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Investigation into wrinkling behavior of thin-walled 5A02 aluminum alloy tubes under internal and external pressure
International Journal of Mechanical Sciences 92 (2015) 245–258
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Investigation into wrinkling behavior of thin-walled 5A02 aluminum alloy tubes under internal and external pressure S.J. Yuan n, Xiao-Lei Cui, Xiao-Song Wang National Key Laboratory of Precision Hot Processing of Metals, Harbin Institute of Technology, Harbin 150001, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 26 June 2014 Received in revised form 27 November 2014 Accepted 3 December 2014 Available online 30 December 2014
In order to investigate the wrinkling behavior of thin-walled tubes under complex stress states, a dedicated experimental setup was designed and manufactured. In this experimental setup, the internal pressure, the external pressure and the axial feeding of left and right punches were coupled together through accurate closed-loop servo control. Thin-walled 5A02 aluminum alloy tubes were pushed into the die cavity under different internal pressures or under the combined actions of internal and external pressures with the same amount of axial feeding. Meanwhile, the wrinkle formation, as well as the stress state, was predicted numerically using the FEA Abaqus/Explicit solver. It has been found that the number and shape of wrinkles are strongly dependent on the internal pressure when only internal pressure is applied to the inner surface of the tube. Furthermore, the wrinkle formation is closely related to the inhomogeneous stress distribution induced by local bending and axial displacement. Moreover, the shape of wrinkles can be fitted effectively using the GaussAmp function when the internal pressures of 1.2ps , 1.6ps and 1.8ps (ps represents the initial yield pressure for the tube) are applied. In addition, the effect of external pressure has been discussed under three cases and it is shown that the wrinkling behavior exhibits hardly any dependence on the external pressure for the constant pressure difference. Finally, the formation of middle wrinkle can be prevented by higher external pressure both for the constant internal pressure and the variable internal pressure. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Wrinkling behavior Tube hydroforming Internal pressure External pressure Aluminum alloy tubes Finite element
1. Introduction
about the prediction of necking or fracture in tube hydroforming have been reported in the literatures [1–4]. On the other hand,
In the past few years, with the rapid development of the automobile and aerospace industries, tube hydroforming technology has attracted considerable attention and has become one of the irreplaceable manufacturing techniques for the fabrication of lightweight hollow-structure components. The forming of thinwalled tubes into different complex-shaped cross-sections can be performed under an appropriate combined action of internal pressure and axial feeding. However, when the combination of internal pressure and axial feeding is not optimal, three major modes of failure, i.e. bursting, localized wrinkling and global buckling, may take place in the tubes. Bursting is caused by excessive circumferential tensile stress which will lead to fracture of the tube along the longitudinal direction. Since fracture in hydroforming processes is a consequence of necking, many works
n
Corresponding author. Tel./fax: þ 86 451 86418776. E-mail addresses: [email protected] (S.J. Yuan), [email protected] (X.-L. Cui), [email protected] (X.-S. Wang). http://dx.doi.org/10.1016/j.ijmecsci.2014.12.017 0020-7403/& 2014 Elsevier Ltd. All rights reserved.
Fig. 1. True stress–strain curve of 5A02-O aluminum alloy tubes along the axial direction.
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Nomenclature σs σb δ σ ε K n σ1 σ0 m ps t d l pi pe Δp
T0 T1
yield stress ultimate tensile strength total elongation equivalent stress equivalent strain strength coefficient strain hardening exponent asymptotic stress attained after severe deformation threshold stress at which plastic deformation begins material constant in Voce equation initial yield pressure for the tube based on Tresca yield criterion, ps ¼ 2t=dσ s initial thickness of tube initial diameter of tube the length of the deformation area internal pressure external pressure pressure difference between internal pressure and external pressure, Δp ¼ pi pe
Table 1 Mechanical parameters of 5A02-O aluminum alloy tubes along the axial direction. Mechanical parameters
Value
Yield stress, σs (MPa) Ultimate tensile strength, σb (MPa) Total elongation, δ (%) Holloman equation σ ¼ Kεn
85.9 222.9 26.1 454.1 0.304 275.4 84.1 14.48
Voce equation σ ¼ σ 1 ðσ 1 σ 0 Þe mε
K (MPa) n σ1 (MPa) σ0 (MPa) m
T2 T3 α xi, yi Δaij Δrij Faxial α Fn Fy y y0, A,w
initial time of the loading path moment when the external pressure is increased to the target value moment when the axial feeding is finished end of the loading path ratio of external pressure to internal pressure, α ¼ pe =pi horizontal and vertical coordinates of the wave top, i, j¼ L, M or R axial distance between two different wave tops, i, j¼L, M or R radial distance between two different wave tops, i, j¼ L, M or R axial force applied on the tube ends tilt angle of the transition zone force acting on the left and right wrinkles by the transition zone horizontal component of the Fn GaussAmp function parameters in the GaussAmp function
either wrinkling or buckling of tubes can be caused by excessive axial compressive stress due to axial feeding [1,5,6], or by low internal pressure [7]. Wrinkle was always considered as one of the defects of tubes in the past [6,8–12]. However, Yuan et al. [13] proposed the concept that wrinkles can be controlled and used in tube hydroforming, so as to improve the formability of tubes. Moreover, Yuan et al. [14–16] conducted many experimental and numerical analyses of wrinkling behavior in tube hydroforming. The “useful” wrinkles should be formed in advance to accumulate materials in the expansion zone, then they can be flattened in the subsequent calibration stage.
Fig. 2. Experimental setup for the investigation of wrinkling behavior: (a) schematic diagram of experimental setup; (b) experimental die; and (c) shape and dimensions of die cavity.
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Fig. 3. Experimental equipment: (a) control strategy of double-sided tube hydroforming and (b) the 20 MN hydroforming press at Harbin Institute of Technology.
Therefore, the shape and dimensions of the wrinkles should be controlled according to the optimal relationship between the internal pressure and axial feeding. Up till now, the wrinkling behavior under the combined actions of internal and external pressures has never been investigated due to the difficulty of using the experimental equipment, in which the internal pressure, external pressure and two punches must be coupled together and controlled simultaneously. As far as the plastic wrinkling behavior of tubes was concerned, Kyriakides et al. [17–20] investigated the plastic wrinkles of circular tubes in detail under different kinds of external load, such as axial compression, axial compression and internal pressure, axial cycling loading and internal pressure. However, their research mainly focused on the wrinkling behavior of small plastic deformation, which varies from the wrinkling behavior in tube hydroforming with a higher wrinkle height. Tang et al. [21] investigated the
wrinkling behavior of an AZ31B magnesium alloy tube experimentally with different loading paths at different temperatures. In their research, the features of wrinkles, including shape, radius and width, as well as the thickness distribution, were obtained from the experiments. Rahmani et al. [22] introduced a new metal forming process for creating a fold with a controllable and certain wavelength in the circular metal tubes during the axial compression. In their investigation a new analytical model for predicting plastic deformation of the circular metal tubes constrained between two rigid punches subjected to quasi-static axial loading was introduced. The theoretical predictions were compared with the experimental results and showed reasonable agreement. As discussed above, the wrinkling behavior of thin-walled tubes was mainly induced by the axial compression or internal pressure/axial compression. When the internal pressure and external pressure are introduced to both inner and outer surfaces of the tube
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simultaneously, the stress state applied on the tube cannot be simplified as plane stress state, which is usually adopted in conventional single-sided tube hydroforming. It is therefore necessary to investigate experimentally whether the wrinkling behavior has a dependence on the external pressure.
In this paper, in order to conduct further research on the wrinkling behavior of thin-walled tubes under internal pressure (single-sided tube hydroforming), and to carry out a preliminary study on the comprehensive effect of internal pressure and external pressure (double-sided tube hydroforming), a dedicated experimental setup has been developed. It can be used for both singlesided tube hydroforming and double-sided tube hydroforming processes with axial feeding. The effects of internal and external pressure on the wrinkling behavior of 5A02-O aluminum alloy tubes were investigated based on the experimental and numerical results.
2. Preparation 2.1. Materials The as-received tube material used in this study is 5A02 aluminum alloy tube with outer diameter of 63 mm and nominal thickness of 2 mm, drawn from extruded tube and then annealed at 380 1C for approximately 2 h. The uniaxial tensile test was conducted on an Instron 5569 machine along the axial direction of the tube. The true stress–strain curve is shown in Fig. 1 and the mechanical parameters are listed in Table 1. The initial yield pressure ps , at which the tube begins to undergo plastic deformation, is often obtained by the Tresca yield
Fig. 4. Schematic of loading paths for (a) loading mode 1 and (b) loading mode 2.
Fig. 5. Finite element model for the investigation of wrinkling behavior.
Table 2 Loading paths for the investigation into wrinkling behavior. Factor
Loading paths
Internal pressure, pi (MPa)
External pressure, pe (MPa)
Displacement of the punches (mm) Left punch
Right punch
The effect of internal pressure
Fig. 4(a)
4.5 (0.8ps ) 5.6 (1.0ps ) 6.8 (1.2ps ) 9.0 (1.6ps ) 10.1 (1.8ps )
0 0 0 0 0
15 15 15 15 15
15 15 15 15 15
The effect of external pressure
Case 1, Fig. 4(a)
6.8 (1.2ps ) 49.3 (0.5σ s þ 1.2ps ) 91.8 (1.0σ s þ 1.2)ps 49.3 49.3 49.3 49.3 0-85 0-85 0-85 0-85
0 42.5 (0.5σ s ) 85.0 (1.0σ s ) 39.2 (49.3 1.8ps ) 40.3 (49.3 1.6ps ) 42.5 (49.3 1.2ps ) 44.8 (49.3 0.8ps ) 0.82pi 0.86pi 0.90pi 0.94pi
15 15 15 15 15 15 15 15 15 15 15
15 15 15 15 15 15 15 15 15 15 15
Case 2, Fig. 4(a)
Case 3, Fig. 4(b)
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Fig. 6. Experimental results of the wrinkles under different internal pressures: (a) pi ¼0.8ps ; (b) pi ¼1.0ps ; (c) pi ¼ 1.2ps ; (d) pi ¼1.6ps ; and (e) pi ¼ 1.8ps .
function, as shown by the following equation: ps ¼
2t σs d
ð1Þ
where t is the initial thickness of the tube, d is the initial diameter of the tube and σ s is the yield stress of the tube material. The initial yield pressure for the tube used in this experiment is 5.63 MPa according to Eq. (1).
2.2. Experimental setup A dedicated experimental setup for the investigation of wrinkling behavior was designed and manufactured, as shown in Fig. 2. To carry out an experimental investigation of the wrinkling behavior of tubes under double-sided pressures, the first difficulty of sealing the external pressure was addressed. So the experimental die used in the present research, varies from the conventional hydroforming die with a flat parting surface, has been designed into embedded structure in order to guarantee the sealing of external pressure by the polyurethane ring embedded into the space between upper and lower die. The working components of the experimental die are the upper die, lower die, left punch and right punch. Fig. 2(b) shows the experimental die, while Fig. 2(c) gives the corresponding shape and dimensions of the die cavity. The maximum diameter of the die cavity is 85 mm with a length of 63 mm. The semi-conical angle between the transition zone and the expansion zone (maximum diameter zone) of the die cavity is 201, thus a total length of 123.5 mm (l=d 2) for the deformation area can be achieved. For the thin-walled tube, a difference between internal and external pressures slightly greater than the set value (or negative), would cause bursting or crashing of the tube. As such, the second problem is the control accuracy of internal and external pressures. When the internal pressure is established, the tube is bulged. Since the volume of the external liquid is compressed, the external pressure increases with the tube bulging process. Therefore, the internal pressure is regarded as a disturbing variable for the external pressure control. In the present research, a digital control strategy was used in the double-sided pressure control, as shown in Fig. 3(a).
Fig. 7. Definition of the Cartesian coordinate on the tube.
The experimental investigations into wrinkling behavior were carried out on the 20 MN hydroforming press at Harbin Institute of Technology, as shown in Fig. 3(b). To carry out the experiment, the hydroforming press was upgraded by introducing another intensifier (intensifier B) to supply the external pressure once the external pressure was required. The upgraded 20 MN press could hence control two intensifiers and four axial cylinders simultaneously. The control system was redesigned and a new doublepressure control mode was developed. The two independent intensifiers, as well as the axial cylinders, whose signals were supplied by the control system, could be coupled together to achieve accurate closed-loop servo control. 2.3. Experimental procedure Two loading modes were used for the experimentation of this study. For loading mode 1, the internal pressure and/or the external pressure were increased to the target value by intensifier A and/or B firstly. It was ensured that the internal pressure was greater than the external pressure by a certain value during the building up of external pressure owing to the geometric feature of the tube. Then the tube ends were pushed inward for a certain distance by the left punch and right punch, while the internal pressure and/or the external pressure remain unchanged, as shown in Fig. 4(a). For loading mode 2, the internal pressure was increased linearly up to the target value, during this period the external pressure was proportional to the internal pressure and was also increased linearly along with the internal pressure. At the
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Table 3 Parameters of wrinkles under different internal pressures. Internal pressure
xL
yL
xC
yC
xR
yR
ΔaLC
ΔaCR
ΔaLR
Δr LC
Δr CR
Δr LR
0.8ps 1.0ps 1.2ps 1.6ps 1.8ps
42.18 41.17 39.36 39.22 34.66
38.57 38.88 39.44 39.83 40.74
– – 0 0 0
– – 41.90 42.30 42.32
42.18 41.17 39.58 38.74 35.26
38.46 38.75 39.35 39.52 40.61
– – 39.36 39.22 34.66
– – 39.58 38.74 35.26
84.35 82.34 78.94 77.96 69.92
– – 2.46 2.47 1.58
– – 2.55 2.78 1.71
0.11 0.13 0.09 0.31 0.13
Fig. 8. Development of wrinkles at the internal pressure of 1.2ps .
same time, the two ends of the tube were pushed into the die cavity by the left and right punch, as is shown in Fig. 4(b). The pressure difference was increased during the formation of wrinkles. Different wrinkles could be obtained through the change of the relationship between pressures and axial feeding, i.e. the loading path, as shown in Fig. 4. In the end, the internal and external pressures were unloaded linearly at the same time. In this investigation, the tube had a total length of 210 mm, and both punches were advanced to a maximum stroke of 15 mm in all experiments, i.e. a total axial feeding of 30 mm. The specific loading paths used in this study are shown in Table 2.
2.4. Finite element model To investigate the shape and dimensions of wrinkles, as well as the stress state, an axisymmetric model for the tube was developed in Abaqus/Explicit, as shown in Fig. 5. The CAX4R elements (a 4-node bilinear axisymmetric quadrilateral, reduced integration, hourglass) were used and ten elements were assigned in thickness direction of the tube with the element size of 0.2 mm. The die was defined as an analytical rigid shell. The internal pressure and external pressure were introduced to both sides of the tube at the same time according to the loading paths presented in Table 2. The Mises isotropic yield function and the elastic-plastic material model were adopted in these simulations. A constant
friction coefficients value of 0.125 was applied on the of tube/die and tube/punches interfaces.
3. Results and discussion 3.1. Effect of internal pressure on the shape and dimensions of wrinkles Fig. 6 shows the wrinkles occured at different internal pressures with a total punch stroke of 30 mm, which is approximately 80% of the ideal feeding length [13]. It can be found from Fig. 6 that the wrinkles only appear on both ends of the deformation zone of the tube when the internal pressure was lower (0.8ps ). Moreover, the two wrinkles are distorted, which could result in a nonaxisymmetric wave shape in the circumferential direction. In this case the tube may fracture before making contact with the die in the calibration stage due to the lack of sufficient materials in the middle of the tube [13]. When the internal pressure was increased to 1.0ps the wrinkles on both sides of the tube change into an axisymmetric pattern. Meanwhile, another wrinkle, called middle wrinkle, starts to develop between the two previous wrinkles, although as shown in Fig. 6(b), it does not locate at the centroid of the tube. Moreover, the middle wrinkle is semi-developed and incomplete in the circumferential direction. For the cases with higher internal pressure (1.2ps , 1.6ps and 1.8ps ), three axisymmetric wrinkles were formed. One small difference is that the
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Fig. 9. Stress distribution on the inner and outer surfaces of the tubes when the total feeding of the punches is 0 (time T1): (a) circumferential stress; (b) axial stress; and (c) Mises stress.
bigger diameter wrinkle occurs in the middle of the tube. In addition, the wrinkles became fatter with increasing internal pressure. In order to analyze the wrinkling behavior of the tube conveniently, the tubes shown in Fig. 6 was rotated 901 anticlockwise and then a Cartesian coordinate system was defined on the tube, as shown in Fig. 7. The central axis of the tube is defined as the x-axis while the radial direction is specified as the y-axis which is located in the middle of the tube. Moreover, the coordinates of the left wave top are defined as (xL, yL), the coordinates of the middle wave top are defined as (xC, yC) and the coordinates of the right wave top are defined as (xR, yR). Therefore, the axial and radial distances between two different wave tops can be expressed as
follows: Δaij ¼ jxi xj j;
i; j ¼ L; M or R
ð2Þ
Δr ij ¼ jyi yj j;
i; j ¼ L; M or R
ð3Þ
Table 3 gives the parameters of the wrinkles under different internal pressures. It can be seen in Table 3 that both the axial and radial distances between two different wave tops decrease with the increasing internal pressure. This indicates that the wrinkles become congested and the radius of the wave bottom will be increased inevitably due to the higher internal pressure.
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Fig. 10. Analysis of the development of wrinkles.
Fig. 12. Outer radius of the wrinkles along the axial direction under different internal pressures.
3.2. Stress analysis during the development of wrinkles
Fig. 11. Effective stress distribution of the tube's inner surface and outer surface when the total feeding of the punches is 15 mm: (a) inner surface and (b) outer surface.
Fig. 8 shows the development process of wrinkles when the internal pressure of 1.2ps was applied, during which two wrinkles first develop at both ends of the die cavity followed by the development of the middle wrinkle with the increasing axial feeding. The reason for this phenomenon will be discussed in detail from the aspect of stress distribution in following sections. Fig. 9 shows the stress distribution of the tube at time T1 (shown in Fig. 4(a)). Several conclusions can be drawn from Fig. 9: (a) the circumferential stress, the axial stress and the Mises stress all increased with the increase of internal pressure on both inner and outer surfaces; (b) the circumferential stress has a peak value at the transition zone of the die cavity on both inner and outer surfaces; (c) the axial stress has its lowest value at the transition zone on the inner surface but has a peak value at the transition zone on the outer surface; (d) more importantly, the Mises stress always has a peak value at the transition zone both on the inner
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where α is the semi-conical angle of the transition zone. The direction of the horizontal component, Fy, is opposite to the internal pressure. Therefore, the horizontal component has an obstructive effect on the further development of the two wrinkles. The smaller the semi-conical angle, the stronger the obstructive effect. As the axial feeding increases, the two end wrinkles are developed steadily due to the obstructive effect of the transition zone. At the same time, the middle segment L1 can be treated as a shorter rod under the combined effects of axial compression and lateral pressure loading. When the lateral pressure is sufficient to make the L1 segment more unstable than the two end wrinkles, the middle wrinkle will be developed in the L1 segment. Fig. 11 shows the Mises stress distribution along the axial direction of the tube when the total feeding of the punches is 15 mm. It can be seen that the third stress peak appears in the middle section of the two end wrinkles. Moreover, the Mises stresses on the inner surface and outer surface all increase with the increasing internal pressure. When the internal pressure is relatively low (0.8ps), it is insufficient to lead to the instability of the L1 segment in the expansion zone. Increasing internal pressure will produce higher Mises stress in the middle section of the two end wrinkles, so that the middle wrinkle occurs. From the above discussion, it can be found that the semiconical angle α of the transition zone may be crucial to the wrinkling behavior. When α is increased to a larger value or to the maximum value of 901, the development of a middle wrinkle is impossible due to the sustained development of the two end wrinkles. In addition, when the L1 segment, the expansion zone, is of sufficient length, two or more middle wrinkles may be developed in the L1 segment following the two end wrinkles. However, the investigation of the effect of angle α and length of the L1 segment is beyond the scope of this paper and will be discussed in further research. 3.3. Geometrical analysis of wrinkles wave
Fig. 13. Fitted shape of wrinkles using the GaussAmp function: (a) pi ¼1.2ps ; (b) pi ¼ 1.6ps ; and (c) pi ¼ 1.8ps .
surface and outer surface. The inhomogeneous stress distribution may be induced by the local bending effect at the border of the feed zone and transition zone. In other words, the higher Mises stress in the transition zone than in the expansion zone indicates that the transition zone will favor plastic bulking once the axial displacement is imposed after time T1. Due to the specificity of the tube's geometric configuration, a small axisymmetric segment was selected to analysis the instability of the tube under the axial feeding, as shown in Fig. 10. The small axisymmetric segment could be regarded as a rod (L0) that is bearing a lateral uniform pressure loading, and there are two stress peaks located at the transition zone due to the local bending effect. When the axial force, Faxial, is imposed on the two ends, the rod L0 will deflect excessively at the two stress peaks and as a result two local wrinkles will firstly be produced at the transition zone. After the wrinkles make contact with the conical surface of the transition zone, the conical surface must has vertical force Fn on the wrinkles. The horizontal component of the vertical force can be expressed as below: F y ¼ F n cos α
ð4Þ
Fig. 12 shows the outer radius distribution of the wrinkles along the axial direction for internal pressures of 1.2ps , 1.6ps and 1.8ps . In previous work, the shape of a single wrinkle wave was described by the sine curve in order to characterize the wave shape and calculate the feeding amount [6]. However, the sine curve cannot be used to describe the whole wave shape including the wave top and wave bottom. It can be seen in Fig. 12 that the wave shape, especially for the middle wrinkle, is more likely to be fitted using the GaussAmp function, which is shown as follows: y ¼ y0 þ Ae ððx xi Þ
2
=2w2 Þ
ð5Þ
where y0 can be used to represent the radius of the wave bottom, xi is the axial coordinate of the wave top, A is the amplitude and can be used to represent the height of the wrinkle, and w is a parameter that is related to the full width at half maximum (FWHM). Fig. 13 shows the fitted shape of wrinkles using Eq. (5) for internal pressures of 1.2ps , 1.6ps and 1.8ps . It can be clearly seen in Fig. 13 that the middle wrinkle can be fitted perfectly using the GaussAmp function regardless of the imposed internal pressure. On the other hand, the fitting accuracy for the left wrinkle and right wrinkle is reduced with the increasing internal pressure. Therefore, it is significant that the fitted functions for the wrinkles can be used to predict the shape and dimensions of the wrinkles, the relative positions between two wrinkles and the effect of internal pressure on the shape and dimensions of the wrinkles. For example, when the internal pressure is 1.2ps , as shown in Fig. 13(a), the radius of wave bottom of the middle
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Fig. 14. Effect of external pressure (pe) on wrinkling behavior in the case of constant pressure difference (Δp ¼ 6.8 MPa): (a) pe ¼ 0 MPa; (b) pe ¼ 42.5 MPa; and (c) pe ¼85.0 MPa.
Table 4 Effect of external pressure on the radius and distance of wrinkle waves in case 1. External pressure (MPa) Radius of the wave top (mm)
yL pe ¼ 0 pe ¼ 42.5 pe ¼ 85.0
yC
yR
Distance between the wave tops (mm)
Radius of the wave bottom (mm)
ΔaLC
ΔaCR
Left
Right
39.58 39.31 38.93
32.54 33.55 33.39
32.51 33.49 33.56
39.44 41.90 39.35 39.36 39.61 42.34 39.31 39.15 39.63 42.47 39.89 39.39
wrinkle is 32.454 mm, the radius of its wave top is 41.994 mm, the distance between the left wave top and middle wrinkle top is 38.298 mm, and the distance between the right wave top and middle wrinkle top is 38.533 mm. Furthermore, the distance between the left wrinkle top and right wrinkle top decreases with the increasing internal pressure. On the other hand, the radius of the wave bottom increases with the increasing internal pressure.
tube. Fig. 14 shows the wrinkled tubes under different external pressures. The radius of the wave top and wave bottom, as well as the distance between the two wave tops under different external pressures, were all measured and are listed in Table 4. It can be seen from Fig. 14 and Table 4 that the external pressure has no obvious effect on the wrinkling behavior of the tube. The slight difference between the wrinkles under different external pressures is likely to be caused by the control accuracy of the pressure sensor and the inevitable variations in the mechanical properties between the tubes. Fig. 15 shows the stress distribution on the outer surface of the tube at time T1 (shown in Fig. 4(a)), while Fig. 16 shows the stress distribution when the total feeding of the punches is 15 mm. It is obvious that the circumferential stress and axial stress are both reduced by the higher external pressure, and that the effective stress has hardly any dependence on the external pressure. However, the trend of the stress distribution along the axial direction is the same regardless of the magnitude of external pressure applied. Therefore, the introduction of external pressure can only change the stress state of the tube, but cannot change the development sequence of wrinkles and their shape.
3.4. Effect of external pressure on wrinkling behavior The external pressure was introduced to the tube outside simultaneously accompanied by the internal pressure when the tube was pushed into the die cavity, in order to investigate the wrinkling behavior under the combined effects of internal and external pressure. When this issue is concerned, the effect of external pressure is discussed from the following three cases in this paper. 3.4.1. Case 1: constant pressure difference In this case, the pressure difference between internal and external pressure was kept constant during the wrinkle development process. In addition, the pressure difference was the same regardless of the magnitude of external pressure applied on the
3.4.2. Case 2: constant internal pressure In this case, the internal pressure remained constant with different levels of external pressure imposed on the tubes. Therefore, the pressure difference decreased with the increasing external pressure. Fig. 17 shows the wrinkled tubes under different external pressures with the constant internal pressure of 49.3 MPa applied in this case. It can be found from Fig. 17 that the wrinkling behavior has the opposite tendency to Fig. 6 with the increase of external pressure. When the external pressure increases from 39.2 MPa to 44.8 MPa, the wrinkles evolve from three thick axisymmetric wrinkles to three slender axisymmetric wrinkles, and then to two end non-axisymmetric wrinkles. It is demonstrated in Section 3.2 that the lateral pressure should be sufficient to make the L1 segment more unstable than the two end wrinkles,
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Fig. 15. Effect of external pressure on the stress distribution along the outer surface of the tube when the total feeding of the punches is 0 (time T1): (a) circumferential stress; (b) axial stress; and (c) Mises stress.
Fig. 16. Effect of external pressure on the stress distribution along the outer surface of the tube when the total feeding of the punches is 15 mm: (a) circumferential stress; (b) axial stress; and (c) Mises stress.
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Fig. 17. Effect of external pressure (pe) on the wrinkling behavior in the case of constant internal pressure (pi ¼ 49.3 MPa): (a) pe ¼ 39.2 MPa; (b) pe ¼40.3 MPa; (c) pe ¼42.5 MPa; and (d) pe ¼44.8 MPa.
Table 5 Effect of external pressure on the radius and distance of wrinkle waves in case 2. External pressure (MPa)
pe ¼ 39.2 pe ¼ 40.3 pe ¼ 42.5 pe ¼ 44.8
Radius of the wave top (mm)
Distance between the wave tops (mm)
Radius of the wave bottom (mm)
yL
yC
yR
ΔaLC
ΔaCR
Left
Right
41.42 40.25 39.61 38.27
42.36 42.50 42.34 –
41.45 40.10 39.31 37.21
33.40 36.71 39.15 –
32.92 37.66 39.31 –
34.25 33.49 33.55 –
34.29 33.43 33.49 –
Fig. 18. Effect of external pressure (pe) on the wrinkling behavior in the case of variable internal pressure (0-85 MPa): (a) pe ¼ 0.82pi; (b) pe ¼0.86pi; (c) pe ¼ 0.90pi; (d) pe ¼0.94pi.
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so that the L1 segment could develop into the middle wrinkle. Similarly, in case 2, the lateral pressure, i.e. the pressure difference, decreased with the increasing external pressure. When the external pressure is large enough, the pressure difference would be too small for the development of the middle wrinkle. In other words, higher external pressure may prevent the development of a middle wrinkle. Likewise, the effect of external pressure on the parameters of the wrinkles, such as the radius of the wave top and wave bottom and the distance between them in case 2, is listed in Table 5. It is demonstrated that the radius of the wave top decreases with the increasing external pressure. At the same time, the distance between the wave tops increases.
3.4.3. Case 3: variable internal pressure In this case, the internal pressure and the external pressure were increased linearly according to a proportion relationship, during which the two ends of the tube were pushed into the die cavity. Moreover, the pressure difference is a variable during the development process of wrinkles. Fig. 18 illustrates the wrinkled tubes under different proportions of external pressure with the fixed internal pressure of 0-85 MPa in this case. When the external pressurepe ¼ 0:82pi , three axisymmetric wrinkles can be formed where one of the wave bottoms has bulged due to the high pressure difference (Δp ¼ 15.3 MPa) at the end of the loading path, as is shown in Fig. 18(a). For the ratio of external pressure to internal pressure of 0.86, two middle wrinkles appear between the two end axisymmetric wrinkles. One of the middle wrinkles develops preferentially into a complete wrinkle whereas another is in the initial state, as shown in Fig. 18(b). If the external pressure increases to 0:90pi only one incomplete middle wrinkle is formed near to one end wrinkle (see Fig. 18(c)). Moreover, for the case of pe ¼ 0:94pi (Fig. 18(d)), only two end wrinkles can be obtained and the folding phenomenon can be found at the position of the end wrinkles, which is possibly due to insufficient supporting pressure. It can be seen clearly from the above discussion and Fig. 18 that a higher ratio of external pressure to internal pressure could prevent the formation of middle wrinkles, which is in agreement with case 2. However, the development of wrinkles between the two cases is very different. In case 2, the three wrinkles are all axisymmetric when the external pressures are 39.2 MPa, 40.3 MPa and 42.5 MPa. On the other hand, all the middle wrinkles are asymmetrical and incomplete in case 3.
4. Conclusions In the present work, a new double-sided tube hydroforming setup was developed. With this experimental setup, the investigation into the wrinkling behavior of thin-walled tubes under the combined action of internal and external pressures was realized. The thin-walled 5A02 aluminum alloy tubes were tested using this experimental setup and their wrinkling behavior were analyzed from the aspects of experiment and numerical simulation. Conclusions can be drawn from the above discussion as follows: (1) When the internal pressure is applied just to the inner surface of the tube, the number and shape of wrinkles have a strong dependence on the internal pressure. The wrinkles appear only in the transition zone when the internal pressure is lower at 0.8ps . However, three axisymmetric wrinkles will be formed when the internal pressure is higher at 1.2ps , 1.6ps and 1.8ps .
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(2) The wrinkle formation is closely related to the inhomogeneous stress distribution along the axial direction, which has two peaks for the effective stress in the transition zone as the internal pressure is increased to the target value. (3) When the internal pressures are 1.2ps , 1.6ps and 1.8ps , the shape of the middle wrinkles can be fitted perfectly using the GaussAmp function. On the other hand, the fitting accuracy for the left wrinkle and right wrinkle is reduced with increasing internal pressure. (4) The effect of external pressure on the wrinkling behavior was discussed as the tube was pushed into the die cavity under the combined action of internal pressure and external pressure. It is shown that the wrinkling behavior has hardly any dependence on the external pressure for the constant pressure difference, which is attributed to the similar distribution trend of stresses regardless of applied external pressure. In contrast, the formation of a middle wrinkle could be prevented by higher external pressure both for the constant internal pressure and the variable internal pressure. However, the configuration of wrinkles exhibits large differences between these two cases.
Acknowledgments This study was financially supported by the Program for Changjiang Scholars and Innovative Research Team in University (No. IRT1229), the National Natural Science Foundation of China (No. 51175111) and the Fundamental Research Funds for the Central Universities (HIT.NSRIF.201134). The authors wish to express their gratitude for these funding.
References [1] Xing HL, Makinouchi A. Numerical analysis and design for tubular hydroforming. Int J Mech Sci 2001;43(4):1009–26. [2] Kim J, Kim SW, Song WJ, Kang BS. Analytical approach to bursting in tube hydroforming using diffuse plastic instability. Int J Mech Sci 2005;47 (7):1023–37. [3] Hashemi R, Assempour A, Masoumi E. Implementation of the forming limit stress diagram to obtain suitable load path in tube hydroforming. Mater Des 2009;30(9):3545–53. [4] Hashemi R, Abrinia K, Assempour A. The strain gradient approach to predict necking in tube hydroforming. J Manuf Processes 2013;15(1):51–5. [5] Asnafi N. Analytical modelling of tube hydroforming. Thin Walled Struct 1999;34:295–330. [6] Chu E, Xu Y. Hydroforming of aluminum extrusion tubes for automotive applications. Part I: buckling, wrinkling and bursting analyses of aluminum tubes. Int J Mech Sci 2004;46:263–83. [7] Wang Y, Han C, Yuan SJ. Effect of internal pressure on corner radius and thickness distribution of shear hydro-bending of 5A02 aluminum alloy tube. Trans Nonferr Metal Soc 2012;22:S376–81. [8] Abrantes JP, de Lima CEC, Batalha GF. Numerical simulation of an aluminum alloy tube hydroforming. J Mater Process Technol 2006;179:67–73. [9] Kim S, Kim Y. Analytical study for tube hydroforming. J Mater Process Technol 2002;128:232–9. [10] Nefussi G, Combescure A. Coupled buckling and plastic instability for tube hydroforming. Int J Mech Sci 2002;44:899–914. [11] Strano M, Jirathearanat S, Altan T. Adaptive FEM simulation for tube hydroforming: a geometry-based approach for wrinkle detection. CIRP Ann – Manuf Technol 2001;50:185–90. [12] Kami A, Dariani BM. Prediction of wrinkling in thin-walled tube push-bending process using artificial neural network and finite element method. Proc Inst Mech Eng B – J Eng 2011;225:1801–12. [13] Yuan S, Wang X, Liu G, Wang ZR. Control and use of wrinkles in tube hydroforming. J Mater Process Technol 2007;182:6–11. [14] Lang L, Yuan S, Wang X, Wang ZR, Fu Z, Danckert J, et al. A study on numerical simulation of hydroforming of aluminum alloy tube. J Mater Process Technol 2004;146:377–88. [15] Lang L, Li H, Yuan S, Danckert J, Nielsen KB. Investigation into the preforming's effect during multi-stages of tube hydroforming of aluminum alloy tube by using useful wrinkles. J Mater Process Technol 2009;209:2553–63.
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[16] Yuan S, Yuan W, Wang X. Effect of wrinkling behavior on formability and thickness distribution in tube hydroforming. J Mater Process Technol 2006;177:668–71. [17] Paquette JA, Kyriakides S. Plastic buckling of tubes under axial compression and internal pressure. Int J Mech Sci 2006;48:855–67. [18] Bardi FC, Kyriakides S. Plastic buckling of circular tubes under axial compression – Part I: experiments. Int J Mech Sci 2006;48:830–41. [19] Bardi FC, Kyriakides S, Yun HD. Plastic buckling of circular tubes under axial compression – Part II: analysis. Int J Mech Sci 2006;48:842–54.
[20] Jiao R, Kyriakides S. Ratcheting, wrinkling and collapse of tubes under axial cycling. Int J Solids Struct 2009;46:2856–70. [21] Tang Z, Liu G, He Z, Yuan S. Wrinkling behavior of magnesium alloy tube in warm hydroforming. Trans Nonferr Metal Soc 2010;20:1288–93. [22] Rahmani DM, Niknejad A. Introducing a new cold metal forming method for producing a fold in the circular tubes with a certain wavelength. Proc Inst Mech Eng B – J Eng 2014. http://dx.doi.org/10.1177/0954405414542114.
Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Investigation into wrinkling behavior of thin-walled 5A02 aluminum alloy tubes under internal and external pressure S.J. Yuan n, Xiao-Lei Cui, Xiao-Song Wang National Key Laboratory of Precision Hot Processing of Metals, Harbin Institute of Technology, Harbin 150001, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 26 June 2014 Received in revised form 27 November 2014 Accepted 3 December 2014 Available online 30 December 2014
In order to investigate the wrinkling behavior of thin-walled tubes under complex stress states, a dedicated experimental setup was designed and manufactured. In this experimental setup, the internal pressure, the external pressure and the axial feeding of left and right punches were coupled together through accurate closed-loop servo control. Thin-walled 5A02 aluminum alloy tubes were pushed into the die cavity under different internal pressures or under the combined actions of internal and external pressures with the same amount of axial feeding. Meanwhile, the wrinkle formation, as well as the stress state, was predicted numerically using the FEA Abaqus/Explicit solver. It has been found that the number and shape of wrinkles are strongly dependent on the internal pressure when only internal pressure is applied to the inner surface of the tube. Furthermore, the wrinkle formation is closely related to the inhomogeneous stress distribution induced by local bending and axial displacement. Moreover, the shape of wrinkles can be fitted effectively using the GaussAmp function when the internal pressures of 1.2ps , 1.6ps and 1.8ps (ps represents the initial yield pressure for the tube) are applied. In addition, the effect of external pressure has been discussed under three cases and it is shown that the wrinkling behavior exhibits hardly any dependence on the external pressure for the constant pressure difference. Finally, the formation of middle wrinkle can be prevented by higher external pressure both for the constant internal pressure and the variable internal pressure. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Wrinkling behavior Tube hydroforming Internal pressure External pressure Aluminum alloy tubes Finite element
1. Introduction
about the prediction of necking or fracture in tube hydroforming have been reported in the literatures [1–4]. On the other hand,
In the past few years, with the rapid development of the automobile and aerospace industries, tube hydroforming technology has attracted considerable attention and has become one of the irreplaceable manufacturing techniques for the fabrication of lightweight hollow-structure components. The forming of thinwalled tubes into different complex-shaped cross-sections can be performed under an appropriate combined action of internal pressure and axial feeding. However, when the combination of internal pressure and axial feeding is not optimal, three major modes of failure, i.e. bursting, localized wrinkling and global buckling, may take place in the tubes. Bursting is caused by excessive circumferential tensile stress which will lead to fracture of the tube along the longitudinal direction. Since fracture in hydroforming processes is a consequence of necking, many works
n
Corresponding author. Tel./fax: þ 86 451 86418776. E-mail addresses: [email protected] (S.J. Yuan), [email protected] (X.-L. Cui), [email protected] (X.-S. Wang). http://dx.doi.org/10.1016/j.ijmecsci.2014.12.017 0020-7403/& 2014 Elsevier Ltd. All rights reserved.
Fig. 1. True stress–strain curve of 5A02-O aluminum alloy tubes along the axial direction.
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Nomenclature σs σb δ σ ε K n σ1 σ0 m ps t d l pi pe Δp
T0 T1
yield stress ultimate tensile strength total elongation equivalent stress equivalent strain strength coefficient strain hardening exponent asymptotic stress attained after severe deformation threshold stress at which plastic deformation begins material constant in Voce equation initial yield pressure for the tube based on Tresca yield criterion, ps ¼ 2t=dσ s initial thickness of tube initial diameter of tube the length of the deformation area internal pressure external pressure pressure difference between internal pressure and external pressure, Δp ¼ pi pe
Table 1 Mechanical parameters of 5A02-O aluminum alloy tubes along the axial direction. Mechanical parameters
Value
Yield stress, σs (MPa) Ultimate tensile strength, σb (MPa) Total elongation, δ (%) Holloman equation σ ¼ Kεn
85.9 222.9 26.1 454.1 0.304 275.4 84.1 14.48
Voce equation σ ¼ σ 1 ðσ 1 σ 0 Þe mε
K (MPa) n σ1 (MPa) σ0 (MPa) m
T2 T3 α xi, yi Δaij Δrij Faxial α Fn Fy y y0, A,w
initial time of the loading path moment when the external pressure is increased to the target value moment when the axial feeding is finished end of the loading path ratio of external pressure to internal pressure, α ¼ pe =pi horizontal and vertical coordinates of the wave top, i, j¼ L, M or R axial distance between two different wave tops, i, j¼L, M or R radial distance between two different wave tops, i, j¼ L, M or R axial force applied on the tube ends tilt angle of the transition zone force acting on the left and right wrinkles by the transition zone horizontal component of the Fn GaussAmp function parameters in the GaussAmp function
either wrinkling or buckling of tubes can be caused by excessive axial compressive stress due to axial feeding [1,5,6], or by low internal pressure [7]. Wrinkle was always considered as one of the defects of tubes in the past [6,8–12]. However, Yuan et al. [13] proposed the concept that wrinkles can be controlled and used in tube hydroforming, so as to improve the formability of tubes. Moreover, Yuan et al. [14–16] conducted many experimental and numerical analyses of wrinkling behavior in tube hydroforming. The “useful” wrinkles should be formed in advance to accumulate materials in the expansion zone, then they can be flattened in the subsequent calibration stage.
Fig. 2. Experimental setup for the investigation of wrinkling behavior: (a) schematic diagram of experimental setup; (b) experimental die; and (c) shape and dimensions of die cavity.
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Fig. 3. Experimental equipment: (a) control strategy of double-sided tube hydroforming and (b) the 20 MN hydroforming press at Harbin Institute of Technology.
Therefore, the shape and dimensions of the wrinkles should be controlled according to the optimal relationship between the internal pressure and axial feeding. Up till now, the wrinkling behavior under the combined actions of internal and external pressures has never been investigated due to the difficulty of using the experimental equipment, in which the internal pressure, external pressure and two punches must be coupled together and controlled simultaneously. As far as the plastic wrinkling behavior of tubes was concerned, Kyriakides et al. [17–20] investigated the plastic wrinkles of circular tubes in detail under different kinds of external load, such as axial compression, axial compression and internal pressure, axial cycling loading and internal pressure. However, their research mainly focused on the wrinkling behavior of small plastic deformation, which varies from the wrinkling behavior in tube hydroforming with a higher wrinkle height. Tang et al. [21] investigated the
wrinkling behavior of an AZ31B magnesium alloy tube experimentally with different loading paths at different temperatures. In their research, the features of wrinkles, including shape, radius and width, as well as the thickness distribution, were obtained from the experiments. Rahmani et al. [22] introduced a new metal forming process for creating a fold with a controllable and certain wavelength in the circular metal tubes during the axial compression. In their investigation a new analytical model for predicting plastic deformation of the circular metal tubes constrained between two rigid punches subjected to quasi-static axial loading was introduced. The theoretical predictions were compared with the experimental results and showed reasonable agreement. As discussed above, the wrinkling behavior of thin-walled tubes was mainly induced by the axial compression or internal pressure/axial compression. When the internal pressure and external pressure are introduced to both inner and outer surfaces of the tube
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simultaneously, the stress state applied on the tube cannot be simplified as plane stress state, which is usually adopted in conventional single-sided tube hydroforming. It is therefore necessary to investigate experimentally whether the wrinkling behavior has a dependence on the external pressure.
In this paper, in order to conduct further research on the wrinkling behavior of thin-walled tubes under internal pressure (single-sided tube hydroforming), and to carry out a preliminary study on the comprehensive effect of internal pressure and external pressure (double-sided tube hydroforming), a dedicated experimental setup has been developed. It can be used for both singlesided tube hydroforming and double-sided tube hydroforming processes with axial feeding. The effects of internal and external pressure on the wrinkling behavior of 5A02-O aluminum alloy tubes were investigated based on the experimental and numerical results.
2. Preparation 2.1. Materials The as-received tube material used in this study is 5A02 aluminum alloy tube with outer diameter of 63 mm and nominal thickness of 2 mm, drawn from extruded tube and then annealed at 380 1C for approximately 2 h. The uniaxial tensile test was conducted on an Instron 5569 machine along the axial direction of the tube. The true stress–strain curve is shown in Fig. 1 and the mechanical parameters are listed in Table 1. The initial yield pressure ps , at which the tube begins to undergo plastic deformation, is often obtained by the Tresca yield
Fig. 4. Schematic of loading paths for (a) loading mode 1 and (b) loading mode 2.
Fig. 5. Finite element model for the investigation of wrinkling behavior.
Table 2 Loading paths for the investigation into wrinkling behavior. Factor
Loading paths
Internal pressure, pi (MPa)
External pressure, pe (MPa)
Displacement of the punches (mm) Left punch
Right punch
The effect of internal pressure
Fig. 4(a)
4.5 (0.8ps ) 5.6 (1.0ps ) 6.8 (1.2ps ) 9.0 (1.6ps ) 10.1 (1.8ps )
0 0 0 0 0
15 15 15 15 15
15 15 15 15 15
The effect of external pressure
Case 1, Fig. 4(a)
6.8 (1.2ps ) 49.3 (0.5σ s þ 1.2ps ) 91.8 (1.0σ s þ 1.2)ps 49.3 49.3 49.3 49.3 0-85 0-85 0-85 0-85
0 42.5 (0.5σ s ) 85.0 (1.0σ s ) 39.2 (49.3 1.8ps ) 40.3 (49.3 1.6ps ) 42.5 (49.3 1.2ps ) 44.8 (49.3 0.8ps ) 0.82pi 0.86pi 0.90pi 0.94pi
15 15 15 15 15 15 15 15 15 15 15
15 15 15 15 15 15 15 15 15 15 15
Case 2, Fig. 4(a)
Case 3, Fig. 4(b)
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Fig. 6. Experimental results of the wrinkles under different internal pressures: (a) pi ¼0.8ps ; (b) pi ¼1.0ps ; (c) pi ¼ 1.2ps ; (d) pi ¼1.6ps ; and (e) pi ¼ 1.8ps .
function, as shown by the following equation: ps ¼
2t σs d
ð1Þ
where t is the initial thickness of the tube, d is the initial diameter of the tube and σ s is the yield stress of the tube material. The initial yield pressure for the tube used in this experiment is 5.63 MPa according to Eq. (1).
2.2. Experimental setup A dedicated experimental setup for the investigation of wrinkling behavior was designed and manufactured, as shown in Fig. 2. To carry out an experimental investigation of the wrinkling behavior of tubes under double-sided pressures, the first difficulty of sealing the external pressure was addressed. So the experimental die used in the present research, varies from the conventional hydroforming die with a flat parting surface, has been designed into embedded structure in order to guarantee the sealing of external pressure by the polyurethane ring embedded into the space between upper and lower die. The working components of the experimental die are the upper die, lower die, left punch and right punch. Fig. 2(b) shows the experimental die, while Fig. 2(c) gives the corresponding shape and dimensions of the die cavity. The maximum diameter of the die cavity is 85 mm with a length of 63 mm. The semi-conical angle between the transition zone and the expansion zone (maximum diameter zone) of the die cavity is 201, thus a total length of 123.5 mm (l=d 2) for the deformation area can be achieved. For the thin-walled tube, a difference between internal and external pressures slightly greater than the set value (or negative), would cause bursting or crashing of the tube. As such, the second problem is the control accuracy of internal and external pressures. When the internal pressure is established, the tube is bulged. Since the volume of the external liquid is compressed, the external pressure increases with the tube bulging process. Therefore, the internal pressure is regarded as a disturbing variable for the external pressure control. In the present research, a digital control strategy was used in the double-sided pressure control, as shown in Fig. 3(a).
Fig. 7. Definition of the Cartesian coordinate on the tube.
The experimental investigations into wrinkling behavior were carried out on the 20 MN hydroforming press at Harbin Institute of Technology, as shown in Fig. 3(b). To carry out the experiment, the hydroforming press was upgraded by introducing another intensifier (intensifier B) to supply the external pressure once the external pressure was required. The upgraded 20 MN press could hence control two intensifiers and four axial cylinders simultaneously. The control system was redesigned and a new doublepressure control mode was developed. The two independent intensifiers, as well as the axial cylinders, whose signals were supplied by the control system, could be coupled together to achieve accurate closed-loop servo control. 2.3. Experimental procedure Two loading modes were used for the experimentation of this study. For loading mode 1, the internal pressure and/or the external pressure were increased to the target value by intensifier A and/or B firstly. It was ensured that the internal pressure was greater than the external pressure by a certain value during the building up of external pressure owing to the geometric feature of the tube. Then the tube ends were pushed inward for a certain distance by the left punch and right punch, while the internal pressure and/or the external pressure remain unchanged, as shown in Fig. 4(a). For loading mode 2, the internal pressure was increased linearly up to the target value, during this period the external pressure was proportional to the internal pressure and was also increased linearly along with the internal pressure. At the
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Table 3 Parameters of wrinkles under different internal pressures. Internal pressure
xL
yL
xC
yC
xR
yR
ΔaLC
ΔaCR
ΔaLR
Δr LC
Δr CR
Δr LR
0.8ps 1.0ps 1.2ps 1.6ps 1.8ps
42.18 41.17 39.36 39.22 34.66
38.57 38.88 39.44 39.83 40.74
– – 0 0 0
– – 41.90 42.30 42.32
42.18 41.17 39.58 38.74 35.26
38.46 38.75 39.35 39.52 40.61
– – 39.36 39.22 34.66
– – 39.58 38.74 35.26
84.35 82.34 78.94 77.96 69.92
– – 2.46 2.47 1.58
– – 2.55 2.78 1.71
0.11 0.13 0.09 0.31 0.13
Fig. 8. Development of wrinkles at the internal pressure of 1.2ps .
same time, the two ends of the tube were pushed into the die cavity by the left and right punch, as is shown in Fig. 4(b). The pressure difference was increased during the formation of wrinkles. Different wrinkles could be obtained through the change of the relationship between pressures and axial feeding, i.e. the loading path, as shown in Fig. 4. In the end, the internal and external pressures were unloaded linearly at the same time. In this investigation, the tube had a total length of 210 mm, and both punches were advanced to a maximum stroke of 15 mm in all experiments, i.e. a total axial feeding of 30 mm. The specific loading paths used in this study are shown in Table 2.
2.4. Finite element model To investigate the shape and dimensions of wrinkles, as well as the stress state, an axisymmetric model for the tube was developed in Abaqus/Explicit, as shown in Fig. 5. The CAX4R elements (a 4-node bilinear axisymmetric quadrilateral, reduced integration, hourglass) were used and ten elements were assigned in thickness direction of the tube with the element size of 0.2 mm. The die was defined as an analytical rigid shell. The internal pressure and external pressure were introduced to both sides of the tube at the same time according to the loading paths presented in Table 2. The Mises isotropic yield function and the elastic-plastic material model were adopted in these simulations. A constant
friction coefficients value of 0.125 was applied on the of tube/die and tube/punches interfaces.
3. Results and discussion 3.1. Effect of internal pressure on the shape and dimensions of wrinkles Fig. 6 shows the wrinkles occured at different internal pressures with a total punch stroke of 30 mm, which is approximately 80% of the ideal feeding length [13]. It can be found from Fig. 6 that the wrinkles only appear on both ends of the deformation zone of the tube when the internal pressure was lower (0.8ps ). Moreover, the two wrinkles are distorted, which could result in a nonaxisymmetric wave shape in the circumferential direction. In this case the tube may fracture before making contact with the die in the calibration stage due to the lack of sufficient materials in the middle of the tube [13]. When the internal pressure was increased to 1.0ps the wrinkles on both sides of the tube change into an axisymmetric pattern. Meanwhile, another wrinkle, called middle wrinkle, starts to develop between the two previous wrinkles, although as shown in Fig. 6(b), it does not locate at the centroid of the tube. Moreover, the middle wrinkle is semi-developed and incomplete in the circumferential direction. For the cases with higher internal pressure (1.2ps , 1.6ps and 1.8ps ), three axisymmetric wrinkles were formed. One small difference is that the
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Fig. 9. Stress distribution on the inner and outer surfaces of the tubes when the total feeding of the punches is 0 (time T1): (a) circumferential stress; (b) axial stress; and (c) Mises stress.
bigger diameter wrinkle occurs in the middle of the tube. In addition, the wrinkles became fatter with increasing internal pressure. In order to analyze the wrinkling behavior of the tube conveniently, the tubes shown in Fig. 6 was rotated 901 anticlockwise and then a Cartesian coordinate system was defined on the tube, as shown in Fig. 7. The central axis of the tube is defined as the x-axis while the radial direction is specified as the y-axis which is located in the middle of the tube. Moreover, the coordinates of the left wave top are defined as (xL, yL), the coordinates of the middle wave top are defined as (xC, yC) and the coordinates of the right wave top are defined as (xR, yR). Therefore, the axial and radial distances between two different wave tops can be expressed as
follows: Δaij ¼ jxi xj j;
i; j ¼ L; M or R
ð2Þ
Δr ij ¼ jyi yj j;
i; j ¼ L; M or R
ð3Þ
Table 3 gives the parameters of the wrinkles under different internal pressures. It can be seen in Table 3 that both the axial and radial distances between two different wave tops decrease with the increasing internal pressure. This indicates that the wrinkles become congested and the radius of the wave bottom will be increased inevitably due to the higher internal pressure.
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Fig. 10. Analysis of the development of wrinkles.
Fig. 12. Outer radius of the wrinkles along the axial direction under different internal pressures.
3.2. Stress analysis during the development of wrinkles
Fig. 11. Effective stress distribution of the tube's inner surface and outer surface when the total feeding of the punches is 15 mm: (a) inner surface and (b) outer surface.
Fig. 8 shows the development process of wrinkles when the internal pressure of 1.2ps was applied, during which two wrinkles first develop at both ends of the die cavity followed by the development of the middle wrinkle with the increasing axial feeding. The reason for this phenomenon will be discussed in detail from the aspect of stress distribution in following sections. Fig. 9 shows the stress distribution of the tube at time T1 (shown in Fig. 4(a)). Several conclusions can be drawn from Fig. 9: (a) the circumferential stress, the axial stress and the Mises stress all increased with the increase of internal pressure on both inner and outer surfaces; (b) the circumferential stress has a peak value at the transition zone of the die cavity on both inner and outer surfaces; (c) the axial stress has its lowest value at the transition zone on the inner surface but has a peak value at the transition zone on the outer surface; (d) more importantly, the Mises stress always has a peak value at the transition zone both on the inner
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where α is the semi-conical angle of the transition zone. The direction of the horizontal component, Fy, is opposite to the internal pressure. Therefore, the horizontal component has an obstructive effect on the further development of the two wrinkles. The smaller the semi-conical angle, the stronger the obstructive effect. As the axial feeding increases, the two end wrinkles are developed steadily due to the obstructive effect of the transition zone. At the same time, the middle segment L1 can be treated as a shorter rod under the combined effects of axial compression and lateral pressure loading. When the lateral pressure is sufficient to make the L1 segment more unstable than the two end wrinkles, the middle wrinkle will be developed in the L1 segment. Fig. 11 shows the Mises stress distribution along the axial direction of the tube when the total feeding of the punches is 15 mm. It can be seen that the third stress peak appears in the middle section of the two end wrinkles. Moreover, the Mises stresses on the inner surface and outer surface all increase with the increasing internal pressure. When the internal pressure is relatively low (0.8ps), it is insufficient to lead to the instability of the L1 segment in the expansion zone. Increasing internal pressure will produce higher Mises stress in the middle section of the two end wrinkles, so that the middle wrinkle occurs. From the above discussion, it can be found that the semiconical angle α of the transition zone may be crucial to the wrinkling behavior. When α is increased to a larger value or to the maximum value of 901, the development of a middle wrinkle is impossible due to the sustained development of the two end wrinkles. In addition, when the L1 segment, the expansion zone, is of sufficient length, two or more middle wrinkles may be developed in the L1 segment following the two end wrinkles. However, the investigation of the effect of angle α and length of the L1 segment is beyond the scope of this paper and will be discussed in further research. 3.3. Geometrical analysis of wrinkles wave
Fig. 13. Fitted shape of wrinkles using the GaussAmp function: (a) pi ¼1.2ps ; (b) pi ¼ 1.6ps ; and (c) pi ¼ 1.8ps .
surface and outer surface. The inhomogeneous stress distribution may be induced by the local bending effect at the border of the feed zone and transition zone. In other words, the higher Mises stress in the transition zone than in the expansion zone indicates that the transition zone will favor plastic bulking once the axial displacement is imposed after time T1. Due to the specificity of the tube's geometric configuration, a small axisymmetric segment was selected to analysis the instability of the tube under the axial feeding, as shown in Fig. 10. The small axisymmetric segment could be regarded as a rod (L0) that is bearing a lateral uniform pressure loading, and there are two stress peaks located at the transition zone due to the local bending effect. When the axial force, Faxial, is imposed on the two ends, the rod L0 will deflect excessively at the two stress peaks and as a result two local wrinkles will firstly be produced at the transition zone. After the wrinkles make contact with the conical surface of the transition zone, the conical surface must has vertical force Fn on the wrinkles. The horizontal component of the vertical force can be expressed as below: F y ¼ F n cos α
ð4Þ
Fig. 12 shows the outer radius distribution of the wrinkles along the axial direction for internal pressures of 1.2ps , 1.6ps and 1.8ps . In previous work, the shape of a single wrinkle wave was described by the sine curve in order to characterize the wave shape and calculate the feeding amount [6]. However, the sine curve cannot be used to describe the whole wave shape including the wave top and wave bottom. It can be seen in Fig. 12 that the wave shape, especially for the middle wrinkle, is more likely to be fitted using the GaussAmp function, which is shown as follows: y ¼ y0 þ Ae ððx xi Þ
2
=2w2 Þ
ð5Þ
where y0 can be used to represent the radius of the wave bottom, xi is the axial coordinate of the wave top, A is the amplitude and can be used to represent the height of the wrinkle, and w is a parameter that is related to the full width at half maximum (FWHM). Fig. 13 shows the fitted shape of wrinkles using Eq. (5) for internal pressures of 1.2ps , 1.6ps and 1.8ps . It can be clearly seen in Fig. 13 that the middle wrinkle can be fitted perfectly using the GaussAmp function regardless of the imposed internal pressure. On the other hand, the fitting accuracy for the left wrinkle and right wrinkle is reduced with the increasing internal pressure. Therefore, it is significant that the fitted functions for the wrinkles can be used to predict the shape and dimensions of the wrinkles, the relative positions between two wrinkles and the effect of internal pressure on the shape and dimensions of the wrinkles. For example, when the internal pressure is 1.2ps , as shown in Fig. 13(a), the radius of wave bottom of the middle
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Fig. 14. Effect of external pressure (pe) on wrinkling behavior in the case of constant pressure difference (Δp ¼ 6.8 MPa): (a) pe ¼ 0 MPa; (b) pe ¼ 42.5 MPa; and (c) pe ¼85.0 MPa.
Table 4 Effect of external pressure on the radius and distance of wrinkle waves in case 1. External pressure (MPa) Radius of the wave top (mm)
yL pe ¼ 0 pe ¼ 42.5 pe ¼ 85.0
yC
yR
Distance between the wave tops (mm)
Radius of the wave bottom (mm)
ΔaLC
ΔaCR
Left
Right
39.58 39.31 38.93
32.54 33.55 33.39
32.51 33.49 33.56
39.44 41.90 39.35 39.36 39.61 42.34 39.31 39.15 39.63 42.47 39.89 39.39
wrinkle is 32.454 mm, the radius of its wave top is 41.994 mm, the distance between the left wave top and middle wrinkle top is 38.298 mm, and the distance between the right wave top and middle wrinkle top is 38.533 mm. Furthermore, the distance between the left wrinkle top and right wrinkle top decreases with the increasing internal pressure. On the other hand, the radius of the wave bottom increases with the increasing internal pressure.
tube. Fig. 14 shows the wrinkled tubes under different external pressures. The radius of the wave top and wave bottom, as well as the distance between the two wave tops under different external pressures, were all measured and are listed in Table 4. It can be seen from Fig. 14 and Table 4 that the external pressure has no obvious effect on the wrinkling behavior of the tube. The slight difference between the wrinkles under different external pressures is likely to be caused by the control accuracy of the pressure sensor and the inevitable variations in the mechanical properties between the tubes. Fig. 15 shows the stress distribution on the outer surface of the tube at time T1 (shown in Fig. 4(a)), while Fig. 16 shows the stress distribution when the total feeding of the punches is 15 mm. It is obvious that the circumferential stress and axial stress are both reduced by the higher external pressure, and that the effective stress has hardly any dependence on the external pressure. However, the trend of the stress distribution along the axial direction is the same regardless of the magnitude of external pressure applied. Therefore, the introduction of external pressure can only change the stress state of the tube, but cannot change the development sequence of wrinkles and their shape.
3.4. Effect of external pressure on wrinkling behavior The external pressure was introduced to the tube outside simultaneously accompanied by the internal pressure when the tube was pushed into the die cavity, in order to investigate the wrinkling behavior under the combined effects of internal and external pressure. When this issue is concerned, the effect of external pressure is discussed from the following three cases in this paper. 3.4.1. Case 1: constant pressure difference In this case, the pressure difference between internal and external pressure was kept constant during the wrinkle development process. In addition, the pressure difference was the same regardless of the magnitude of external pressure applied on the
3.4.2. Case 2: constant internal pressure In this case, the internal pressure remained constant with different levels of external pressure imposed on the tubes. Therefore, the pressure difference decreased with the increasing external pressure. Fig. 17 shows the wrinkled tubes under different external pressures with the constant internal pressure of 49.3 MPa applied in this case. It can be found from Fig. 17 that the wrinkling behavior has the opposite tendency to Fig. 6 with the increase of external pressure. When the external pressure increases from 39.2 MPa to 44.8 MPa, the wrinkles evolve from three thick axisymmetric wrinkles to three slender axisymmetric wrinkles, and then to two end non-axisymmetric wrinkles. It is demonstrated in Section 3.2 that the lateral pressure should be sufficient to make the L1 segment more unstable than the two end wrinkles,
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Fig. 15. Effect of external pressure on the stress distribution along the outer surface of the tube when the total feeding of the punches is 0 (time T1): (a) circumferential stress; (b) axial stress; and (c) Mises stress.
Fig. 16. Effect of external pressure on the stress distribution along the outer surface of the tube when the total feeding of the punches is 15 mm: (a) circumferential stress; (b) axial stress; and (c) Mises stress.
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Fig. 17. Effect of external pressure (pe) on the wrinkling behavior in the case of constant internal pressure (pi ¼ 49.3 MPa): (a) pe ¼ 39.2 MPa; (b) pe ¼40.3 MPa; (c) pe ¼42.5 MPa; and (d) pe ¼44.8 MPa.
Table 5 Effect of external pressure on the radius and distance of wrinkle waves in case 2. External pressure (MPa)
pe ¼ 39.2 pe ¼ 40.3 pe ¼ 42.5 pe ¼ 44.8
Radius of the wave top (mm)
Distance between the wave tops (mm)
Radius of the wave bottom (mm)
yL
yC
yR
ΔaLC
ΔaCR
Left
Right
41.42 40.25 39.61 38.27
42.36 42.50 42.34 –
41.45 40.10 39.31 37.21
33.40 36.71 39.15 –
32.92 37.66 39.31 –
34.25 33.49 33.55 –
34.29 33.43 33.49 –
Fig. 18. Effect of external pressure (pe) on the wrinkling behavior in the case of variable internal pressure (0-85 MPa): (a) pe ¼ 0.82pi; (b) pe ¼0.86pi; (c) pe ¼ 0.90pi; (d) pe ¼0.94pi.
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so that the L1 segment could develop into the middle wrinkle. Similarly, in case 2, the lateral pressure, i.e. the pressure difference, decreased with the increasing external pressure. When the external pressure is large enough, the pressure difference would be too small for the development of the middle wrinkle. In other words, higher external pressure may prevent the development of a middle wrinkle. Likewise, the effect of external pressure on the parameters of the wrinkles, such as the radius of the wave top and wave bottom and the distance between them in case 2, is listed in Table 5. It is demonstrated that the radius of the wave top decreases with the increasing external pressure. At the same time, the distance between the wave tops increases.
3.4.3. Case 3: variable internal pressure In this case, the internal pressure and the external pressure were increased linearly according to a proportion relationship, during which the two ends of the tube were pushed into the die cavity. Moreover, the pressure difference is a variable during the development process of wrinkles. Fig. 18 illustrates the wrinkled tubes under different proportions of external pressure with the fixed internal pressure of 0-85 MPa in this case. When the external pressurepe ¼ 0:82pi , three axisymmetric wrinkles can be formed where one of the wave bottoms has bulged due to the high pressure difference (Δp ¼ 15.3 MPa) at the end of the loading path, as is shown in Fig. 18(a). For the ratio of external pressure to internal pressure of 0.86, two middle wrinkles appear between the two end axisymmetric wrinkles. One of the middle wrinkles develops preferentially into a complete wrinkle whereas another is in the initial state, as shown in Fig. 18(b). If the external pressure increases to 0:90pi only one incomplete middle wrinkle is formed near to one end wrinkle (see Fig. 18(c)). Moreover, for the case of pe ¼ 0:94pi (Fig. 18(d)), only two end wrinkles can be obtained and the folding phenomenon can be found at the position of the end wrinkles, which is possibly due to insufficient supporting pressure. It can be seen clearly from the above discussion and Fig. 18 that a higher ratio of external pressure to internal pressure could prevent the formation of middle wrinkles, which is in agreement with case 2. However, the development of wrinkles between the two cases is very different. In case 2, the three wrinkles are all axisymmetric when the external pressures are 39.2 MPa, 40.3 MPa and 42.5 MPa. On the other hand, all the middle wrinkles are asymmetrical and incomplete in case 3.
4. Conclusions In the present work, a new double-sided tube hydroforming setup was developed. With this experimental setup, the investigation into the wrinkling behavior of thin-walled tubes under the combined action of internal and external pressures was realized. The thin-walled 5A02 aluminum alloy tubes were tested using this experimental setup and their wrinkling behavior were analyzed from the aspects of experiment and numerical simulation. Conclusions can be drawn from the above discussion as follows: (1) When the internal pressure is applied just to the inner surface of the tube, the number and shape of wrinkles have a strong dependence on the internal pressure. The wrinkles appear only in the transition zone when the internal pressure is lower at 0.8ps . However, three axisymmetric wrinkles will be formed when the internal pressure is higher at 1.2ps , 1.6ps and 1.8ps .
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(2) The wrinkle formation is closely related to the inhomogeneous stress distribution along the axial direction, which has two peaks for the effective stress in the transition zone as the internal pressure is increased to the target value. (3) When the internal pressures are 1.2ps , 1.6ps and 1.8ps , the shape of the middle wrinkles can be fitted perfectly using the GaussAmp function. On the other hand, the fitting accuracy for the left wrinkle and right wrinkle is reduced with increasing internal pressure. (4) The effect of external pressure on the wrinkling behavior was discussed as the tube was pushed into the die cavity under the combined action of internal pressure and external pressure. It is shown that the wrinkling behavior has hardly any dependence on the external pressure for the constant pressure difference, which is attributed to the similar distribution trend of stresses regardless of applied external pressure. In contrast, the formation of a middle wrinkle could be prevented by higher external pressure both for the constant internal pressure and the variable internal pressure. However, the configuration of wrinkles exhibits large differences between these two cases.
Acknowledgments This study was financially supported by the Program for Changjiang Scholars and Innovative Research Team in University (No. IRT1229), the National Natural Science Foundation of China (No. 51175111) and the Fundamental Research Funds for the Central Universities (HIT.NSRIF.201134). The authors wish to express their gratitude for these funding.
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