Materials Science and Engineering A366 (2004) 104–113

A synchrotron study of residual stresses in a Al6022 deep drawn cup

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

Timothy J. Foecke c, Z. Cedric Xia d, Ulrich Lienert e

a Department of Materials and Nuclear Engineering, University of Maryland, College Park, MD 20742-2115, USA

b National Institute of Standards and Technology, Center for Neutron Research, Gaithersburg, MD 20899, USA c National Institute of Standards and Technology, Metallurgy Division, Gaithersburg, MD 20899, USA d FORD Scientific Research Laboratory, Dearborn, MI 48121, USA e Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439-4815, USA

Received 8 March 2003; received in revised form 28 August 2003

Abstract

Fueled by pressures to reduce scrap and tooling costs, the modeling and prediction of springback has become a major focus of interest in sheet

metal forming. Finite element codes and packages are being developed or improved but face the demand for higher predictive accuracy which, in

turn, requires accurate property data and a more complete understanding of the stresses that are responsible for the elastic part of the springback.

In order to provide experimental data for these calculations, synchrotron X-ray diffraction measurements were carried out in order to determine

the through-thickness distribution of axial and tangential residual stresses in an Al6022 deep drawn cup. The technique is able to provide true

spatial resolutions of 0.05 mm for a strain measurement on a cup with 0.92 mm wall thickness. It is found that both axial and tangential stresses

exhibit non-linear gradients through thickness and both exhibit a pronounced dependency on the axial position. The springback measured on

a split ring cut from the cup agrees within 3% accuracy with the value predicted from the average of measured through-thickness stresses.

© 2003 Elsevier B.V. All rights reserved.

Keywords: Springback; Sheet metal forming; Stress; Synchrotron

1. Introduction

The stamping of sheet metal induces complex deformations, which in different regions of the sheet metal can have

vastly different accumulated plastic strains. Upon unloading

by removing the die, springback occurs which changes the

dimensions of the part. Springback is caused by non-uniform

residual stresses through the thickness of the sheet metal

that create a bending moment which causes a distortion of

the part upon unloading. Springback strains are almost completely elastic; non-linear recovery strains are usually small

but, in extreme cases, they can amount up to 10% of the total recovery [1]. The accurate prediction of the springback

is of major interest for small part tolerances and to avoid a

costly redesign of the stamping tool. In the past a number

of studies have looked into ways of reducing springback,

particularly the role of pre-straining, blankholder force and

∗ Corresponding author. Tel.: +1-301-975-5380;

fax: +1-301-921-9847.

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

yield strength. Still, even in this case springback cannot be

disregarded completely but it can be reduced which allows

a more accurate prediction with smaller absolute tolerances

(not necessarily relative tolerances). Experimental data of

springback stresses are scarce in the available literature; instead, most publications deal with the modeling aspect of

the problem, thus highlighting the need for data that can

benchmark the model predictions.

The stresses, which are generally bi-axial, are created

when sheet metal is subjected to bending operations during which locations with different thickness coordinates

undergo different strain paths, thus producing stress differentials across sheet metal thickness and leading to bending moment upon tool removal. An accurate modeling of

springback thus requires accurate prediction of pointwise

stresses throughout.

In an attempt to simplify the problem, various studies have

dealt with symmetric, generic shapes among which circular

cups have received the most attention [3–7] as a promising way to characterize a material’s springback properties

in a quick and consistent way. For a standardizable test, it

0921-5093/$ – see front matter © 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.msea.2003.08.059
T. Gnaeupel-Herold et al. / Materials Science and Engineering A366 (2004) 104–113 105

is desirable that it be fairly easy, 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 consists of a ring sample taken from the

sidewall of a flat bottom, deep drawn cup. The main advantage of such a simple geometry is that the springback can be

measured very easily by cutting off rings and subsequently

splitting them. Using this operation, the through-thickness

stress change can be calculated very easily, thus providing a

means to follow stress variations induced by changing certain stamping parameters such as ironing [2,6].

For this to be useful also for non-generic parts with more

complicated strain paths the successful simulation of the

springback must be based on an accurate prediction of the

depth distribution of stresses. However, because of the small

thicknesses involved a pointwise stress measurement to validate the models has been very difficult and there are few

studies that provide even approximate experimental results.

Saito and Shimahashi [3] employed Sachs’ boring method

on a mild steel deep drawn cup and obtained both axial and

tangential stresses that indicate a highly non-linear gradient

of stresses both in the axial and radial direction. Their results are based on data from a single strain gage with no distinction between strain gage location and the original rolling

direction (RD) of the blank. Because this method of stress

depth profiling is essentially based on layer removal the results depend on the sequence of the removal, i.e. if layers

are removed from the inner or the outer surface. The reconstruction of the undisturbed stress state was not possible because one prerequisite of the boring-out theory, namely that

the stresses are constant along the length, was clearly not

applicable. Danckert [2] used a variation of layer removal

together with X-ray diffraction for monitoring the stress

changes. Yuying et al. [5] used a similar method of layer removal together with X-ray diffraction on stainless steel cups

having varying degrees of strain-induced martensite. Their

findings revealed vastly different through-thickness tangential stresses depending on the content of induced martensite.

In contrast to the layer-removal methods, diffraction techniques offer the only possibility for mapping stress variations

through the thickness, circumferentially, and axially. For example, early advances in neutron diffraction instrumentation

in combination with a high neutron flux made it possible to

achieve spatial resolutions in the sub-millimeter range, thus

enabling the first non-destructive measurements on ≈1 mm

thick cups by Lange and Bruckner [4]. They confirmed that

the through-thickness stress gradients in a 1 mm thick brass

cup are non-linear but concluded that higher spatial resolutions were required to elucidate the stress distribution with

sufficient detail.

In order to avoid the aforementioned difficulties in destructive stress determination there is no alternative to

diffraction methods, either as neutron or as synchrotron

diffraction. In practice, even tighter restrictions are imposed by sheet metal thicknesses of ≤1 mm where neutron

diffraction reaches the limit of its capability. Synchrotron

diffraction is basically the only method capable both of

penetrating 1 mm of aluminum and providing spatial resolutions <0.1 mm. In this paper we describe the results of

such measurements on an intact and a split ring cut from

an Al6022 cup. The investigation included measurements

of axial and tangential stresses on different axial locations

and around the circumference.

It is conceivable that the stress state in the initial blank

as induced by rolling would have some effect on the springback of the rings. However, any pre-existing stresses in the

blank have a zero bending moment (or the blanks would

not be flat), thus they cannot add to the springback. More

important are the initial plastic strains due to rolling. These

plastic strains tend to be present as a gradient symmetrical to

the mid-plane of the sheet and, especially for thicker blanks,

the strains can be substantial near the sheet surfaces. As a

result, the initial yield stress of the blank may not be uniform through the thickness, thus possibly affecting the stress

differential after forming. In an experiment described in [8]

through-thickness stress measurements were performed on

cup rings made of as-received and stress-relieved AISI-1010

blanks. It was found that the through-thickness residual

stresses in both cups were virtually identical. When comparing the blank thickness of 3 mm used in [8] to the sheet

thickness used here (0.92 mm) we can assume that the effect

of the initial plastic strain from rolling both on the residual

stresses due to forming and on the springback is small.

2. Experimental

A ring with 110 mm in diameter, 15.0 mm in height and

0.92 mm wall thickness was cut by EDM from the center

section of a deep-drawn cup (Fig. 1a and b). Except for

the height (±0.1 mm), the dimensions are approximate due

to the cup being slightly conical and because of thickness

variations of about 0.1 mm due to flow anisotropy. The procedure was then repeated on a second cup made under the

same forming conditions and the ring was split open. No

pre-strain was applied. As shown by the results in Table 1,

the level of control over the conditions of the deep drawing

is such that the springback values of four different rings are

within 3% of their average, thus justifying the comparison

of stresses in different rings.

Table 1

Ring openings and radii of rings made under identical conditions as the

ones investigated here

Specimen

no.

Opening

2y (mm)

Radius of the

split ring R

(mm)

Calculated radius R of the

split ring (calculated from

measured stresses) (mm)

6022-9 80.2 68.5 70.8

6022-10 82.5 68.9

6022-11 79.25 68.32

6022-12 84.33 69.25

The value in the last column was calculated from the through-thickness

stress profiles.
106 T. Gnaeupel-Herold et al. / Materials Science and Engineering A366 (2004) 104–113

Fig. 1. Sectioned cup (a) and split ring (b).

On the intact ring, through thickness measurements were

performed on two axial locations, one in the middle of the

ring, the other 5 mm towards the cup bottom. On the split

ring, measurements were performed in the middle, 4.5 mm

towards the top, and 5 mm towards the bottom. The measurement locations around the circumference (intact: 0, 180◦ =

RD, split: 90◦ = TD, 180◦ = RD) were set with respect to

the rolling direction of the original blank material. A sketch

is shown in Fig. 2. Because of the tight restrictions of the

beamtime at the synchrotron it was not possible to measure

more locations.

In sheet metal forming one has to deal with plane stress

states for which the normal stress (“radial” in Fig. 2) can

be safely set to zero everywhere. This allows the use of a

scheme in which the unknowns tangential stress and axial

stress are obtained from three measured strains in the axial,

tangential and radial directions [9,10]. For this method an

estimate for the unstressed d-spacing is sufficient [10].

Residual stress measurements were performed at the 1IDB

beamline at the Advanced Photon Source at the Argonne National Lab. Because the basic principles of diffraction residual stress analysis are well known [10], only the specifics of

this measurement will be described. Fig. 3 shows a schematic

of the experimental setup.

Despite the relatively low photon energy of 20 keV the

high intensity of the primary beam allowed the use of the

radial

source detector

axial

Fig. 2. Specimen orientations for the three principal strains.

Al(4 2 2) sample reflection, thus increasing the Bragg angle

to approx. 2θ ≈ 44◦. This choice decreases the notorious

“diamond” distortion of the sampling volume and maintains

the nominal spatial resolution. In order to minimize the intrinsic blurring of the shape of the sampling volume due

to cold-work induced peak broadening a Si(1 1 1) analyzer

crystal was used to restrict the acceptance angle of the detector. This measure removed the instrumental line broadening. A typical peak profile is shown in Fig. 4. The counting time was chosen such that the typical uncertainty of the

peak position was ≤ 5 × 10−5.

Despite the small sampling volume (≈0.003 mm3), and

the fact that aluminum alloys virtually always contain large

grains, for the same orientation but different locations the

intensity fluctuations of the reflected peak were within 62%

of the average. Preferred orientation was investigated but it

was found to be of little impact on the measurement and

on the data analysis because the reflected intensity was essentially the same in the three measurement directions. This

and the preferred orientation data indicate that the reflecting grains belong to the isotropically oriented fraction of the

grains. Sheet metal textures due to rolling do not change significantly the elastic constants in the directions of the principal axis [14]. With the fact that aluminum is mechanically

rather isotropic, this also allows the use of diffraction elastic

constants calculated according to a self consistent scheme
T. Gnaeupel-Herold et al. / Materials Science and Engineering A366 (2004) 104–113 107

Fig. 3. Experimental arrangement of the synchrotron experiment. The incident beam penetrating the sample is diffracted along the entire path through

the sample. The reflected beam is broadened due to microstrains in the sample. The low angular acceptance of the analyzer prevents the counting of

photons that were reflected from regions outside the nominal sampling volume.

[11,12]. The stresses are obtained from the basic equation

of diffractive stress analysis

εφψ = dφψ − d0

d0

= 1

2 s2(hkl) × [(σ11 cos2 φ

+ σ22 sin2 φ + σ12 sin2φ)sin2 ψ

+ (σ13 cos φ + σ23 sin φ)sin2ψ + σ33 cos2 ψ]

+ s1(hkl)(σ11 + σ22 + σ33) (1)

By setting all shear stress components and σ33 equal to 0,

one finds that

ε0,90 − ε0,0 = d0,90 − d0,0

d0

= 1

2 s2(hkl)σ11 (2)

and

ε90,90 − ε0,0 = d90,90 − d0,0

d0

= 1

2 s2(hkl)σ22 (3)

Using σ11 = σaxial and σ22 = σtangential, d0,90 = daxial,

d0,0 = dradial, and d90,90 = dtangential, we obtain for the

0

50

100

150

200

250

300

350

400

450

43.6 43.7 43.8 43.9 44.0 44.1 44.2 44.3

2 theta [deg.]

intensity (a.u.)

Fig. 4. Typical peak profile of the Al (4 2 2) reflection.

stresses

σtangential = dtangential − dradial

d0

× 1

1/2s2(hkl) (4)

σaxial = daxial − dradial

d0

× 1

1/2s2(hkl) (5)

For the diffraction elastic constant 1/2s2(hkl) we calculated

1.9096 × 10−5 MPa−1 [11]. The value of d0 was estimated

from the average of all measured d-spacings. It can be seen

in Eqs. (4) and (5) that the uncertainty of this estimate for

d0 (approximately 10−5) has no impact on the stress uncertainties (typically ±5 MPa).

3. Results and discussion

We collected data from 10 different positions through

the wall. The experimental uncertainties are plotted in the

graphs but they are in most cases of the size of the data

points. The innermost and outermost locations were chosen
108 T. Gnaeupel-Herold et al. / Materials Science and Engineering A366 (2004) 104–113

180 ̊,RD

0 ̊,RD

90 ̊,TD axial

tangential

radial

Fig. 5. Measurement locations on the ring. The axial direction is constant

while the radial and tangential directions are rotated depending on the

circumferential position: (a) 0◦ = RD; (b) 180◦ = RD.

such that the sampling volume was half immersed into the

specimen. Due to the small beam size and the low acceptance angle of the analyzer/detector system, abberations that

could affect the stresses at these two points are minimized.

The use of laboratory X-rays for measurements directly at

the surfaces was not considered because the stresses within

the penetration depth (≤10m) are affected by frictional

surface damage [13]. While removal of the damaged surface layer with subsequent surface stress measurement is

possible it would not add anything to the data because the

new depths after removal are about the same as the depths

(center of gravity) of the first (near the outer surface) and

last (near the inner surface) sampling volume in our measurements. Also, unlike laboratory XRD, synchrotron radiation allows the direct measurement of in-plane strains (axial and tangential). For medium strength textures in rolled

and deep drawn materials, the differences of elastic constants in the three measurement directions (principal axis

system of the sheet) can be neglected [14], thus avoiding the difficulties involved in dealing with out-of-plane

diffraction elastic constants as it is necessary for laboratory

XRD [11].

3.1. Intact ring

We have investigated the depth distribution of axial and

tangential stresses at two axial and two positions around the

circumference for which the geometry is shown in Fig. 5.

The tangential stresses were measured at the “0◦” position

and at the “180◦” position, both of which correspond to the

original rolling direction of the blank. The plots are shown in

Fig. 6. The agreement between stresses at “0◦” and “180◦” is

very good as was expected from the initial sheet symmetry.

Circumferential coordinate measurement machine (CMM)

scans of the wall thickness show that the thickness alternates

similar to a sine wave with a 180◦ period and a maximum

thickness difference between maximum and minimum of

about 3%; however, this appears to be too small a variation

to cause a difference in the stresses at both positions [15].

Similar CMM measurements on other cups have shown that

the positions of the thickness minima and maxima are not

correlated with the rolling direction or transverse direction

(TD) [15]. The initial symmetry of the blank is given by

rolling direction and transverse direction, thus suggesting a

90◦ period; the complete absence of this symmetry is a clear

indication for an asymmetric material flow during the cup

forming caused by an inhomogeneous blankholder force or

variation from centering the blank (a misalignment of the

die can be excluded). Compared to this combined effect the

initial plastic anisotropy (due to preferred orientation) of the

blank has a negligible impact on the cup wall thickness.

The change of the tangential stresses with the axial position has been pointed out in the literature [3,5,6], in this case

we find a difference of appoximately 50 MPa over an axial

distance of 5 mm, increasing from the mid-line towards the

cup bottom. Because of the axial gradient there will be also

a slight twist of the ring, i.e. both sides of the split ring will

not align. As pointed out by Saito and Shimahashi [3], the

origin of these stresses is the combination of a bending of

the sheet metal at the radius of the die and the subsequent

unbending at the radius of the punch. The intact ring is not

in the tangentially unloaded state and the stress distribution

is analogous to that of a plastically bent beam in a loaded

configuration. Stress balance is neither demanded nor given

for any of the individual curves of the tangential stresses.

However, the equilibrium condition  σt dA = 0 demands

that the integral of the tangential stresses over the wall cross

section of the ring vanishes. By extrapolating the stresses in

Fig. 6 over the full wall cross section, i.e. by assuming that

the tangential stresses at 5 mm up from the ring middle can

be obtained by linear extrapolation it becomes clear that the

equilibrium is not fulfilled which means that the variation of

tangential stresses is non-linear in the axial direction. The

through-thickness axial stresses, on the other hand, show little change in the axial direction (Fig. 7).

The comparison of the axial stresses in the ring middle

and 5 mm down shows a gradient of about 5 MPa/mm decrease towards the bottom. Otherwise, the stresses in the

ring middle both at the “0◦” and at the “180◦” mark are

almost identical as was the case for the tangential stresses,

thus confirming the accuracy of the data. The axial stresses

show a similar behavior as the tangential stresses but they

are well balanced with the resulting axial bending moment

≈0. This is a consequence of cutting the ring from the cup

bottom (Fig. 1), thus setting the axial bending moment to

zero everywhere in the ring.

3.2. Split ring

Splitting the ring partially releases the tangential stresses

and it also resets the total tangential bending moment to zero.

The splitting is analogous to an unloading in the example of a plastically bent beam shown schematically in

Fig. 8. Fig. 9 shows the axial and tangential stress distributions measured on the split ring. None of the individual through-thickness stress curves necessarily have a
T. Gnaeupel-Herold et al. / Materials Science and Engineering A366 (2004) 104–113 109

tangential, "0 ̊" mark

-250

-200

-150

-100

-50

0

50

100

150

0 0.2 0.4 0.6 0.8 1

dist. from outer surface [mm]

tangential stress [MPa]

middle

5 mm down

 0° = RD

tangential, "180 ̊" mark

-250

-200

-150

-100

-50

0

50

100

150

0 0.2 0.4 0.6 0.8 1

dist. from outer surface

tangential stress [MPa]

middle

5 mm down

 180° = RD 

(a)

(b)

Fig. 6. Tangential stresses in the intact ring for the initial rolling direction. For an explanation of the labels see Fig. 4.

axial stresses

-200

-150

-100

-50

0

50

100

150

0 0.2 0.4 0.6 0.8 1

dist. from outer surface [mm]

stress [MPa]

180 ̊, middle

180 ̊, 5 mm down

0 ̊, middle

Fig. 7. Comparison of axial stresses: (a) loaded configuration; (b) stress

change upon unloading; (c) residual stresses.

zero tangential bending moment (Fig. 9a); however, at a

fixed angle (circumferential position) it must be true for the

through-thickness tangential stress averaged over the width

of the ring.

Unfortunately the synchrotron measurements could not

be fully completed which is why the data at the “180◦”

mark were not taken at the same axial position as the others. Nonetheless, the tangential stresses all show the same

behavior and the depth curve taken at “180◦” is different

from the ones at “90◦” only by a small constant shift which

means that there are only minor stress differences between

locations at the RD and at the TD. The axial stresses are

close to being identical. Significant changes with respect

to the intact ring are found only for the tangential stresses

which now exhibit the typical “S”-curve shape as shown

in Fig. 8c. The direct comparison of stresses in the intact

and split ring in Fig. 10 supports the interpretation given

above.
110 T. Gnaeupel-Herold et al. / Materials Science and Engineering A366 (2004) 104–113

σt

neutral surface thickness

coordinate

σt σt

thickness

coordinate

(a) loaded configuration (b) stress change upon unloading (c) residual stresses 

Fig. 8. Stress distributions in elastic-plastic beam bending. Outer and inner surfaces have a negative and positive thickness coordinates: (a) tangential

stresses; (b) axial stresses, respectively.

3.3. Springback

By averaging the measured tangential stresses from all

locations we can estimate the total tangential stress change

from the intact ring to the split ring which is shown in Fig. 11.

Using the intercept from the linear regression result in

Fig. 11 we can calculate the curvature of the split ring. The

equations needed are obtained as follows. First we calculate

the new radius of the split ring from actual data of ring

openings. An analysis of flatbed scanner images of split

rings (accuracy ±0.05 mm) confirmed that the outline of the

split ring is circular within ±1%, and the new radius R is

obtained from the ring opening 2y, the arc length c, and the

split ring, tangential

-200

-150

-100

-50

0

50

100

150

0 0.2 0.4 0.6 0.8 1

depth below outer surface [mm]

tangential stress [MPa]

90 ̊, middle

90 ̊, 5 mm down

180 ̊, 4.5 mm up

split ring, axial

-200

-150

-100

-50

0

50

100

150

0 0.2 0.4 0.6 0.8 1

depth below outer surface [mm]

axial stress [MPa]

90 ̊, middle

90 ̊, 5 mm down

180 ̊, 4.5 mm up

(a)

(b)

Fig. 9. Tangential and axial stresses in the split ring: (a) tangential stresses;

(b) axial stresses.

old radius R0 by means of two equations

c = 2R arcsin y

R



c = 2π(R − R0)

(6)

which have to be solved for R numerically. The meaning of

the variables in Eq. (6) is shown in Fig. 12.

The linear stress change shown in Fig. 11 is associated

with the change in curvature by

σt = Eεt (7)

-250

-200

-150

-100

-50

0

50

100

150

0 0.2 0.4 0.6 0.8 1

depth below outer surface [mm]

hoop stress [MPa]

intact ring (180 ̊)

intact ring (0 ̊)

split ring (90 ̊)

split ring (180 ̊)

 tangential stresses

-250

-200

-150

-100

-50

0

50

100

150

200

0 0.2 0.4 0.6 0.8 1

depth below outer surface [mm]

axial stress [MPa]

intact ring (180 ̊)

intact ring (0 ̊)

split ring (180 ̊)

split ring (90 ̊)

 axial stresses 

(a)

(b)

Fig. 10. Comparison of stresses in the split and intact ring.
T. Gnaeupel-Herold et al. / Materials Science and Engineering A366 (2004) 104–113 111

depth coordinate [mm]

50

100

-200

-150

-100

-50

0

150

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

stress [MPa]

split

intact

difference

Linear (difference)

Fig. 11. Average tangential stresses in the intact and split ring as well as the difference between them plotted. The linear regression line is the average

stress change from the loaded configuration to the unloaded configuration.

in which σt is the stress on the outer surface (this is the

intercept in Fig. 11 with 130 MPa at y = −0.46 mm). For

the bulk value of Young’s modulus E a value of 70 GPa was

used. The strain εt can be written in terms of the thickness

d and the two radii R and R0 as

εt = d

2

 1

R0

− 1

R



(8)

from which R can be easily calculated. The results of the

calculation and openings of several rings made under the

same conditions are listed in Table 1.

The agreement between the calculated radius and the experimental results is good with a relative difference of just

3%. Keeping in mind that diffraction can inherently measure only elastic strains we find that within the accuracy of

the measurement (the standard error of the intercept is 8%)

there is no indication of plastic recovery which would appear

as a difference between the calculated radius (purely elastic,

calculated from through-thickness stress) and the measured

radius of the split ring. The latter contains both the elastic

effect and the plastic recovery.

R

2y

R0

c

Fig. 12. Intact and split ring with radii R0 and R.

3.4. Peak shape analysis

In addition to the shifts in position diffraction peaks also

contain information about grain size and dislocation densities. These contribute to the full-width at half-maximum

(FWHM) of a peak. The actual FWHM is a convolution of

instrumental broadening and physical line broadening. One

of the advantages of synchrotron radiation when used with

an analyzer crystal is that for cold worked metals the contribution of the instrumental broadening is small compared

to the physical line width. Because of that, changes in the

FWHM can be tracked with high precision, as is shown in

Fig. 13.

As shown by Wilkens [16], the line width increases with

the dislocation density and inversely with domain size, and

each effect has a different contribution depending on the

diffraction angle. Our measurements are based on a single

peak at a particular diffraction angle so that the domain size

and dislocation contributions cannot be separated without

additional information. However, a measure of grain size

is available in the intensity fluctuations of the diffraction

peaks in the axial and tangential configurations in which the

absorption is constant for all locations. The two main contributing factors to the intensity fluctuations are the photon

counting statistics and fluctuations of the diffraction grain

volume from one position of the sampling volume to the

next (under identical absorption conditions). The sampling

volume is determined by the beam dimensions and it is constant. At different locations, the total volume of the grains

within the sampling volume that contribute to the diffracted

intensity varies depending on how many grains have the

right orientation to meet the diffraction condition. Because

of the high base intensity we can disregard the influence of

the counting statistics (which would become apparent when

measuring repeatedly at the same location under the same
112 T. Gnaeupel-Herold et al. / Materials Science and Engineering A366 (2004) 104–113

0.0011

0.0012

0.0013

0.0014

0.0015

0.0016

0.0017

0 0.2 0.4 0.6 0.8 1

depth below outer surface [mm]

∆d/d

radial

axial

tangential

Fig. 13. Normalized FWHM for all three directions. Each individual curve is the average of all measurements at different axial or circumferential positions

in the respective direction.

conditions) in which the relative standard deviation is <4%.

For example, at one location the integrated intensity may be

1200(35) while at the neighboring location the integrated intensity is 800(28). The values in brackets indicate the standard deviation given by the counting statistics. Clearly, the

difference between both intensities is far outside these limits. Thus, these large fluctuations are produced by variations

of the diffracting grain volume.

In the following, a simple method to estimate grain dimensions is utilized. For a constant specimen orientation but different through-thickness locations, the standard deviations

of the intensities are a measure of how much the diffracting volume varies from location to location. By disregarding

preferred orientation and by assuming that the number of

grains per sampling volume is Poisson-distributed, we obtain for n, the average number of reflecting grains, that

n =

E(I)

σ

2

(9)

in which E(I) is the expectation value of the intensity and

σ is the standard deviation of the intensity. E(I) and σ are

obtained from the intensities of through-thickness measurements at a given circumferential position. It can be assumed

that no through-thickness texture gradients are left after the

forming process. Using Eq. (9) we find for the axial orientation that σ/E(I) = 0.33 (relative standard deviation), and

the number of relecting grains is n = 9; in the tangential

orientation we obtain n = 3 with σ/E(I) = 0.62.

The different n in the two orientations indicate elongated

grains. It is important to note that these numbers are independent of the multiplicity of the reflection (hkl). Furthermore,

by assuming a uniform grain size and using the detector coverage (1.2 × 10−5 of the sphere determined by the sample

to detector distance) together with the sampling volume of

3 × 106 m3 we calculate the size of the diffracting volume

to be 36m3 (for an isotropic orientation distribution). We

obtain an average grain volume of 12 m3 in the tangential orientation and an average grain volume of 4 m3 in the

axial orientation. Even with large grain aspect ratios commonly found in rolled materials (approximately 520) these

grain volumes point to grain sizes that will not contribute to

a size broadening of the diffraction peak. As a consequence,

the observed broadening effects are only due to dislocation

related microstrains. Thus, the data in Fig. 13 are a measure

of the accumulated plastic strain, distributed in symmetric

curves with a minimum at the location of the neutral surface

at 0.46 mm depth. The existence of the minumum is a consequence of the existence of a neutral plane in the sequence

of bending and unbending.

4. Conclusions

The through-thickness residual stress distributions in a

ring cut from a deep drawn Al-6022 cup in the loaded and

unloaded state were measured non-destructively using synchrotron radiation. The measurements were performed with

high spatial resolution and small step width in order not to

lose details of the steep stress gradients in a wall <1 mm

thick.

The tangential stresses both in the loaded (intact ring)

and in the unloaded (split ring) state were measured at several locations and were found to be largely identical except

for a constant shift. This shift depends mostly on the axial

position and only to a small extent on the circumferential

position, i.e. the position with respect to the original rolling

direction. The axial stresses also showed some dependence

on the axial position with a smaller constant shift than the

tangential stresses. Unloading (splitting the ring) had only a

small effect on the axial stresses.
T. Gnaeupel-Herold et al. / Materials Science and Engineering A366 (2004) 104–113 113

A simple model was developed by which the opening of

the split ring could be predicted from the stresses measured

in the intact ring. The measured radius of the split ring was

found to be within 3% of the calculated one.

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