Influence of forming effects on the axial crush response of hydroformed aluminum alloy tubes

International Journal of Impact Engineering 37 (2010) 1008e1020

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Influence of forming effects on the axial crush response of hydroformed aluminum alloy tubes B.W. Williams a, M.J. Worswick a, *, G. D’Amours b, A. Rahem b, R. Mayer c a

Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada Aluminum Technology Centre, Industrial Materials Institute, National Research Council of Canada, Saguenay (Chicoutimi), Quebec G7H 8C3, Canada c Vehicle Development Research Lab, GM R&D Center, MC 480-106-256, Warren, MI, USA b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 24 September 2009 Received in revised form 18 March 2010 Accepted 29 April 2010 Available online 21 May 2010

The impact behaviour of tubular hydroformed axial crush tubes is examined. The results of dynamic axial crush tests performed with both non-hydroformed and hydroformed AA5754 aluminum alloy tubes were compared to predictions from finite element models. Explicit dynamic finite element simulations of the hydroforming and crash events were carried out with particular attention to the transfer of forming history from the hydroforming simulations to the crash models. The values of tube thickness, work hardening, and residual stresses at the end of the hydroforming simulations were used as the initial state for the crash models. In general, simulations performed using the von Mises yield criterion with isotropic material behaviour gave reasonable predictions when compared to experimental data. It was found that it was important to account for the forming history of the hydroforming operation in the axial crush models. The results showed that work hardening resulting from hydroforming is beneficial to increasing the energy absorption during crash, whereas thickness reduction decreased the energy absorption. Residual stresses had little effect on the energy absorption characteristics. It was also shown that the energy absorption characteristics of tubes with the same mass could vary greatly by adjusting the geometry of the tube and the amount of work hardening experienced by the tube during hydroforming. Ó 2010 Elsevier Ltd. All rights reserved.

Keywords: Hydroforming Axial crush Aluminum Energy absorption

1. Introduction The reduction of vehicle weight, and the corresponding reduction in fuel consumption, is of great concern to the automotive industry. In recent years, lightweight materials such as aluminum and magnesium alloys, high-strength steels, and composites have been considered as alternatives to mild steel in structural components that could be subject to crash events [1e4]. Aluminum is attractive due to its low-weight, good corrosion resistance, and the fact that it can be recycled with much less energy than that required to produce primary aluminum. One disadvantage to using aluminum is its lower formability compared to mild steel. An axial crush structure is designed to absorb energy by progressive axial folding. Several investigations have compared experimental axial crush results to predictions from finite element simulations. Bardi et al. [5] compared results from finite element simulations to experimental and theoretical predictions, showing good agreement for circular AA6061 aluminum alloy and mild steel

* Corresponding author. Tel.: þ1 519 888 4567. E-mail address: [email protected] (M.J. Worswick). 0734-743X/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2010.04.006

tubes. Langseth et al. [6,7] have also shown good agreement between simulation and experiment for square AA6060 aluminum alloy tubes. Otubushin [8] validated simulations incorporating the CowpereSymonds constitutive equation for strain-rate effects against experimental mild steel axial crush data. More recently, Tarigopula et al. [4] have studied the axial crush behaviour of thinwalled square tubes and spot-welded top-hat sections of DP800 steel showing agreement between simulation and experiment. Generally, the structures studied in these investigations did not undergo previous forming operations. In fact, there is very little available data which considers the effect of forming history on the axial crush response. In recent years, tube hydroforming has been utilized to fabricate structural components on a wide range of vehicles using mild steel tubes. There are several advantages to hydroforming over conventional processes, such as stamping and welding that include a reduction in the number of parts and the overall assembly weight. In addition, hydroformed tubes offer increased strength and stiffness and more precise component dimensions, all of which lead to lower manufacturing cost. It has been shown that it is important to incorporate the forming history (thicknesses, residual stress, and work hardening) in subsequent crash simulations of stamped

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components [9]. Consequently, there is a requirement to determine whether it is important to account for forming history when studying energy absorption characteristics of hydroformed aluminum alloys. Previously, Kellicut et al. [10] and Oliveira et al. [3] have shown that work hardening resulting from hydroforming is important to consider during impact of S-shaped structures. The purpose of the current work is to isolate and study the influence of thickness changes, residual stress, and work hardening resulting from the hydroforming operation on the subsequent axial crush response of the material. This objective is accomplished through both experimentation and finite element analysis. Experiments are performed in which hydroforming process parameters are varied in a parametric fashion after which the axial crush response is measured. Experimental parameters include the tube thickness and the hydroformed corner-fill radii of the tubes, as well as consideration of the so-called ‘low’ versus ‘high’ pressure hydroforming processes, as will be outlined in the following sections. Numerical studies were carried out using explicit dynamic finite element models to capture the entire forming and axial crush history. Particular attention was given to the transfer of the forming history, comprising of thicknesses, residual stress, and work hardening from the hydroforming operation to the axial crush simulations. The finite element models were performed using isotropic yield criteria and isotropic hardening to describe the behaviour of the material. The research was performed using an AA5754 aluminum alloy, which is a candidate for automotive structural applications. 2. Experiments 2.1. Material AA5754 aluminum alloy tube with a diameter of 76.2 mm, supplied by VAW, a German corporation currently owned by Hydro Aluminum, was considered in the study. This alloy is non-heattreatable and composed of 2.8% magnesium and 0.3% manganese as principal alloying elements. The tubes were fabricated from sheet using a roll forming process, induction seam welded and then annealed to the O-temper. There was some variation in the tube thickness due to the rolling process. The thickness was measured at various positions around the radius of the tube and ranged from 3.01 mm to 3.12 mm with an average thickness of 3.07 mm. Tensile tests were performed on specimens taken from the 3, 6, and 9 o’clock positions around the tube (the weld seam was taken as the 12 o’clock position), with the longitudinal axis corresponding to the loading direction. Both the engineering stress versus strain and true stress versus true plastic strain curves are shown in Fig. 1.

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The yield strength of the AA5754 alloy was 100 MPa with an ultimate tensile strength of 220 MPa and a total elongation to failure of 22%. From this data, the effective stress, s, versus the effective plastic strain, ep , was determined which was required to describe the isotropic hardening behaviour in the finite element models. Based on tensile and hydraulic bulge testing, it has been shown that the AA5754 alloy follows a Voce-type hardening [2]. In this work, a 4-parameter Voce model was considered, given by,








s ¼ a  a  sy exp  b3p

c



(1)

with a ¼ 315 MPa, b ¼ 5.5, c ¼ 0.77, and sy ¼ 100 MPa determined for the AA5754 tube alloy. 2.2. Hydroforming During the low-pressure hydroforming process, which is depicted in Fig. 2, the tube was slightly pressurized during the die closing stage such that pinching and wrinkling of the tube was avoided. The perimeter of the die was approximately equal to the circumference of the tube, thus avoiding circumferential stretching of the tube. The pressure in the tube during the die closure was held at about 3 MPa. Once the die was closed, a higher pressure was applied to fully form the tube. The hydroforming die cavity is shown in Fig. 3. In order to seal the tube during the low-pressure hydroforming operation, so-called “floating” end-plugs were used which incorporated a high-pressure polymeric seal, as shown in Fig. 3c. The hydroforming die adjacent to the end-plugs counteracted the axial force induced by the internal pressure acting on the end-plugs. In the current research, three low-pressure hydroforming inserts were fabricated with corner-fill radii equal to 6, 12, and 18 mm. The cross-sectional dimensions of the dies are given in Table 1. For the high-pressure hydroforming operation, the width of the cross-sections of the hydroforming inserts was 76.7 mm which required circumferential expansion of the tube leading to thickness reduction during hydroforming. Dimensions of the three highpressure inserts fabricated with 6, 12, and 18 mm corner-fill radii are also given in Table 1. All high-pressure hydroforming was conducted using end-feed actuators that acted directly on the tube ends and pushed material into the die, promoting greater formability. The tube was sealed through metal-to-metal contact between the end-feed actuators and tube. For each of the three corner-fill radii, tubes were formed utilizing about 60 mm of endfeed at each end of the tube. The internal pressure versus end-feed displacement profile used for the high-pressure hydroforming process is given in Fig. 4. The same profile was used to produce tubes with 6, 12, and 18 mm corner radii except that the tests were

Fig. 1. (a) Engineering stress versus strain and (b) true stress versus plastic strain curve for as-tubed AA5754 alloy.

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Fig. 2. Depiction of low-pressure hydroforming process.

terminated prior to 60 mm end-feed displacement for the tubes formed with 12 or 18 mm corner radius. The actual level of endfeed specified for the 6, 12, and 18 mm corner radii cases were 60.0, 59.9, and 57.8 mm with maximum pressures of 43, 102, and 140 MPa, respectively. Prior to the hydroforming process, circle grids were electrochemically etched onto the surface of the tube. After hydroforming, an optical system was used to measure the change in shape of the circles and software would then calculate the corresponding circumferential and longitudinal strain based on the deformation of the circle relative to the original size. The circumferential and longitudinal strains were measured at the half-length of the tube. A non-destructive ultrasonic thickness gauge was used to measure the variation in thickness of tubes formed using the high-pressure hydroforming process. The thickness measurements were taken at the half-length of the tube. The circumferential strain measurements from both the low- and high-pressure processes and the

longitudinal strain measurements and thickness variation from the high-pressure process are presented in Section 4 and compared to results from simulation. For the low-pressure process, the length of the tube did not change significantly such that the longitudinal strains were approximately zero. Also, the circumference of tube remained constant during the test such that the thickness of the tube remained relatively constant at an average of about 3.07 mm. D’Amours et al. [11] provide further details regarding the highpressure hydroforming process. HydroDraw 625, a solid-film lubricant, was used in all hydroforming operations. The lubricant was first sprayed onto the tube and allowed to dry. A coefficient of 0.05 was determined for the lubricant based on twist-compression friction tests [11,12] which was then adopted in finite element simulations of the hydroforming operation. After hydroforming, all tubes were trimmed to a 400 mm length for axial crush testing. Some tubes formed using the high-pressure process were annealed after the hydroforming

Fig. 3. (a) Schematic of low-pressure hydroforming die assembly (lower die half) and (b) cross-section of hydroforming die assembly (lower die half with 6mm corner radius), and (c) end-plugs used to seal tubes during low-pressure hydroforming.

B.W. Williams et al. / International Journal of Impact Engineering 37 (2010) 1008e1020 Table 1 Dimensions of hydroforming inserts.

Insert

Hydroforming process

w (mm)

r (mm)

Percent change in perimeter (relative to non-hydroformed tube)

#1 #2 #3 #4 #5 #6

LP e low pressure LP e low pressure LP e low pressure HP e high pressure HP e high pressure HP e high pressure

62.9 65.5 68.1 76.7 76.7 76.7

6 12 18 6 12 18

0.8 0.8 0.8 23.9 19.6 15.3

process, prior to impact, to compare energy absorption characteristics with the corresponding as-formed tubes. Tensile specimens were not tested from the annealed sections, but rather it was assumed that Eq. (1) was still valid to describe the behaviour of the annealed tubes. Fold initiators were incorporated in some of the tubes to reduce the peak load obtained during the axial crush experiments. All tubes formed using the high-pressure process incorporated fold initiators. Tubes formed using the low-pressure process were produced both with and without initiators. The width of each initiator, located on two opposite sides of the tube, was 38.1 mm and the depth was approximately 5 mm from the surface of the tube [12]. 2.3. Axial crush testing The tubes, tested two at a time, were clamped to support plates and impacted on a sled-track, as depicted in Fig. 5. Two clamps, one

Fig. 4. Internal tube pressure versus end-feed displacement applied during the highpressure hydroforming process.

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at the top and one at the bottom, each engaged a 25 mm length of the tube, leaving a 350 mm length available for crush. The location of the fold initiators was 50 mm from the top of the tube or when clamped prior to testing, 25 mm from the upper clamp. The mass of the sled was 1120 kg. A 5 mm thick rubber pad was placed between the impact support plate and the face of the sled to prevent metalto-metal contact, which reduces the degree of high frequency oscillation obtained in the data. Load cells were arranged in a “triangular plus one” pattern at the bottom support plate, such that the pressure distribution during impact was centred about the load cells [13]. The impact velocity of the sled was selected to ensure about 200 mm of deformation. If too high an impact velocity were selected, then the tube might have completely deformed which could have led to overloading and damage to the load cells. Tubes formed using the low-pressure process, as well as the annealed high-pressure tubes, were impacted at 6.3 m/s. Tubes formed using the high-pressure process (as-formed) were impacted at 7.0 m/s while the non-hydroformed, circular tubes were impacted at 7.5 m/s. The non-hydroformed, circular tubes did not incorporate fold initiators. The hydroforming and axial crush test program is summarized in Table 2. 3. Finite element models A simulation was performed for each of the experimental conditions outlined in Table 2. Additional simulations were performed to isolate the influence of work hardening, residual stresses, and thickness changes resulting from hydroforming, on the axial crush response. Numerical studies were also carried out to compare the axial crush response when using different element sizes and element types. All simulations were performed using the von Mises yield criterion with Eq. (1) used to describe the hardening behaviour of the AA5754 aluminum alloy tubes. The explicit dynamic finite element code, LS-DYNA [14], was used for all simulations. There were three components required for the low-pressure hydroforming simulations: the lower die half, the upper die half, and the tube. The tube, with a nominal thickness of 3.07 mm, was meshed using approximately 4  4 mm shell elements with seven integration points through the thickness. Both die surfaces were meshed using rigid shell elements of approximately 1 1 mm size, with the properties of steel being specified for use in the contact treatment. This smaller element size was selected to capture the finer details of the die surfaces in the models, including the corner radius of the die and the two 50.8 mm long transitions from square to circular cross-sections (Fig. 3a). The smaller element size also ensured adequate contact detection between the tube and tooling surfaces, which was prescribed using a penalty function-based approach. All nodes for the bottom die were constrained and the top die was prescribed to move in the vertical direction in order to simulate the die closure during the low-pressure hydroforming operation. The mesh geometry for the low-pressure process is shown in Fig. 6a. The floating end-plugs, shown in Fig. 3c were not modeled since there was no end-feeding and the end-plugs would not have contributed to the deformation of the tube in the lowpressure operation. A control-volume approach was used to apply the hydroforming pressure using material properties of water for the fluid, as opposed to defining the applied pressure directly on the surface of the elements. The high-pressure simulations were similar to the low-pressure simulations, except that instead of two die halves, the high-pressure dies were meshed as a single surface, as shown in Fig. 6b. Since the tube fits into the closed die it was not necessary to model the die closure operation. The actuators used for end-feeding were modeled using the mesh shown in Fig. 6b. Penalty function-based contact definitions were again used to simulate the metal-to-metal

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Fig. 5. (a) Schematic of sled-track test apparatus and (b) photograph of experimental setup.

contact between the end-feed actuators and tube. During the simulations, the end-feed displacement and internal tube pressure were applied as a function of time and matched the experimental profiles presented in Fig. 4. The forming operations and impact event were simulated using an explicit dynamic formulation. In addition, static implicit analysis steps were used to simulate springback after forming. The changes in thickness, work hardening, and residual stress from the hydroforming simulation (explicit) were carried forward to a springback simulation (implicit). A simulation was then performed to model forming of the fold initiators (explicit) followed by a second springback simulation (implicit) after which the forming history was carried forward to the axial crush simulation (explicit). The axial crush simulations were performed on a single tube with an impact mass of 560 kg using a moving rigid wall treatment to simulate the impacting sled-track platform. The initial velocity specified for the rigid wall corresponded to the measured impact velocity for each test simulated. A second rigid wall was used at the bottom of the tube to prevent the tube from deforming past the bottom plane during the simulation. Contact was prescribed such that the tube would detect contact with itself and the rigid walls. All nodes in the clamped region on the bottom of the tube were fully constrained and all nodes in the clamped region on the top of the tube were only allowed to move in the vertical direction. The influence of the rubber pad and support clamps on the predicted crush response was considered in [12], where it was found that there was a slight difference in the initial peak load response predicted compared to when the rubber pad and support clamps were not considered in the finite element models. Overall, it was found that there was little difference in the predicted crush response between the two cases such that the effects of the rubber pad and support clamps were not considered in the current work.

4. Comparison of simulation and experiment e hydroforming Fig. 7 shows the predicted and measured circumferential engineering strain distribution for the outer surface of tubes hydroformed to a corner radius of 6 mm using the low-pressure process. The results show good agreement between simulation and experiment, showing that the simulations accurately captured the forming behaviour of the tube. There was little change in the thickness of the tube during the low-pressure operation because there was no end-feeding and the perimeter of the tube did not change significantly relative to the circumference of the nonhydroformed tube (Table 1). Fig. 8a shows the predicted and measured engineering strains, circumferential and longitudinal, for tubes hydroformed using the high-pressure process to a corner radius of 6 mm with 60 mm of end-feed. Fig. 8b shows the predicted and measured percent change in thickness, also for tubes formed to a 6 mm corner radius with 60 mm of end-feed. The measured results show that the thickness reduction, of about 17%, occurred in the area directly on either side of the corners of the tube. There was almost no thickness reduction in the flat regions of the tube. A cross-sectional image at the mid-length of a tube hydroformed using the high-pressure process to a corner radius of 6 mm is shown in Fig. 9. The regions of thinning on either side of the corner are an indication of the onset of strain localization. The results presented in Fig. 8 show that the predicted circumferential strains in the corners were greater than the measured strains and that the predicted thicknesses in the corners did not agree entirely with experiment. Also, there was a slight over-prediction of the thickness in the flat regions of the tube. The longitudinal strains are in agreement with the measured values. Potentially, these discrepancies could be attributed to the onset of

Table 2 AA574 axial crush test matrix. Hydroforming process

Corner radius (mm)

e HP e high pressure HP e high pressure HP e high pressure HP e high pressure HP e high pressure HP e high pressure LP e low pressure LP e low pressure LP e low pressure LP e low pressure LP e low pressure LP e low pressure

Circular 6 12 18 6 12 18 6 12 18 6 12 18

End-feed displacement e per end (mm)

Annealed (yes/no)

Crush initiators

Number of tubes impacted

60 mm 60 mm 60 mm 60 mm 60 mm 60 mm None None None None None None

No No No Yes Yes Yes No No No No No No

No Yes Yes Yes Yes Yes Yes No No No Yes Yes Yes

4 4 4 4 4 4 4 2 2 2 4 4 4

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Fig. 6. Mesh geometry required for (a) low- and (b) high-pressure hydroforming simulations.

strain localization at the locations of thinning in the corners of the tube during experiment which was not captured in the simulations. These differences could also be attributed to through-thickness stresses resulting from contact with the die wall. Through-thickness stresses are not captured when using the current shell element formulation, but could be accounted for by using solid elements in the simulations [11]. As mentioned above, strain and thickness measurements were taken only at the mid-length of the tube and do not provide an indication of the variation along the length of the tube. Fig. 10 shows contour plots of the predicted effective plastic strain and thickness changes for a tube with a 6 mm corner radius formed using the high-pressure process. The predicted results show that the strains and thickness remain reasonably uniform along the length of the tube, with a slight increase in circumference strain and decrease in thickness at the mid-length of the tube compared to the ends of the tube. The largest effective plastic strain predicted was about 0.67 while the thinnest point of the cross-section was predicted to be 2.6 mm. The strain and thickness measurements for tubes formed using the low- or high-pressure processes with 12 and 18 mm corner radius, which are not presented for brevity, exhibited trends similar to those obtained for the 6 mm corner radius case [12]. Although there are some discrepancies between predicted and measured data, the simulations reasonably captured the hydroforming response of the tubes. 5. Comparison of simulation and experiment e axial crush The predicted and measured crush force versus displacement responses for a tube formed to a corner radius of 12 mm using the

low-pressure process are shown in Fig. 11. Predicted and measured mean loads of 54.1 kN and 50.5 kN were obtained by taking the area under the curve up to 200 mm of crush distance. The measured crush response shows higher frequency oscillations, generally below 50 mm of crush distance, which were attributed to structural vibrations measured in the load cells of the sled-track system. These oscillations were not expected to significantly affect the overall energy absorption calculated for the tubes. The measured and predicted crush response is compared between a non-hydroformed (circular) tube and a tube (with initiators) formed using the low-pressure hydroforming process in Fig. 12a and between tubes formed using the high-pressure hydroforming process in the asformed and annealed conditions in Fig. 12b. The sled-track velocity versus crush distance for the measured cases is presented in Fig. 12c, indicating a reasonably consistent decrease in velocity during testing. The circular tube had the highest crush loads of all tubes. Fig. 12b shows that there was a large initial peak load of approximately 225 kN for the as-formed tube formed using the high-pressure process compared to the annealed tube with an initial peak load of approximately 100 kN. In some cases, there were slight differences between the locations of the peaks and valleys of the crush load, but generally the results show good agreement between measurement and prediction. Fig. 13 shows the predicted and observed crush patterns of tubes formed to a 6 mm corner radius using the low- and high-pressure hydroforming operations, showing that the predicted response was similar to the experimental behaviour. As demonstrated in the figure, the hydroformed tubes deformed with a symmetric mode of crush in both experiment and simulation. A more detailed discussion of crush modes from experiment, including comparison to theoretical predictions, is provided in Section 8.

Fig. 7. Predicted and measured strains for tubes formed to a corner radius of 6 mm using low-pressure hydroforming process.

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Fig. 8. Predicted and measured response for tubes formed to a corner radius of 6 mm using high-pressure hydroforming process (a) engineering strain and (b) percent change in thickness.

In the following sections, the results are mainly presented based on the mean loads obtained at approximately 200 mm of crush distance. The actual force versus crush distance responses for all cases presented can be found in [12]. The predicted and measured mean crush loads for the AA5754 tubes formed using the lowpressure process are plotted versus the radius ratio (tube radius/ initial radius of 38.1 mm) in Fig. 14a. The simulations were performed using a tube mesh comprising of 4 mm BelytschkoeLineTsay shell elements [14] for both the hydroforming and impact simulations with isotropic yielding behaviour described by the von Mises yield criterion and isotropic hardening described by Eq. (1). The measured mean loads for tubes with initiators were about 13% less than for tubes without initiators. The predicted results are about 6% greater than the measured results for tubes with crush initiators and about 13% greater for tubes without initiators. These results showed that slightly more accurate predictions of the experimental data were obtained when incorporating fold initiators. Also shown in Fig. 14a, is the predicted and measured response for the non-hydroformed, circular tubes showing reasonable agreement, with about a 7% over-prediction. The predicted and measured mean crush loads for the AA5754 tubes formed using the high-pressure hydroforming process are presented in Fig. 14b. The predicted values for the as-formed tubes were slightly less than the measured values, except for the 18 mm corner radius for which there was an over-prediction. On average, the predicted results for the as-formed tubes were 1.1% greater than the measured values. The predicted values were about 8.6% greater than the measured values for the tubes that were annealed after the hydroforming process. The results show that the energy absorption capabilities of the tubes decreases slightly for smaller corner-fill radius which is due to the increased thickness reduction in the corners due to hydroforming. Annealing the tubes after the hydroforming significantly reduced the energy absorption capabilities by over 30% compared to the as-formed tubes, showing that

it is important to account for work hardening from previous forming operations. A detailed discussion on the influences of anisotropy yielding, kinematic hardening, and strain-rate sensitivity on the predicted axial crush response of the hydroformed tubes is provided in Ref. [15]. The percent difference between predicted and measured mean crush loads obtained from the various descriptions of the material behaviour in Ref. [15] are summarized in Table 3 for tubes formed using the high-pressure process in the as-formed and annealed (after hydroforming) conditions. The work identified that capturing the kinematic hardening behaviour of the material reduced the predicted mean crush loads, particularly for the asformed tubes by about 4% relative to the von Mises predictions. Capturing the anisotropic yielding behaviour of the AA5754 aluminum alloy in the simulations significantly decreased the predicted mean crush load of the axial crush structures relative to the isotropic (von Mises) predictions and resulted in an underprediction of the measured mean loads by an average of 8.7% and 3.9% for the as-formed and annealed tubes, respectively. Results from simulations capturing the strain-rate sensitivity of the material predicted mean crush loads 3.2% and 10.9% higher than the measured mean loads for the as-formed and annealed tubes, respectively, counteracting some of the decreases in mean load predicted by including the anisotropic and kinematic behaviour of the material. Results from simulations combining all three effects produced average predicted mean loads of 9.1% and 1.4% less than measured values for the as-formed and annealed tubes, respectively. In summary, results from simulations combining the effects of anisotropic yielding, kinematic hardening, and strain-rate sensitivity resulted in a slight under-prediction of the measured mean crush loads, while simulations performed using isotropic yielding (von Mises) and hardening behaviour (presented in Fig. 14b) resulted in a slight over-prediction of the measured mean loads. Given the rather small potential error associated with the adoption of a rate insensitive von Mises formulation, the results presented in the following sections of this paper which isolate the influences of element formulation, mesh refinement, and forming history on the axial crush response are based on simulations performed using a rate insensitive von Mises yield surface with isotropic hardening behaviour. 6. Effect of element formulation and mesh refinement

Fig. 9. Cross-section of tube with a 6 mm corner radius formed using the high-pressure hydroforming process.

It has been shown that different energy absorption characteristics for S-shaped structures made of AA5754 can be predicted when using different mesh sizes and element types in the simulations [3]. Particularly, Oliveira et al. [3] observed a shear-locking phenomenon when considering fully integrated shell elements

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Fig. 10. Contour plot of predicted (a) effective plastic strain and (b) thickness for a tube formed to a 6 mm corner radius using the high-pressure hydroforming process.

under buckling of S-shaped structures. Consequently, mesh and element type sensitivity studies were carried out in the current research for the axial crush structures. Two meshes with element sizes of 2 and 4 mm were studied with either the BelytschkoeLineTsay (BLT) elements (Type 2 in LS-DYNA) or fully integrated (FI) shell elements (Type 16 in LS-DYNA). Simulations were performed based on tubes formed using both the low- and highpressure processes. It is noted that only the size of the tube mesh varied in the finite element models, while all other aspects remained the same as those described in Section 3. Also, suitable hourglassing control was prescribed in the models depending on whether BLT or FI elements were employed [14]. Fig. 15a shows the predicted mean loads for simulations in which the tube was defined using BLT or FI elements of 4 mm mesh size. The figure shows that the FI elements gave a higher prediction than the BLT elements, by an average of about 7%. Fig. 15b shows the predicted response based on simulations performed using BLT elements with either a 2 or 4 mm mesh size, showing that the force predictions with the 2 mm mesh size were less than the predictions with the 4 mm mesh size by an average of about 5%. Fig. 15c shows an average under-prediction of about 6% between 2 and 4 mm meshes when FI elements were used. Fig. 15d shows that there was almost no difference obtained in the crush response when BLT elements were used for the impact simulations when using either BLT or FI elements for the hydroforming simulations. This suggests that the difference in crush responses obtained between FI and BLT elements arises from the axial crush simulations, and not the hydroforming simulations. This discrepancy could be attributed to the shear-locking phenomenon observed with fully integrated shell elements under buckling, as discussed by Oliveira et al. [3]. The

results presented in the balance of this paper were performed using a 4 mm mesh size with BLT elements for both the hydroforming and axial crush operations, mainly because these were the most computationally efficient simulations. Overall, the simulations appeared to capture the forming history and produce accurate predictions of the axial crush response for cases in which the tubes incorporated fold initiators. 7. Influence of forming history on the crush response The forming history carried forward from the hydroforming simulation incorporated the thickness changes, work hardening (or strain hardening given by the plastic strains and effective plastic strain), and residual stresses. In order to isolate the effect of each of these characteristics on the energy absorption during axial crush, additional simulations were performed. For tubes formed using the low-pressure hydroforming operation, three additional impact simulation conditions were carried out: cases in which the tubes had a nominal thickness of 3.07 mm and no prior forming history; cases in which the thickness changes from hydroforming were included, but the work hardening and residual stresses were not included; and cases in which only the residual stresses were not included. Fig. 16a shows the results from these simulations, comparing them to measured data and simulations in which the entire forming history was included. The results show that there was no significant difference in the mean load from simulations with a nominal thickness of 3.07 mm and simulations in which the thickness changes from hydroforming were included. This was expected since during the low-pressure hydroforming operation the thickness of the tube did not change significantly. The results

Fig. 11. Axial crush response of tube formed to 12 mm corner radius using low-pressure hydroforming operation (a) predicted and (b) measured.

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also show residual stresses did not affect the energy absorption characteristics during impact. The most significant changes result from whether or not work hardening was included. The mean load from simulations with the entire forming history included was greater, by about 16%, than cases for which work hardening was not included (similar to annealing the as-formed tubes). Similar impact simulations were performed based on tubes formed using the high-pressure hydroforming operation. However, an additional condition was analysed, in which the tube had a nominal thickness of 3.07 mm with work hardening included. These simulations would allow the effects of thickness changes to be isolated. The results are presented in Fig. 16b and compared with experimental data. Again, residual stresses had little influence on the energy absorption characteristics. However, including the thickness changes did affect the crush response of the tubes compared to cases in which a nominal thickness of 3.07 mm was used. The simulation for a tube with a 6 mm corner radius and nominal thickness of 3.07 mm in which work hardening was included had a mean load which was 11% greater compared to the simulation in which the predicted thickness changes were included. This showed that thinning in the corners of the tube during hydroforming could decrease the energy absorption during axial crush. The most significant decreases in energy absorption occurred when the work hardening was removed from the simulations (similar to annealing the tubes). The predicted mean loads decreased by over 30% compared to the cases in which work hardening was included, which is similar to the results obtained when comparing the measured as-formed and annealed crush responses.

8. The influence of mass and geometry on the crush response The measured mean loads were about 83 kN for the nonhydroformed, circular tubes (Fig. 14a), ranged from 50 to 55 kN for tubes formed using the low-pressure process (Fig. 14a), and varied from 65 to 75 kN for tubes formed using the high-pressure process (Fig. 14b). The masses of the non-hydroformed and low-pressure tubes were about 750 g, whereas the masses of the high-pressure tubes were about 920 g, all based on a 400 mm length tube. The large reduction in mean load, when comparing the crush response of the non-hydroformed, circular tubes with tubes formed using the low-pressure process resulted mainly from the difference in deformation mode, offset somewhat by the work hardening resulting from the hydroforming process. The circular tubes deformed with a non-axisymmetric (diamond mode) of crush and the hydroformed tubes deformed with a symmetric mode of crush. The following theoretical equation was developed by Abramowicz and Jones [16] to predict the mean load (Pm) for the diamond crush mode of circular tubes,

Pm ¼ 24:87s0 D1=3 t 5=3

(2)

where, D is the mean diameter of the tube, t is the tube thickness, and s0 is the energy equivalent flow stress. Similarly, a theoretical equation used to predict the mean load for a square tube undergoing symmetric crush, developed by Abramowicz and Jones [17], is given by,

Pm ¼ 9:53s0 w1=3 t 5=3

(3)

Fig. 12. (a) Comparison of measured and predicted crush load versus distance response for non-hydroformed, circular tube and tube formed using low-pressure hydroforming process, (b) comparison between as-formed and annealed tubes formed using high-pressure hydroforming process, and (c) sled-track velocity versus crush distance for measured cases.

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Fig. 13. (a) Predicted and (b) experimental folding behaviour for tubes formed using the low (left) and high (right) pressure hydroforming operation.

where, w is the cross-sectional width of the square tube. Alternatively, Eq. (4) can also be utilized to describe the mean crushing force of a rectangular column [18],

Pm ¼ 12:16s0 t 2

w0:37 t

:

(4)

The tubes formed using the low-pressure process had a crosssectional width that ranged from about 83e90% of the original diameter of the tube and as mentioned above, the tube thickness did not change during forming. Eqs. (2) and (3) can be used to show that, with no work hardening effects due to forming considered in the calculation of the flow stress, the expected reduction of mean load due to change of crush mode would be from 83 kN to about 30 kN. If Eq. (4) is used instead, then the predicted mean loads of the low-pressure hydroformed tubes would be between 40 and 44 kN. However, the previous section demonstrated that work hardening would increase the mean crush loads by about 16% (Fig. 16a), suggesting theoretical mean loads of about 35 kN, when using Eq. (3), and between 46 and 51 kN, when using Eq. (4) for the hydroformed tubes. The measured values of 50e55 kN are slightly greater than the predictions obtained using Eq. (4). These results indicate that the energy absorption characteristics of a hydroformed tube can significantly decrease compared to a non-hydroformed, circular tube of the same mass due to change of crush mode. However, the loss of energy absorption is partially compensated for by the increase in energy absorption due to work hardening during forming. It is noted that the theoretical predictions are limited because equations developed for square columns were applied to describe the crushing behaviour of square tubes with rounded corners.

End-feeding during the high-pressure hydroforming process increased the mass of the tubes formed using the high-pressure process to about 920 g (for a 400 mm length). Thus, even though the mean loads for tubes formed using the high-pressure process were 30e36% greater than those for tubes formed using the lowpressure process, there was also a 23% increase in the mass of the tubes. In order to compare the energy absorption characteristics of tubes formed using the two processes, finite element simulations were performed using tubes of the same final mass after hydroforming. This comparison would be expensive to accomplish experimentally because tubes with varying initial diameter and thickness would be required to produce tubes with the same mass after forming (and trimming) and would require additional hydroforming tooling. Table 4 describes five simulations developed to model the crush response of a tube with a 12 mm corner radius and mass (after forming and trimming) of approximately 920 g for a 400 mm length tube. In all cases, after the hydroforming simulation another simulation was performed to incorporate fold initiators on the tube, followed by the impact simulation. Simulations #1 and #3 correspond to the predicted response presented in Fig. 14b for tubes hydroformed to a 12 mm corner radius with 60 mm of end-feed, with and without annealing. The energy absorption characteristics of a tube annealed after hydroforming would be similar to that of an extruded (and annealed) tube in which no prior forming history would exist. The current experiments performed using the low-pressure hydroforming operation were conducted in a die of cross-section 65.5 mm and corner radius of 12 mm. In order to produce a tube of 920 g using this process, an initial tube thickness of about 3.7 mm was required. A simulation was performed based on this initial

Fig. 14. Mean crush loads for AA5754 tubes formed using (a) low-pressure and (b) high-pressure hydroforming process.

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Table 3 Average percent change of predicted mean crush load relative to measured mean crush load for tubes formed using the high-pressure hydroforming process based on various descriptions of the constitutive material behaviour. Description

Rate insensitive von Mises yielding with isotropic hardening (%)

Kinematic hardening with von Mises yielding (%)

Strain-rate sensitive hardening with von Mises yielding (%)

Anisotropic yielding (%)

Combined e anisotropic yielding with kinematic and rate sensitive hardening (%)

As-formed tubes Tubes that were annealed after hydroforming

1.1 8.6

4.7 4.8

3.2 10.9

8.7 3.9

9.1 1.4

thickness and the result is given in Table 4 as Simulation #4. In order to produce a 400 mm length tube of about 920 g with a 76.2 mm cross-section when using the low-pressure operation without end-feeding, a tube with an initial diameter of 87.3 mm and a thickness of 3.11 mm was required. The result corresponding to this tube geometry is given as Simulation #2 in Table 4. Simulation #5 was performed based on a tube with an initial diameter of 65.5 mm and thickness of 3.7 mm, formed using the high-pressure process with 60 mm of end-feed at each end of the tube. This was required in order to produce a tube of 65.5 mm width with a mass of approximately 920 g. Table 4 shows that a 97% increase in the mean loads between Simulations #1 and #5 was obtained, even though the mass of the tubes was the same. Eq. (3) can be used to help explain these results. In order to obtain higher mean loads during axial crush, the cross-sectional width should be decreased to increase the thickness, while maintaining the same mass. Also, a greater amount of work hardening (s0) would increase the mean load, which can be obtained by using the high-pressure hydroforming process compared to the low-pressure process for which there is less work

hardening. This behaviour is confirmed with the results presented in Table 4 where Simulation #5 resulted in the highest mean load. Of all the simulations in Table 4, Simulation #5 had the greatest mean load with the smallest cross-sectional width, largest thickness, and greatest amount of work hardening. Future work should explore methods to determine the optimal geometry and work hardening in order to maximize the energy absorption characteristics of the structure. It should also be noted that the constitutive model (von Mises) utilized in these simulations did not account for the fracture behaviour of the material. When high levels of work hardening are predicted during forming (such as for Simulation #5) and the material is near the point of failure, it becomes important to capture the fracture behaviour of the material in the models. Future work should consider utilizing a fracture or damage-based model in both the hydroforming and impact simulations under such conditions. Since it was shown that a large range in axial crush energy absorption could be obtained for tubes of the same mass by varying the geometry, thickness, and amount of work hardening, caution should be exercised when trying to compare the energy absorption

Fig. 15. Predicted axial crush response of hydroformed tubes. (a) BLT or FI elements with 4 mm mesh. (b) 2 mm or 4 mm mesh sizes with BLT elements. (c) 2 mm or 4 mm mesh sizes with FI elements. (d) BLT elements for axial crush when using either BLT or FI elements for hydroforming.

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Fig. 16. Effect of forming history on crush response for tubes formed using (a) low- and (b) high-pressure hydroforming process.

Table 4 Predicted crush response of AA5754 tubes with a mass of 920 g. Simulation

Hydroforming End-feed displacement Initial tube process e per end (mm) diameter (mm)

Initial tube thickness (mm)

Tube width (mm)

Mean crush load (kN)

#1

High pressure 60 e annealed Low pressure 0 High pressure 60 Low pressure 0 High pressure 60

76.2

3.07

76.2

49.2

0.0

87.3 76.2 76.2 65.5

3.11 3.07 3.70 3.70

76.2 76.2 65.5 65.5

58.1 67.2 77.7 97.1

18.1 36.6 57.9 97.4

#2 #3 #4 #5

characteristics of tubes with different geometry and materials. For materials of different density, the following approach based on the theoretical equations is suggested to compare the energy absorption capabilities. For a square tube with no corner radius, the mass, m, can be approximated by,

m ¼ 4wlt r

(5)

where, l is the length of the tube, w is the width of the tube, t is the thickness of the tube, and r is the density of the material. For two tube materials with different density, but equal mass, length, and width, Eq. (5) can be used to show that,

t2 ¼ t1

r1 r2

(6)

The energy absorption, EA, of the tube can be calculated as, EA ¼ Pml, where the mean load for a square tube can be described by Eq. (3). The predicted energy absorption for two materials with different density can be compared using Eqs. (3) and (6), resulting in the following expression,

 EA2 ¼

s02 r2 5=3



!

r1 5=3 EA1 s01

(7)

This relationship shows that the material with the highest energy absorption capabilities (neglecting effects such as strainrate sensitivity and anisotropy) can be determined by selecting the material with the highest value for the parameter, s0 =r5=3 . This relationship should be validated in future work by comparing the axial crush response from testing of materials such as high-strength steel to that of aluminum.

9. Conclusions In general, finite element simulations performed using the von Mises yield criterion with isotropic material behaviour gave

Percent change of mean load relative to Simulation #1

reasonable predictions of the measured crush response. From these simulations, it was found that the work hardening resulting from the hydroforming operation was the most important factor to be considered in the axial crush simulations, followed by the thinning in the corners of the tube resulting from the hydroforming operation. It is therefore important to transfer the predicted forming history to models of axial crush. The energy absorption during axial crush increased when including work hardening and decreased when including thickness changes. Residual stresses from the hydroforming operation were shown to have little effect on the energy absorption. For the range of tube geometry considered in this work, it was found that a tube with the smallest width, largest thickness, and highest amount of work hardening due to the hydroforming operation had the greatest energy absorption characteristics during axial crush. Acknowledgements The authors wish to thank the Aluminum Technology Centre of the National Research Council of Canada for performing the hydroforming and the General Motors Technical Center for performing the axial crush experiments. The authors would also like to acknowledge many useful discussions with C.H.M. Simha and D. Oliveira from the University of Waterloo. This research was funded by General Motors of Canada Limited, the Natural Sciences and Engineering Research Council of Canada, and the Aluminum Technology Centre of the National Research Council of Canada. References [1] Hanssen AG, Langseth M, Hopperstad OS. Static and dynamic crushing of square aluminum extrusions with aluminum foam filler. Int J Impact Eng 2000;24:347e83. [2] Lee M, Kim D, Kim C, Wenner ML, Wagoner RH, Chung K. Spring-back evaluation of automotive sheets based on isotropic-kinematic hardening laws and non-quadratic anisotropic yield functions. Part II: characterization of material properties. Int J Plasticity 2005;21:883e914.

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[3] Oliveira DA, Worswick MJ, Grantab R, Williams BW, Mayer R. Effect of forming process variables on the crashworthiness of aluminum alloy tubes. Int J Impact Eng 2006;32:826e46. [4] Tarigopula V, Langseth M, Hopperstad OS, Clausen AH. Axial crushing of thinwalled high-strength steel sections. Int J Impact Eng 2006;32:847e82. [5] Bardi FC, Yun HD, Kyriakides S. On the axisymmetric progressive crushing of circular tubes under axial compression. Int J Solids Struct 2003;40:3137e55. [6] Langseth M, Hopperstad OS, Hanssen AG. Crash behaviour of thin-walled aluminium members. Thin Wall Struct 1998;32:127e50. [7] Langseth M, Hopperstad OS, Berstad T. Crashworthiness of aluminium extrusions: validation of numerical simulations, effect of mass ratio and impact velocity. Int J Impact Eng 1999;22:829e54. [8] Otubushin A. Detailed validation of a non-linear finite element code using dynamic axial crushing of a square tube. Int J Impact Eng 1998;21:349e68. [9] Ryou H, Chung K, Yoon J, Han C, Youn IR, Kang TJ. Incorporation of sheetforming effects in crash simulations using ideal forming theory and hybrid membrane and shell method. J Manuf Sci E Trans ASME 2005;127:182e92. [10] Kellicut A, Cowell B, Kavikondala K, Dutton T, Iregbu S, Sturt R. Application of the results of forming simulation in crash models. In: Gelin JC, Picart P, editors. Proceedings of numisheet ’99; 1999. p. 509e16.

[11] D’Amours G, Rahem A, Mayer R, Williams B, Worswick M. Crashworthiness of aluminum tubes; part 1: hydroforming at different corner-fill radii and end feeding levels. In: Cesar de Sa JMA, Santos AD, editors. Proceedings of numiform; 2007, p. 787e792. [12] Williams BW. A study on the axial crush response of hydroformed aluminum tubes. Ph.D. thesis, University of Waterloo; 2007. [13] Williams BW, Oliveira DA, Simha CHM, Worswick MJ, Mayer R. Crashworthiness of straight section hydroformed aluminium tubes. Int J Impact Eng 2007;34:1451e64. [14] Ls-Dyna. LS-DYNA theory manual. LSTC; 2006. [15] Williams BW, Simha CHM, Abedrabbo N, Worswick MJ, Mayer R. Effect of anisotropy, kinematic hardening, and strain-rate sensitivity on the predicted axial crush response of hydroformed aluminum alloy tubes. Int J Impact Eng 2010;37:652e61. [16] Abramowicz W, Jones N. Dynamic axial crushing of circular tubes. Int J Impact Eng 1984;2:263e81. [17] Abramowicz W, Jones N. Dynamic axial crushing of square tubes. Int J Impact Eng 1984;2:179e208. [18] Wierzbicki T, Abramowicz W. The mechanics of deep collapse of thin-walled structures, structure failure. Wiley; 1989.