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Hydroforming of aluminum extrusion tubes for automotive applications. Part II: process window diagram
International Journal of Mechanical Sciences 46 (2004) 285 – 297
Hydroforming of aluminum extrusion tubes for automotive applications. Part II: process window diagram E. Chu∗ , Yu Xu Product Manufacturing Division, Alcoa Inc., Alcoa Technical Center, Alcoa Center, PA 15069-0001, USA Received 22 November 2002; received in revised form 7 January 2004; accepted 4 February 2004
Abstract Based on the mathematical formulations for predicting forming limits induced by buckling, wrinkling and bursting of free-expansion tube hydroforming, a theoretical “Process Window Diagram” (PWD) is proposed and established in this paper. The theory developed in the ;rst part of the present work was formulated within the context of free-expansion tube hydroforming with both combined internal pressure and end feeding. The PWD is designed to provide a quick assessment of part producibility for tube hydroforming. The predicted PWD is validated against experimental results conducted for 6260-T4 60 × 2 × 320 (mm) aluminum tubes. An optimal loading path is also proposed in the PWD with an attempt to de;ne the ideal forming process for aluminum tube hydroforming. Parametric studies show that the PWD has a strong dependency on tube geometry, material property and process parameters. To the authors’ knowledge, this is the ;rst attempt that a PWD is being formulated theoretically. Such a concept can be advantageous in deriving design solutions and determining optimal process parameters for tube hydroforming processes. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Tube hydroforming; Forming limits; Wrinkling; Buckling; Bursting; Process Window Diagram (PWD); Aluminum tubes
1. Introduction Hydroforming of tubular components presents an excellent opportunity for lightweight manufacturing of structural automotive components. As the hydroforming process becomes more widely used for automotive applications, issues regarding process control, design Aexibility and material utilizations must be addressed in order to provide adequate supports for implementing the technology in the automotive industry. In tube hydroforming, a critical aspect during the process design stage is to accurately determine of the trajectory of internal pressure versus end feeding sequence. Too much pressure will cause bursting ∗
Corresponding author.
0020-7403/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2004.02.013
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while too much end feeding will cause buckling or wrinkling. It is apparent that a diagram showing a safe working zone within the plane of certain process parameters is desirable for application engineers. Some investigators have discussed the implications of such a concept from an experimental perspective [1,2], however, according to the authors’ knowledge, there has not been any noticeable eIort to formulate this kind of diagram theoretically. In view of the need to provide a better analytical tool for Alcoa engineers to quickly assess the appropriate formability limits for tube hydroforming and to advance the use of aluminum extrusions in structural applications, an attempt is made herein to theoretically formulate the “Process Window” within which an aluminum tube can operate safely without failure under a free-expansion hydroforming process. Theoretically, the “Process Window” should be bounded by the forming limits (failure limits), imposed by the previously discussed aforementioned failure modes such as buckling, wrinkling and bursting [3]. To establish such a “Process Window”, the onset condition for each failure mode must be ;rst formulated theoretically. In the ;rst part of the present work [3], two distinct modes of failures in tube hydroforming have been identi;ed: (1) failures due to either global buckling or local axisymmetric wrinkling, and (2) bursting of tubes due to localized necking of the material. A unique mathematical foundation for bifurcation analysis of tube hydroforming under combined internal pressure and end feeding has also been established for predicting forming limits due to buckling and wrinkling in the free-expansion tube hydroforming [3]. Numerical results revealed that: local wrinkling is the dominant mode for short tubes while global buckling occurs for long slender tubes; the onset of asymmetric wrinkling always requires a higher critical axial compressive stress than the axisymmetric one and hence this mode can be ignored in the hydroforming analysis. Closed-form analytical expressions for the onset condition of bursting and for maximum internal pressure have been derived for predicting bursting failure in the free-expansion tube hydroforming [3]. The conditions of incipient failure established in the ;rst part of the present work provide us a complete set of theoretical tools to construct the “Process Window” after which application engineers have been seeking for many years. The development of a general Process Window Diagram (PWD) for the free-expansion tube hydroforming will be discussed within the context of these theoretical tools. Subsequently, the PWD diagram will be validated with experimental measurements provided by the scientists at the Battelle Paci;c Northwest National Laboratory (PNNL) [4]. An optimal load path for tube hydroforming is then proposed using the PWD. Finally, the inAuence of geometric parameters and the material properties on the PWD is investigated. The failure mode of folding, which usually occurs in the intake zone at the end of process [1], is not considered in the present study. Additionally, the sealing limit generally represented by a straight line on the plane of the internal pressure versus the axial compressive force diagram [1,2], will also be omitted in the present study. Other investigations have already documented these ;ndings and will not be repeated here. However, the expression for the sealing limit will be included in this paper for the sake of completeness. 2. The theoretical PWD As discussed previously any of the three possible modes such as buckling, wrinkling and bursting, could potentially cause failures in the free-expansion tube hydroforming. The onset conditions of
E. Chu, Y. Xu / International Journal of Mechanical Sciences 46 (2004) 285 – 297
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these failure modes have already been established theoretically in the ;rst part of the present work [3]. These onset conditions can be plotted either as forming limit (failure limit) curves in the strain space (axial strain versus circumferential strain plot), or on the stress space (axial stress versus circumferential stress plot), or in the load space consisting of the internal pressure and the axial compressive force. Any of these failure limit plots presented on diIerent space can be used to describe the PWD for tube hydroforming process. The process parameters such as internal pressure and axial compression (or end-feeding) are commonly used to control the tube hydroforming process in practice [1,2]. Hence, it is logical to establish the “Process Window” in the load space for practical purposes. The PWD, bounded by the theoretical buckling, wrinkling and bursting forming limits, is dependent on the material property, tube geometry and loading conditions. As such, the PWD represents the incipient failure conditions or the forming limits for a given aluminum tube. The theoretical PWD will encompass the failure limits under which a tube can operate safely in hydroforming operations. The validity of such a diagram will be demonstrated in comparison with experimental measurements conducted by PNNL scientists. The development of a general “Process Window” or formability window for this process requires the bifurcation theory developed in Part I of the study. For the sake of completeness, the equations that will be used for the current purpose are repeated herein without further explanation and the interest reader is referred to the ;rst part of the present work [3] for more details of the mathematical formulations.
2.1. The onset of incipient buckling and wrinkling of a tube The condition for the onset of global buckling of a tube is given by ∗
cr = L1111
2 2
r0 ‘0
2
e2(22 −11 )
(1)
and the condition for the onset of axisymmetric wrinkling of the tube in hydroforming is written as ∗
cr = L1111
(m) L 2 12
t0 ‘0
2
e
−2(22 +211 )
+ L2222
1 m L
2
‘0 r0
2
e2(11 −22 ) :
(2)
To make it self-explanatory, a brief description of the symbols used is given here. A cylindrical coordinate system is used with subscripts 1, 2 and 3 referred to the axial, circumferential and ra∗ is the critical axial compressive stress, i.e. ∗ = − . dial directions, respectively. cr 11 11 and cr 22 are axial stress and circumferential stress, respectively. 11 and 22 are axial and circumferential strain. L1111 and L2222 are components of the plane stress moduli, which are functions of 11 and 22 . mL is the number of wrinkle waves in the axial direction of the tube. ‘0 , t0 , and r0 are the initial length, thickness and radius (measured from the middle surface) of the tube, respectively. In the free-expansion tube hydroforming, a tube can either buckle as a column, wrinkles axisymmetrically or wrinkles asymmetrically [3]. The mode that requires the least amount of axial
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compressive load is considered as the actual occurrence of the bifurcation phenomenon for the tube. Similarly, the wave number corresponding to the least amount of axial compressive load is considered as the actual wave number. It has been demonstrated that the axisymmetric wrinkling mode is the weaker mode, which generally requires a smaller compressive load than the asymmetric wrinkling mode [3]. Hence, the asymmetric wrinkling mode is excluded in constructing the PWD for the free-expansion tube hydroforming. 2.2. The onset of incipient bursting of a tube The onset condition of bursting can be expressed as [3] 11 + 222 = n;
(3)
where n is the strain-hardening index of the tube material. In deriving the above equations, the tube is assumed to obey a piecewise power law and undergoes a proportional stress path. Eqs. (1)–(3) provide a general framework for constructing the forming limit curves or the PWD. The detailed procedures for such a construction have been discussed in the ;rst part of the present work [3] for an aluminum tube obeying a speci;c yield criterion. The forming limits established previously in the stress or strain space, could easily be recast into the load space in terms of the internal pressure p and the critical axial compressive force, Fx =2r0 t0 , via the following equations: t0 p = 22 e(33 −22 ) ; (4) r0 and
Fx 22 − 11 e(22 +33 ) : = 2r0 t0 2
(5)
Theoretically, the current mathematical framework can accommodate any yield criteria to be used for predicting the forming limit curves. Note that it is extremely diNcult to correctly characterize anisotropy of a circular tube along the circumferential and 45 degree directions. With this constraint in mind, certain simpli;cations are adopted herein in constructing the PWD. Hill’s 1948 yield criterion seems to better serve the purpose for demonstrating the validity of this work and hence will be used to model the tube material. The components of the plane stress moduli, such as L1111 and L2222 , will be calculated based on the deformation theory of plasticity [3]. The tube material is assumed to obey the piecewise power law equation of the form 1=n L L L L for L ¡ y in the elastic range; = for L ¿ y and = (6) y y y where y and y are the uniaxial yield stress and yield strain, respectively. As mentioned in the introduction, sealing of tubes is also a requirement in tube hydroforming. The axial compressive force required to seal a tube with the internal pressure is approximately expressed as Fx = pr02 [1], which is further simpli;ed to a straight line in the PWD as Fx r0 = p: (7) 2r0 t0 2t0
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Numerical procedures have been developed and implemented for generating the operating window for a given aluminum tube, in order to demonstrate the applicability of the concept. This theoretical PWD provides a means for the design and process engineers to determine the forming limits of a given tube a priori before hydroforming. A few investigators [1,2] have discussed the implications of such a concept from an experimental perspective or some rules of thumb, however, to the authors’ knowledge, this is the ;rst time that anyone ever attempt to formulate the “Process Window” theoretically.
3. Validation of proposed PWD Scientists at PNNL conducted the experimental study of the free-expansion tube hydroforming process, which is under a Cooperative Research and Development Agreement (CRADA) between Alcoa and PNNL. A special system, which provides the simultaneous application of internal pressure and axial load, was developed at PNNL and the apparatus [4], as shown in Fig. 1, was used to perform the experiment. The system was ;t with a 222 kN load cell and unique grips to support both axial and internal pressurization. The load frame’s actuator controlled the axial position to a maximum travel of 152 mm. The specimen grips consisted of a locking device, an internal mandrel, and seal assembly that is typically used for hydrostatic testing of tubular products. The internal pressurization system consisted of an air driven water pump that was capable of achieving a pressure of 69 MPa. A digital servohydraulic controller directed the axial load and actuator position, internal pressure and collected data during experiments. Additionally, the controller can be programmed to apply various ratios of internal pressure to axial load to simulate actual hydroforming control. Both the axial position and internal pressurization systems consisted
Fig. 1. Photograph illustrating the apparatus developed at the Battelle PNNL for the free-expansion hydroforming study [4].
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of servohydraulic control valves that maintained testing conditions throughout the duration of the tests. The focus of this section is the comparison between the predicted PWD and the experimental measurements conducted for round 6260-T4 structural extruded tubes with a 60 mm outside diameter and a 2 mm wall thickness. The overall length of the tubes tested was 460 mm. The unsupported length of the tube between grips was 320 mm, with 70 mm inserted into each grip. The aluminum tubes were subjected to a series of hydroforming experiments for determining the strains that can be developed under various proportional loading conditions. During the experiments, selected ratios of axial load and internal pressure were maintained proportionally and Fig. 2 illustrates a sampling of the testing population used in the hydroforming experiments [4]. In this ;gure, the samples are arranged in the order of increasing axial compression from top to bottom, such that the bottom sample received the most axial compressive load compared to other samples in the same ;gure. Generally, increasing axial compression tends to increase the achievable circumferential expansion. If excessive axial compression is imposed on the tube way beyond its critical axial compressive load, buckling or wrinkling begins to occur. This is observable from Fig. 2, which shows increasing expansion with increasing axial compression until the axial load becomes too excessive that wrinkling of the tube sets in. It also indicates that the modes of failure will change with respect to the applied amount of axial compression as well as internal pressure used in hydroforming. Fig. 3 shows trajectories of the internal pressure versus the axial displacement curves for various proportional loading conditions during hydroforming. It can be seen that the failure modes change
Fig. 2. Samples of hydroformed tubes showing fracture and wrinkling under various proportional loading conditions [4].
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291
Fig. 3. Diagram showing the trajectories of the pressure load versus the axial displacement for various proportional loading conditions during hydroforming. Fracture and wrinkling of tubes are clearly visible (Courtesy of PNNL).
from bursting to wrinkling at increasing axial compression at high internal pressures [4]. At relative low internal pressures, axisymmetric wrinkling of the tube as end-feeding increases mainly controls failure. To measure strain localization during forming, square grids were electrochemically etched onto the tubes before forming. The grids surrounding the failure region of the specimen were then measured using a grid strain analysis system. The system determined the magnitude of the diIused and localized failure strains that developed around a failure site. Strains determined from grids that were within the necked region of the failure site were identi;ed as localized or unsafe strains, while those that were measured away from the fracture region were identi;ed as diIused or safe strains. A note of caution is worth mentioning here, since the determination of the diIused and localized strains can be extremely subjective in nature. A small error in selecting the right grids to measure can result in a relatively large error in failure strain. Furthermore, strain grids were measured after bursting of the tube already happened, and the abrupt burst opening of the tube could potentially induced some additional strains around the failure regions, and curvature of the tube was also not accounted for in the measurements, thus introduce further errors. Fig. 4 provides a comparison between experimental measurements conducted by PNNL and the predicted “Process Window” in the stress space, which is bounded by predicted buckling, wrinkling and bursting (fracture) limits. The mechanical property of the 6260-T4 aluminum tube used for the experiment is summarized in Table 1. The measured bursting limits as well as the buckling and wrinkling limits for 6260-T4 aluminum tubes under various proportional loading conditions were plotted in the ;gure. It shows that the overall predicted results compared very well with experimental
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Fig. 4. The circumferential stress is plotted as a function of the axial stress for 6260-T4 60 × 2 × 320 (mm) aluminum tubes. Both wrinkling and fracture limits generated from experiments are compared with theoretical predictions.
Table 1 Mechanical properties of 6260-T4 60 × 2 × 320 (mm) aluminum tubes Material
r-value L
n-value
K-value (MPa)
E (MPa)
Yield strength (MPa)
6260-T4
1.0a
0.26
397
62 000
67.3b
a Not measured due to diNculty in getting right measurements for the tube. Generally, r-value L is less than unity for aluminum and results in smaller fracture limits, but it has a small eIect on buckling/wrinkling. b This is calculated based on the intersection between the elastic portion and the plastic portion represented by the power law.
;ndings. Particularly, the theoretical predictions correlated exceptional well with experimental data in both trends and magnitudes for wrinkling bifurcation. Historically, it has been diNcult to correlate experimental results with theoretical predictions correctly. Any slight imperfection that exists in a tube could potentially aIect the onset of buckle/wrinkle stress. Therefore, the comparison here has given us con;dence in our theoretical model. It was also con;rmed from both experiment and predictions that axisymmetric wrinkling was a weaker mode which required a lower axial compressive load than the asymmetric one. In order to determine the bursting limits from experiments, the average data points for each loading path were extracted from the forming limit curve that developed by PNNL scientists. The average data that represented localization of the tubes were plotted along with the theoretical predictions in the ;gure. Generally, the agreement between predicted and experimental results compared very well.
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293
4. Determination of an optimal path in PWD Fig. 4 shows the predicted “Process Window” in the stress space for 6260-T4 60 × 2 × 320 (mm) aluminum tubes. Now, Fig. 4 can be recast in the loading space commonly used in the hydroforming process as shown in Fig. 5. In the ;gure, the critical axial compressive force, Fx =2r0 t0 [3], though plotted positively is considered negative in magnitude, and is given as a function of the internal pressure applied to the tube. Such a PWD provides a useful means for guiding future development of tube hydroforming processes. Finding an optimal loading path within the PWD shown in Fig. 5 is of practical interest to the industry. It is therefore necessary to brieAy discuss how to apply such a PWD for industrial applications. To begin, one must de;ne an objective function that is required to achieve the “optimal path” for the forming process. Consider the PWD given in the strain space as shown in Fig. 6(a), an objective function must be assumed in order to de;ne a path for the tube to achieve a maximum expansion ratio, r=r0 . It is understood that an in;nite number of paths can be used to describe this deformation; some can be complex while others are simple, and some may even be diNcult to implement practically. Hence, in an eIort to avoid complexity in implementing an objective function for equipment control, the simplest possible function is assumed in this work—a proportional loading path. Under this assumption, the loading path is a straight line originated from the origin in the strain space by keeping the ratio between the axial strain and the circumferential strain proportional at all time. Without invoking any types of failure models, the maximum circumferential strain, ‘n(r=r0 ), could be attained which is represented by the abscissa of point A shown in the ;gure. Therefore, the straight line from the origin to point A de;nes the optimal loading path plotted in the strain space. This optimal path can be recast in terms of the critical axial compressive force, Fx =2r0 t0 , and the internal pressure, p, as illustrated in Fig. 6(b). Such an optimal path plotted in the load
Fig. 5. Illustrating an example of the PWD that governs the theoretical buckling/wrinkling and bursting limits of the free-expansion tube hydroforming process.
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Fig. 6. (a) Diagram showing an example of utilizing the PWD for designing a desirable optimal loading path (OA) for tube hydroforming process. (b) Illustrating an optimal path (OA) de;ned within the PWD in the load space.
space can be very useful for designers or forming engineers to develop an optimal loading trajectory for controlling the pressure and load relationship in the hydroforming equipment. This path allows an engineer to control the equipment to maximize part performance without excessive wrinkling and bursting of the tube. The authors are well aware of the possibility that using a piece-wise linear combination of strain paths might enable the process to attain a larger expansion ratio for the tube hydroforming process. However, such a curved loading path will result in shifting the boundary of the Process Window very much like shifting of a kinematic hardening yield surface. The path dependency of the PWD is beyond the scope of the present study, which will not be discussed in this paper.
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5. Material and geometrical e!ects on PWD The eIects of material properties on the variations of the PWD are presented in Figs. 7 and 8. The strain-hardening index, n, ranging from 0.06 to 0.26, and the material plastic anisotropy, r, L ranging from 0.5 to 1.5 were used to generate these ;gures. It is apparent from the ;gures that both material
Fig. 7. Showing the eIect of material plastic anisotropy on the variation of the PWD for the ratios of r0 =‘0 = 0:05 and t0 =r0 = 0:04.
Fig. 8. Showing the eIect of strain-hardening index on the variation of the PWD for the ratios of r0 =‘0 = 0:05 and t0 =r0 = 0:04.
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Fig. 9. Illustrating eIects of wall thicknesses on the variation of the PWD indicating thicker tubes sustain higher formability for r0 =‘0 = 0:1.
parameters have more eIects on bursting failure than on buckling/wrinkling bifurcation. It is also evident from the ;gures that the strain-hardening index has a much more signi;cant inAuence on both bifurcation and bursting (fracture) failure limits of the tube hydroforming process than the plastic anisotropy of the material, which has only a small inAuence. Nevertheless, the “Process Window” expands with the strain-hardening index and plastic anisotropy of the tube material. Fig. 9 shows the eIect of metal thickness on tube hydroforming. It indicated that thicker tubes have higher buckling or wrinkling resistance as well as higher bursting failure limits in tube hydroforming. This is also consistent with practical applications observed in the industry. However, the theory showed that metal thickness seems to bene;t the localization much more than the bifurcation limits. 6. Conclusions Based on the theoretical formulations for predicting buckling, wrinkling and bursting in the ;rst part of the present work [3], a PWD is proposed and established in this paper. The PWD provides a theoretical assessment of part producibility a priori during the product and process design stages. It describes the forming limits imposed by buckling, wrinkling and bursting of a tube undergoing the free-expansion tube hydroforming operation. The predicted PWD correlates very well with experimental measurements conducted by PNNL. The theory was able to predict strain localization and its trends to within statistically acceptable limits. The PWD, similar in nature to the well-known Forming Limit Diagram (FLD) commonly used in the sheet metal forming processes, can be a powerful tool in providing insight in assessing part feasibility for tool designers or process engineers during the initial design stage. Unlike the FLD, the PWD is plotted in a load space. The PWD can also be used to provide some general forming guidelines for application engineers who assist OEM customers in troubleshooting part design issues.
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It enables the engineer to navigate through the hydroforming process for maximizing part performance while avoiding both buckling=wrinkling and bursting. Acknowledgements The authors would like to thank scientists from Battelle Paci;c Northwest National Laboratory, particularly Messieurs R.W. Davies and G.J. Grant, for conducting the experiments and providing the test data for our use in this study. References [1] Dohmann F. Introduction of the process of hydroforming. In: Siegert K, editor. Hydroforming of tubes, extrusions and sheet metals. Proceedings of the International Conference on Hydroforming, Fellbach/Stuttgart, Germany, vol. 1, 12–13 October 1999. p. 1–21. [2] Asna; N. Analytical modeling of tube hydroforming. Thin-Walled Structures 1999;34:295–330. [3] Chu E, Xu Y. Hydroforming of aluminum extrusion tubes for automotive applications. Part I: buckling, wrinkling and bursting analyses of aluminum tubes. International Journal of Mechanical Sciences 2004, doi:10.1016/j.ijmecsci.2004.02.014. [4] Davies RW, Grant GJ, Herling DR, Smith MT, Khaleel MA. Evaluation of the Forming Limit Diagrams of 6061 extrusions during free hydroforming. Technical Report for CRADA PNNL/145, 26 January 2000.
Hydroforming of aluminum extrusion tubes for automotive applications. Part II: process window diagram E. Chu∗ , Yu Xu Product Manufacturing Division, Alcoa Inc., Alcoa Technical Center, Alcoa Center, PA 15069-0001, USA Received 22 November 2002; received in revised form 7 January 2004; accepted 4 February 2004
Abstract Based on the mathematical formulations for predicting forming limits induced by buckling, wrinkling and bursting of free-expansion tube hydroforming, a theoretical “Process Window Diagram” (PWD) is proposed and established in this paper. The theory developed in the ;rst part of the present work was formulated within the context of free-expansion tube hydroforming with both combined internal pressure and end feeding. The PWD is designed to provide a quick assessment of part producibility for tube hydroforming. The predicted PWD is validated against experimental results conducted for 6260-T4 60 × 2 × 320 (mm) aluminum tubes. An optimal loading path is also proposed in the PWD with an attempt to de;ne the ideal forming process for aluminum tube hydroforming. Parametric studies show that the PWD has a strong dependency on tube geometry, material property and process parameters. To the authors’ knowledge, this is the ;rst attempt that a PWD is being formulated theoretically. Such a concept can be advantageous in deriving design solutions and determining optimal process parameters for tube hydroforming processes. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Tube hydroforming; Forming limits; Wrinkling; Buckling; Bursting; Process Window Diagram (PWD); Aluminum tubes
1. Introduction Hydroforming of tubular components presents an excellent opportunity for lightweight manufacturing of structural automotive components. As the hydroforming process becomes more widely used for automotive applications, issues regarding process control, design Aexibility and material utilizations must be addressed in order to provide adequate supports for implementing the technology in the automotive industry. In tube hydroforming, a critical aspect during the process design stage is to accurately determine of the trajectory of internal pressure versus end feeding sequence. Too much pressure will cause bursting ∗
Corresponding author.
0020-7403/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2004.02.013
286
E. Chu, Y. Xu / International Journal of Mechanical Sciences 46 (2004) 285 – 297
while too much end feeding will cause buckling or wrinkling. It is apparent that a diagram showing a safe working zone within the plane of certain process parameters is desirable for application engineers. Some investigators have discussed the implications of such a concept from an experimental perspective [1,2], however, according to the authors’ knowledge, there has not been any noticeable eIort to formulate this kind of diagram theoretically. In view of the need to provide a better analytical tool for Alcoa engineers to quickly assess the appropriate formability limits for tube hydroforming and to advance the use of aluminum extrusions in structural applications, an attempt is made herein to theoretically formulate the “Process Window” within which an aluminum tube can operate safely without failure under a free-expansion hydroforming process. Theoretically, the “Process Window” should be bounded by the forming limits (failure limits), imposed by the previously discussed aforementioned failure modes such as buckling, wrinkling and bursting [3]. To establish such a “Process Window”, the onset condition for each failure mode must be ;rst formulated theoretically. In the ;rst part of the present work [3], two distinct modes of failures in tube hydroforming have been identi;ed: (1) failures due to either global buckling or local axisymmetric wrinkling, and (2) bursting of tubes due to localized necking of the material. A unique mathematical foundation for bifurcation analysis of tube hydroforming under combined internal pressure and end feeding has also been established for predicting forming limits due to buckling and wrinkling in the free-expansion tube hydroforming [3]. Numerical results revealed that: local wrinkling is the dominant mode for short tubes while global buckling occurs for long slender tubes; the onset of asymmetric wrinkling always requires a higher critical axial compressive stress than the axisymmetric one and hence this mode can be ignored in the hydroforming analysis. Closed-form analytical expressions for the onset condition of bursting and for maximum internal pressure have been derived for predicting bursting failure in the free-expansion tube hydroforming [3]. The conditions of incipient failure established in the ;rst part of the present work provide us a complete set of theoretical tools to construct the “Process Window” after which application engineers have been seeking for many years. The development of a general Process Window Diagram (PWD) for the free-expansion tube hydroforming will be discussed within the context of these theoretical tools. Subsequently, the PWD diagram will be validated with experimental measurements provided by the scientists at the Battelle Paci;c Northwest National Laboratory (PNNL) [4]. An optimal load path for tube hydroforming is then proposed using the PWD. Finally, the inAuence of geometric parameters and the material properties on the PWD is investigated. The failure mode of folding, which usually occurs in the intake zone at the end of process [1], is not considered in the present study. Additionally, the sealing limit generally represented by a straight line on the plane of the internal pressure versus the axial compressive force diagram [1,2], will also be omitted in the present study. Other investigations have already documented these ;ndings and will not be repeated here. However, the expression for the sealing limit will be included in this paper for the sake of completeness. 2. The theoretical PWD As discussed previously any of the three possible modes such as buckling, wrinkling and bursting, could potentially cause failures in the free-expansion tube hydroforming. The onset conditions of
E. Chu, Y. Xu / International Journal of Mechanical Sciences 46 (2004) 285 – 297
287
these failure modes have already been established theoretically in the ;rst part of the present work [3]. These onset conditions can be plotted either as forming limit (failure limit) curves in the strain space (axial strain versus circumferential strain plot), or on the stress space (axial stress versus circumferential stress plot), or in the load space consisting of the internal pressure and the axial compressive force. Any of these failure limit plots presented on diIerent space can be used to describe the PWD for tube hydroforming process. The process parameters such as internal pressure and axial compression (or end-feeding) are commonly used to control the tube hydroforming process in practice [1,2]. Hence, it is logical to establish the “Process Window” in the load space for practical purposes. The PWD, bounded by the theoretical buckling, wrinkling and bursting forming limits, is dependent on the material property, tube geometry and loading conditions. As such, the PWD represents the incipient failure conditions or the forming limits for a given aluminum tube. The theoretical PWD will encompass the failure limits under which a tube can operate safely in hydroforming operations. The validity of such a diagram will be demonstrated in comparison with experimental measurements conducted by PNNL scientists. The development of a general “Process Window” or formability window for this process requires the bifurcation theory developed in Part I of the study. For the sake of completeness, the equations that will be used for the current purpose are repeated herein without further explanation and the interest reader is referred to the ;rst part of the present work [3] for more details of the mathematical formulations.
2.1. The onset of incipient buckling and wrinkling of a tube The condition for the onset of global buckling of a tube is given by ∗
cr = L1111
2 2
r0 ‘0
2
e2(22 −11 )
(1)
and the condition for the onset of axisymmetric wrinkling of the tube in hydroforming is written as ∗
cr = L1111
(m) L 2 12
t0 ‘0
2
e
−2(22 +211 )
+ L2222
1 m L
2
‘0 r0
2
e2(11 −22 ) :
(2)
To make it self-explanatory, a brief description of the symbols used is given here. A cylindrical coordinate system is used with subscripts 1, 2 and 3 referred to the axial, circumferential and ra∗ is the critical axial compressive stress, i.e. ∗ = − . dial directions, respectively. cr 11 11 and cr 22 are axial stress and circumferential stress, respectively. 11 and 22 are axial and circumferential strain. L1111 and L2222 are components of the plane stress moduli, which are functions of 11 and 22 . mL is the number of wrinkle waves in the axial direction of the tube. ‘0 , t0 , and r0 are the initial length, thickness and radius (measured from the middle surface) of the tube, respectively. In the free-expansion tube hydroforming, a tube can either buckle as a column, wrinkles axisymmetrically or wrinkles asymmetrically [3]. The mode that requires the least amount of axial
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compressive load is considered as the actual occurrence of the bifurcation phenomenon for the tube. Similarly, the wave number corresponding to the least amount of axial compressive load is considered as the actual wave number. It has been demonstrated that the axisymmetric wrinkling mode is the weaker mode, which generally requires a smaller compressive load than the asymmetric wrinkling mode [3]. Hence, the asymmetric wrinkling mode is excluded in constructing the PWD for the free-expansion tube hydroforming. 2.2. The onset of incipient bursting of a tube The onset condition of bursting can be expressed as [3] 11 + 222 = n;
(3)
where n is the strain-hardening index of the tube material. In deriving the above equations, the tube is assumed to obey a piecewise power law and undergoes a proportional stress path. Eqs. (1)–(3) provide a general framework for constructing the forming limit curves or the PWD. The detailed procedures for such a construction have been discussed in the ;rst part of the present work [3] for an aluminum tube obeying a speci;c yield criterion. The forming limits established previously in the stress or strain space, could easily be recast into the load space in terms of the internal pressure p and the critical axial compressive force, Fx =2r0 t0 , via the following equations: t0 p = 22 e(33 −22 ) ; (4) r0 and
Fx 22 − 11 e(22 +33 ) : = 2r0 t0 2
(5)
Theoretically, the current mathematical framework can accommodate any yield criteria to be used for predicting the forming limit curves. Note that it is extremely diNcult to correctly characterize anisotropy of a circular tube along the circumferential and 45 degree directions. With this constraint in mind, certain simpli;cations are adopted herein in constructing the PWD. Hill’s 1948 yield criterion seems to better serve the purpose for demonstrating the validity of this work and hence will be used to model the tube material. The components of the plane stress moduli, such as L1111 and L2222 , will be calculated based on the deformation theory of plasticity [3]. The tube material is assumed to obey the piecewise power law equation of the form 1=n L L L L for L ¡ y in the elastic range; = for L ¿ y and = (6) y y y where y and y are the uniaxial yield stress and yield strain, respectively. As mentioned in the introduction, sealing of tubes is also a requirement in tube hydroforming. The axial compressive force required to seal a tube with the internal pressure is approximately expressed as Fx = pr02 [1], which is further simpli;ed to a straight line in the PWD as Fx r0 = p: (7) 2r0 t0 2t0
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Numerical procedures have been developed and implemented for generating the operating window for a given aluminum tube, in order to demonstrate the applicability of the concept. This theoretical PWD provides a means for the design and process engineers to determine the forming limits of a given tube a priori before hydroforming. A few investigators [1,2] have discussed the implications of such a concept from an experimental perspective or some rules of thumb, however, to the authors’ knowledge, this is the ;rst time that anyone ever attempt to formulate the “Process Window” theoretically.
3. Validation of proposed PWD Scientists at PNNL conducted the experimental study of the free-expansion tube hydroforming process, which is under a Cooperative Research and Development Agreement (CRADA) between Alcoa and PNNL. A special system, which provides the simultaneous application of internal pressure and axial load, was developed at PNNL and the apparatus [4], as shown in Fig. 1, was used to perform the experiment. The system was ;t with a 222 kN load cell and unique grips to support both axial and internal pressurization. The load frame’s actuator controlled the axial position to a maximum travel of 152 mm. The specimen grips consisted of a locking device, an internal mandrel, and seal assembly that is typically used for hydrostatic testing of tubular products. The internal pressurization system consisted of an air driven water pump that was capable of achieving a pressure of 69 MPa. A digital servohydraulic controller directed the axial load and actuator position, internal pressure and collected data during experiments. Additionally, the controller can be programmed to apply various ratios of internal pressure to axial load to simulate actual hydroforming control. Both the axial position and internal pressurization systems consisted
Fig. 1. Photograph illustrating the apparatus developed at the Battelle PNNL for the free-expansion hydroforming study [4].
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of servohydraulic control valves that maintained testing conditions throughout the duration of the tests. The focus of this section is the comparison between the predicted PWD and the experimental measurements conducted for round 6260-T4 structural extruded tubes with a 60 mm outside diameter and a 2 mm wall thickness. The overall length of the tubes tested was 460 mm. The unsupported length of the tube between grips was 320 mm, with 70 mm inserted into each grip. The aluminum tubes were subjected to a series of hydroforming experiments for determining the strains that can be developed under various proportional loading conditions. During the experiments, selected ratios of axial load and internal pressure were maintained proportionally and Fig. 2 illustrates a sampling of the testing population used in the hydroforming experiments [4]. In this ;gure, the samples are arranged in the order of increasing axial compression from top to bottom, such that the bottom sample received the most axial compressive load compared to other samples in the same ;gure. Generally, increasing axial compression tends to increase the achievable circumferential expansion. If excessive axial compression is imposed on the tube way beyond its critical axial compressive load, buckling or wrinkling begins to occur. This is observable from Fig. 2, which shows increasing expansion with increasing axial compression until the axial load becomes too excessive that wrinkling of the tube sets in. It also indicates that the modes of failure will change with respect to the applied amount of axial compression as well as internal pressure used in hydroforming. Fig. 3 shows trajectories of the internal pressure versus the axial displacement curves for various proportional loading conditions during hydroforming. It can be seen that the failure modes change
Fig. 2. Samples of hydroformed tubes showing fracture and wrinkling under various proportional loading conditions [4].
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Fig. 3. Diagram showing the trajectories of the pressure load versus the axial displacement for various proportional loading conditions during hydroforming. Fracture and wrinkling of tubes are clearly visible (Courtesy of PNNL).
from bursting to wrinkling at increasing axial compression at high internal pressures [4]. At relative low internal pressures, axisymmetric wrinkling of the tube as end-feeding increases mainly controls failure. To measure strain localization during forming, square grids were electrochemically etched onto the tubes before forming. The grids surrounding the failure region of the specimen were then measured using a grid strain analysis system. The system determined the magnitude of the diIused and localized failure strains that developed around a failure site. Strains determined from grids that were within the necked region of the failure site were identi;ed as localized or unsafe strains, while those that were measured away from the fracture region were identi;ed as diIused or safe strains. A note of caution is worth mentioning here, since the determination of the diIused and localized strains can be extremely subjective in nature. A small error in selecting the right grids to measure can result in a relatively large error in failure strain. Furthermore, strain grids were measured after bursting of the tube already happened, and the abrupt burst opening of the tube could potentially induced some additional strains around the failure regions, and curvature of the tube was also not accounted for in the measurements, thus introduce further errors. Fig. 4 provides a comparison between experimental measurements conducted by PNNL and the predicted “Process Window” in the stress space, which is bounded by predicted buckling, wrinkling and bursting (fracture) limits. The mechanical property of the 6260-T4 aluminum tube used for the experiment is summarized in Table 1. The measured bursting limits as well as the buckling and wrinkling limits for 6260-T4 aluminum tubes under various proportional loading conditions were plotted in the ;gure. It shows that the overall predicted results compared very well with experimental
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Fig. 4. The circumferential stress is plotted as a function of the axial stress for 6260-T4 60 × 2 × 320 (mm) aluminum tubes. Both wrinkling and fracture limits generated from experiments are compared with theoretical predictions.
Table 1 Mechanical properties of 6260-T4 60 × 2 × 320 (mm) aluminum tubes Material
r-value L
n-value
K-value (MPa)
E (MPa)
Yield strength (MPa)
6260-T4
1.0a
0.26
397
62 000
67.3b
a Not measured due to diNculty in getting right measurements for the tube. Generally, r-value L is less than unity for aluminum and results in smaller fracture limits, but it has a small eIect on buckling/wrinkling. b This is calculated based on the intersection between the elastic portion and the plastic portion represented by the power law.
;ndings. Particularly, the theoretical predictions correlated exceptional well with experimental data in both trends and magnitudes for wrinkling bifurcation. Historically, it has been diNcult to correlate experimental results with theoretical predictions correctly. Any slight imperfection that exists in a tube could potentially aIect the onset of buckle/wrinkle stress. Therefore, the comparison here has given us con;dence in our theoretical model. It was also con;rmed from both experiment and predictions that axisymmetric wrinkling was a weaker mode which required a lower axial compressive load than the asymmetric one. In order to determine the bursting limits from experiments, the average data points for each loading path were extracted from the forming limit curve that developed by PNNL scientists. The average data that represented localization of the tubes were plotted along with the theoretical predictions in the ;gure. Generally, the agreement between predicted and experimental results compared very well.
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4. Determination of an optimal path in PWD Fig. 4 shows the predicted “Process Window” in the stress space for 6260-T4 60 × 2 × 320 (mm) aluminum tubes. Now, Fig. 4 can be recast in the loading space commonly used in the hydroforming process as shown in Fig. 5. In the ;gure, the critical axial compressive force, Fx =2r0 t0 [3], though plotted positively is considered negative in magnitude, and is given as a function of the internal pressure applied to the tube. Such a PWD provides a useful means for guiding future development of tube hydroforming processes. Finding an optimal loading path within the PWD shown in Fig. 5 is of practical interest to the industry. It is therefore necessary to brieAy discuss how to apply such a PWD for industrial applications. To begin, one must de;ne an objective function that is required to achieve the “optimal path” for the forming process. Consider the PWD given in the strain space as shown in Fig. 6(a), an objective function must be assumed in order to de;ne a path for the tube to achieve a maximum expansion ratio, r=r0 . It is understood that an in;nite number of paths can be used to describe this deformation; some can be complex while others are simple, and some may even be diNcult to implement practically. Hence, in an eIort to avoid complexity in implementing an objective function for equipment control, the simplest possible function is assumed in this work—a proportional loading path. Under this assumption, the loading path is a straight line originated from the origin in the strain space by keeping the ratio between the axial strain and the circumferential strain proportional at all time. Without invoking any types of failure models, the maximum circumferential strain, ‘n(r=r0 ), could be attained which is represented by the abscissa of point A shown in the ;gure. Therefore, the straight line from the origin to point A de;nes the optimal loading path plotted in the strain space. This optimal path can be recast in terms of the critical axial compressive force, Fx =2r0 t0 , and the internal pressure, p, as illustrated in Fig. 6(b). Such an optimal path plotted in the load
Fig. 5. Illustrating an example of the PWD that governs the theoretical buckling/wrinkling and bursting limits of the free-expansion tube hydroforming process.
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Fig. 6. (a) Diagram showing an example of utilizing the PWD for designing a desirable optimal loading path (OA) for tube hydroforming process. (b) Illustrating an optimal path (OA) de;ned within the PWD in the load space.
space can be very useful for designers or forming engineers to develop an optimal loading trajectory for controlling the pressure and load relationship in the hydroforming equipment. This path allows an engineer to control the equipment to maximize part performance without excessive wrinkling and bursting of the tube. The authors are well aware of the possibility that using a piece-wise linear combination of strain paths might enable the process to attain a larger expansion ratio for the tube hydroforming process. However, such a curved loading path will result in shifting the boundary of the Process Window very much like shifting of a kinematic hardening yield surface. The path dependency of the PWD is beyond the scope of the present study, which will not be discussed in this paper.
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5. Material and geometrical e!ects on PWD The eIects of material properties on the variations of the PWD are presented in Figs. 7 and 8. The strain-hardening index, n, ranging from 0.06 to 0.26, and the material plastic anisotropy, r, L ranging from 0.5 to 1.5 were used to generate these ;gures. It is apparent from the ;gures that both material
Fig. 7. Showing the eIect of material plastic anisotropy on the variation of the PWD for the ratios of r0 =‘0 = 0:05 and t0 =r0 = 0:04.
Fig. 8. Showing the eIect of strain-hardening index on the variation of the PWD for the ratios of r0 =‘0 = 0:05 and t0 =r0 = 0:04.
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Fig. 9. Illustrating eIects of wall thicknesses on the variation of the PWD indicating thicker tubes sustain higher formability for r0 =‘0 = 0:1.
parameters have more eIects on bursting failure than on buckling/wrinkling bifurcation. It is also evident from the ;gures that the strain-hardening index has a much more signi;cant inAuence on both bifurcation and bursting (fracture) failure limits of the tube hydroforming process than the plastic anisotropy of the material, which has only a small inAuence. Nevertheless, the “Process Window” expands with the strain-hardening index and plastic anisotropy of the tube material. Fig. 9 shows the eIect of metal thickness on tube hydroforming. It indicated that thicker tubes have higher buckling or wrinkling resistance as well as higher bursting failure limits in tube hydroforming. This is also consistent with practical applications observed in the industry. However, the theory showed that metal thickness seems to bene;t the localization much more than the bifurcation limits. 6. Conclusions Based on the theoretical formulations for predicting buckling, wrinkling and bursting in the ;rst part of the present work [3], a PWD is proposed and established in this paper. The PWD provides a theoretical assessment of part producibility a priori during the product and process design stages. It describes the forming limits imposed by buckling, wrinkling and bursting of a tube undergoing the free-expansion tube hydroforming operation. The predicted PWD correlates very well with experimental measurements conducted by PNNL. The theory was able to predict strain localization and its trends to within statistically acceptable limits. The PWD, similar in nature to the well-known Forming Limit Diagram (FLD) commonly used in the sheet metal forming processes, can be a powerful tool in providing insight in assessing part feasibility for tool designers or process engineers during the initial design stage. Unlike the FLD, the PWD is plotted in a load space. The PWD can also be used to provide some general forming guidelines for application engineers who assist OEM customers in troubleshooting part design issues.
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It enables the engineer to navigate through the hydroforming process for maximizing part performance while avoiding both buckling=wrinkling and bursting. Acknowledgements The authors would like to thank scientists from Battelle Paci;c Northwest National Laboratory, particularly Messieurs R.W. Davies and G.J. Grant, for conducting the experiments and providing the test data for our use in this study. References [1] Dohmann F. Introduction of the process of hydroforming. In: Siegert K, editor. Hydroforming of tubes, extrusions and sheet metals. Proceedings of the International Conference on Hydroforming, Fellbach/Stuttgart, Germany, vol. 1, 12–13 October 1999. p. 1–21. [2] Asna; N. Analytical modeling of tube hydroforming. Thin-Walled Structures 1999;34:295–330. [3] Chu E, Xu Y. Hydroforming of aluminum extrusion tubes for automotive applications. Part I: buckling, wrinkling and bursting analyses of aluminum tubes. International Journal of Mechanical Sciences 2004, doi:10.1016/j.ijmecsci.2004.02.014. [4] Davies RW, Grant GJ, Herling DR, Smith MT, Khaleel MA. Evaluation of the Forming Limit Diagrams of 6061 extrusions during free hydroforming. Technical Report for CRADA PNNL/145, 26 January 2000.