , r45 and r90 . According to literature,16 Lankford’s coefficient r-value of copper alloy is basically in the range of 0.5–2. So the r-value range of H96 brass is chosen to be 0.5–2. Then the range of R22, R33 and R12 was determined according to Eq. (9), as shown in Eq. (13): 8 > < 0:70 R22 1:50 0:75 R33 1:23 ð13Þ > : 0:60 R12 1:50
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Flow chart of parameters determination of Hill’48 yield criterion by using inverse method.
Please cite this article in press as: XIA Y et al. Cross-sectional deformation of H96 brass double-ridged rectangular tube in rotary draw bending process with different yield criteria, Chin J Aeronaut (2019), https://doi.org/10.1016/j.cja.2019.08.006
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Cross-sectional deformation of H96 brass double-ridged rectangular tube 301 302 303 304 305 306 307 308 309 310 312 313 314 315 316 317 318
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According to the BBD method, three levels are selected for each parameter. And the three levels value of each parameter is determined as shown in Table 1. Therefore, 17 FE simulations of uniaxial tensile test of DRRT need to be executed as shown in Table 2. Fig. 3 shows the specimen and FE model of uniaxial tensile test of H96 brass DRRT, and the mechanical parameters of H96 brass DRRT obtained by uniaxial tensile test were shown in Table 3. The constitutive equation in Eq. (14) is used to describe the plastic deformation of DRRT.
So the objective function is constructed by substituting Eqs. (15)–(17) into the Eq. (11). Then the value of R22, R33 and R12 can be determined by minimizing the objective function shown in Eq. (11). And the parameters of Hill’48 yield criterion can be obtained by plugging value of R22, R33 and R12 into Eq. (8), as shown in Table 4. The parameters of Barlat/Lian yield criterion can be determined by using same method, as shown in Table 4 too.
r ¼ Kðe þ bÞn
3. Influence of yield criteria on prediction accuracy of crosssectional deformation of DRRT
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3.1. Description of cross-sectional deformation and selection of characteristic section of DRRT
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Fig. 4(a) shows the section of tube before and after deformation. In order to describe the cross-sectional deformation of the DRRT during RDB process, Eqs. (18)–(21) are introduced to describe the different part deformation of characteristic sections.
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ð14Þ
The 17 FE simulation results of the uniaxial tensile test of DRRT are also recorded in Table 2. Based on these simulation results, the mathematical models (Eq. (10)) relating three responses DW10, DW20 and DW25 to the parameters R22, R33 and R12 were established, as shown in Eqs. (15)–(17): DW10 ¼ 2:71759 34:46865 R22 þ 41:09587 R33 2:16405 R12 2:31887 R22 R33 þ 12:60032 13:90879
320
R233
þ 1:00590
R222 ð15Þ
R212
321
DW20 ¼ 5:63682 65:67314 R22 þ 74:55923 R33
DHi ¼
0:26228 R12 þ 22:30118 R222 27:87448 R233 ð16Þ
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DW25 ¼ 4:18154 78:03289 R22 þ 93:60387 R33 0:14614 R12 1:74463 R22 R33 þ 27:03655
326
34:20049
ð18Þ
ð17Þ
R233
ð19Þ
Parameters
Level 1
Level 2
Level 3
R22 R33 R12
0.70 0.75 0.60
1.10 0.99 1.05
1.50 1.23 1.50
Table 2
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349 350
Bi B DBi ¼ B
ð20Þ
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bi b Dbi ¼ b Table 1 Three parameters R22, R33 and R12 and their respective levels.
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hi h Dhi ¼ h
324
R222
Hi H H
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ð21Þ
where DHi and DBi are the height and width deformation ratio of the tube, respectively, and Dhi and Dbi are the height and spacing deformation ratio of ridge groove of tube, respectively. H and B are the values of height and width of cross-section, while h and b are the values of height and spacing of ridge groove before bending deformation. Hi, Bi, hi and bi are the corresponding values after bending deformation. Meantime, in order to systematically study the effect of different yield criteria on the cross-sectional deformation of tube,
The experiment arrangement of BBD and the simulation results of response.
Number
R22
R33
R12
4W10(%)
4W20(%)
4W25(%)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
0.7 0.7 1.1 1.1 1.5 1.1 1.5 1.1 1.1 1.5 1.1 1.1 1.5 1.1 0.7 1.1 0.7
0.99 0.99 1.23 0.99 0.75 0.99 0.99 0.75 0.99 1.23 0.99 1.23 0.99 0.75 0.75 0.99 1.23
0.6 1.5 0.6 1.05 1.05 1.05 1.5 1.5 1.05 1.05 1.05 1.5 0.6 0.6 1.05 1.05 1.05
9.417 9.293 5.417 3.413 1.383 3.413 1.915 0.257 3.413 3.614 3.413 5.338 1.905 0.250 5.196 3.413 11.084
17.333 15.951 9.230 6.631 2.719 6.631 3.609 0.463 6.631 6.374 6.631 9.668 3.618 0.455 10.320 6.631 20.432
20.872 20.326 11.833 8.139 3.321 8.139 4.386 0.557 8.139 8.055 8.139 11.873 4.413 0.551 12.592 8.139 24.637
Please cite this article in press as: XIA Y et al. Cross-sectional deformation of H96 brass double-ridged rectangular tube in rotary draw bending process with different yield criteria, Chin J Aeronaut (2019), https://doi.org/10.1016/j.cja.2019.08.006
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Fig. 3
Table 3
Specimen and FE model of uniaxial tensile test of DRRT.
Mechanical parameters of H96 brass double-ridged rectangular tube.
Elasticmodulus E (GPa)
Poisson ratio m
Initial yield stress rs (MPa)
Hardening exponent n
Strength coefficient K (MPa)
Material constant b
92.82
0.324
70
0.425
588.17
0.0058
Table 4 Parameters of Hill’48 and Barlat/Lian yield criteria determined by the inverse method. Hill’48 Barlat/Lian
F 0.481
G 0.515
H 0.485
N 1.559
a 1.013
k 0.983
p 1.009
m 8
370
the two characteristic sections along the bending direction are selected as shown in Fig. 4(b), which are section S2 in the mandrel support zone and section S6 in the non-mandrel support zone. And the distribution of nodes along the height and width direction on characteristic section has been given in Fig. 5(a) and (b).
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3.2. Establishment of FE model of RDB process of DRRT
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Based on the mechanical properties of the H96 brass DRRT in Table 3 and the parameters of yield criterion in Table 4, the constitutive models of different yield criteria of H96 brass DRRT are constructed and introduced in the FE model. In order to improve the computational efficiency and accuracy of the FE model, the rigid dies are modeled with 4-node 3D bilinear quadrilateral elements C3D4, meantime the tube is defined as deformable body and described by 4-node doubly curved thin shell elements S4R. And mesh sizes of tube and
Fig. 4
mandrels are all 1 mm 1 mm, the other rigid dies are 3 mm 3 mm. The friction between tube and dies is modeled with classical Coulomb friction model. Based on the above solved key technologies, the RDB process of DRRT is carried out based on ABAQUS/Explicit, the springback process of DRRT is carried out based on ABAQUS/Standard, and the established FE models considering different yield criteria are shown in Fig. 6. And the tube bender and dies for RDB process are shown in Fig. 7. The process parameters of RDB of DRRT in FE simulation and experiment are shown in Table 5.
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3.3. Deformation analysis of characteristic section of DRRT during RDB with different yield criteria
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3.3.1. Analysis of height deformation ratio of characteristic section
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Fig. 8 shows the distribution of the height deformation of characteristic sections S2 and S6 with Mises, Hill’48 and Barlat/Lian yield criteria. It can be seen that no matter which yield criterion is employed, the height deformation ratio (DHi) of characteristic section is symmetrical about the centerline 5-50 (Fig. 5(a)) and presents the variation trend of the ‘‘M” type. DHi is all negative values with the three yield criteria, which shows that the inner and outer flanges of the tube have collapsed during RDB process. The height deformation of section S6 is greater than that of section S2 for different yield criteria. DHi by using Mises yield criterion is close to that by using Hill’48 yield criterion. However, DHi by using Barlat/Lian
395
Sketch of characteristic section before and after deformation and position of characteristic section.
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Cross-sectional deformation of H96 brass double-ridged rectangular tube
Fig. 5
7
Distribution of nodes on characteristic section.
Table 5
Process parameters of RDB of DRRT.
Parameters
Experiment
Simulation
Bending radius (mm) Bending angle (°) Boosting velocity (mm/s) Bending velocity (rad/s) Friction between tube and mandrel
60 90 35.21 0.5 Hydraulic oil Dry friction Dry friction 0.2
60 90 35.21 0.5 0.05
0
0
Friction between tube and clamp die Friction between tube and other dies
Fig. 6
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FE model for RDB of DRRT.
yield criterion is quite different from DHi by using Mises and Hill’48 yield criterion, and prediction value of the height deformation of DRRT by using Barlat/Lian yield criterion is maximal, which is minimal when using Hill’48 yield criterion. 3.3.2. Analysis of height deformation ratio of ridge groove of characteristic section Distribution of the height deformation of ridge groove in sections S2 and S6 with Mises, Hill’48 and Barlat/Lian yield criteria has been shown in Fig. 9. The height deformation of ridge
Fig. 7
Clearances between tube and mandrel (mm) Clearances between tube and other dies (mm)
Rough 0.14 0.2
groove of section S6 is bigger than that of section S2 with the different yield criteria. The height deformation of ridge groove ratio (Dhi) is also systematical about the centerline 5-50 and presents the variation trend of inverse ‘‘V” type. Dhi is all negative values with these three yield criteria, which shows that the height of ridge groove produced compression deformation,
Tube bender and dies.
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Fig. 8 Distribution of the height deformation ratio of sections S2 and S6 with different yield criteria.
Fig. 9 Distribution of the height deformation ratio of ridge groove of sections S2 and S6 with different yield criteria.
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and maximum Dhi is located at nodes 10(10’) and 19(19’), namely at the opening of ridge grooves. This is because the ridge groove is not directly supported by the mandrel, and its stiffness is insufficient during bending process, the ridge groove is severely deformed by radial extrusion from the sidewall of the tube. Like the law of cross-sectional height deformation ratio, the height deformation ratio of ridge groove of characteristic section is very close with Mises and Hill’48 yield criteria, and which is biggest with Barlat/Lian yield criterion.
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Fig. 10 Distribution of the width deformation ratio of sections S2 and S6 with different yield criteria.
tion of characteristic section is located at the intersection of side wall and ridge groove. This is because the ridge groove part has been undergone severe contraction deformation during bending process, and the inner and outer flanges of ridge groove collapsed. Due to the deformation coordination effect, the sidewall of the tube is also greatly deformed. 4Bi by using Mises yield criterion is very close to that by using Hill’48 yield criterion, and prediction value of the width deformation of characteristic section by using Mises and Hill’48 yield criteria is bigger than that by using Barlat/Lian yield criterion.
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3.3.4. Analysis of spacing deformation ratio of ridge groove of characteristic section
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Distribution of the spacing deformation of ridge groove of sections S2 and S6 with different yield criteria has been shown in Fig. 11. The spacing deformation ratio of ridge groove of characteristic section (Dbi) is all negative values with three yield criteria, which shows that spacing of ridge groove produced compression deformation. The absolute value of Dbi from node 29(290 ) to node 33(330 ) decreases after an initial increase, the maximum is reached at node 30(300 ) which is more than 14%. Due to no-mandrel filling, spacing of ridge groove of tube has undergone severe deformation during bending process. Spacing deformation of ridge groove of section S2 with Barlat/Lian yield criterion are smaller than with Mises and Hill’48 yield criteria, and spacing deformation of ridge groove
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3.3.3. Analysis of width deformation ratio of characteristic section Fig. 10 shows the distribution of the width deformation of sections S2 and S6 with different yield criteria. The width deformation ratio (DBi) of characteristic section is all negative values with three yield criteria, which shows that the area also produced compression deformation, and width deformation of characteristic section in the outer side of neutral layer is larger than that in the inner side of neutral layer. The absolute value of DBi in the outer side of neutral layer from node 20(200 ) to node 28(280 ) increases after an initial decrease, the maximum deformation is located at node 28(280 ), namely the ridge of the outer flange of the ridge groove. The absolute value of DBi in the inner side of neutral layer from node 34(340 ) to node 42 (420 ) also increases after an initial decrease, the maximum is reached at node 34(340 ), namely the ridge of the inner web of the ridge groove. In conclusion, the maximum width deforma-
Fig. 11 Distribution of the spacing deformation ratio of ridge groove of sections S2 and S6 with different yield criteria.
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of section S6 with Barlat/Lian yield criterion are larger than with Mises and Hill’48 yield criteria. Regardless of section S2 or section S6, the predicted values of the spacing deformation of ridge groove are close with Mises and Hill’48 yield criteria. 3.4. Applicability evaluation of three yielding criteria for describing cross-sectional deformation of DDRT In order to evaluate the accuracy of the cross-sectional deformation of DRRT predicted by different yield criteria, Fig. 12 (a) shows the experimental result of the DRRT bent part adopting conditions shown in Table 5, and Fig. 12(b), (c) and (d) gives simulation results of the DRRT bent part by using different yield criteria under the same conditions. From Fig. 12, it can be qualitatively found that the tube bent by experiment is coincide with the tube bent by using different yield criteria. However, from this figure, it is difficult to directly judge from the intuitional shape which yield criterion is suitable for describing the cross-sectional deformation of DRRT. So Fig. 13 shows the quantitative experiment and simulation results corresponding to Fig. 12, which is crosssectional height deformation located at node 1(10 ) and crosssectional width deformation located at node 20(200 ) along the bending direction.
Fig. 12
Fig. 13
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From Fig. 13(a), it can be seen that the variation trend of cross-sectional height deformation of tube is almost same by using different yield criteria, and the predicted values by using these three yield criteria all underestimate the cross-sectional height deformation of the DRRT. The predicted values with Mises and Hill’48 yield criterion are very close, which are different from that with Barlat/Lian yield criterion, but the predicted value with Barlat/Lian yield criterion are the closest to the experimental values. The average relative errors between experiment value and simulated values of Mises, Hill’48 and Barlat/Lian yield criteria are 14%, 14.1% and 6.8%, respectively. It is visible that Barlat/Lian yield criterion can better predict the cross-sectional height deformation of the DRRT. From Fig. 13(b), it can be seen that the variation trend of cross-sectional width deformation of tube is also same by using different yield criteria. Different from the height crosssectional deformation, the predicted values by using three yield criteria all overestimate the width cross-sectional deformation of the DRRT, and the predicted values by using Barlat/Lian yield criterion are the closest to the experimental values among three yield criteria, the average relative errors is only 4.8%. However, the average relative errors between simulated values by using Mises and Hill’48 yield criteria and experiment value are 26.5% and 23.9%, respectively, and the maximum relative errors of the experiment and simulation values by using these
DRRT bent comparison of simulation results by using different yield criteria and experiment result.
Simulation results of cross-sectional deformation with three yield criteria and experiment result.
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two yield criteria reach up to 35.8% and 32.5%, respectively. It can be seen that Mises and Hill’48 yield criteria have a poor simulation precision for the cross-sectional width deformation of H96 brass DRRT, and Barlat/Lian yield criterion can better predict cross-sectional width deformation of the DRRT. From the above analysis, it can be seen that different yielding criteria have different prediction accuracy for the crosssectional deformation of DRRT, which is mainly caused by the difference of yield criterion in describing plastic anisotropic behavior and predicting the shape of yield locus.5 In order to further analyze the differences of predicting the crosssectional deformation of the DRRT by using different yield criteria, the r-value distribution and yield function shape of Mises, Hill’48 and Barlat/Lian yield criteria are given in Fig. 14. Compared with Mises yield criterion, the plastic anisotropy of the material is considered by Hill’48 yield criterion, but the difference between the functional shapes of two yield criteria is small, which leads that the prediction values of cross-sectional deformation of the DRRT are relatively close. From Fig. 14(a), it can be seen that the difference of normal anisotropy r-value of the Hill’48 and Barlat/Lian yield criteria is small in different directions, but the functional shapes of the two yield criteria are quite different (as shown in Fig. 14(b)), which leads to a larger difference in the predicted value of the cross-sectional deformation of the DRRT. Therefore, com-
Fig. 14
Fig. 15
pared with the anisotropy r-value, the difference of yield locus can be used to better explain the difference of prediction accuracy of cross-sectional deformation of DRRT. The different yielding criteria have different prediction accuracy for the cross-sectional deformation of DRRT, which also can be explained from the tangential stress distribution of the tube bent by using different yield criteria, as shown in Fig. 15. Compared with Fig. 15(a) and (b), the distribution and magnitude of tangential stress of tube is very close by using Mises and Hill’48 yield criterion. But the distribution and magnitude of tangential stress of tube by using Barlat/ Lian yield criterion as shown in Fig. 15(c) is obviously different from those as shown in Fig. 15(a) and (b). In summary, the error between the experimental value and the simulated one by using Barlat/Lian yield criterion is the smallest, which can more accurately predict the crosssectional deformation of the DRRT among three yield criteria.
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4. Conclusion
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(1) Mises, Hill’48 and Barlat/Lian yield criteria are selected to describe the RDB process of small-sized H96 brass DRRT, and the parameters of the Hill’48 and Barlat/ Lian anisotropic yield criteria are obtained by using
r-value distribution and functional shape of the three yield criteria.
Distribution of tangential stress of DRRT under different yield criteria.
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the inverse method combining uniaxial tensile test of whole tube and response surface methodology. On the basis of solving the key problems of model, 3D FE models of DRRT considering different yield criteria are established. (2) The variation trend of cross-sectional deformation ratio of tube is same when using different yield criteria. The deformation of each part of the characteristic section located in non-mandrel support zone is greater than that mandrel support zone. The cross-sectional deformation by using Mises yield criterion is close to that by using Hill’48 yield criterion. However, when using Barlat/Lian yield criterion, the cross-sectional deformation is greatly different. (3) The prediction values of cross-sectional height deformation by using three yield criteria all underestimate the experiment ones, and the prediction values of crosssectional width deformation overestimate the experiment ones. The prediction values of cross-sectional deformation by Barlat/Lian yield criterion agree well with the experimental ones, while those by using Mises and Hill’48 yield criteria have a larger error with the experiment ones. Acknowledgments
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The research is supporting by the Science and Technology Project of Shenzhen of China (Nos. JCYJ20170306160003433 and JCYJ20180306171058717) and the National Natural Science Foundation of China (No. 51375392).
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1. Xue QZ, Meng X, Wang SJ, Li K, Liu WX. Analysis of two-beam ka band folded waveguide traveling-wave tube. J Infrared Millim Te 2015;36(1):31–41. 2. Yang H, Li H, Zhang ZY, Zhan M, Liu J, Li GJ. Advances and trends on tube bending forming technologies. Chin J Aeronaut 2012;25(1):1–12. 3. Miller JE, Kyriakides S. Three-dimensional effects of the bend– stretch forming of aluminum tubes. Int J Mech Sci 2003;45 (1):115–40. 4. Zhu QX, Xu Y, Zhang SH, Zhang GJ, Song HW. The influence of yield criteria on the prediction accuracy of stainless steel tube bending. Mater Sci Technol 2013;21(6):124–9 [Chinese].
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5. Hopperstad OS, Berstad T, Ilstad H. Effects of the yield criterion on local deformations in numerical simulation of profile forming. J Mater Process Technol 1998;s80–81(98):551–5. 6. Szeliga D, Gawad J, Pietrzyk M. Inverse analysis for identification of rheological and friction models in metal forming. Comput Method Appl M 2006;195(48–49):6778–98. 7. Yan Y, Wang HB, Li Q. The inverse parameter identification of Hill48 yield criterion and its verification in press bending and roll forming process simulations. J Manuf Processes 2015;20:46–53. 8. Liu DH, Wang C, Li GY. Identification of material parameters of thin-walled steel tube by inverse method. China Mech Eng 2008;19 (6):688–90 [Chinese]. 9. Chamekh A, Salah HBH, Hambli R. Inverse technique identification of material parameters using finite element and neural network computation. Int J Adv Manuf Tech 2009;44(1–2):173–9. 10. Malakizadi A, Cedergren S, Sadik I, Nyborg L. Inverse identification of flow stress in metal cutting process using Response Surface Methodology. Simul Model Pract Th 2016;60(3):40–53. 11. Xue X, Liao J. Assessment of yield criterion for deformation prediction in cross-die deep drawing of DC05 steel sheet. J Plasticity Eng 2018;25(4):217–24 [Chinese]. 12. Wang HB, Wang M, Yan Y, Wu XD. Effect of the solving method of parameters on the description ability of the yield criterion about the anisotropic behavior. J Mech Eng 2013;49(24):45–53 [Chinese]. 13. Xiao YH, Liu YL, Yang H. Research on cross-sectional deformation of double-ridgedd rectangular tube during H-typed rotary draw bending process. Int J Manul Tech 2014;73(9–12):1789–98. 14. Liu CM, Liu YL, Yang H. Influence of different mandrels on cross-sectional deformation of the double-ridged rectangular tube in rotary draw bending process. Int J Manul Tech 2016;91(1– 4):1–12. 15. Zhang HL, Liu YL, Yang H. Deformation behaviors of doubleridged rectangular H96 tube in rotary draw bending under different mandrel types. Int J Manul Tech 2016;82(9–12):1569–80. 16. Zhang HL, Liu YL. The inverse parameter identification of Hill’48 yield function for small-sized tube combining response surface methodology and three-point bending. J Mater Res 2017;32 (12):1–9. 17. Barlat F, Lian K. Plastic behavior and stretchability of sheet metals. Part I: a yield function for orthotropic sheets under plane stress conditions. Int J Plasticity 1989;5(1):51–66. 18. Cao J, Lin J. A study on formulation of objective functions for determining material models. Int J Mech Sci 2008;50(2):193–204. 19. Xue XY, Chen JY, Li HW, Gao EZ, Li JS, Zhou L. Anisotropic effects on cold deep drawing of TA1 spherical part. J Plasticity Eng 2010;17(2):11–6 [Chinese].
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Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
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Cross-sectional deformation of H96 brass doubleridged rectangular tube in rotary draw bending process with different yield criteria
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Yangyang XIA a,b, Yuli LIU a,b,*, Mengmeng LIU a,b
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a
Research & Development Institute of Northwestern Polytechnical University in Shenzhen, Shenzhen 518057, China State Key Laboratory Solidification Process, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an 710072, China b
Received 28 March 2019; revised 19 June 2019; accepted 15 July 2019
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KEYWORDS
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Anisotropy yield criterion; Cross-sectional deformation; H96 double-ridged rectangular tube; H-type rotary draw bending; Inverse method
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Abstract Different yield criterion has great difference in predicting the deformation of tube with different material. In order to improve the prediction accuracy of the cross-sectional deformation of the double-ridged rectangular tube (DRRT) during rotary draw bending (RDB) process, Mises isotropic yield criterion, Hill’48 and Barlat/Lian anisotropic yield criteria commonly used in practical engineering are introduced to simulate RDB of DRRT. The inverse method combining uniaxial tensile test of whole tube and response surface methodology was proposed to identify the parameters of Hill’48 and Barlat/Lian yield criteria of small-sized H96 brass extrusion DRRT as well. Then based on ABAQUS/Explicit platform, the FE models of RDB process of DRRT considering Mises, Hill’48 and Barlat/Lian yield criteria were built. The results show that: The variation trend of cross-sectional deformation ratio is same when using different yield criteria. The cross-sectional deformation ratio by using Mises yield criterion is close to that by using Hill’48 yield criterion. However, there is a quite difference between by using Barlat/Lian yield criterion and by using Mises or Hill’48 yield criteria. The prediction values of cross-sectional height deformation by using three yield criteria all underestimate the experiment ones, and the prediction values of cross-sectional width deformation overestimate the experiment ones. By comparing the simulation results of cross-sectional deformation of the DRRT with different yield criteria and experiment ones, Barlat/Lian yield criterion is found to be suitable for describing the RDB process of DRRT. Ó 2019 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
* Corresponding author at: Research & Development Institute of Northwestern Polytechnical University in Shenzhen, Shenzhen 518057, China. E-mail address: [email protected] (Y. LIU). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier https://doi.org/10.1016/j.cja.2019.08.006 1000-9361 Ó 2019 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: XIA Y et al. Cross-sectional deformation of H96 brass double-ridged rectangular tube in rotary draw bending process with different yield criteria, Chin J Aeronaut (2019), https://doi.org/10.1016/j.cja.2019.08.006
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1. Introduction
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Due to the merits of wider operation bandwidth, lower characteristic impedance and lower cutoff frequency, the bent double-ridged rectangular tube (DRRT) has been widely applied in the fields of the microwave communication, radar and electronic countermeasure.1 Rotary draw bending (RDB) is commonly used to manufacture DRRT bending parts because of its high bending accuracy, stable forming quality and high productivity.2 However, the RDB of DRRT is a high-nonlinearity complex physical process, and defects such as cross-sectional deformation, springback and wrinkling occur easily during RDB process. Moreover, the crosssectional deformation will seriously affect the microwave transmission efficiency, so controlling it becomes a prime priority to improve the precision of DRRT during RDB process. The FE simulation method has been widely used in tube bending research because of its simplicity, high efficiency and accuracy. However, since the tube is generally formed by drawing, extrusion, rolling, etc., which appears to be obvious anisotropy. In order to improve the prediction accuracy of the cross-sectional deformation of DRRT during RDB process, it is very important to select the yield criterion that can accurately describe the anisotropic behavior of the material. Therefore, it is of great significance to study the influence of different yield criteria on prediction accuracy of the cross-sectional deformation of RDB process of DRRT. At present, domestic and foreign scholars have done a lot of research about the influence of yield criterion on the prediction accuracy of cross-sectional deformation of bending tube. Miller and Kyriakides3 used Mises isotropic yield criterion to simulate the bending process of aluminum alloy rectangular tube, the results showed that the prediction value of crosssectional deformation of the tube was seriously underestimated by about 30%. Zhu et al.4 introduced Mises, Hill’48 and Barlat91 yield criteria into the bending process of 304 austenitic stainless steel tube, the parameters of these yield criteria were determined by uniaxial tensile tests at 0°, 45° and 90°. Then by comparing section flatness of tube, it was found that the simulated value obtained by Hill’48 yield criterion is more consistent with the experimental value. Hopperstad et al.5 obtained the parameters of Hill’48 and Barlat/Lian yield criteria by uniaxial tensile tests at 0°, 45° and 90°, and the above yield criteria were introduced into the bending process of tube, then found that the latter is more accurate for predicting the height deformation of the bending tube. From the above research, different yield criteria have great difference in predicting the deformation of different material tubes. So far, no one yield criterion is suitable for forming process of all material tubes. Moreover, the parameters of the anisotropic yield criterion in the above literatures are determined directly by using uniaxial tensile tests with different angles of tube extrusion direction. H96 brass DRRT widely used in microwave transmission is small in size and has many edges, so it is difficult to determine the parameters of the anisotropic yield criterion by using uniaxial tensile tests at 45° and 90° with respect to tube extrusion direction. However, the reverse method is to obtain the material parameters backwardly by using FE method to simulate the deformation process of the specimen, and restric-
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Y. XIA et al. tions on the size and deformation of the specimen are relaxed,6 therefore, the reverse method is widely applied to determine parameters of anisotropic yield criterion. Yan et al.7 determined the parameters of Hill’48 yield criterion of low carbon steel by the inverse method combining plate bending test and orthogonal experiment, and discovered that Hill’48 yield criterion can better describe the deformation behavior of plate during roll bending. Because it is difficult to obtain the specimen that meets the dimensional requirements, Liu et al.8 obtained the strength coefficient and hardening exponent of steel tube by the reverse method and flattening test. And the reliability of the reverse parameters was verified by the tube collision test. Chamekh et al.9 adopted the inverse method combining the cylindrical cup drawing test to obtain the parameters of the steel’s Hill’48 yield criterion. The result showed that Hill’48 yield criterion determined by the reverse method can well describe the earing phenomenon. Material performance parameters in the above studies were obtained based on the inverse method combining the actual forming process, and the reliability of the reverse parameters was verified by experiment. However, a large number of iterative calculations were required in the reverse process, resulting in lower efficiency. In order to improve the efficiency of determining parameters of yield criterion, inverse method combining response surface methodology (RSM) was proposed,10 in which the FE simulations can be executed simultaneously and iterations can be avoided, so the computing time was reduced sharply. Moreover, the accuracy of parameters of yield criterion is largely determined by the suitability of experimental measurements in the inverse method. Two commonly used methods to determine the parameters of anisotropy yield criterion are based on yield stresses and r-values, respectively. Xue and Liao11 determined the parameters of Hill’48 yield criterion by the above two methods of DC05 steel plate, the result showed that Hill’48 yield criterion determined by r-values could describe in-plane principal strain and outline of crossdie deep drawing specimen better. Wang et al.12 found the earing phenomenon of 5754O aluminum alloy can be predicted well by Hill’48 yield criterion based on r-values. The above studies showed that the parameters of yield criterion through r-values was more suitable for describing the workpiece deformation problem. Our research group13–16 has done a lot of research about the cross-sectional deformation of H96 brass DRRT in H-typed bending and E-typed bending by the FE method. The establishment of the FE models are based on Mises isotropic yield13–15 and Hill’48 anisotropy yield criterion,16 while the parameters of Hill’48 yield criterion are determined based on stress measurements. Therefore, in order to confirm the yield criterion which is suitable for predicting the cross-sectional deformation of RDB process of H96 brass DRRT, Mises isotropic yield criterion, Hill’48 and Barlat/Lian anisotropic yield criteria widely used in practical forming are selected and the parameters of Hill’48 and Barlat/Lian anisotropic yield criteria are determined based on deformation measurements in inverse method combining RSM. The prediction accuracy of the three yield criteria on the cross-sectional deformation of RDB process of DRRT was evaluated by comparing simulation results and experiment ones.
Please cite this article in press as: XIA Y et al. Cross-sectional deformation of H96 brass double-ridged rectangular tube in rotary draw bending process with different yield criteria, Chin J Aeronaut (2019), https://doi.org/10.1016/j.cja.2019.08.006
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Cross-sectional deformation of H96 brass double-ridged rectangular tube
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2. Selection of yield criterion and obtainment of the corresponding parameters
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2.1. Selection of yield criterion
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Mises, Hill’48 and Barlat/Lian yield criteria with wide application and high practicability are selected to study the bending process of the DRRT. In bending process, DRRT is usually considered to be in plane stress state due to its wall thickness is very thin. Therefore, the three yield criteria can be described as follows.
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m
gðpÞ ¼
2mr
1 r45 þ @r@fyy r45
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ð7Þ 202 203
2.2. Obtainment of parameters of anisotropy yield criteria
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where rxx and ryy are the normal stress in the rolling and transversal directions, respectively. rxy is the shear stress, r is equivalent stress, and C is a constant related to material properties.
2.2.1. The determination procedure of the parameters of anisotropic yield criterion
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Because the size of H96 DRRT is small, the parameters of the anisotropic yield criterion can’t be determined by the conventional method. So the inverse method combining the actual forming process of the workpiece is proposed. In view of the fact that uniaxial tensile test of the whole tube is simple and the measurement of the deformation process is convenient, the inverse method combining uniaxial tensile test of whole tube and RSM to determine the parameters of Hill’48 and Barlat/Lian yield criteria was proposed. In the reverse method, some responses needed to be determined to quantify deformation degree of DRRT in uniaxial tensile test. The normal anisotropy is usually characterized by the r-value of plastic strain at 15% or 20% of uniaxial tensile test.18 Therefore, the transverse strains of middle position of calibration section DW10, DW20 and DW25 (as shown in Fig. 1) are taken as the responses when the tensile deformation degree of tube are 10%, 20% and 25%, respectively, which can be measured by digital image correlation (DIC). The parameter determination of Hill’48 yield criterion is taken as an example to explain the proposed reverse method in detail. When Hill’48 yield criterion is applied in ABAQUS, the anisotropic parameters F, G, H, and N of Hill’48 yield function is determined by the yield stress ratios R11, R22, R33 and R12 embedded in the software, as shown in Eq. (8)19 in plane stress state: 8 > r 1 1 1 > þ 2 2 ¼ 12 R12 þ R12 R12 > F ¼ 2 2 > r22 r33 r11 22 33 11 > > > > > > > < G ¼ r2 21 þ 21 21 ¼ 12 R12 þ R12 R12 r33 r11 r22 33 11 22 ð8Þ > > > r 1 1 1 1 1 1 1 >H ¼ þ 2 2 ¼ 2 R2 þ R2 R2 > 2 2 > r11 r22 r33 11 22 33 > > > > > : N ¼ 3 s ¼ 32 2 r 2R
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(2) Hill’48 yield criterion Hill’48 yield criterion is expressed in plane stress state in Eq. (2), which can describe the yield behavior of orthotropic materials:
ð2Þ
where the meanings of rxx , ryy , rxy and r are the same with those in Eq. (1). F, G, H, and N are parameters that characterize material anisotropy, which can be expressed by Lankford’s coefficients r0, r45 and r90 as shown in Eq. (3). 8 F ¼ r90 ðrr00 þ1Þ > > > > < G ¼ r01þ1 ð3Þ 0 H ¼ r0rþ1 > > > > : 90 Þð1þ2r45 Þ N ¼ ðr0 þr 2r90 ðr0 þ1Þ where r0, r45 and r90 are the r-values at 0°, 45°, and 90° with respect to rolling direction in uniaxial tensile test, respectively. (3) Barlat/Lian yield criterion
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where r45 is yield stress of 45° to the rolling direction.
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where the value of p has a certain relationship with rh when a and k are known. Many studies show that rh increases with the increase of p when h ¼ 45 , as shown in Eq. (7).
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Mises yield criterion is expressed in plane stress state, as shown in Eq. (1): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ r ¼ r2xx rxx ryy þ r2yy þ 3r2xy ¼ C
2
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@f @rxx
fðrij Þ ¼ ðG þ HÞr2xx þ ðH þ FÞr2yy 2Hrxx ryy þ 2Nr2xy r ¼ 0
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a, c, k and p are material parameters, and a = 2-c, their relationship with Lankford’s coefficient is shown in Eq. (6). 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r0 r90 > 1þr < a ¼ 2 2 1þr 0 90 ð6Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > r 1þr 0 90 :k ¼ r90 1þr0
(1) Mises yield criterion
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Barlat and Lian17 proposed a three-parameter anisotropic yield criterion for solving plane stress state problems in 1989. The expression is shown in Eq. (4): m
fðrij Þ ¼ ajK1 þ K2 jm þ ajK1 K2 jm þ cj2K2 jm 2r ¼ 0 where 8 rxx þkryy > < K1 ¼ 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 rxx kryy > : K2 ¼ þ p2 r2xy 2
ð4Þ
12
ð5Þ
where m is related to the crystal structure, recommended values of the exponent are m = 6 for BCC materials and m = 8 for FCC materials. For H96 brass tube in this paper, m = 8.
12
where rij is the nonzero yield stress of rij , and s ¼ r =3. The yield stress ratios R11, R22, R33 and R12 are defined with respect to the equivalent stress r, and when the equivalent stress r is equal to r11 (the yield stress in the rolling direction), R11 is equal to 1. And R22, R33, and R12 are parameters that need to be determined. Combining the Eqs. (3) and (8), the relation-
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Fig. 1
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ship between the parameters R22, R33, and R12 and Lankford’s coefficients r0 , r45 and r90 is shown in Eq. (9): qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 1þð1=r0 Þ > > R22 ¼ 1þð1=r90 Þ > > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0Þ ð9Þ R33 ¼ ð1=r1þð1=r 0 Þþð1=r90 Þ > > q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 3½1þð1=r0 Þ :R ¼ 12 ½ð1=r0 Þþð1=r90 Þ½1þ2r45 The flow chart of determining the parameters R22, R33, and R12 of Hill’48 yield criterion of H96 brass DRRT based on inverse method combining RSM is shown in Fig. 2, which consists of the following three main steps. Firstly, the uniaxial tensile of H96 brass DRRT was conducted and three responses DW10, DW20 and DW25 were measured by DIC. Then the parameters (R22, R33, and R12) combination of Hill’48 yield criterion were created by using Box-Behnken design (BBD). Taking these parameters combination as the material data of FE simulations, the FE simulations of uniaxial tensile of H96 brass DRRT were performed and the simulation results of responses were recorded. Based on these simulation results, the mathematical models between each response DW10, DW20 and DW25 and the parameters R22, R33, and R12 were established by using RSM, and its polynomial equation is expressed in Eq. (10): Yk ¼ b0 þ
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Sketch of position of calibration section.
3 X
bm xm þ
3 X
m¼1
m¼1
bmm x2m þ
2 3 X X
bmn xm xn
ð10Þ
m¼1 n¼mþ1
where Yk represents response, that is transverse strain DW10, DW20 and DW25 of the calibration section under different tensile deformation of tube. xm and xn are influence factors, these refers to the parameters R22, R33, and R12. b0, bm, bmm and bmn are the coefficients of intercept, linear, quadratic and interactive terms, respectively. Finally, an objective function describ-
Fig. 2
ing the gap between the simulated results and the experimental values of the response was established as shown in Eq. (11):18 " RSM 2 # T 1 X Yk YEXP k ð11Þ Fobj ¼ uk T k¼1 YEXP k where T is number of responses, in this paper, T = 3. YRSM k RSM represents the response value of DWRSM and DWRSM 10 , DW20 25 calculated from mathematical models shown in Eq. (10). EXP YEXP is response of DWEXP and DWEXP measured by k 10 , DW20 25 DIC. Uk is the weight coefficient representing the relative importance of each term, as shown in Eq. (12): uk ¼ YEXP = k
T X
YEXP k
ð12Þ
k¼1
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275 276 277 278 279 280 281 282
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2.2.2. The determination of parameters of anisotropic yield criterion
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Before creating the FE models of uniaxial tensile test of DRRT with different combination of three parameters R22, R33 and R12 by using Box-Behnken Design (BBD) as shown in Fig. 2, the reasonable range of three parameters needs to be determined, firstly. According to Eq. (9), the values of the three parameters R22, R33 and R12 can be determined according to r0 , r45 and r90 . According to literature,16 Lankford’s coefficient r-value of copper alloy is basically in the range of 0.5–2. So the r-value range of H96 brass is chosen to be 0.5–2. Then the range of R22, R33 and R12 was determined according to Eq. (9), as shown in Eq. (13): 8 > < 0:70 R22 1:50 0:75 R33 1:23 ð13Þ > : 0:60 R12 1:50
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Flow chart of parameters determination of Hill’48 yield criterion by using inverse method.
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According to the BBD method, three levels are selected for each parameter. And the three levels value of each parameter is determined as shown in Table 1. Therefore, 17 FE simulations of uniaxial tensile test of DRRT need to be executed as shown in Table 2. Fig. 3 shows the specimen and FE model of uniaxial tensile test of H96 brass DRRT, and the mechanical parameters of H96 brass DRRT obtained by uniaxial tensile test were shown in Table 3. The constitutive equation in Eq. (14) is used to describe the plastic deformation of DRRT.
So the objective function is constructed by substituting Eqs. (15)–(17) into the Eq. (11). Then the value of R22, R33 and R12 can be determined by minimizing the objective function shown in Eq. (11). And the parameters of Hill’48 yield criterion can be obtained by plugging value of R22, R33 and R12 into Eq. (8), as shown in Table 4. The parameters of Barlat/Lian yield criterion can be determined by using same method, as shown in Table 4 too.
r ¼ Kðe þ bÞn
3. Influence of yield criteria on prediction accuracy of crosssectional deformation of DRRT
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3.1. Description of cross-sectional deformation and selection of characteristic section of DRRT
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Fig. 4(a) shows the section of tube before and after deformation. In order to describe the cross-sectional deformation of the DRRT during RDB process, Eqs. (18)–(21) are introduced to describe the different part deformation of characteristic sections.
339
ð14Þ
The 17 FE simulation results of the uniaxial tensile test of DRRT are also recorded in Table 2. Based on these simulation results, the mathematical models (Eq. (10)) relating three responses DW10, DW20 and DW25 to the parameters R22, R33 and R12 were established, as shown in Eqs. (15)–(17): DW10 ¼ 2:71759 34:46865 R22 þ 41:09587 R33 2:16405 R12 2:31887 R22 R33 þ 12:60032 13:90879
320
R233
þ 1:00590
R222 ð15Þ
R212
321
DW20 ¼ 5:63682 65:67314 R22 þ 74:55923 R33
DHi ¼
0:26228 R12 þ 22:30118 R222 27:87448 R233 ð16Þ
323
DW25 ¼ 4:18154 78:03289 R22 þ 93:60387 R33 0:14614 R12 1:74463 R22 R33 þ 27:03655
326
34:20049
ð18Þ
ð17Þ
R233
ð19Þ
Parameters
Level 1
Level 2
Level 3
R22 R33 R12
0.70 0.75 0.60
1.10 0.99 1.05
1.50 1.23 1.50
Table 2
330 331 332 333 334
336
338
340 341 342 343 344 346
349 350
Bi B DBi ¼ B
ð20Þ
352 353
bi b Dbi ¼ b Table 1 Three parameters R22, R33 and R12 and their respective levels.
329
347
hi h Dhi ¼ h
324
R222
Hi H H
327 328
ð21Þ
where DHi and DBi are the height and width deformation ratio of the tube, respectively, and Dhi and Dbi are the height and spacing deformation ratio of ridge groove of tube, respectively. H and B are the values of height and width of cross-section, while h and b are the values of height and spacing of ridge groove before bending deformation. Hi, Bi, hi and bi are the corresponding values after bending deformation. Meantime, in order to systematically study the effect of different yield criteria on the cross-sectional deformation of tube,
The experiment arrangement of BBD and the simulation results of response.
Number
R22
R33
R12
4W10(%)
4W20(%)
4W25(%)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
0.7 0.7 1.1 1.1 1.5 1.1 1.5 1.1 1.1 1.5 1.1 1.1 1.5 1.1 0.7 1.1 0.7
0.99 0.99 1.23 0.99 0.75 0.99 0.99 0.75 0.99 1.23 0.99 1.23 0.99 0.75 0.75 0.99 1.23
0.6 1.5 0.6 1.05 1.05 1.05 1.5 1.5 1.05 1.05 1.05 1.5 0.6 0.6 1.05 1.05 1.05
9.417 9.293 5.417 3.413 1.383 3.413 1.915 0.257 3.413 3.614 3.413 5.338 1.905 0.250 5.196 3.413 11.084
17.333 15.951 9.230 6.631 2.719 6.631 3.609 0.463 6.631 6.374 6.631 9.668 3.618 0.455 10.320 6.631 20.432
20.872 20.326 11.833 8.139 3.321 8.139 4.386 0.557 8.139 8.055 8.139 11.873 4.413 0.551 12.592 8.139 24.637
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Fig. 3
Table 3
Specimen and FE model of uniaxial tensile test of DRRT.
Mechanical parameters of H96 brass double-ridged rectangular tube.
Elasticmodulus E (GPa)
Poisson ratio m
Initial yield stress rs (MPa)
Hardening exponent n
Strength coefficient K (MPa)
Material constant b
92.82
0.324
70
0.425
588.17
0.0058
Table 4 Parameters of Hill’48 and Barlat/Lian yield criteria determined by the inverse method. Hill’48 Barlat/Lian
F 0.481
G 0.515
H 0.485
N 1.559
a 1.013
k 0.983
p 1.009
m 8
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the two characteristic sections along the bending direction are selected as shown in Fig. 4(b), which are section S2 in the mandrel support zone and section S6 in the non-mandrel support zone. And the distribution of nodes along the height and width direction on characteristic section has been given in Fig. 5(a) and (b).
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3.2. Establishment of FE model of RDB process of DRRT
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Based on the mechanical properties of the H96 brass DRRT in Table 3 and the parameters of yield criterion in Table 4, the constitutive models of different yield criteria of H96 brass DRRT are constructed and introduced in the FE model. In order to improve the computational efficiency and accuracy of the FE model, the rigid dies are modeled with 4-node 3D bilinear quadrilateral elements C3D4, meantime the tube is defined as deformable body and described by 4-node doubly curved thin shell elements S4R. And mesh sizes of tube and
Fig. 4
mandrels are all 1 mm 1 mm, the other rigid dies are 3 mm 3 mm. The friction between tube and dies is modeled with classical Coulomb friction model. Based on the above solved key technologies, the RDB process of DRRT is carried out based on ABAQUS/Explicit, the springback process of DRRT is carried out based on ABAQUS/Standard, and the established FE models considering different yield criteria are shown in Fig. 6. And the tube bender and dies for RDB process are shown in Fig. 7. The process parameters of RDB of DRRT in FE simulation and experiment are shown in Table 5.
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3.3. Deformation analysis of characteristic section of DRRT during RDB with different yield criteria
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3.3.1. Analysis of height deformation ratio of characteristic section
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Fig. 8 shows the distribution of the height deformation of characteristic sections S2 and S6 with Mises, Hill’48 and Barlat/Lian yield criteria. It can be seen that no matter which yield criterion is employed, the height deformation ratio (DHi) of characteristic section is symmetrical about the centerline 5-50 (Fig. 5(a)) and presents the variation trend of the ‘‘M” type. DHi is all negative values with the three yield criteria, which shows that the inner and outer flanges of the tube have collapsed during RDB process. The height deformation of section S6 is greater than that of section S2 for different yield criteria. DHi by using Mises yield criterion is close to that by using Hill’48 yield criterion. However, DHi by using Barlat/Lian
395
Sketch of characteristic section before and after deformation and position of characteristic section.
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Cross-sectional deformation of H96 brass double-ridged rectangular tube
Fig. 5
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Distribution of nodes on characteristic section.
Table 5
Process parameters of RDB of DRRT.
Parameters
Experiment
Simulation
Bending radius (mm) Bending angle (°) Boosting velocity (mm/s) Bending velocity (rad/s) Friction between tube and mandrel
60 90 35.21 0.5 Hydraulic oil Dry friction Dry friction 0.2
60 90 35.21 0.5 0.05
0
0
Friction between tube and clamp die Friction between tube and other dies
Fig. 6
407 408 409 410
411 412 413 414 415
FE model for RDB of DRRT.
yield criterion is quite different from DHi by using Mises and Hill’48 yield criterion, and prediction value of the height deformation of DRRT by using Barlat/Lian yield criterion is maximal, which is minimal when using Hill’48 yield criterion. 3.3.2. Analysis of height deformation ratio of ridge groove of characteristic section Distribution of the height deformation of ridge groove in sections S2 and S6 with Mises, Hill’48 and Barlat/Lian yield criteria has been shown in Fig. 9. The height deformation of ridge
Fig. 7
Clearances between tube and mandrel (mm) Clearances between tube and other dies (mm)
Rough 0.14 0.2
groove of section S6 is bigger than that of section S2 with the different yield criteria. The height deformation of ridge groove ratio (Dhi) is also systematical about the centerline 5-50 and presents the variation trend of inverse ‘‘V” type. Dhi is all negative values with these three yield criteria, which shows that the height of ridge groove produced compression deformation,
Tube bender and dies.
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Fig. 8 Distribution of the height deformation ratio of sections S2 and S6 with different yield criteria.
Fig. 9 Distribution of the height deformation ratio of ridge groove of sections S2 and S6 with different yield criteria.
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and maximum Dhi is located at nodes 10(10’) and 19(19’), namely at the opening of ridge grooves. This is because the ridge groove is not directly supported by the mandrel, and its stiffness is insufficient during bending process, the ridge groove is severely deformed by radial extrusion from the sidewall of the tube. Like the law of cross-sectional height deformation ratio, the height deformation ratio of ridge groove of characteristic section is very close with Mises and Hill’48 yield criteria, and which is biggest with Barlat/Lian yield criterion.
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Fig. 10 Distribution of the width deformation ratio of sections S2 and S6 with different yield criteria.
tion of characteristic section is located at the intersection of side wall and ridge groove. This is because the ridge groove part has been undergone severe contraction deformation during bending process, and the inner and outer flanges of ridge groove collapsed. Due to the deformation coordination effect, the sidewall of the tube is also greatly deformed. 4Bi by using Mises yield criterion is very close to that by using Hill’48 yield criterion, and prediction value of the width deformation of characteristic section by using Mises and Hill’48 yield criteria is bigger than that by using Barlat/Lian yield criterion.
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3.3.4. Analysis of spacing deformation ratio of ridge groove of characteristic section
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Distribution of the spacing deformation of ridge groove of sections S2 and S6 with different yield criteria has been shown in Fig. 11. The spacing deformation ratio of ridge groove of characteristic section (Dbi) is all negative values with three yield criteria, which shows that spacing of ridge groove produced compression deformation. The absolute value of Dbi from node 29(290 ) to node 33(330 ) decreases after an initial increase, the maximum is reached at node 30(300 ) which is more than 14%. Due to no-mandrel filling, spacing of ridge groove of tube has undergone severe deformation during bending process. Spacing deformation of ridge groove of section S2 with Barlat/Lian yield criterion are smaller than with Mises and Hill’48 yield criteria, and spacing deformation of ridge groove
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3.3.3. Analysis of width deformation ratio of characteristic section Fig. 10 shows the distribution of the width deformation of sections S2 and S6 with different yield criteria. The width deformation ratio (DBi) of characteristic section is all negative values with three yield criteria, which shows that the area also produced compression deformation, and width deformation of characteristic section in the outer side of neutral layer is larger than that in the inner side of neutral layer. The absolute value of DBi in the outer side of neutral layer from node 20(200 ) to node 28(280 ) increases after an initial decrease, the maximum deformation is located at node 28(280 ), namely the ridge of the outer flange of the ridge groove. The absolute value of DBi in the inner side of neutral layer from node 34(340 ) to node 42 (420 ) also increases after an initial decrease, the maximum is reached at node 34(340 ), namely the ridge of the inner web of the ridge groove. In conclusion, the maximum width deforma-
Fig. 11 Distribution of the spacing deformation ratio of ridge groove of sections S2 and S6 with different yield criteria.
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of section S6 with Barlat/Lian yield criterion are larger than with Mises and Hill’48 yield criteria. Regardless of section S2 or section S6, the predicted values of the spacing deformation of ridge groove are close with Mises and Hill’48 yield criteria. 3.4. Applicability evaluation of three yielding criteria for describing cross-sectional deformation of DDRT In order to evaluate the accuracy of the cross-sectional deformation of DRRT predicted by different yield criteria, Fig. 12 (a) shows the experimental result of the DRRT bent part adopting conditions shown in Table 5, and Fig. 12(b), (c) and (d) gives simulation results of the DRRT bent part by using different yield criteria under the same conditions. From Fig. 12, it can be qualitatively found that the tube bent by experiment is coincide with the tube bent by using different yield criteria. However, from this figure, it is difficult to directly judge from the intuitional shape which yield criterion is suitable for describing the cross-sectional deformation of DRRT. So Fig. 13 shows the quantitative experiment and simulation results corresponding to Fig. 12, which is crosssectional height deformation located at node 1(10 ) and crosssectional width deformation located at node 20(200 ) along the bending direction.
Fig. 12
Fig. 13
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From Fig. 13(a), it can be seen that the variation trend of cross-sectional height deformation of tube is almost same by using different yield criteria, and the predicted values by using these three yield criteria all underestimate the cross-sectional height deformation of the DRRT. The predicted values with Mises and Hill’48 yield criterion are very close, which are different from that with Barlat/Lian yield criterion, but the predicted value with Barlat/Lian yield criterion are the closest to the experimental values. The average relative errors between experiment value and simulated values of Mises, Hill’48 and Barlat/Lian yield criteria are 14%, 14.1% and 6.8%, respectively. It is visible that Barlat/Lian yield criterion can better predict the cross-sectional height deformation of the DRRT. From Fig. 13(b), it can be seen that the variation trend of cross-sectional width deformation of tube is also same by using different yield criteria. Different from the height crosssectional deformation, the predicted values by using three yield criteria all overestimate the width cross-sectional deformation of the DRRT, and the predicted values by using Barlat/Lian yield criterion are the closest to the experimental values among three yield criteria, the average relative errors is only 4.8%. However, the average relative errors between simulated values by using Mises and Hill’48 yield criteria and experiment value are 26.5% and 23.9%, respectively, and the maximum relative errors of the experiment and simulation values by using these
DRRT bent comparison of simulation results by using different yield criteria and experiment result.
Simulation results of cross-sectional deformation with three yield criteria and experiment result.
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two yield criteria reach up to 35.8% and 32.5%, respectively. It can be seen that Mises and Hill’48 yield criteria have a poor simulation precision for the cross-sectional width deformation of H96 brass DRRT, and Barlat/Lian yield criterion can better predict cross-sectional width deformation of the DRRT. From the above analysis, it can be seen that different yielding criteria have different prediction accuracy for the crosssectional deformation of DRRT, which is mainly caused by the difference of yield criterion in describing plastic anisotropic behavior and predicting the shape of yield locus.5 In order to further analyze the differences of predicting the crosssectional deformation of the DRRT by using different yield criteria, the r-value distribution and yield function shape of Mises, Hill’48 and Barlat/Lian yield criteria are given in Fig. 14. Compared with Mises yield criterion, the plastic anisotropy of the material is considered by Hill’48 yield criterion, but the difference between the functional shapes of two yield criteria is small, which leads that the prediction values of cross-sectional deformation of the DRRT are relatively close. From Fig. 14(a), it can be seen that the difference of normal anisotropy r-value of the Hill’48 and Barlat/Lian yield criteria is small in different directions, but the functional shapes of the two yield criteria are quite different (as shown in Fig. 14(b)), which leads to a larger difference in the predicted value of the cross-sectional deformation of the DRRT. Therefore, com-
Fig. 14
Fig. 15
pared with the anisotropy r-value, the difference of yield locus can be used to better explain the difference of prediction accuracy of cross-sectional deformation of DRRT. The different yielding criteria have different prediction accuracy for the cross-sectional deformation of DRRT, which also can be explained from the tangential stress distribution of the tube bent by using different yield criteria, as shown in Fig. 15. Compared with Fig. 15(a) and (b), the distribution and magnitude of tangential stress of tube is very close by using Mises and Hill’48 yield criterion. But the distribution and magnitude of tangential stress of tube by using Barlat/ Lian yield criterion as shown in Fig. 15(c) is obviously different from those as shown in Fig. 15(a) and (b). In summary, the error between the experimental value and the simulated one by using Barlat/Lian yield criterion is the smallest, which can more accurately predict the crosssectional deformation of the DRRT among three yield criteria.
546
4. Conclusion
563
(1) Mises, Hill’48 and Barlat/Lian yield criteria are selected to describe the RDB process of small-sized H96 brass DRRT, and the parameters of the Hill’48 and Barlat/ Lian anisotropic yield criteria are obtained by using
r-value distribution and functional shape of the three yield criteria.
Distribution of tangential stress of DRRT under different yield criteria.
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the inverse method combining uniaxial tensile test of whole tube and response surface methodology. On the basis of solving the key problems of model, 3D FE models of DRRT considering different yield criteria are established. (2) The variation trend of cross-sectional deformation ratio of tube is same when using different yield criteria. The deformation of each part of the characteristic section located in non-mandrel support zone is greater than that mandrel support zone. The cross-sectional deformation by using Mises yield criterion is close to that by using Hill’48 yield criterion. However, when using Barlat/Lian yield criterion, the cross-sectional deformation is greatly different. (3) The prediction values of cross-sectional height deformation by using three yield criteria all underestimate the experiment ones, and the prediction values of crosssectional width deformation overestimate the experiment ones. The prediction values of cross-sectional deformation by Barlat/Lian yield criterion agree well with the experimental ones, while those by using Mises and Hill’48 yield criteria have a larger error with the experiment ones. Acknowledgments
598
The research is supporting by the Science and Technology Project of Shenzhen of China (Nos. JCYJ20170306160003433 and JCYJ20180306171058717) and the National Natural Science Foundation of China (No. 51375392).
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Please cite this article in press as: XIA Y et al. Cross-sectional deformation of H96 brass double-ridged rectangular tube in rotary draw bending process with different yield criteria, Chin J Aeronaut (2019), https://doi.org/10.1016/j.cja.2019.08.006
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