Materials Science and Engineering A 399 (2005) 26–32

An investigation of springback stresses in AISI-1010 deep drawn cups

Thomas Gnaeupel-Herold a,b,∗, Timothy Foecke c, Henry J. Prask b, Richard J. Fields c

a University of Maryland, Department of Materials and Nuclear Engineering, College Park, MD 20742-2115, USA b National Institute of Standards and Technology, Center for Neutron Research, 100 Bureau Dr. Stop 8562, Gaithersburg, MD 20899, USA c National Institute of Standards and Technology, Metallurgy Division, Gaithersburg, MD 20899, USA

Abstract

The residual stresses created by deep drawing of AISI-1010 cups were investigated by means of neutron diffraction. The wall thickness

of the cups was below 3.0 mm, for which 8–10 through-thickness measurements were conducted. It was found that the axial, and tangential

stress profiles are similar to what is expected from a bending–unbending operation, but it also exhibits a strong dependence on the axial

and, to a minor extent, on the circumferential position. The strong axial stress dependence reflects the effect of stretching while the weaker

circumferential stress variations are due to sheet anisotropy. We also found that the thermal relief of cold work in the initial AISI-1010 cold

rolled blank has no discernable effect both on the residual stresses and the springback.

© 2005 Elsevier B.V. All rights reserved.

Keywords: Springback; Deep drawing; Residual stress

1. Introduction

Springback is the elastic shape change of sheet metal 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. Due to the increasing 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 springback magnitudes,

thus making an accurate prediction all the more necessary in

order to avoid costly re-design of the stamping tool.

Current modeling efforts concentrate on the development

of 3D computer models to predict stress, strain, and fracture

in sheet metal, with the goal of a significant reduction in tool

design time. For the first stage of springback modeling with

the accompanying measurements, deep drawn cups are chosen as a generic part. They are easily produced and provide

a simple measure of springback with a large magnitude displacement. Furthermore, the formed cups exhibit many features found in production parts, such as large biaxial strains,

thickness variations, and texture.

∗ Corresponding author. Tel.: +1 301 975 5380; fax: +1 301 921 9847.

E-mail address: tg-h@nist.gov (T. Gnaeupel-Herold).

Residual stresses in the cup exist because different locations in the cup have accumulated different magnitudes

of plastic strain. These stresses are key for the modeling

of springback because their integral over the wall thickness

yields a non-zero bending moment and thus a shape change.

Most of the stress-related experimental and theoretical

work done on deep drawn cups has focused on the actual

value of the springback, which is measured as the opening of

a ring cut from the center section of the cup and subsequently

split open [1–7]. The difficulty in modeling the opening as an

integral effect is that a multitude of stress distributions can

produce the same opening, thus making it more difficult to

optimize the computer model in a way that it reproduces the

actual stresses in the cup wall. However, even with the extensive research done on deep drawn cups, only few publications have dealt with the measurement of springback stresses.

These include X-ray surface measurements [1], an early neutron diffraction measurement done at low spatial resolution

[3], and a very recent synchrotron diffraction study on a Al6022 cup [7]. While the latter has provided detailed insight

into the axial and tangential stresses in the cup wall, questions

remain about the dependence of the stresses on the axial and

circumferential position. It is also long established that external processes and material parameters such as the strength

of the material, sheet thickness, pre-strain and flow control

0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.msea.2005.02.017
T. Gnaeupel-Herold et al. / Materials Science and Engineering A 399 (2005) 26–32 27

measures have a large effect on the springback. However,

more data are needed to clarify how the stress distributions

are affected by these parameters.

In this work, we examine in detail the stresses in the cup

rings before and after cutting operations, i.e. after cutting the

cup into rings and after splitting the ring. In order to study the

effect of pre-strain a second cup was made from a blank where

the cold work from cold rolling was removed by annealing.

2. Experimental

2.1. Forming

The cups were made in the Forming Laboratory of the

NIST Metallurgy Division using a hydraulic deep drawing

machine with 500 kN capacity. The AISI-1010 blanks were

circular disks of 200 mm diameter and 3.00 mm thickness.

The diameter of the die bore was 110 mm with a rim radius

of 10 mm. The stamping tool had a diameter of 100 mm with

an edge radius of 10 mm. The depth of the cups was 55 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. A detailed description of the forming process can be

found in Ref. [8].

2.2. Neutron diffraction measurements

All strain measurements were done using the BT8 Residual Stress Diffractometer at the NIST Center for Neutron

Research using a wavelength of 2.26 A and the (0 1 1) re-  ̊

flection. The (0 1 1) reflection is best suitable for this experiment because of the texture-induced raise in scattered

neutron intensity in the axial, radial and hoop directions.

Because of the specimen geometry involved, and in order

to ensure the best possible spatial resolution, two different

gauge volumes were defined with apertures for the incident

beam (width × height) and the diffracted beam (opening).

A schematic of the neutron beam geometry and the respective specimen orientations is shown in Fig. 1. Both for the

tangential and the axial strain component a gage volume of

3 mm × 0.5 mm × 3 mm (w × h × o) was used while for the

radial direction a gage volume of 0.5 mm × 7 mm × 0.5 mm

was defined. This way, the depth resolution for each of the

three directions was 0.5 mm. Data were collected in eight

steps of 0.4 mm through-thickness, in five steps of 45◦ around

the circumference in the midsection, and at three different

axial locations. The d-spacings both the innermost and the

outermost point of each scan were affected by partial illumination, i.e. the gauge volume was half immersed in the

sample. This effect is equivalent to a specimen displacement,

and it was corrected by rotating the specimen by 180◦ for

which the shift of the d-spacing is opposite. For small displacements as in our case the correction is obtained by the

average of both measurements.

Two cups were investigated. The first cup was made from a

blank in the as-received condition. After EDM cutting three

rings from the cups, stress measurements were made both

for the intact and split rings in the mid-ring at positions (0◦,

45◦, 90◦, 135◦ and 180◦) and at the (0◦) position in the topring and the bottom-ring. The dimensions of the rings and

measurement locations are illustrated in Fig. 11.

In order to relieve internal stresses from cold rolling,

the blank for the second cup was annealed for 40 min at

630 ◦C in an argon atmosphere. At this temperature, the

yield strength of a mild steel such as AISI-1010 is <30 MPa

[9] and virtually all cold work is removed after 40 min. Then,

a cup was drawn from this blank using the same forming

conditions as for the cup made from the as-received blank.

Diffraction measurements on the second were done on the

mid-ring at the (0◦) position.

The low thickness of the cup wall permitted the application

of the σradial ≡ 0 boundary condition. This increases the overall accuracy of the stress calculations because σradial ≡ 0 is

generally very well fulfilled for such a thin-walled structure,

and the condition applies for every point on the surface. The

use of this condition removes the influence of d0-variations.

It also simplifies the stress equations to

σhoop = dhoop − dradial

d0

× Ehkl

1 + νhkl

σaxial = daxial − dradial

d0

× Ehkl

1 + νhkl

(1)

where dradial, dhoop and daxial are the measured d-spacings,

d0 is the reference d-spacing (for which the average of all

d-spacings was chosen) and Ehkl, νhkl are the diffraction

elastic constants calculated according to [10]. The Eq. (1)

are insensitive to the choice of d0 because the variations of

measured d-spacings are well below 1%. The uncertainties

of the stresses in Figs. 4–6, 8–10 were obtained from the

standard deviations of the fitted peak positions.

Fig. 1. Neutron beam and specimen orientation for the radial (a), axial (b) and hoop (c) measurements. The arrows point out the direction of the scattering

vector.
28 T. Gnaeupel-Herold et al. / Materials Science and Engineering A 399 (2005) 26–32

Fig. 2. (0 1 1) pole figures for the initial blank (a), the 45◦ location (b) and the 135◦ location (c) of the mid-ring, the tensile sample after 35% deformation in

the TD direction (d) and the tensile sample after 40% deformation in the RD direction.

In order to identify the initial rolling direction (RD) and

transverse direction (TD) of the blank (0 1 1)-pole figures

were measured every 45◦ on the cup wall and on tensile samples cut from a second blank. The texture measurements on

the cup showed distinctive differences between pole figures

at locations at 45◦, 90◦ and 135◦. Then, tensile samples cut

from an undeformed blank in RD and TD orientation were

subjected to a uniaxial strain of 35% and 40%, and pole figures were measured again. The pole figure of the deformed

RD specimen was very similar to the one at the 135◦ location,

and the pole figure of the deformed TD specimen was similarly close to the pole figure at the 45◦ location on the cup

Fig. 3. Surface roundness profiles (a) and thickness profiles (b) for the top

and bottom rings of the cup made from the as-rolled blank. The thickness of

the original blank was 3.00 mm.

wall. This way both RD and TD could be clearly identified

as shown in Fig. 2.

2.3. Thickness measurements

Thickness measurements were done in 10◦ steps on all the

three rings using a dial gage with ±0.005 mm accuracy by

measuring the variation of the roundness both on the inside

and the outside of a ring, and normalizing both measurements

to the thickness at one location. This procedure yields the radii

on the inside and on the outside. The subtraction from one

another provides the thickness distribution.

Fig. 4. Tangential (a) and axial (b) stresses in the intact rings at the “0◦”

position at different axial positions (see Fig. 11). The x-axis (depth) starts at

the outer surface.
T. Gnaeupel-Herold et al. / Materials Science and Engineering A 399 (2005) 26–32 29

Fig. 5. Tangential (a) and axial (b) stresses in the intact mid-ring measured

at the original transverse direction (TD) and at the rolling direction (RD).

The x-axis (depth) starts at the outer surface.

3. Results and discussion

The measurements on the rings were made on locations associated with the symmetry of the original blank: the rolling

direction, the transverse direction, and the bi-sector lines between RD and TD. The influence of that symmetry is still

visible in the profiles of radial deviation in Fig. 3a where local maxima of increased radius are visible along TD (45◦ and

225◦), and along RD (135◦ and 315◦). Minima are found at

the bi-sector positions. However, the comparison of Fig. 3a

and the thickness profiles in Fig. 3b shows that the maxima

and minima of the radial deviation do not coincide with those

of the thickness profiles in Fig. 3b, hence the radial deviation

is more of an indicator for a radial shape distortion and it does

not allow conclusions about thickness variations.

There is a direct link between the wall thickness and the

accumulated plastic strain because any variation from the

original thickness of the blank is the result of a plastic flow

of material both in the axial direction and around the circumference. One way of characterizing forming strains is

the R-value, i.e. the ratio of the width-strain in the in-plane

direction perpendicular to the applied stress to the thickness

strain (ND direction), also perpendicular to the applied stress.

From the tensile samples we found εRD/εND = 1.50 for a 33%

Fig. 6. Tangential (a) and axial (b) stresses in the intact mid-ring measured

at positions (0◦, 90◦, 180◦; see Fig. 11). The x-axis (depth) starts at the outer

surface.

strain in the RD direction, and εRD/εND = 1.88 for a 36%

strain in the TD direction. However, we do not find the 180◦

(RD–TD–RD–TD) symmetry in the thickness profile of the

cup wall. The absence of the symmetry of the original blank

in the wall thickness was consistently found also in other

cups, and it can be attributed to variations of the blankholder

force as well as friction.

Overall, the circumferential thickness differences are

<3%, which is small compared to the decrease in thickness

Fig. 7. Stress–strain curve for specimens cut from the original blank in the

RD and TD directions.
30 T. Gnaeupel-Herold et al. / Materials Science and Engineering A 399 (2005) 26–32

Fig. 8. Average stresses for the intact and split mid-ring and their difference.

The x-axis (depth) starts at the outer surface.

Fig. 9. Comparison of stresses in the cups made from the annealed blank

(HT) and made from the as rolled blank (AS). The x-axis (depth) starts at

the outer surface.

from the top-ring to the bottom-ring (≈10%). As a result, the

overall variations of plastic strain are much bigger in the axial

direction than around the circumference. The effect of overall

increased plastic strain magnitudes is that the strain coordinates of every location for a given axial coordinate are moved

Fig. 10. Average axial and tangential stresses in the split ring. The x-axis

(depth) starts at the outer surface.

Fig. 11. Schematic of the cup with the locations of the rings. The rolling

direction (RD) and the transverse direction (TD) of the initial blank are

indicated.

to higher values in the stress–strain curve. The stress–strain

curve becomes increasingly flat in the plastic region, thus reducing the possible stress differences between points in the

cup wall that have different magnitudes of true strain. This

effect is analogous to a stress relief by a homogeneous plastic

deformation.

Consequently, stress magnitudes decrease together with

the thickness towards the cup bottom. This is well reflected

both for the tangential stresses and axial stresses in the intact

top, middle and bottom rings in Fig. 4 showing that the residual stresses become increasingly flat towards the cup bottom.

The axial stresses are affected more strongly because the true

strain increases in the axial direction.

Around the circumference the stress variations for different specimen directions are small with the most notable

differences occuring between the RD and TD directions

(Figs. 5 and 6). The positions (0◦, 90◦ and 180◦) are bi-sector

directions that are equivalent with respect to the symmetry

of the blank. In Fig. 5, the difference between maximum

and minimum stresses is slightly smaller in the TD direction which points towards an increased level of plastic strain.

However, there is no obvious correlation to the thickness profile, and, as shown in Fig. 7, the similar tensile properties

along RD and TD give no indication of strength related effects.

Other possible explanations are different grain–grain interaction stresses due to grain orientation dependent yield

points in the RD and TD directions, and differences in the

local elastic constants. Both effects are correlated through

preferred grain orientation, which is different in both directions. The in-plane elastic anisotropy of the initial blank is

<5% and it was neglected in the stress calculations. However, the deep drawing produces substantial changes of the

grain orientations (Fig. 2), which is currently under further

investigation.

The split ring test is one of the most common springback measurements, in which the opening of the split midring is measured. It also offers an opportunity to evaluate

the quality of the residual stress measurements by comparing
T. Gnaeupel-Herold et al. / Materials Science and Engineering A 399 (2005) 26–32 31

Table 1

Measured and calculated ring openings

Top Middle Bottom

Measured opening (mm) 12.9 (2) 10.4 (2) 7.4 (2)

Calculated (mm) – 11.8 (4) –

the measured opening to the numerical value calculated from

the measured stresses. If anelastic effects (non-elastic recovery strains) are disregarded then the through-thickness stress

change is a linear function of the depth. The stress change

together with the stresses in the intact and the split ring is

shown in Fig. 8. The observed stress change is linear to a

good approximation. The procedure for calculating the ring

opening from the stress change is described in [7]; here we

list in Table 1 only the measured openings for the three rings

and the calculated opening for the mid-ring.

The agreement between measurement and calculation is

reasonably good, considering that the averages are taken only

over one half of the rings at the positions (0◦, 45◦, 90◦, 135◦

and 180◦). As an effect of the large thickness, the springback

is comparatively small.

The effect of the stress relief in the blank on the residual

stresses in the ring is shown in Fig. 9. The stresses are virtually

identical, thus indicating the negligible effect of the initial

coldwork on the springback.

Some insight into the stresses in the intact cup before cutting can be obtained by comparing the average axial and tangential stresses in the split ring in which the springback has

already taken place both in the tangential and in the axial

direction as shown in Fig. 10.

The operation of cutting the cup into rings induces axial

recovery, i.e. after cutting, each ring is in the axially unloaded

state and there is both stress balance ( σax dA = 0, where A

is the wall cross section) and balance of the bending moment

(

 σax z dz = 0, where z is the thickness coordinate).

After splitting the ring, the same applies to the tangential

stresses. The ring is in the unloaded state in both directions,

and as a consequence, both stress distributions become very

similar. Their characteristic shape is caused by a sequence of

bending at the die radius and unbending at the punch radius.

At a more detailed level, the stresses depend on the specific

combination of forming parameters for the deep drawing operation, specifically on the depth of the cup, the cup diameter

and on the bending radii of the tooling. An increased depth

of the cup increases homogeneous stretching which ‘flattens’

the tangential stresses and, even more so, the axial stress distributions (Fig. 4). A decreased bending radius of the tooling

can cause excessive strains on both sides, thus actually reducing springback at an increased risk of cracking on the

outside.

Data for a direct analysis of the effect of the bending radius are not available. However, there are some results from

a previous work [7] on an Al-6022 cup with the same forming parameters but with lower wall thickness of 0.92 mm for

which the ratio of bending radius (ring radius for tangential

Fig. 12. Average tangential (a) and axial (b) stresses in the intact mid-rings

of an AISI-1010 cup and an Al-6022 cup. The x-axis (depth) starts at the

outer surface.

stresses) to thickness is three times smaller. The stress–strain

curves of Al-6022 and AISI-1010 are very similar which enables a direct comparison as shown in Fig. 12.

For the tangential stresses, the only notable dissimilarities

appear close to the inner surface because here the magnitude

of the true strain increases faster with the thickness coordinate

in compression than in tension. Here, the bending radius is

the diameter of the cup, and the ratio of radius to thickness

is large for both Al-6022 and AISI-1010.

The differences become larger for the axial stresses where

the bending radius is only 10 mm compared to sheet thicknesses of 3 and 1 mm, respectively. In the case of AISI-1010

cup the thickness effect becomes visible, and both for axial

and tangential stresses the neutral plane is shifted towards

the inside, which is consistent with predictions from plastic

bending theory.

4. Conclusions

Overall, both tangential and axial stress distributions exhibit the general features characteristic for the plastic bending

of a material with work hardening. At a more detailed level,

the stresses are generally non-symmetric with respect to the

neutral plane which generally appears to be shifted towards

the inner surface as a result of higher true strains on the inside
32 T. Gnaeupel-Herold et al. / Materials Science and Engineering A 399 (2005) 26–32

of the cup. The true strains are also higher towards the cup

bottom due to increased stretching which acts as a stress relief

and it reduces stress magnitudes. The circumferential variations of both axial and hoop stresses appear to be associated

with the RD/TD symmetry of the original blank. The thickness effect points out the influence of the ratio of bending

radius to thickness which, for small values of the ratio, can

produce large changes in the stress with smaller springback

due to higher true strain magnitudes. A heat treatment of the

blank material has no visible effect on the stress distributions.

Acknowledgement

The authors wish to thank Dr. Vladimir Luzin for providing the estimate of the elastic anisotropy of the original

blank.

References

[1] J. Danckert, Ann. CIRP 43 (1994) 249.

[2] K. Saito, Y. Shimahashi, in: H. Lippman (Ed.), Proceedings of the

Conference on Metal Forming Plasticity, Tutzing, Germany, August

1978, Springer-Verlag, Berlin, 1979, p. 53.

[3] K. Lange, L. Bruckner, Trans. NAMRI/SME (1990) 71.

[4] Y. Yuying, L. Chunfeng, X. Hongzhi, J. Mater. Process. Technol. 30

(1992) 167.

[5] M.S. Ragab, H.Z. Orban, J. Mater. Process. Technol. 99 (2000) 54.

[6] F. Morestin, M. Boivin, C. Silva, J. Mater. Process. Technol. 56

(1996) 619.

[7] T. Gnaeupel-Herold, H.J. Prask, R.J. Fields, T.J. Foecke, Z.C. Xia,

U. Lienert, Mater. Sci. Eng. A 366 (2004) 104–113.

[8] T. Foecke, S.W. Banovic, R.J. Fields, J. Miner., Met. Mater. Soc. 53

(2) (2001) 27–30.

[9] R.W. Cahn, P. Haasen (Eds.), Physical Metallurgy, Part II, NorthHolland Physics Publishing, New York, 1983.

[10] F. Bollenrath, V. Hauk, E.H. Muller, Z. Metallk. 58 (1) (1967) 76–82.  ̈