05M-507

A High Resolution Technique for Measuring Thru-Thickness 

Stresses in Sheet Metal 

Thomas Gnaeupel-Herold

NIST Center for Neutron Research, University of Maryland

Tim Foecke, Mark Iadicola, Steve Banovic 

National Institute of Standards and Technology, Metallurgy Division

Copyright © 2005 SAE International

ABSTRACT

In the Numisheet 2005 benchmark 3 four different 

automotive materials - AKDQ, HSLA 50, DP 600, and 

AA6022-T43 - were subjected to a channel draw process 

with different levels of draw bead penetrations, resulting 

in varying degrees of springback. Resulting specimen 

thicknesses were < 1 mm. The numerical prediction of 

the springback amplitudes was an important part of the 

benchmark; however, only limited experimental results 

were available to verify the residual stresses used to 

predict the springback. In this work, we present a high 

resolution X-ray diffraction technique that was used to 

determine the thru-thickness residual stresses 

responsible for springback. Using standard equipment, 

spatial resolutions down to 0.05 mm were achieved with 

data acquisition times of several hours or more, 

depending on the material. Specimen size limitations 

exist, and cross section surfaces need to be polished or 

electro-polished. Thus, the technique is particularly 

suited for residual stress determination on sheet metal in 

the thickness range > 0.5 mm where specimen 

dimensions can be as small as 25 mm to 50 mm without 

affecting the stress component of interest.

INTRODUCTION

The push to improve the accuracy of finite element 

predictions in sheet metal forming invites the comparison 

with experimental data that is intermediately related to 

the materials behavior as goal of the FEM modeling. An

example is springback which is known for the 

considerable cost it causes in the tryout phase for 

forming tool development. Insufficient accuracy of 

springback prediction for a given tool geometry causes 

another round of refinement of the tool geometry until the 

final and desired shape is reached. Recent advances in 

finite element modeling together with enormous

increases in computing power have enabled the 

application of this technology to daily production work. 

The results have been mixed, however, with good 

accuracy for some parts and insufficient accuracy for 

others. This is in no small part due to the complex nature 

of springback which is a scalar property originating from 

the equilibration of a bending moment induced by 

through-thickness residual stresses. The latter stem from

plastic strains that are non-uniform though-thickness and 

generally in all directions of space. These strains are 

predicted based on the stress-strain properties of the 

material. It is obvious that improving the springback 

prediction is difficult without additional experimental data, 

and with so many parameters affecting the final 

outcome. The additional data can be obtained from the 

through-thickness residual stresses which are

experimentally accessible and closely related to 

springback. Techniques for measuring the residual 

stress profiles non-destructively have been developed 

recently. However, due to the spatial resolution required 

(≈ 0.1 mm) there is a degree of experimental 

sophistication necessary including neutron diffraction 

and synchrotron x-ray diffraction [7,8] that has limited 

such measurements to a few selected samples.

Improvements over these techniques are presented here 

using a very recently developed formalism that, using 

simple x-ray diffraction equipment, allows the 

determination of stress profiles on the length scale 

≈ 1 mm with spatial resolutions of the order of 0.05 mm.

This method was applied to small strips that were cut

from the NUMISheet side wall of channel drawn panels 

in such a way to preserve the stresses of interest.

Spatial resolutions in the x-ray measurements were 

nominally 0.05 mm which is sufficient to provide between 

10 and 20 through-thickness stress values. This work

presents results of such measurements on selected 

specimens of the Numisheet ʼ05 Benchmark 3.

BENCHMARK AND SAMPLES

The details and the purpose of the benchmark, as well 

as materials property data, were presented in earlier 

publications [1,2,3,4], thus only an outline together is 
presented here. The forming geometry is shown in Figure 1.

x

z

y

die

blank

holder

punch

draw

bead

 

Figure 1. Setup of the channel draw process as presented in [3].

A number of steps were necessary to produce samples 

suitable for x-ray diffraction. The side panels were

trimmed, and smaller specimens (x=25 mm, y=12 mm, 

z ≤ 1 mm) were removed by means of electro-discharge 

machining. The specimens were cast in a block of epoxy

and their cross section surfaces polished to remove all 

influence of the EDM cutting (total removal ≈ 1 mm). The 

final polishing step was a 0.05 μm SiC suspension 

intended to provide a surface undisturbed by mechanical 

damage other than the forming process. Figure 2 shows 

the steps involved.

Figure 2. After the channel draw, the side panels 

(bottom) are cut from the u-channel, and smaller 

specimens are removed from the side panels.

The dimensions of the specimens were many multiples 

of the thickness, and it is a reasonable assumption that 

the stresses were left undisturbed by the removal 

procedure. The specimens used in the diffraction 

experiment are shown in Figure 3. 

Figure 3. Example for the specimens used in the 

diffraction tests. The white rectangle indicates the 

measurement field with areas outside screened from 

diffraction by lead foil. The white line inside is an 

exaggerated depiction of the beam spot which is thinner 

in reality.

X-RAY MEASUREMENT PROCEDURE

X-ray stress evaluation is based on the measurement of

changes of interplanar lattice spacings of atoms in 

crystalline materials through Braggʼs equation
λ = 2dhkl sinθ eq. 1

λ is the x-ray wavelength, dhkl is the lattice spacing for 

the plane defined by the Miller indices (hkl), and θ is the 

Bragg angle. As with any strain gage based stress 

measurement, strains have to measured in more than 

one direction, thus the need to rotate or tilt a specimen. 

Through-thickness stresses are not an obvious subject 

of an x-ray (surface limited) stress measurement. The

choice for measuring such sub-surface stresses is

usually to use a bulk-penetrating neutron or synchrotron 

x-ray technique as discussed in [4,7,8,11]. However, the 

same stresses are “visible” to surface limited x-ray 

techniques on the cross section surface, provided that

strains on a sufficiently small area can be measured. 

Thus, the task is to measure the stresses in the xdirection (see Figure 1, Figure 2, top) on the polished 

cross sections shown in Figure 3. It is clear that in order 

to resolve the stress profile sufficiently an x-ray beam 

having a width in the z-direction < 0.1 mm has to be 

employed. The impact of the small beam on data 

collection time is a prolonged counting time (5-10 times 

longer). On the positive side it should be noted that x-ray 

equipment costs and availability put the technique 

presented here ahead of other methods.

The formalism used here differs in some crucial aspects 

from commonly used procedures for X-ray diffraction

stress analysis [9,10]. It is common practice to use large 

primary beams that are, depending on the tilt mode,

either circular with > 1 mm diameter (psi-mode) or 

rectangular beams with several millimeters of height and 

≈ 1 mm horizontal width (omega-mode). In both tilt 

modes the beam spot increases in width or covers 

different portions of the specimen surface with 

increasing tilt angle. It is obvious that in these cases 

strains are measured and averaged over areas several 

millimeters across. Consequently, the beam spot on the 

specimen surface is between one and two orders of 

magnitude too big for through-thickness stress fields in 

sheet metal. The solution to the insufficient spatial 

resolution is to define, through slit systems, a x-ray 

beam that is narrow in one dimension but wide in the 

other dimension. The first requirement is derived directly 

from the desired spatial resolution (e.g. 0.05 mm for 

sheet metal with ≈ 1 mm thickness), and the second 

requirement is a concession to the data acquisition time

which is inversely proportional to the size of the 

illuminated area.

Ψ

primary 

beam

q

ys

xs

zs

Φ=0

Ψ

δ

q

ys

xs

zs

Φ

δ=0

ys

xs

zs

Ψ

Φ

δ

χ=90° χ=45° χ=15°

ID Φ

Figure 4. Measurement of a through-thickness stresses using a narrow beam. The reference frame used here adheres to 

the one usually used in x-ray diffraction with the z-axis perpendicular to the surface. Note that the reference frame of the 

forming ram has the z-direction parallel to the through-thickness direction. The vector q

r

denotes the direction in which the 

strain is measured, and Φ, Ψ describe the orientation of q

r

within the specimenʼs reference frame. The label ʻIDʼ refers to 

the inner diameter.

The tilt mode used here makes use of the fact that the specimen can be orientated in such a way that a long, narrow, 

rectangular beam spot has the same orientation on the specimen surface at every azimuth and tilt angle, thus providing 

constant spatial resolution at all angles. The spatial resolution is optimal if the plane defined by the primary beam is 

perpendicular to the specimen surface. The implementation is described in Figure 4. Generally, a tilt or rotation of the 

specimen causes a rotation of the beam spot described by the angle δ. This rotation can be corrected for by rotating the 

specimen to -δ. An explicit expression for δ is given in [5]. Note that with increasing sample tilt the beam spot increases in 

length. The change in length can be undesirable for curved specimens because different depths are averaged over with 

increasing length of the beam spot. This is dealt with by using absorbing masks attached directly on the specimen 

surface. For spatial resolutions near the nominal 0.05 mm slit opening masks of 2.5 mm were used for the specimens 

investigated here. Stresses are considered constant over the length of the beam spot. It should be noted that for reasons 

both of optimal spatial resolution (narrowest beam spot) and optimal strain resolution (uncertainty of the d-spacing 

measured) the Bragg angle of the interplanar lattice spacing measured should be as large as possible, generally > 140 o. 

As a drawback, the choice of possible reflections (hkl) is very limited.
The rotational degrees of freedom necessary for orienting the specimen are found in Euler goniometers commonly used in 

x-ray diffraction as shown in Figure 5. The angle calculations for obtaining the azimuth angle Φ and the tilt angle Ψ from 

the Euler angles are given elsewhere [5]. The result of a measurement with a series of orientations (Φ,Ψ) yields lattice 

strains ε (hkl,Φ,Ψ) that relate to the residual stresses through [9]

( )

( )

( )

( )( ) 1 11 22 33

2

13 23 33

2

12

2

22

2

11

2 2

1

0

0

 cos sin sin 2 cos

cos sin sin 2 sin

σ σ σ

σ σ σ

σ σ σ

ε

+ + +

#

#

$

%

&

&

'

(

Φ + Φ Ψ + Ψ

Φ + Φ + Φ Ψ +

= × − = ΦΨ

ΦΨ

s hkl

s hkl

d

d d

eq. 1

dΦΨ is the interplanar lattice spacing measured at the azimuth angle Φ and the tilt angle Ψ, 0 d is the unstressed spacing, 

s (hkl) 1 and s (hkl) 2 2

1 are diffraction elastic constants, hkl denote the Miller indices of the reflection, and σij are the stress 

tensor components. Eq. 1 is linearized and solved by a least squares method.

Figure 5. X-ray diffractometer in four circle configuration. 

RESULTS

Results are presented for the minimum (25 %) and maximum (100 %) draw bead penetrations. Note that the stress 

distributions at 100 % draw bead penetrations have fewer data points due to the thickness reduction of ≈ 10 %. The 

thicknesses are listed in table 1. With the exception of AA6022-T4, the samples were zinc coated on both sides which is 

included in the thickness values.

Table 1. Sheet thickness in millimeters for samples with 25 % and 100 % draw bead penetrations. The accuracy is ± 0.01 

mm.

DB 

penetration

AA6022-T4 AKDQ HSLA 50 DP 600

25 % 0.98 0.94 0.79 1.00

100 % 0.85 0.81 0.72 0.90
AA6061-T4

-60

-40

-20

0

20

40

60

0 0.2 0.4 0.6 0.8 1

depth from ID [mm]

residual stress [MPa]

R25

R100

AKDQ

-100

-50

0

50

100

150

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

depth from ID [mm]

residual stress [MPa]

R25

R100

HSLA

-150

-100

-50

0

50

100

150

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

depth from ID [mm]

residual stress [MPa]

R25

R100

DP600

-200

-150

-100

-50

0

50

100

150

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

depth from ID [mm]

residual stress [MPa]

R25

R90

Figure 6. Through-thickness stresses in AA6022, AKDQ, HSLA 50 and DP600 (top to bottom) for 25 % and 100 % draw 

bead penetration. 

The results shown in Figure 6 can be understood by considering the individual components of the plastic strains that 

comprise the forming process of the panels. By pulling the sheet through the draw bead the sheet undergoes the principal 

steps of bending, backbending, stretching, and, upon tool release, elastic springback. As residual stresses arise from 

non-uniform plastic strains, bending is the fundamental process responsible for through-thickness stresses as shown for 

HSLA in Figure 7.
-200

-150

-100

-50

0

50

100

150

200

250

300

350

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

depth from ID [mm]

stress [MPa]

unloaded

bent

??? ???

Figure 7. Through-thickness stresses for plastic bending of a HSLA 50 sheet with 0.80 mm thickness. The bending radius 

is r=46.8 mm, equivalent to ±0.86 % strain at the surfaces. In the unloaded state the radius is r=58.5 mm from which the 

stress difference Δσ = ±370 MPa is calculated.

Deformation through stretching is uniform through thickness, thus the main effect of stretching on the residual stresses is

a reduction of the stress maxima similar to a stress relief by plastic deformation. Bending-backbending sequences do not 

alter the principal shape but rather reverse the stresses with respect to the neutral plane. Figure 7 indicates also the 

presence of the Bauschinger effect through the visibly smaller compressive stress region in the bent state which can be 

explained by a decreased compressive yield strength.

INTERGRANULAR STRESSES

In order to obtain reliable results it is essential to consider intergranular stresses that arise in neighboring grains with 

different orientations and orientation dependent yield points. Diffraction measures changes in inter-planar atomic lattice 

spacings (hkl) on grains in specific orientations, and it is therefore subject to effects of this type of stress. Intergranular 

stresses can be subtracted my measuring a small coupon that is relieved from all long-range stresses (i.e. stresses that 

extend over the specimens dimensions), thus retaining only the intergranular stresses. However, this is not a viable 

approach because of the difficulties involved in producing samples small enough to achieve sufficient stress relief without 

introducing new stresses by the extraction procedure.

Here, two measures were taken to mitigate the effect of intergranular stresses: First, the stresses measured on the

undeformed blank are subtracted, thus removing all preexisting intergranular stresses. This step also removes stress 

effects from the surface polishing, if any. Second, a portion of the intergranular stresses is introduced during forming, thus 

affecting different draw bead penetrations differently. The effects can be minimized by measuring stresses with different 

lattice planes (hkl) and subsequent averaging of the stresses. This is based on the equilibrium requirement that, in the 

absence of external forces, the sum of intergranular stresses for all grain orientations is zero. Two reflections ((310/(211)) 

were used for the steel samples, and three reflections were used for the AA6022 specimens ((311)/(331)/(420)). The use 

of more independent (hkl) is desirable especially for steel; however, with the available x-ray wavelengths only the two 

reflections (211) and (310) can be measured in the backscatter region (2θ > 140°). Measurements should be performed in 

this region both for accuracy and for the low sensitivity to misalignment.

Stress balance and balance of bending moments are heuristic measures of the effect of intergranular stresses on the total 

stresses. Both are fulfilled satisfactorily for the lower strength, single phase materials AA6022, AKDQ and HSLA. DP600 

differs with the two most likely reasons: first, the spread of intergranular stresses tends to increase with the yield stress,

and second, intergranular stresses can be expected to increase with the difference between ultimate tensile strength

(UTS) and yield strength. Both are the largest for DP600 with stress differentials between grains in the (310) and (211) 

orientation reaching 200 MPa (see Figure 8). Figure 8 also shows that the intergranular stresses are not uniform through 

the thickness as demonstrated by the fact that there is no simple offset/difference between the stresses measured using 

the (211) lattice planes and the (310) lattice planes.
-450

-400

-350

-300

-250

-200

-150

-100

-50

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

depth from ID [mm]

stress [MPa]

(310)

(211)

blank,(310)

blank,(211)

caption on same page?

Figure 8. Comparison of stresses measured on DP600 using different lattice planes in the undeformed state and 100 % 

draw bead penetration.

On the opposite end, the differential of intergranular stresses is smallest for AA6022 which has both the lowest yield 

stress and the lowest UTS. Stresses measured on three different lattice planes are shown in Figure 9.

-100

-80

-60

-40

-20

0

20

40

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

depth from ID [mm]

stress [MPa]

(331)

(420)

(311)

Figure 9. Stresses in the AA6022 panel with 100% penetration for three different lattice planes (hkl). The stresses in 

Figure 6, top were obtained by averaging the stresses shown here. ?

CONCLUSIONS

The through-thickness stresses on Numisheet 2005 benchmark 3 samples were measured for different materials and 

draw bead penetrations. The measurement procedure used offers new levels of resolution and detail, and it reveals that 

similar for all materials and draw bead penetrations the stress distributions can be explained by sequences of plastic 

bending. The results also demonstrate the presence of intergranular stresses which affect the stresses presented here to 

a minor degree.

REFERENCES

1. M. A. Iadicola, T. Foecke, and T. B. Stoughton (2005): “Experimental Procedures and Results for Benchmark 3: Stage 

2 Forming Process”, Proceedings of Numisheet 2005, edited by J. Cao, M. F. Shi, T. B. Stoughton, C.-T. Wang, and 

L. Zhang, American Institute of Physics, CP778 Volume B, pp. 905-915.

2. D.E. Green, T.B. Stoughton, T. Gnaeupel-Herold, M.A. Iadicola, T.Foecke (2006): “Influence of drawbeads in deep 

drawing of plane-strain channel sections”, Proceedings of the International Deep Drawing Research Group 2006 

Conference, June 19-21, 2006 - Porto, Portugal, edited by A.D. Santos and A. Barata da Rocha, pp. 559-566
3. M.C. Oliveira, A.J. Baptista, J.L. Alves, L.F. Menezes, D.E. Green, T. Gnaeupel-Herold, M.A. Iadicola, T. Foecke, T.B. 

Stoughton, Two Stage Forming: Experimental and FE Analysis, Proceedings of IDDRG, June 2006, Porto (Portugal), 

p. 279-286

4. Z. Cedric Xia, Craig E. Miller, Maurice Lou, Ming F. Shi, Alex A. Konieczny, X.M.Chen, Thomas Gnaeupel-Herold, A 

Benchmark Test for Springback: Experimental Procedures and Results of a Slit-Ring Test, SAE Trans.: J. Mater. 

Manufacturing 114, 19 (2006).

5. T. Gnaupel-Herold, Formalism for the Determination of Intermediate Stress Gradients using X-ray Diffraction, 

submitted to Journal of Applied Crystallography

6. M.E. Hilley, SAE J784a: Residual Stress Measurement by X-Ray Diffraction, Society of Automotive Engineers,

Warrendale, PA 1971

7. T. Gnaeupel-Herold, T.J. Foecke, H.J. Prask, R.J. Fields (2005), An Investigation of Springback Stresses in an AISI1010 Deep Drawn Cup, Mat. Sci. Eng. A399, 26-32.

8. T. Gnaeupel-Herold, H. J. Prask, R.J. Fields, T.J. Foecke, Z.C. Xia and U. Lienert (2004) "A synchrotron study of 

residual stresses in a Al6022 deep drawn cup", Materials Science and Engineering A366, 104-113.

9. V. Hauk (ed.), Structural and Residual Stress Analysis by Nondestructive Methods, Elsevier Science B.V. Amsterdam 

(1997)

10. I. C. Noyan, J. B. Cohen, Residual Stress Measurement by Diffraction and Interpretation, Springer, New York (1987)

11. T.Gnaupel-Herold, T.J.Foecke, H.J.Prask, R.J. Fields, L.E. Levine, Z. Cedric Xia, U. Lienert, An Investigation of 

Springback Stresses in Deep Drawn Cups Using Diffraction Techniques, SAE Trans.: J. Mater. Manufacturing 114, 

175 (2006).

CONTACT

For further information or questions and comments regarding this study, please contact Thomas Gnäupel-Herold at the 

NIST Center for Neutron Research at tg-h@nist.gov.