Robustness of the Sheet Metal Springback Cup Test

T. FOECKE and T. GNAEUPEL-HEROLD

The robustness of a proposed test for elastic springback characterization of sheet metal has been

examined using a matrix of defined experimental errors. A series of flat bottom deep drawn cups

made from AISI 1010 steel sheet were examined. It was found that misalignment of the blank over

the forming tool and error in the vertical location where the springback ring was cut from the cup

sidewall had the largest effect on the resulting springback opening. Other experimental errors involving cup height and ring width were found to be less important. The effect of in-plane anisotropy of

mechanical properties on springback was negligible. The results are examined in terms of measured

through thickness residual stresses and elastic bending of beams with circumferential thickness

gradients.

I. INTRODUCTION

SPRINGBACK is the elastic shape change of a sheet

metal part after forming and removal from the die. In the

automotive industry, the poor predictability of springback

has always been a problem, but it was somewhat alleviated

by the use of rather thick walled, low strength steels and

by the accumulation of empirical methods of compensation

during die design and rework. Due to the need for weight

reduction, there is a major shift underway to use thinner

high-strength steels and aluminum alloys. In both cases,

these materials exhibit larger and unpredictable springback

magnitudes, thus making an accurate prediction all the

more necessary in order to avoid a costly redesign of the

stamping tool.

Modeling efforts in die design are beginning to treat

springback in the same way that other phenomena such

as forming limits and friction have been handled, through

the modeling of a standard test geometry to verify theoretical predictions. There have been attempts to measure

springback deflections in near-production parts,[1] but often

these measurements are difficult to perform because the displacements can be small and body forces (gravity) often distort the part, so that fixturing becomes very important and

is difficult to do in a similar manner for each differently

shaped part. There is a need for a standard test that will characterize a material’s springback properties in a quick and

consistent way.

For a standard test, it is desirable that it be easy to perform, repeatable and in the case of this particular need,

have a large springback component that can be easily measured. A test that has received considerable attention[2–6]

consists of a ring sample taken from the sidewall of a flat

bottom, deep drawn cup. The test geometry is presented

graphically below in Figure 1. This geometry provides an

axisymmetric sample with a large residual hoop stress that,

when the ring is split, provides a large opening displacement. Residual stresses in the cup exist because different

locations in the cup have accumulated different magnitudes

of plastic strain during the draw-bend-unbend process.

These stresses are key for the modeling of the springback

because, integrated over the thickness, they create a bending moment in the split ring, and thus the shape change.

In preparation for submitting this test to the standardization process, a variational analysis has been performed

to study the robustness of the springback measurement

to experimental errors. A number of different experimental errors that might be expected during forming and slicing

the samples were introduced systematically, and the resulting change in springback opening was measured. Likely

causes for any changes in observed springback opening in

the rings are examined and analyzed in terms of measured

residual stress profiles and ring thickness variations.

II. EXPERIMENTAL PROCEDURE

The cups were made in the Forming Laboratory of the

NIST Metallurgy Division using a hydraulic deep drawing

machine with 500 kN capacity.[7,8] The AISI 1010 steel

blanks, with mechanical properties listed in Table I, were

circular disks of 200-mm diameter and 0.88-mm nominal

thickness. The diameter of the die bore was 110 mm with a

rim radius of 10 mm. The forming tool had a diameter of

100 mm with an edge radius of 10 mm and a recessed center. The depth of the cups was nominally 56 mm. Vegetable

oil was used as a lubricant. The forming rate was constant

at 5 mm/s, and the hold-down load was constant at 90 kN

with no draw or lock bead in use. The forming rate and

hold-down load were not varied as one of the sources of

possible error, as it is possible to accurately calibrate modern servohydraulic equipment for both of these inputs and

apply them accurately. The current study was designed to

measure the effects of likely errors introduced by experimenters and technicians in performing the test.

The errors in forming the cups that were examined were

thought to be among the more common that someone performing the test might commit. These can be divided

broadly into sectioning variables, or how the ring is isolated

from the cup and sliced open, and forming variables, which

are errors in how the cup is formed. The former include the

method used to cut the ring open, any errors in the vertical

location or width of the ring taken from the sidewall, and

where along the ring’s circumference (with respect to the

rolling direction of the sheet) the ring is cut open. The latter

T. FOECKE is with the Metallurgy Division, National Institute of Standards and Technology, Technology Administration, United States Department of Commerce, Gaithersburg, MD 20899-8553. Contact e-mail:

tfoecke@nist.gov T. GNAEUPEL-HEROLD is with the Department of

Materials and Nuclear Engineering, University of Maryland, College Park,

MD 20742-2115, and the Center for Neutron Research, National Institute

of Standards and Technology, Technology Administration, United States

Department of Commerce.

Manuscript submitted March 21, 2006.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 37A, DECEMBER 2006—3503U.S. GOVERNMENT WORK

NOT PROTECTED BY U.S. COPYRIGHT
include forming the cup too deeply or shallowly and forming an asymmetric cup because the circular blank was not

properly aligned with the punch. These variables were

tested by systematically changing each independently on

each test. As an example, when examining the effect of a

misaligned blank, the sheet sample was displaced by a

number of millimeters from a centered circle scribed on

the tooling using a micrometer, in either the rolling or

transverse direction, before forming the cup.

After forming, each cup was cleaned and placed on a

flatbed scanner so that an image of the earing profile could

be taken. Subsequent to this, the thickness profiles were

measured along the centerline of the ring to be cut from

the sidewall using a coordinate measuring machine. The

inner and outer profiles of the cups were measured at a

number of points, fit with a spline, and the spline values

were calculated every 10 deg and subtracted, giving a circumferential plot of the thickness. Prior to sectioning, the

total cup height was measured. The rings were then cut

from the sidewalls using wire electrical discharge machining (EDM), ensuring excellent location of the cuts and

parallel edges of the rings. The rings were given fiduciary

marks along the edge on both sides of the intended slice

location, approximately 1 cm apart. The rings were imaged

on the flatbed scanner, and initial fiduciary mark and diameter measurements were made. Finally, the rings were split

and imaged again, with measurements made of the final

fiducial mark spacing and ring diameter. Calculations were

made of ring opening and diameteral strain. In total, 202

cups with various intentional errors were produced and

analyzed. Standard deviations reported subsequently were

calculated using

s 5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

+x2

n  x2

r

and the correlation coefficients reported in the figure captions of the data plots were calculated using

r 5

+

n

i 5 1

ð Þ xi  x yð Þ i  y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

+

n

i 5 1

ð Þ xi  x

2 +

n

i 5 1

yð Þ i  y

2

s

III. DISCUSSION

Drawing of flat-bottomed cups is considered to be at one

end of the forming spectrum in terms of strain state,[10]

where one principal strain is positive and the other negative. During the cupping operation, the change in thickness

is small. The Swift cup test[11] was devised to determine the

limiting draw ratio (LDR) that a material can withstand,

whereby the diameter of the cup tooling was decreased to

a point where a given material would fail when partially

drawn. The parameters and geometry in this test protocol

were chosen such that materials with low LDR, and thus

low formability in general, will not produce useful test

specimens, but that materials with higher LDR and formability, and thus usefulness for automotive forming applications, could be characterized using this test. This was

primarily done through the proper choice of punch edge

radius and tooling gap between the punch and hold-down

ring. The protocol proposed that gave input to the present

investigation dictates the forming of a flat bottomed cup

56-mm tall from a 200-mm round blank. The ring is then

Fig. 1—Schematic of the sheet metal springback cup test. The top image

shows the important geometric variable for the sidewall ring, and the

bottom image denotes the rolling direction (RD) and transverse direction

(TD).

Table I. Mechanical Properties of 1010 Steel Used in

this Study; Values are Averages and Standard Deviations

over Five Tests

Mechanical Property

Rolling

Direction

Transverse

Direction

Yield strength (MPa) 217.0 6 6.7 220.8 6 7.8

Ultimate tensile strength (MPa) 331.3 6 4.1 328.3 6 2.8

Tensile strain

(1-in.gage) (pct) 41.7 6 1.9 41.2 6 1.1

Strain hardening Exponent 0.24 6 0.008 0.25 6 0.005

Strain ratio (R) 1.74 6 0.02 1.79 6 0.04

3504—VOLUME 37A, DECEMBER 2006 METALLURGICAL AND MATERIALS TRANSACTIONS A
cut from the sidewall, having a width of 15 mm centered on

a circumference located 37.5 mm down the cup from the

outside flat bottom. Following this procedure, a springback

opening of 48.2 6 2.1 mm was found for these ‘‘perfect’’

cups of 1010 steel sheet. More cups were made for which

deliberate systematic experimental errors were introduced

into how the cup was formed or how the ring was cut from

the sidewall, and the effect on the resulting springback

opening was observed. One error was introduced at a time,

and their effects are discussed in the following sections.

In the cases of geometric parameter errors, the reader is

referred to Figure 1 for visualization purposes.

For samples of all types to be discussed in this study,

there was a consistent scatter of about 10 pct about the

mean value of the ring opening (with forming/slicing

parameters constant). A possible cause of this persistent

error was found during an investigation of how the thickness profile of the sidewall might be affected by the inplane anisotropy of the mechanical properties of the sheet.

If the sheet has significant mechanical anisotropy, finite

element modeling predicts a sinusoidal thickness variation

around the circumference of the sidewall of the cup. The

cup would thin (draw more) in regions where the material is

softer, and vice versa where stronger. If the anisotropy

showed typical fourfold symmetry in the plane of the sheet,

the thickness variation measured should be sinusoidal with

a wavelength of pi radians with no phase shifts. What was

found was various samples exhibited roughly sinusoidal

thickness profiles with a wavelength of 2 pi radians, with

random phase shifts from one sample to another, as seen in

Figure 3.

The observed thickness variations are likely an artifact of

how the test is performed even under ‘‘perfect’’ conditions.

Forming a circular blank into a flat bottomed cup requires

the material to undergo a draw/bend/unbend operation. As

this is occurring, the diameter of the starting blank is

reduced as the material is drawn into the die. If the blankholder force is sufficiently large, the formation of wrinkles

will be suppressed and the frictional holding force will be

uniform. However, during actual operation, depending on

material, blankholder forces sufficient to completely suppress wrinkling are often too large to allow the cup to form

without splitting. In practice, small wrinkles form around

the periphery of the blank and are ironed out as they are

drawn. These effects translate into an unpredictable variation in the thickness of the sidewall from point to point, as

the material in the wrinkle is thicker than the neighboring

material, and results in a slightly thicker section of the

sidewall. In this study of 1010 steel, the thickness variations

due to hold-down inhomogeneities overwhelm any variations due to in plane mechanical anisotropy. This may not

be the case for a material exhibiting more mechanical anisotropy. In addition, neither the blank nor the tooling are

perfectly smooth, and local variations in frictional force

and physical contact exist, allowing more gross variations

in thickness as random locations within the blankholder

stick and release.

A. Model Considerations

The driving force for ring opening is the residual hoop

stress induced in the sidewall of the cup during the forming

process. This stress has been investigated[9] and found to

differ significantly from formalisms that model through

thickness stresses for work hardening materials.[10] It was

also found that there are variations in the hoop stress both

around the circumference and in the vertical direction, both

of which affect the springback opening. A plot of the average hoop stress measured in a thick-walled AISI-1010 cup

is seen in Figure 2. Plots are presented showing both intact

and cut rings. The component of the hoop stress that drives

opening is actually the amount that is released after slicing,

or the difference between the intact and sliced through

thickness stress profiles. This is also plotted in Figure 2.

This linear stress change allows the use of a simple model

that relates the hoop stress, the thickness, the ring radius,

and the springback opening.

In this model, we can express the stress on the outside of the

ring (which determines the stress change completely) as[10,13]

st 5 E

t

2

(r0  r) [1]

(r0 is the curvature of the intact ring, r is the curvature of

the split ring, t is the thickness, E is the Young’s modulus,

and st is the change of hoop stress on the outside of the

ring; this is identical to the intersection of the linear stress

change with the stress axis in Figure 2). A simple geometric

consideration shows that the curvature of the split ring is

related to the gap g of the split ring by

g 5 2

r

sin p  pr

r0

 [2]

Using the series expansion for the sine-function for

r=r0 # 1, we obtain for g

g ffi 2p 1

r  1r0

 5 2p 1

r0  2st=Et  1r0

 [3]

Fig. 2—Plot of residual stresses through the thickness of a ring taken from

a 1010 steel cup, showing loaded and unloaded states, and the difference.

Some error bars are not visible because they are smaller than the symbol

size. The straight line is the linear regression of the stress difference split

intact (Eq. [1]).

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 37A, DECEMBER 2006—3505
It is conceivable that preferred orientation due to rolling

and deep drawing produces circumferential variations of

the elastic modulus E, thus affecting the ring opening g.

However, as shown in Reference 12, for rolling textures, the

in-plane variations of the elastic moduli are usually small

and insufficient to produce significant stress changes. Thus,

the two variables determining the magnitude of the ring

opening are the hoop stress and the wall thickness. Evidence for stress variations was found in Reference 9, and

Eq. [1] shows that local stress variations can affect the local

curvature of the split rings. Thus, the curvature r of the split

ring may not be uniform and the variations are given by the

ratio of the local hoop stress and the local thickness.

B. Effect of Thickness Variations

The spread of local thickness around the average value

reaches 12 pct, therefore suggesting variations in the local

curvature due to the thickness effect. What was found is

that the local diameters of selected split rings (with thickness variations shown in Figure 3) are not correlated with

the thickness but rather that the split rings are near elliptical

in their shape with the largest diameters always close to

the transverse direction (thus implying a lower curvature in

the regions of the original rolling direction), as shown in

Figure 4. Relative diameter variations were ,2 pct for split

rings, while for intact rings, the variations were ,0.1 pct.

These findings suggest that local variations in thickness

are accompanied by variations in the hoop stresses such

that the curvature stays relatively unaffected. A lower thickness by itself would lower the curvature and increase the

ring opening, but this also means higher levels of plastic

strain at this location. The latter implies deformations farther beyond the yield stress, thus reducing locally the hoop

stress, which, in turn, increases the local curvature. This is

supported by the findings in Reference 9, which show

lower-than-average hoop stresses in the region of the ring

around the transverse direction and higher-than-average

hoop stresses in the region of the rolling direction.

Our findings regarding the thickness effect also provide

an answer to the question of whether the location of the cut

affects the springback opening. The material immediately

adjacent to the slice does not contribute to the bending

moment of the beam, because it is not stressed. Thus, when

considering the thickness not locally but as an effective

thickness for the entire ring, there could be an effect

whether the cut is located in a thin region or a thick region.

This is not the case because, first, there is no indication for

a thickness effect, and, second, according to Saint Venant’s

principle, these edge effects subside at a distance from the

slice comparable to the thickness (which is a small quantity).

C. Effect of Sample Sectioning Parameters

The EDM was chosen to isolate the ring from the cup

sidewall for the reasons stated in the previous section: precision and parallelism of the cuts. However, slitting open

the rings to produce the springback component can be

accomplished in a number of ways, and several of these

were compared to determine their effects on the final measurement. Slitting methods were selected based on two factors: the ease of use and the minimization of mechanical or

thermal effect on the sample. The techniques chosen and

the measured values of springback opening for each technique are presented in Table II. As seen, there was no observable influence of the chosen slitting method on the

resultant opening. It was found that as long as the slitting

method selected localized the deformation to a small section of the ring, whether by shear or material removal, and

did not introduce bending moments on the rest of the ring,

the resulting opening was unaffected.

Fig. 3—Plot of collection of circumferential thickness profiles. The uncertainty of the thickness measurement is 60.005 mm

Fig. 4—Relative diameter variations of selected split rings. Slitting of the

rings was performed using EDM. The uncertainty of the relative diameter

is 4 3 104

.

Table II. Springback Openings Resulting from Various

Slicing Methods; Values are Averages and Standard

Deviations over Five Tests

Slicing Method

Ring Opening 6

Standard Deviation (mm)

Jeweler’s saw (80 teeth per inch) 47.8 6 1.7

Shear 48.2 6 2.2

High-speed cutoff saw 47.6 6 2.4

Low-speed diamond saw 48.4 6 1.8

EDM 48.3 6 1.8

3506—VOLUME 37A, DECEMBER 2006 METALLURGICAL AND MATERIALS TRANSACTIONS A
D. Effect of Geometric Parameters

Effect of error in vertical location of ring and

ring width

The largest variation in springback opening displacement

is seen when the location where the ring is cut from the

sidewall is displaced vertically. This can be visualized in

Figure 1 as a change of the same magnitude and sign of the

distances a and b, and the resulting data are presented in

Figure 5. Again, the considerable vertical decrease of the

ring opening confirms the findings in Reference 9, where a

corresponding vertical decrease of the hoop stresses toward

the cup bottom was measured. This was attributed to a

rising level of accumulated plastic strain as the rings are

closer to the bottom. The forming parameters in Reference

9 were essentially the same as here; however, a major discrepancy exists in the vertical thickness distribution. While

the wall thickness for cups made of 3-mm-thick blanks

decreases toward the bottom, here, it increases slightly

(but is still lower than the blank thickness), as shown in

Figure 6. This is most likely a result of the tooling gap-tothickness ratio (5 to 3 mm vs 5 to 0.88 mm). The different

behavior of vertical thickness variations represents a problem insofar as, for the thick walled cup, a decrease in thickness is a clear indication of uniform vertical stretching

(rising plastic strain), and it explains the lower springback

at the cup bottom. The slight increase in thickness toward

the bottom found for the thin walled cup seemingly contradicts that, because with no rise in vertical stretching, the

springback is still lower at the bottom. The explanation lies

in the different circumferential plastic flow of material during the deep drawing process. For example, the thickness

variations of the thick walled cup are approximately 3 pct,

while the thin walled cups exhibit variations up to 12 pct,

which indicates that the level of tangential plastic strain is

generally higher in the thin walled cups as compared to

thick walled cups where higher strains are predominantly

caused by vertical stretching. More evidence for higher

circumferential strains in thin walled cups is apparent in

the standard deviations of the thickness profiles, which increase toward the cup bottom, as shown in Figure 7. The

thickness standard deviations are a measure for how much

the wall thickness varies at a given vertical position. Higher

standard deviations mean larger thickness differences, which

implies an increased circumferential material flow and

higher strain levels. The latter have the effect of reducing

springback as more material is deformed beyond the yield

stress.

In the same manner as the vertical ring position, an error

in the width of the ring can also affect the resulting springback, but to a much smaller degree. In this case, the ring cut

from the sidewall is still centered on 37.5 mm down from

the flat outside of the cup, but the width is varied (in Figure

1, change a and b the same amount, with opposite sign).

The effect is plotted in Figure 8. The slight decrease in

opening for increasing ring width is a result of the more

pronounced conical shape of the ring. Both the wall thickness and the initial curvature increase toward the cup bottom with the result that the ring opening is not simply the

average of values from Figure 5, but it is somewhat dominated by the near-bottom portions of the ring.

E. Effect of Blank Misplacement

Due to the overall configuration of the tooling needed to

perform the test, one of the more likely expected errors

might be the misplacement of the blank with respect to a

Fig. 5—Plot of effect of vertical ring location displacement on springback

opening. Each value for the ring opening represents the average of nine

rings. The correlation coefficient is –0.99.

Fig. 6—Plot of thickness standard deviations over the vertical position.

The correlation coefficient is 0.97.

Fig. 7—Thicknesses at different vertical locations on cup sidewall. Each

data point represents the average of thicknesses of eight rings sectioned at

the same locations. The correlation coefficient is 0.88.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 37A, DECEMBER 2006—3507
position centered on the punch. This would result in several

effects on the formed cups. The earing pattern from a misplaced blank shows the fourfold symmetry seen in the correctly formed cups described previously (Figure 9(a)), but

superimposed upon this is a twofold symmetry from the

misalignment (Figure 9(b)). In cases of severe misalignment (in excess of 5 mm), one side of the blank escapes

the hold-down platen before the other side, and the cup is

severely misformed. A series of cups were produced with

prescribed misalignments over the center of the forming

ram of up to 4 mm, displaced at 0, 45, and 90 deg with

respect to the rolling direction of the blank. The misalignment was limited to 4 mm, as beyond this the shape of the

cup is so asymmetric as to not produce a useable ring from

the sidewall.

The resulting springback measurements are presented in

Figure 10. There appears to be a small drop-off in springback within the first 2 mm of misalignment, and then the

value plateaus. Also, there is seemingly no effect of changing the direction of misalignment with respect to the sheet

rolling direction. The thickness distributions and their

standard deviations in Figure 11 show no clear correlation

with the misalignment. Thus, the springback as shown in

Figure 10 appears to be unaffected by the misalignment.

F. Effect of Error in Cup Height

Another possible source of error in forming the cups is

an error in the final height of the sample. For this proposed

test, the target value of the total height of the formed cup is

56 mm. A series of cups were made with final heights as

much as 8 mm shorter and 8 mm taller than this, and the

resulting springback values are seen in Figure 12. Note that

again in each case the ring cut from the sidewall of the cup

was still 15-mm wide and centered on 37.5 mm from the

outside of the flat end of the cup. The effect of the error in

total cup height is approximately 1 mm less springback

opening for each millimeter of additional height.

This can be partly analyzed if one considers the vertical

change in residual hoop stresses seen in a previous study.[9]

As the cup is drawn deeper, the fraction of uniform plastic

strain in the cup wall increases. As a result, the material in

all physical locations in the cup wall is deformed significantly past the yield stress, and the magnitude of the

through-thickness strain inhomogeneities (difference in

plastic strain from one location to another) introduced by

the bending and unbending of the blank decreases. As the

uniform portion of the plastic strain increases, the material

at the neutral plane is deformed above the yield point. In

order to produce this effect, the strains involved can be

small, and the latter are accompanied by equally small

thickness changes (Figure 13). Thus, in terms of the

stress-strain curve of the material, the deformation of the

entire cup wall proceeds into the strain region where large

strain differences correspond to relatively small stress differences. The latter translates into reduced residual stresses

and springback.

G. Effect of Direction of Slicing with Respect to

Rolling Direction

Due to in-plane anisotropy of the mechanical properties

of the sheet, it might be expected that slitting the springback ring open at various angles with respect to the rolling

direction might produce different amounts of opening. A

series of samples were prepared that were identical except

that they were slit open at 15 deg increments with respect to

the rolling direction. The results can be seen in Figure 14,

Fig. 8—Plot of effect of error in the width of the ring on springback

opening. The correlation coefficient is 0.95.

Fig. 9—Earing profiles imaged on (a) correctly formed and (b) blank misaligned cups.

3508—VOLUME 37A, DECEMBER 2006 METALLURGICAL AND MATERIALS TRANSACTIONS A
and the effect was found to be negligible in this material.

However, in materials that exhibit much more mechanical

anisotropy, this might become a more important factor.

IV. CONCLUSIONS

The systematic introduction of errors in running the

springback cup test produced systematic errors in ring opening in the sample. The largest effects were found due to

errors in initial blank alignment over the punch and errors

in the vertical location of the sidewall ring. Other errors

produced small or negligible effects on measured springback. The observations can be rationalized by considering

how the error affects the magnitude of plastic strain in the

ring, and how this subsequently changes the springback

opening through the residual stresses in the wall.

ACKNOWLEDGMENTS

The authors thank R. Fields, NIST, M. Demeri, Ford and

M. Shi, US Steel, for helpful discussions.

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Fig. 10—Plot of effect of error in aligning the blank over the center of the

forming ram for three misalignment directions: along the rolling direction

(0 deg), 45 deg between the rolling direction and the transverse direction,

and along the transverse direction (90 deg). The correlation coefficients are

0.12 (0 deg), 0.79 (45 deg), and –0.08 (90 deg).

Fig. 11—Plot of thicknesses and standard deviations with blank

misalignment.

Fig. 12—Plot of effect of error in final cup height on springback opening.

The correlation coefficient is –0.94.

Fig. 13—Plot of ring thickness with cup height.

Fig. 14—Plot of the effect of orientation of slicing direction with respect

to the rolling direction on springback opening. The correlation coefficient

is –0.12.

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 37A, DECEMBER 2006—3509
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3510—VOLUME 37A, DECEMBER 2006 METALLURGICAL AND MATERIALS TRANSACTIONS A