Buckling of beverage cans under axial loading - · PDF fileBuckling of beverage cans under axial loading ... compression of beverage cans while LS-Dyna predicted buckling in a few

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Buckling of beverage cans under axial loading

Vishwanath Hegadekatte1 and Yihai Shi

2

1Novelis Inc. c/o Aditya Birla Science and Technology Company Ltd., Plot No. 1 & 1-A/1, MIDC,

Taloja, Navi Mumbai – 410208, India 2 Novelis Global Technology Centre, 945 Princess Street, Kingston, ON Canada K7L 5L9

Abstract: Wrinkling is one of the major defects in sheet metal forming processes. It may become a

serious obstacle to implementing forming process and assembling the parts, and may also play a

significant role in the wear of the tool. Wrinkling is a local buckling phenomenon that results from

compressive stresses (compressive instability) e.g., in the hoop direction for axisymmetric systems

such as beverage cans. In the present work, we have studied the buckling of ideal (no imposed

imperfections like dents) beverage cans under axial loading both by laboratory testing and finite

element analysis. Our laboratory test showed that 2 out of 11 cans fail by sidewall buckling. We

have developed finite element models to study the effect of a couple of manufacturing parameters

on the buckling of beverage cans. Further we have studied the buckling of dented beverage cans

under axial compression through both laboratory testing and finite element analysis using Abaqus

and LS-Dyna. Our results show that Abaqus did not predict sidewall buckling during axial

compression of beverage cans while LS-Dyna predicted buckling in a few cases.

Keywords: Buckling, Experimental Verification, Forming, Plasticity, Shell Structures, Springback.

1. Introduction

Wrinkling is a local buckling phenomenon that results from compressive stresses (compressive

instability) in the hoop direction for axisymmetric systems. Wrinkling is one of the major defects

in sheet metal forming processes. It may be a serious obstacle to implementing the forming

process and assembling the parts, and may also play a significant role in the wear of the tool.

Therefore a good understanding of the buckling/wrinkling phenomenon is needed to effectively

overcome the issues arising from such defects in sheet metal forming.

This article summarizes the work done within the Novelis global R&D organization on prediction

of wrinkling in axisymmetric (in terms of geometry) systems with material parameters and friction

being assumed to be isotropic. This work was initiated after Mao & Santamaria (2009) published a

paper at the 2009 Simulia Customer Conference where the authors simulated wrinkling using a 3D

model of the deep drawing of a cup and used it to optimize the tooling and forming parameters.

Making use of symmetry, Mao & Santamaria (2009) modeled a 15o segment of the cup drawing

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process with 4 noded reduced integration shell elements and a fine mesh density. They showed

that their model “duplicated wrinkling and thinning seen in real life”.

This paper raises some important questions, for instance:

1. What initiates wrinkling in the model?

2. Is this equivalent to what happens in the actual process?

3. Are the number, wavelength and amplitude of wrinkles predicted correctly?

4. If we were to predict wrinkles in e.g., die-necking process then what should be the

symmetry segment in the model one should use? Is always a 15o segment the right

symmetry to use for such axi-symmetric systems?

5. Is the prediction sensitive to the numerical precision chosen for the simulation and/or

mesh density and/or type of elements?

To find answers to the above important questions we started the current research with the

following aim:

1. Determine the key parameters influencing the predictions of wrinkling using FE

packages.

2. Validate predictions of wrinkling using laboratory testing.

The initiation and growth of wrinkles may be influenced by many factors. Some are purely

numerical in nature like solver type – implicit or explicit, numerical precision, element

formulation, hourglass control, mesh density, mesh regularity – mapper or free, symmetry

boundary conditions etc. while others have a physical basis such as surface roughness,

longitudinal score-lines, initial off-flatness of the sheet, coating effects, tool surface imperfection,

hard particles, wall thickness variation to name a few. In the current article we present a detailed

discussion on the results from modeling and laboratory experiments of axial compression of

beverage cans where a few of the above parameters were tested.

2. Physical testing and modeling of the axial compression of beverage cans

In this section we discuss the results from the physical testing of the axial compression of the

beverage can and subsequently the results from the modeling of this test using both LS-DYNA

and Abaqus/Explicit will be presented. For the physical testing and finite element modeling, both

ideal (without any imposed imperfection) and “imperfect” (dented, varying thickness, slant trim

edge) cans were used.

2.1 Laboratory testing of the axial compression of an ideal beverage can

The axial compression test was carried out on an Instron tensile testing machine (see Figure 1 (a)).

The beverage can was placed on the bottom plate with a preload applied from the top plate to

eliminate any initial gap. Then, the displacement-controlled axial loading was applied to push the

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top plate down at a rate of 6 mm/min (=0.003937 in/s) to compress the beverage can until the can

began to buckle. The reaction force (axial load) as a function of the downward displacement of the

(a)

(b)

Figure 1: (a) Can axial load test setup and (b) typical load displacement curve for ideal geometry of the can.

top plate was recorded. When an ideal can is used in the test, the load- displacement curve is as

shown in Figure 1 (b). This load-displacement curve is characteristic of what is termed as “base

squat” response in can making jargon. Base squat essentially describes the failure of the shoulder

region of the can bottom. The peak load in Figure 1 (b) corresponds to the critical load for base

squat which was measured to be approximately 270 lbf. An ideal (free from imperfections)

beverage can under axial compression would first fail from base squat (see Figure 2) and if the

axial compression is continued further then it would begin to buckle at the side wall. Even though

the thickness of the can side wall is approximately 100 microns compared to approximately 250

micron thickness of the can bottom, for an ideal geometry, the bottom fails first in the shoulder

region which indicates that can bottom design is the key factor for axial strength of the beverage

can. The reason being the side wall is work hardened after the ironing operation and therefore is

comparatively stronger. However if there is any imperfection in the can side wall then it could fail

by side wall buckling before the shoulder region would begin to fail.

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Figure 2: Deformed base of the can after the axial load test that failed by base squat.

During the axial compression of cans with ideal geometry, it was found that 2 out of 11 cans failed

due to side wall buckling while the others failed due to base squat as shown in Figure 3. The

Figure 3: Deformed cans after the axial load test. 2 out of 11 failed due to side wall buckling while the rest failed due to base squat.

failure of cans can be due to any of the following reasons like unintended dents on the side wall,

material anisotropy, slant trimmed edge, thickness variation in the circumferential direction of the

can, inclusions, surface roughness, longitudinal scorelines, transverse surface tears, initial off-

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flatness of sheet, presence of hard particles. The thickness variation along the axial direction of the

can was measured and is shown in Figure 4 (a). It should be noted that this variation in the

(a)

(b)

Figure 4: (a) Variation of wall thickness along the (a) axial direction (in terms of absolute value) and (b) circumferential direction (in terms of percentage values).

thickness along the length of the can is by design. However, the thickness variation in the

circumferential (hoop) direction may be due to imperfect tool geometry during the forming

process. In Figure 4 (b) we present the measured thickness variation in the circumferential

direction. It can be seen from this graph that the maximum circumferential wall thickness variation

is approximately 10% which in absolute terms is approximately 8 – 13 micron.

2.2 Finite element modeling of the axial compression of an ideal can

In this section, we discuss the possible reasons for the 2 ideal cans that failed due to side wall

buckle during physical testing as presented in Figure 3. For our study, we consider two probable

scenarios where we tested the circumferential thickness variation and a slightly slant trim edge.

Generally, three types of elements are employed in sheet metal forming simulations, i.e.

membrane element, continuum element and shell element. Membrane elements have been widely

used to model the forming processes, due to its simplicity and lower computation time, especially

in the inverse and optimization analysis where many iterations of forming are required. However,

it does not include bending stiffness, therefore, it may not be appropriate in cases where one has to

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deal with the buckling phenomena. In general, the bending-dominant processes are simulated by

the continuum or shell elements. In continuum analysis, the bending effect can be taken into

(a)

(b)

Figure 5: (a) Schematic of continuum shell element and its usage in can modeling. (b) Comparison of load-displacement curves obtained using continuum (CS3D)

and conventional axi-symmetric continuum elements (CAX).

account by having multiple layers of elements through the thickness. However, this leads to

extremely large computation time especially for three-dimensional problems. Shell elements may

be considered as the compromise between the continuum and membrane elements. It is possible to

take into account the effect of bending with much less computation time than continuum analysis

although integration in the thickness section is still needed (Wang & Cao, 2000). Therefore, using

shell elements in an explicit code is an attractive proposition for studying wrinkling.

Abaqus provides a new type of shell element which it calls the “continuum shell element” that can

be used to discretize an entire three-dimensional body as shown in Figure 5 (a). The thickness is

determined from the element nodal geometry. Continuum shell elements have only displacement

degrees of freedom. From a modeling point of view continuum shell elements look like three-

dimensional continuum solid elements, but their kinematic and constitutive behavior is similar to

conventional shell elements. User has to take care of properly defining the thickness direction

when using continuum shell elements.

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In our study we used continuum shell elements to model the axial compression of beverage cans.

Since this is a fairly new element, we bench marked this element against the conventional axi-

symmetric continuum element. It should be noted that with the continuum axi-symmetric element

the beverage can was modeled in 2D while with the continuum shell element and the conventional

shell element it was modeled in 3D. It can be seen from Figure 5 (b) that with continuum axi-

symmetric element and continuum shell element model we got a peak load (critical load for base

squat) of approximately 350 lbf (1.58 kN). Further we also modeled the axial compression of the

beverage can with conventional shell elements in 3D. The load-displacement response from the

model showed that the critical load predicted by the conventional shell element is close to 400 lbf

(1.75 kN) It should be noted that the critical load predicted by the model (350 - 400 lbf) is quite

higher than the measured value of approximately 270 lbf presented in Figure 1 (b). It should be

noted that the geometry used for all the three element types is exactly the same. In order to speed

up the simulation we used the time scaling technique by trying different loading rates of 0.25 in/s,

25 in/s and 250 in/s. Time scaling can be appropriate for this case as the material model is not rate

dependent. We found that we got the same predicted response for loading rates of 0.25 and 25 in/s

while with 250 in/s loading rate we obtained a wave propagation response due to increased inertial

forces. Therefore we chose a loading rate of 25 in/s which are approximately 6000 times higher

than the loading rate used in the laboratory test. This loading rate enabled us to speed-up the

calculations while giving a satisfactory response. To further reduce the computational time we

have used one half of the beverage can making use of symmetry.

2.2.1 Circumferential thickness variation

In this section the model prediction for the onset of buckling for different circumferential

thickness variations is presented. The circumferential thickness variation in the model was

achieved by shifting the internal surface of the can in relation to the outer surface as shown in

Figure 6 (a). As a result there is a uniform variation in the thickness around the circumference of

the can where the maximum percentage thickness variation is calculated using the relation in

Figure 6 (a).

When such a can with varying circumferential thickness is compressed in the axial direction under

displacement control, the response is as shown in Figure 6 (b) and Figure 7. It can be

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(a)

(b)

Figure 6: (a) A schematic showing the calculation of the circumferential thickness variation and (b) the predicted load-displacement curves for various thickness

variations.

seen from these two figures that atleast a 50% variation in the circumferential thickness is needed

for the can sidewall to buckle before base squat. The measured circumferential thickness variation

Figure 7: Predicted response of the can under compression for various circumferential thicknesses.

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had a maximum value of 10% which is clearly quite low compared to what is required as per the

model predictions. It should be noted that the stress-strain curve used for the entire model is that

for the work hardened side wall obtained from a tension test on a specimen cut from the side wall.

But a non work hardened material model for the can bottom would have increased the propensity

for the can to fail from base squat. Therefore it can be safely concluded that the circumferential

thickness variation was not the reason for the 2 cans that failed by side wall buckling during the

physical testing.

2.2.2 Trim edge flatness variation

In this section we present the modeling predictions for the onset of buckling under axial

compression of beverage cans with slant trim edges. In Figure 8 (a), the schematic of the axial

load test of a can with a slant trim edge is shown. The slope, m of the trim edge is written in terms

of the thin wall thickness, t and the diameter of the can, D as m=nt/D where n=1, 2, 3, …. The

load-displacement response from the model for such a can with a slant trim edge undergoing axial

compression is presented in Figure 8 (b). It can be seen from this graph and Figure 9 that for trim

(a)

(b)

Figure 8: (a) Schematic showing the can with a slant trim edge and the set up for the axial load test. (b) The predicted load-displacement curves for various slopes

of the trim edge.

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Figure 9: Predicted response of the can under compression for various slopes of the trim edge.

edge slopes of, m>3t/D, the can begins to buckle at the side wall. 3t is approximately 300 microns

which is quite small compared to the typical diameter of a can of 65 mm which means that if the

slope of the trim edge is greater than 1 in 250, an ideal beverage can will buckle at the side wall

under axial compression. This imperfection seems to be a more likely cause (compared to the

circumferential thickness variation) for the 2 cans that failed due to side wall buckling during the

physical testing.

However, it should be noted that in our study we only modeled two of the several possible causes

for such buckling failures. Other notable causes include material anisotropy, surface roughness,

initial off flatness of the sheet etc. which were not selected for our modeling exercise.

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2.3 Laboratory testing of axial compression of dented cans

Axial compression of dented cans was conducted at Novelis Global Technology Centre. For

creating the dent on the beverage can sidewall, a spherical tipped pin with a radius of 1 mm was

used. Dents were created at 1, 2, 3 and 4 inch from the shoulder as shown in Figure 10 (a). No die

(a)

(b)

(c)

Figure 10: (a) Cross section profile of the can as measured using …. along with the spherical tipped pin to create the dent on the side wall. Buckled cans with a side wall dent at 1 inch (b) and 4 inch (c) from the shoulder after the axial load test.

was used from the inside of the can to locate the denting pin and create a perfect spherical dent,

instead from the inside a manual support for creating the dents was adopted. This means that the

created dents were not perfect but we can still use this imposed imperfection to get a trend for the

buckling behavior. In Figure 10 (b), we can see that the buckling initiates at the dent when the dent

location is 1 inch from the shoulder. The video recording of the test also confirm that buckling

initiated at the dent locations for dents at 1, 2 and 3 inches from the shoulder. However, when the

dent was at 4 inches from the shoulder, the buckling initiated at the middle of the can, away from

the dent as shown in Figure 10 (c).

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Figure 11: The measured load-displacement curves for various locations of the dent.

In Figure 11, the measured load-displacement curve for various locations of the dent is shown.

The graph clearly shows that with a dent on the sidewall (imposed imperfection), the can fails

from buckling (drop in the slope of the load-displacement curve) well before the critical load for

base squat is reached indicated by the peak in the load-displacement curve for the ideal can (

“Normal Can”) in the above graph.

2.4 Finite element modeling of the axial load test of dented cans

Load-displacement response from LS Dyna and Abaqus model of the axial compression test of

dented cans will be presented in this section.

In Figure 12 we present the load-displacement response predicted by LS-Dyna for dented cans

with dents at 1, 2 and 3 inches from the shoulder of the can bottom. For comparison the load-

displacement curve predicted for an ideal can (NoDent) is also presented in the same graph. It can

be seen from Figure 12 that LS-Dyna predicts sidewall buckling for dents at 1 in and 2 in from the

shoulder which is qualitatively in agreement with experimental data presented in Figure 11.

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Figure 12: The predicted load-displacement curves from LS-Dyna with dents as 1 in (Pos1), 2 in (Pos2), 3 in (Pos3) from the shoulder of the can bottom

However, when the dent is at 3 in from the shoulder of the can bottom, LS-Dyna does not predict

sidewall buckling. Further the critical load for base squat (peak load in the load-displacement

curve for the ideal can) predicted by LS-Dyna is approximately 0.98 kN which is quite lower than

the experimental value of approximately of 1.2 kN as shown in Figure 11. We have developed a

(a)

(b)

Figure 13: The predicted load-displacement curves from Abaqus for (a) ideal can and (b) for a dented can with the dent at 1 in from the shoulder of the beverage

can.

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python code to translate the LS-Dyna input deck to an Abaqus input file. In Figure 13 (a), the

load-displacement curve using Abaqus for an ideal beverage can under axial compression is

presented. This load-displacement curve compares favorably with the results from LS-Dyna.

(a)

(b)

Figure 14: (a) Finite element model showing the forming of the dent on the sidewall (b) the predicted load-displacement curves for an ideal and dented can.

However, Abaqus did not predict sidewall buckling (no drop in the slope of the load-displacement

curve) for dented beverage can for dents located at any position along the axial direction. It should

be noted that identical mesh, material property, boundary conditions, loading cases and default

solver settings for both Abaqus and LS-Dyna model were used in our studies.

In order to understand the reason for Abaqus not predicting sidewall buckling, we developed a half

symmetry 3D model of the beverage can used in the laboratory tests. The geometry of the

beverage can used in the laboratory test was measured using a CMM co-ordinate measuring

machine. The 3D model was built using continuum shell elements and in order to make the model

more accurate we modeled the denting of the sidewall (see Figure 14 (a)). After extracting the

geometry of the dented can from this model it was then used in the axial compression model. The

predicted load-displacement curve from the axial compression model is shown in Figure 14 (b). It

can be seen in Figure 14 (b) that Abaqus does not predict sidewall buckling even when there is a

dent in the sidewall. The load-displacement curve is infact identical to the corresponding curve

predicted for an idea beverage can. Analyzing the dent in the model we noticed that the perimeter

of the dent

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(a)

(b)

Figure 15: (a) Finite element model incorporating a sharp perimeter for the side wall dent and (b) the predicted load-displacement curves for an ideal and dented

can.

was “smooth”, therefore we decided to change this to a “sharp” perimeter as shown in Figure 15

(a). With this type of dent, when we axially compressed the beverage can, Abaqus predicted side

wall buckling as seen by the drop in the slope of the load-displacement curve presented in Figure

15 (b). This indicates that the predictions from Abaqus are not mesh independent. Since we did

not conduct such tests with LS-Dyna, we do not have anything to comment on LS-Dyna‟

performance in this respect.

3. Conclusion

In this article, we have shown that for ideal beverage cans, a thickness variation of at least 50 %

(w. r. t. thinwall ) is needed for the can to fail from sidewall buckling which is much more than the

measured wall thickness variation in the circumferential direction. However, a slope of slightly in

excess of 1 in 250 for the tilt in the trim edge is sufficient for the can to fail from buckling. We

believe that the latter is a more likely cause for the two out of eleven ideal cans that failed due to

sidewall buckling in our laboratory tests.

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Abaqus did not predict sidewall buckling during axial compression of dented cans while LS-Dyna

predicted buckling in at least a few cases. The load-displacement curve predicted by Abaqus when

tried with different element types yielded different results even when everything else remained

constant. Abaqus consistently predicted higher critical load compared to LS-Dyna for the same

mesh, material property, boundary conditions and loading cases. We are not sure if this is because

of any difference in the default solver settings that we used for the two FE packages.

4. References

1. Mao, K. & Santamaria, A. (2009). Aluminum bottle forming simulation with Abaqus. 2009

Simulia Customer conference.

2. Wang, X., Cao, J. (2000). On the prediction of side wall wrinkling in sheet metal forming

processes. Int. J. Mech. Sci., 42, 2369 – 2394.