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Author's personal copy

Materials Science and Engineering A 545 (2012) 168–175

Contents lists available at SciVerse ScienceDirect

Materials Science and Engineering A

journal hom ep age: www.elsevier.com/locate/msea

Effective X-ray elastic constant measurement for in situ stress measurement of

biaxially strained AA5754-O

Mark A. Iadicolaa,∗, Thomas H. Gnäupel-Heroldb

a Metallurgy Division, 100 Bureau Drive, STOP 8553, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA b NIST Center for Neutron Research, 100 Bureau Drive, STOP 6102, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

a r t i c l e i n f o

Article history:

Received 21 September 2011

Received in revised form 27 February 2012

Accepted 29 February 2012

Available online 14 March 2012

Keywords:

X-ray diffraction

Aluminum alloys

Sheet forming

Plasticity

Residual stresses

Neutron diffraction

Intergranular strain

a b s t r a c t

Accurate measurement of stresses by X-ray diffraction requires accurate X-ray elastic constants. Calibration experiments are one method to determine these for a specific material in a specific condition.

In this paper, uniaxial tension experiments are used to investigate the variation of these constants after

uniaxial and equal-biaxial plastic deformation for an aluminum alloy (AA5754-O) of interest to the automotive industry. These data are critical for accurate measurement of the biaxial mechanical properties of

the material using a recent experimental method combining specialized sheet metal forming equipment

with portable X-ray diffraction equipment. The measured effective X-ray elastic constants show some

minor variation with increased plastic deformation, and this behavior was found to be consistent for both

uniaxially and equal-biaxially strained samples. The use of two average values for effective X-ray elastic

constants, one in the rolling direction and one transverse to the rolling direction of the sheet material,

is shown to be of sufficient accuracy for the combined tests of interest. Comparison of uniaxial data

measured using X-ray diffraction and standard methods show good agreement, and biaxial stress–strain

results show good repeatability. Additionally, the calibration data show some non-linear behavior, which

is analyzed in regards to crystallographic texture and intergranular stress effects. The non-linear behavior

is found to be the result of intergranular stresses based on comparison with additional measurements

using other X-ray diffraction equipment and neutron diffraction.

Published by Elsevier B.V.

1. Introduction

The mechanical response of sheet metals to large strains in biaxial tension (e.g., uniaxial, plane strain, and equal-biaxial strains)

is of interest to the metal forming community. Many testing configurations exist to achieve some or all of the tensile strain states

of interest including cylindrical tension-torsion-inflation, inflation

bulge, hemispherical or flat ram, and cruciform testing, but all have

inadequacies resulting from out of plane bending, application of

non-uniform deformation, limited strain range, or the need for

assumed constitutive laws to determine the stress in the sample.

Iadicola et al. [1] used a combination of biaxial mechanical testing

and X-ray diffraction (XRD) techniques to produce less ambiguous

stress–strain measurements up to large plastic (strains > 15%) inplane (biaxial) stretching of an as-received sheet metal sample. In

this method of testing, XRD is used to measure the in situ interatomic lattice spacing in the sheet which is proportional to the

stress through effective X-ray elastic constants (XECs). This method

∗ Corresponding author. Tel.: +1 301 975 5703; fax: +1 301 975 4553.

E-mail addresses: mark.iadicola@nist.gov (M.A. Iadicola), gnaeupel@nist.gov

(T.H. Gnäupel-Herold).

requires careful calibration, but removes the need for numerical modeling with an assumed constitutive law to determine the

stress–strain behavior. In this paper, the calibration measurements

of effective XECs for the biaxial experiments in Iadicola et al. [1] are

described, with special attention to the possibility of effective XEC

variation due to the macroscopic plastic strain history. Preliminary

work on these measurements was presented briefly in Iadicola and

Gnaeupel-Herold [2], but has been substantially extended here.

In this work, elastic uniaxial loading is used in conjunction

with established XRD techniques to determine effective XECs for

one material of interest. These experiments are performed on asreceived material and material deformed equal-biaxially to various

strain levels in a manner similar to Iadicola et al. [1]. Due to the

orthotropic nature of the rolled sheet metal, the experiments are

performed for samples in both the rolling direction (RD) and transverse to the rolling direction (TD).

2. Material & mechanical testing procedure

The material used in this investigation is commercially available 1 mm thick AA5754-O (which is of interest to the automotive

industry). The microstructure of the as-received sheet shows

recrystallized grains, which are relatively equiaxed in the rolling

0921-5093/$ – see front matter. Published by Elsevier B.V.

doi:10.1016/j.msea.2012.02.100
Author's personal copy

M.A. Iadicola, T.H. Gnäupel-Herold / Materials Science and Engineering A 545 (2012) 168–175 169

(a) (b) (c)

66o

Specimen

Washer

100

12

12

X3

X1

Lower

Binder

Upper

Binder

Ram

Lock Bead

152.4

110

Fig. 1. Specimen and XRD axes defined by (a) through section of axisymmetric Marciniak tooling (dimensions in mm), (b) standard XRD angles, and (c) sketch of the XRD

head.

plane, but slightly elongated when viewed in the transverse plane.

The size of the grains is relatively uniform in each direction, with

an average diameter of 40 m in the normal plane.

The effective XEC calibration experiments presented here use

uniaxial tensile tests, of both as-received and equal-biaxially

strained samples, in conjunction with XRD measurements to calibrate the effective XEC. A portable uniaxial testing frame is used

to test ASTM-E8 sub-sized sheet metal specimen shape samples

with a reduced tab length. During deformation, load (from a calibrated load cell) and strain (from a calibrated axial extensometer)

are measured to determine the true stress in the gage section. At

selected strain levels the specimen is unloaded (to a stress never

less than 80% of the as-received initial uniaxial yield stress, approximately 94 MPa for the RD) and reloaded elastically. During these

elastic unload/reload times, the uniaxial loading is paused at multiple points and XRD is used to measure the current interatomic

lattice spacing. These experiments are performed on as-received

samples and equal-biaxially strained samples (as in Iadicola et al.

[1]) with approximately 5%, 10%, 15%, and 20% equal-biaxial truestrain, in both RD and TD.

The method of specimen deformation used here to perform the

equal-biaxial straining is a variation [3] ofthe Marciniak flat bottom

ram test [4] (Fig. 1a is a through section drawing of the axisymmetric tooling), which imposed near-linear strain paths in a range of

strain states from uniaxial to equal-biaxial tension (under plane

stress conditions on as-received sheets, stretching the sheet like

a drum head). The upper surface of the specimen is exposed for

metrology. In Iadicola et al. [1], biaxial stress–strain behavior was

measured using a combination of this mechanical deformation and

XRD to determine stress. A washer of mild steel is used to reinforce

the specimen where it will bend around the ram radius, thus concentrating the deformation in the center of the specimen directly

above the centerline of the ram. The upper and lower surfaces at

this center section are traction free. The ram force cannot be simply related to the resulting in-plane stresses in this center section of

the sheet, therefore XRD is used to measure the in situ interatomic

lattice spacing, which is proportional to the in situ stresses in the

sheet through an effective XEC.

The procedure used in Iadicola et al. [1] is similar to that used to

calibrate the effective XEC.We briefly review the procedure here. In

Iadicola et al.[1], after as-received sheet and reinforcing washer are

initially clamped in the forming machine binder (or locked in the

grips for the uniaxial experiments), the general procedure includes

three steps: (1) the material is loaded by imposing tensile strains

(by an increase in ram height for biaxial loading or by a crosshead

displacement for uniaxial loading) at a quasi-static rate, (2) the

displacement is held fixed and the XRD system is focused on the

surface of the sheet, and (3) an XRD scan (described below) is performed. This procedure is then repeated for the next data point.

For both the experiments in Iadicola et al. [1] and in this work, the

entire procedure (1 through 3) takes approximately 8 min (due in

large part to the low power of the XRD system and X-ray absorption by the sample). The macroscopic strain during this hold time

is calculated by averaging the output of an extensometer for the

entire scan time (standard deviations in these data were typically

<0.005% strain). The biaxial experiments use a biaxial extensometer with similar performance to the axial extensometer used in the

uniaxial experiments.

3. XRD procedure

A portable low-power (60W) XRD system is mounted to the

back of the biaxial sheet metal forming machine, and is used to

measure the interatomic lattice spacing in the exposed top surface

of the specimen as described in [5]. The system uses the sin2 -

method [[6], pp. 122 ff.] in the -geometry (Fig. 1b and c), with

interchangeable single source X-ray tubes. The two coordinate systems in Fig. 1b are the material system and the XRD-head system.

The material system is oriented with the X1 axis in the RD, X2 in

the TD, and X3 vertical to the sheet. The XRD system (Xˆ) is offset to

the material system first by a rotation () about the sheet normal

and then by a rotation ( ) about the Xˆ2 axis, where the Xˆ3 axis is

the bisector of the X-ray source and reflected beams. Fig. 1c is a

sketch XRD-head (X-ray source beam with left and right detectors

shown) and the axes for one -angle at = 0. In this figure, the Xˆ3

axis is oriented in the direction associated with the measurements

using the right detector. Due to the space constraints, this system

only permits tilting in one direction with respect to the specimen

(= constant) over a limited angle range ( = ±35◦).

In this work, as well as in Iadicola et al. [1], Co K radiation

is used. The beam is collimated and passed through a 5 mm

long and 1.5 mm wide (in the -tilt direction) aperture. The

diffracted/reflected X-rays from the specimen are acquired by two

256 channel 10◦ linear position sensitive scintillation detectors

(one placed symmetrically on each side of the source beam). Note

that Fig. 1c shows both detectors, but the axes as shown are for the

source beam and right detector only. The system is configured to

acquire a single reflection peak profile in each detector at one time,

and here the {4 2 0} family of planes (Bragg angle  = 81.25◦ for

the Co X-ray source) is used exclusively. The profiles are fit using a

two peak (K1 and K2) Pearson’s VII function (with an exponent

of 1.77) to the upper 85% of the total peak after background

subtraction. Lattice spacing (d) is computed from the peak

position through Bragg’s Law. The stress free lattice spacing (do)

is not measured directly for the material, since the sin2 -method

is not very sensitive to error in this value, and a system default

value for aluminum is assumed. To verify this, an XRD measurement of a stress free powder is made, and the assumed do is

considered acceptable if the measured lattice strains are less than
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170 M.A. Iadicola, T.H. Gnäupel-Herold / Materials Science and Engineering A 545 (2012) 168–175

the systematic uncertainty (±9 × 10−5 strain). For all of the XRD

measurements reported here, 26 -angles (13 for each detector)

are used for each XRD scan. To reduce the effect of random noise

and improve the quality of the peak profile in light of grain size

and crystallographic texture, each peak profile is scanned for about

15 s during which a small (±3◦) -angle oscillation is performed.

The approximate penetration depth is 40 m, resulting in about

4500 grains diffracting at any given time, but only a few of these

will be properly oriented for the {4 2 0} reflection at any given tilt.

The standard governing equation [[6], p. 118 Eq. (5.4)] relating the lattice strain (based on the reference and measured lattice

spacing) to the surface strains (εij) rotated to the specimen surface

system (Fig. 1b) is

d− do

do

= ε11 cos2sin2 + ε12 sin2sin2 + ε22 sin2sin2 +

ε33 cos2 + ε13 cossin2 + ε23 sinsin2 . (1)

An isotropic constitutive law is assumed as is typical for the

sin2 -method. Additionally, the top and bottom surfaces of the

specimen are stress free in the center section (this is true for the

uniaxial case andfor the biaxial case above the center ofthe ram,i.e.,

the washer hole),therefore we neglectthe through thickness stress

(33 = 0). When considering the above assumptions and tilting in

the rolling direction (= 0) Eq. (1) becomes,

d − d0

d0

= 1

2S211 sin2 + S1(11 + 22) +

1

2S213 sin2 (2)

where ij are the components of stress in the surface system, and S1

and (1/2)S2 are the effective XECs. If tilting in the transverse direction (= 90◦), the  indices 1 and 2 in Eq. (2) are exchanged. Ideally

S1 = −/E and (1/2)S2 = (1 + )E, but the effective constants actually

vary with the chosen family of planes and the specific material. For

example, reported values of (1/2)S2 [[6], Appendix F] for pure aluminumandAA5083-H23 for the {4 2 0} family of planes usingCoK

radiation differ by 11%. Additionally, it seems reasonable to assume

the effective XECs may differ for RD and TD, due to the orthotropic

nature of rolled sheet metal; therefore we will investigate calibration of the effective XECs in both of these directions. Only the first

term on the right hand side of Eq. (2) is used here to determine 11.

The first term captures this value as proportional to the slope of the

sin2 behavior. Additionally, the XRD system combines the data

from the two detectors by resetting the intercept of the linear lattice strain versus sin2 to zero for both detectors, which prevents

accurate calculation ofthe second term on the right hand side of Eq.

(2). In both this work and Iadicola et al. [1], the stress in the -tilt

direction (11 in Eq. (2)) is of primary interest, so only the (1/2)S2

effective XEC is needed. The third term (‘splitting’term)is expected

to be zero, since the top and bottom surfaces are stress free in the

normal direction and the measurements are made after initial plastic yield throughout the thickness. Potential causes of divergence

from linear behavior, due to texture or intergranular stresses, will

be discussed in the next section.

The calibration experiments permit measurement of the truestress in the axial direction at any given extension hold, while all

other macroscopic stresses are zero. Combining these data with

the XRD data at these extension holds, Eq. (2) can be solved for

(1/2)S2 (the effective XEC of interest). These data are developed for

as-received and equal-biaxial strained samples in both the RD and

TD for a range of macroscopic strain levels to verify orientation and

deformation induced effective XEC variation.

-10 0 10 20 30

 ε RD (%)

-10

0

10

20

30

(%)

ε TD 

Fig. 2. Matrix of effective XEC uniaxial calibration unload/reload cycles (points)

plotted on true-strain axes. Dashed line shows the equal-biaxial strain path from

as-received condition to start of uniaxial tests.

4. Results and analysis

4.1. Effective XEC results

For each of the five equal-biaxial strain levels (including asreceived) and in each direction (RD and TD) a uniaxial effective XEC

experiment is performed with 5 to 9 elastic unload/reload series at

various strain levels resulting in almost 70 independent measurements of effective XEC. Fig. 2 plots the matrix of experiments where

each data point represents one unload/reload calibration measurement set at a given true-strain in RD and TD. The uncertainty of

these strain values is much smaller than the data symbols shown.

The dashed line in the figure represents the equal-biaxial strain

path. Fig. 3a plots the lattice strain versus sin2 behavior for one

series, and the linear fits to these data. The slopes of these fits are

plotted in Fig. 3b versus the measured true-stresses, and this is also

fit with a line where the slope is the effective XEC (1/2)S2 for this

elastic unload/reload. The uncertainty for each (1/2)S2 value is calculated from the propagation of the lattice strain uncertainty, that

is approximately ±9 × 10−5 strain based on the system uncertainty.

Fig. 4a and b plots the calculated (1/2)S2 values (with uncertainties

shown as error bars) versus the hardening stress and plastic strain,

respectively, for the as-received sample (with RD in red filled circles and TD in blue open squares). Large variations in effective XEC

are seen for hardening stress and plastic strain near the yield offset stress of 94 MPa and the initial 0.2% yield offset. At these low

stresses and strains, the calculated uncertainties (from the propagated uncertainties) are quite large, and exceed the variation seen

in effectiveXEC. This is partially due to the small range of stress over

which the linear fit is being calculated. In the figures, calculated

(1/2)S2 values for the uniaxial calibration experiments of previously equal-biaxially strained samples are also shown. Differences

between RD and TD values (solid and open symbols, respectively)

are even less distinct, but the trend for all the points seems to show

a slight trend between effective XEC and hardening stress or plastic strain during plastic deformation, irrespective of the method of

plastic deformation (uniaxial or equal-biaxial), although the higher

stress and plastic strain levels are only achievable with biaxial

deformation. A few outlier points exist, but these are associated

with elastic unload/reload cycles near or just after necking where

the uniaxial stress-state assumption breaks down. The overall
Author's personal copy

M.A. Iadicola, T.H. Gnäupel-Herold / Materials Science and Engineering A 545 (2012) 168–175 171

0 0.1 0.2

(a) (b)

0.3 0.4

 sin ψ2

0

3

6

9

12

(x 10 ) -4 

d ο

_____

d - d ψ ο

 85 MPa

 108 MPa

 135 MPa

 163 MPa

 189 MPa

0 50 100 150 200

 σ (MPa)

0

1

2

3

4

(x 10 ) -3 

[sin ] ψ2

slope

1_

 2 2 S

1

Fig. 3. Calibration experiment results for as-received TD (a) sin2 behavior and (b) the associated effective XEC determination from the slope of linear fits in (a).

average (1/2)S2 values for RD and TD are 19.3 TPa−1 and 19.7 TPa−1,

respectively. Apparently the orthotropic nature of the rolled sheet

does not result in a very substantial difference in the XECs in the

RD and TD.

In Fig. 3a, there is some variation from the linear sin2 behavior, which in part is due to random errors within the systematic

uncertainty. These are present even when measuring the stress

free powder samples, which may be used to bound these errors

to ±9 × 10−5 lattice strain. Some samples in this study, particularly

after equal-biaxial straining, have lattice strain variations that deviate from linearity by values slightly greater then these bounds, and

in some cases seem to follow a pattern (similar to a wave form). This

could be a result of other issues, such as texture or intergranular

stresses so these are now discussed.

4.2. Non-linear sin2 behavior

Although crystallographic texture has been shown to be an

important factor in non-linear sin2 behavior in some materials

(e.g., iron and steels), the non-linear sin2 behavior seen here

cannot be explained by the effects of preferred orientation. The

full extent of the texture effect on the X-ray elastic constants in

aluminum was investigated using the inverse Kröner model [7]

together with the crystal orientation distribution function (ODF).

Details of the calculation of X-ray elastic constants using the ODF

can be found in [7]. The ODF was determined from measured

X-ray pole figures ((2 0 0), (2 2 0), (1 1 1) reflections) using the

program popLA [8]. Fig. 5 plots the calculated lattice strains for an

applied uniaxial stress of 100 MPa for a uniform crystallographic

texture distribution (with (1/2)S2 = 19.3 TPa−1), for the measured

as-received ODF, for the ODF measured after 5% equal-biaxial

straining, and for the ODF measured after 20% equal-biaxial straining. Also plotted in the figure are the measured lattice strain data

for a uniaxial calibration sample in the TD after 5% equal-biaxial

strain at a true-stress of 100 MPa. Recall that the systematic uncertainty is estimated at ±9 × 10−5 lattice strain. The results in Fig. 5

demonstrate the effect of the texture induced non-linear behavior of the XEC is very small even for the sample with the highest

strain (20% equal-biaxial strain). The calculated non-linear behavior is much less than the non-linear behavior seen in the data and

does not follow the same trends.

The other source of variations from linearity considered here

is intergranular stress, which is difficult to measure. Initial assessment of intergranular stresses was performed using the calibration

data developed here. The results led to further study using another

XRD system as well as measurement through neutron diffraction.

We will discuss each method in turn, and compare the results.

First the calibration data was analyzed. If we assume intergranular stresses add an offset strain (εint) to the average lattice strain

(ε ̄) during a single XRD scan then the measured lattice strain is

ε = ε ̄ + εint [9], where ε ̄ equals the right hand side of Eq. (2). This

formulation assumes the εint values are not stress level dependent,

but are added to the average lattice strain (which is dependent on

the average stress). The εint values are dependent on the particular

0 75 150 225 300

(a) (b)

 σ p (MPa)

10

15

20

25

30

1_

 2 2 S

(TPa ) -1

 AR-RD

 AR-TD

 5%-RD

 5%-TD

 10%-RD

 10%-TD

 15%-RD

 15%-TD

 20%-RD

 20%-TD

0 6 12 18 24

 ε p (%)

10

15

20

25

30

1_

 2 2 S

(TPa ) -1

Fig. 4. Variation of effective XEC (1/2)S2 with plastic deformation, shown versus (a) hardening stress and (b) plastic strain.
Author's personal copy

172 M.A. Iadicola, T.H. Gnäupel-Herold / Materials Science and Engineering A 545 (2012) 168–175

0 0.1 0.2 0.3 0.4

 sin ψ2

0

2

4

6

8

(x 10 ) -4 

d ο

_____

d - d ψ ο

 EB 5 % Data

 Uniform Calc.

 AR Calc.

 EB 5 % Calc.

 EB 20 % Calc.

Fig. 5. Inter-atomic lattice strain data (circles) versus sin2 for uniaxial sample

in TD after 5% equal-biaxial straining at applied true-stress of 100 MPa. Calculated

lattice strains (lines) versus sin2 at 100 MPa for uniaxial TD sample with uniform crystallographic texture distribution, as-received (AR) texture, texture after

5% equal-biaxial (EB 5%) and 20% equal-biaxial (EB 20%) straining.

crystallites (and the difference in the yield stress between adjacent

grains) diffracting at the current tilt and rotation angles, therefore

εint would be a function of -angle and -angle. In this work we are

also admitting the possibility of variation with plastic strain (εp) as

well, thus Eq. (2) becomes,

d − d0

d0

= 1

2S211 sin2 + S1(11 + 22) +

1

2S213 sin2 + εint( ,= 0, εp). (3)

Recall that at each sin2 scan, the intercept of the linear fit was

set to zero, this effectively assumes that there is a constant do that

exists for all the -angles (and allows for combination of the data

from the two detectors). In the absence of intergranular strains, the

variation of measured strain with applied stress should intercept

at zero. This does not occur exactly, due to random and systematic

errors (as seen when measuring a stress-free powder sample). For

a uniaxial stress state with intergranular stresses Eq. (3) simplifies

to,

d − d0

d0

=

1

2S2 sin2 + S1



11 + εint( ,= 0, εp). (4)

This suggests thatthe strain associated with intergranular stress

may be determined from the intercept of a linear fit of the lattice

strains to the applied uniaxial stress for a fixed combination of 

and for one elastic unload/reload cycle (εp is constant). Fig. 6

shows the results and analysis for one uniaxial effective XEC experiment in the TD on a 5% equal-biaxial strained sample. Fig. 6a plots

the mechanical response (black line) with circle symbols at the

time in the load history that each XRD scan is performed. Fig. 6b

shows the determination of the intergranular strains as a perspective plot where the black dashed lines are described by Eq. (4) for

two -angles and the intercepts with the 11 = 0 plane are the

εint values for the given TD (= 90◦) and the -angles. The data

in Fig. 6b are color coded to the points in Fig. 6a (above the 94 MPa

threshold) during the first loading after the equal-biaxial straining.

Fig. 7a shows lattice strain versus stress for = −34.7◦ (in the TD

for the 5% equal-biaxial strained sample). The non-zero intercept

of the linear fit is partially due to intergranular stresses. Fig. 7b is

the distribution of these values with -angle for the same sample

(error-bars are the propagated uncertainty). There are points with

slightly (but distinctly) larger amplitude than the powder sample

level (±9 × 10−5), suggesting this behavior is more than random or

systematic error. Analyzing the data from the entire matrix of calibration experiments, similar behavior is seen for the initial loading

0 10 20 30

 ε (%)

0

100

(b)

(a)

200

300

(MPa)

σ

 continuous

 XRD scans

0

3

6

9

12

(x 10 ) -4 

ε

0 0.1 0.2 0.3 0.4

 sin ψ2

0

50

100

150

200

(MPa)

σ

 100 MPa

 125 MPa

 150 MPa

 175 MPa

Fig. 6. (a) Uniaxial true-stress versus true-strain curve in TD after 5% equal-biaxial straining. Data from load cell and extensometer shown as a solid line with circles at the

times of the XRD scans. (b) Inter-atomic lattice strain versus sin2 at various elastic stress levels in (a) plotted along the applied true-stress. Black squares are calculated εint

and dashed lines are fits to Eq. (4) at = −34.7◦ and −28.23◦.
Author's personal copy

M.A. Iadicola, T.H. Gnäupel-Herold / Materials Science and Engineering A 545 (2012) 168–175 173

0 50 100 150 200

(a) (b)

 σ (MPa)

0

3

6

9

12

(x 10 ) -4 

ε

-40 -30 -20 -10 0 10 20 30 40

ψ ( )o

-3.0

-1.5

0

1.5

3.0

(x 10 ) -4

ε int from (a)

 Detector 1

 Detector 2

+/- 9e-05 strain

Fig. 7. Example of intergranular strain data analysis: (a) lattice strains versus stress at = −34.7◦ with linear intercept at the calculated intergranular strain value (from

Fig. 6b) and (b) intergranular strain variation with -angle for TD sample after 5% equal-biaxial straining.

of the equal-biaxially strained samples and not for the as-received

samples. After initial uniaxial plastic deformation, the intergranular strains seem to reduce to below the ±9 × 10−5 bounds for almost

all samples, but some distinctly negative strains seem to develop

near = ±15◦ in multiple cases. Recalculation of the (1/2)S2 values

after subtraction of the calculated intergranular strains results in

values similar (less than 0.5% difference) to those shown in Fig. 4.

The sudden increase in lattice strains at the edge of the tilt

range in Fig. 7b suggested that a review of the intergranular

strains over a larger range was needed to verify the behavior. Therefore measurements of an unloaded sample equal-biaxial

strained to approximately 25% strain were taken using a XRD

system with Euler goniometer that can measure tilts to almost

 = ±70◦, well beyond the limited range of the XRD system

0 15 30 45 60 75 90

 ψ ( )o

-3.0

-1.5

0

1.5

3.0

(x 10 ) -4 

ε int 

0 15 30 45 60 75 90

 ψ ( )o

-3.0

-1.5

0

1.5

3.0

(x 10 ) -4

ε int 

0 15 3

(a) (b)

(c) (d)

0 45 60 75 90

 ψ ( )o

-3.0

-1.5

0

1.5

3.0

(x 10 ) -4 

ε int 

0 15 30 45 60 75 90

 ψ ( )o

-3.0

-1.5

0

1.5

3.0

(x 10 ) -4

ε int 

Fig. 8. Average intergranular strains for {4 2 0} plane spacing after equal-biaxial straining calculated from calibration data (dashed lines for equal-biaxial 20% strain) compared

with those measured by Euler goniometer XRD system (solid lines for equal-biaxial 26% strain) measurements in (a) RD and (b) TD. Neutron diffraction measurements (solid

lines) and average intergranular strains calculated from effective XEC data (dashed lines) in (c) RD and (d) TD for equal-biaxial 5% strained samples. Note the larger range

than Fig. 7.
Author's personal copy

174 M.A. Iadicola, T.H. Gnäupel-Herold / Materials Science and Engineering A 545 (2012) 168–175

0 10 20 30

 ε (%)

0

(a) (b)

(c) (d)

100

200

300

(MPa)

σ

 standard test

 XRD

0 10 20 30

 ε (%)

0

100

200

300

(MPa)

σ

 standard test

 XRD

0 5 10 15 20

 ε (%)

0

100

200

300

400

(MPa)

σ PS in RD

 PS in TD

0 5 10 15 20

 ε (%)

0

100

200

300

400

(MPa)

σ EB in RD

 EB in TD

Fig. 9. Calibrated XRD uniaxial stress–strain results from Iadicola et al. [1] in (a) RD and (b) TD with standard uniaxial curves (solid lines). Biaxial XRD stress–strain results

from Iadicola et al. [1] for (c) plane-strain and (d) equal-biaxial deformation, with RD and TD plotted in closed and open circles, respectively.

incorporated into the sheet metal forming machine ( = ±35◦).

These measurements also use Co K radiation looking at the spacing of the {4 2 0} family of planes, and the data are plotted in

Fig. 8a and b for the RD and TD respectively (solid lines). Symmetry in the ± range is assumed and the measured data are

averaged for equal | | angles. The error bars shown are uncertainties based on the counting statistics of the measurements.

Also plotted in Fig. 8a and b are the values calculated from the

calibration data for the 20% equal-biaxially strained sample in

both directions, where the results have been averaged for ± 

symmetry and for | | angles within the ±3◦ scanning oscillation

of one another. The error bars are the propagated uncertainties of each average value. The quantitative values agree within

the uncertainty of the measurements at only some -angles, but

the qualitative variation is quite similar between the calibration

data and the values measured using the Euler goniometer. The

agreement is quite remarkable considering the Euler goniometer

measurements are for an unloaded sample and the effective XEC

measurements are extrapolated from results on a loaded sample.

The difference could be due in part to the different plastic strain

levels where approximately 25% strain level is near to the failure

strain.

A second set of measurements over an even larger range

(approximately = ±90◦) using neutron diffraction were performed at the NIST Center for Neutron Research. Again the {4 2 0}

family of planes was measured and the results are plotted (solid

lines) in Fig. 8c and d in the RD and TD respectively for an unloaded

5% equal-biaxially strained sample. The error bars are based on

the counting statistics of each diffraction peak. These data are

averaged results assuming symmetry in positive and negative -

axis. The values calculated from the effective XEC calibration data

for a 5% equal-biaxially strained sample in both directions are plotted in dashed lines. Again the calculated values assume symmetry

in± and are averaged for | | angles within±3◦ of one another. The

neutron results quantitatively match the calibration measurement

results for only a small range of angles. However, the qualitative

variation is similar, and the locations of the intergranular strain

minimums are similar. Only the neutron measurements show a

peak at = 0◦, whereas both XRD methods show a leveling off at

this angle.

Considering the uncertainties associated with the effective XEC

XRD measurements and that they are performed over a smaller 

range it is not surprising that the measurements do not capture

the intergranular behavior as well as the other two methods, but

the fact that it does capture the trends using a test method not

specifically intended to measure intergranular stress without a full

unloading is very impressive. It must be emphasized that X-ray and

neutron diffraction differ in one important aspect: depth penetration which is about 103 times larger for neutrons. As a result, in

X-ray diffraction the number of contributing grains is by the same

factor smaller, and, because of the shallow penetration, it is subject

to possible through thickness differences stemming from the initial

rolling texture or a lack of constraining grains at the surface. This

may explain the difference described above between the XRD and

neutron values at = 0◦.

Similar to above, the (1/2)S2 values were recalculated adjusting

the data by subtraction ofthe intergranular strains measured by the

alternate methods. The recalculated values again showed minimal
Author's personal copy

M.A. Iadicola, T.H. Gnäupel-Herold / Materials Science and Engineering A 545 (2012) 168–175 175

change in the effective XECs with plastic strain, therefore intergranular stresses do not seem to significantly affect our effective

XEC determination. One possible reason is that large range of stress

over which the effective XEC is being measured out weighs the

comparatively small intergranular stresses. Additionally since the

biaxial stress–strain measurements are made in a consistent way

with the effective XEC measurements, it is likely that any minor

offsets in strain or angle will effect both sets of data equally

and therefore be corrected through the self-consistent calibration

method described here.

4.3. Macro scale stress–strain data comparison

Stress–strain curves from standard uniaxial fixed rate tests [1]

are shown in Fig. 9a and b for the RD and TD, respectively, along

with data points measured by XRD using the average (1/2)S2 values

described above. The XRD data coincide well with the standard test

results, with some variation atlower strain levels (especially for the

RD). The results for the XRD plane-strain and equal-biaxial experiments [1] calculated using the same (1/2)S2 values are plotted in

Fig. 9c and d. The plane-strain stress data for the near-zero strain

direction are plotted using the strain in the maximum (principal)

strain direction for visibility. Each plot includes data from two data

setsperformedmonths apart bydifferent systemoperators to check

measurement repeatability. With the exception of one RD equalbiaxial data point (at about 2.7% strain) the repeat tests agree well.

See Iadicola et al. [1] for explanation of measured uncertainties. A

more complicated calibration using the variation of (1/2)S2 with

stress level (as shown in Fig. 4a) could be used. This more complicated calibration would slightly increase the measured stress

values at lower stress levels and slightly decrease the measured

stresses at the higher stress levels, which may correct the minor

differences seen in Fig. 9a and b. Overall the calibration method

produces useful biaxial stress–strain data for sheet metal samples

through very high levels of plastic strain, previously unattainable.

5. Summary and conclusion

Combined uniaxial and in situ X-ray diffraction experiments

were performed to calibrate the effective X-ray elastic constants

for use in other experiments. The results of almost 70 uniaxial

effective X-ray elastic constant experiments were presented. These

experiments were performed on a single aluminum alloy ofinterest

that had been plastically deformed either through either uniaxial

or equal-biaxial tension, and the effective X-ray elastic constants

were measured in the sheet rolling and transverse directions. The

results show that the X-ray elastic constant for the rolling direction was slightly higher than that of the transverse direction. They

also show that there is a trend for the elastic constants to increase

slightly with increased uniaxial or equal-biaxial plastic deformation. It was not clear from the data if this trend is more closely

associated with the increase in hardening stress level or the plastic

strain. The possibility of intergranular stresses causing nonlinear

behavior in the sin2 technique was investigated. The measured

intergranular strains seemed to be most prevalent in the equalbiaxial plastically strained samples, which were the only cases with

a large enough effect to be measured using the less sensitive X-ray

system associated with the mechanical testing. The effect of the

intergranular strains on the X-ray elastic constants was found to

be minor. The measured average effective X-ray elastic constants

were used to calibrate separate uniaxialtest of as-received samples,

and the results compared very well with the results from standard testing. These same constants were used to calibrate uniaxial,

plane-strain, and equal-biaxial data associated with another study

[1].

References

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