Copyright ©JCPDS-International Centre for Diffraction Data 2008 ISSN 1097-0002
IN SITU STRESS MEASUREMENT OF BIAXIALLY STRAINED
ALUMINUM ALLOY 5754-O
Mark A. Iadicola1 and Thomas H. Gnaeupel-Herold2
1
Metallurgy Division and 2NIST Center for Neutron Research, National Institute of Standards
and Technology, Gaithersburg, Maryland 20899
ABSTRACT
A method of measuring biaxial stress-strain curves for sheet metal samples subjected to large
plastic strains is described for one example aluminum alloy (AA 5754-O). The method uses a
portable X-ray diffraction (XRD) system to measure in situ stresses during tensile deformations.
Calibration of X-ray elastic constants and the effects of texture and intergranular stresses are
discussed. Uniaxial data measured using XRD and conventional means agree to within 18 MPa,
and biaxial stress-strain results are very repeatable.
INTRODUCTION
The mechanical response of sheet metals to large strains in biaxial tension (e.g., uniaxial, plane
strain, and 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 nonuniform
deformation, limited strain range, or the need for assumed constitutive laws to determine the
stress in the sample. The purpose of this work is to produce less ambiguous stress-strain
measurements up to large plastic (>15% strain) in-plane (biaxial) stretching of an as-received
sheet metal sample. The method of specimen deformation is a variation [1] of the Marciniak flat
bottom ram test [2] [Figure 1(a) is a through section drawing of the axisymmetric tooling], which
can impose near-linear strain paths in a range of strain states from uniaxial to equi-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. 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 center line of the ram. The upper
and lower surfaces at this center section are stress free. The ram force can not be simply related
to the resulting stress in the center section of the sheet, therefore XRD is used to measure the in
situ stress in the sheet (rather than modeling it numerically with an assumed constitutive law).
The material used in this investigation is commercially available, 1 mm thick, AA 5754-O
(which is of interest to the automotive industry). The microstructure of the as-received (AR)
sheet shows recrystallized grains, which are relatively equiaxed in the rolling plane, but slightly
elongated when viewed in the transverse plane. The size of the grains is relatively uniform in all
directions, with a 40 µm average diameter in the normal plane.
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132
Figure 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.
XRD PROCEDURE
A portable low-power (60 W) XRD system is mounted to the back of the forming machine, and
is used to measure the lattice strain in the exposed top surface of the specimen, as described in
reference [3]. The system uses the sin2ψ method in the ψ-geometry [Ω-mode, Figure 1(b) and
1(c)], with interchangeable single source X-ray tubes. Because of space constraints, the current
system only permits tilting in one direction with respect to the specimen over a limited angle
range (ψ = ± 35°). The CoKα beam is collimated and passed through a 5 mm long and 1.5 mm
wide (in the ψ 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). Figure 1(c) is a sketch of the beam and detector
geometry at φ = 0. The system is configured to acquire a single reflection peak profile at a time,
and here the {420} family of planes (Bragg angle 2θ = 162.5°) is used exclusively. The profiles
are fit using a two-peak (Kα1 and Kα2) Pearson’s VII function (with an exponent of 1.77) to the
upper 85% of the total peak after background subtraction. Lattice spacing (dφ ψ) is computed
from the peak position through Bragg’s Law. The stress free lattice spacing (do) is not measured
directly, since the sin2ψ 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 stress is less than the
uncertainty of the measurement.
Since the top and bottom surfaces of the specimen are stress free in the center section (above the
center of the ram, i.e., the washer hole), we neglect the through thickness stress. Using the
standard governing equation (Eq. (5.4) in reference [4]) relating the lattice strain to the surface
strain rotated to the specimen surface system [Figure 1(b)] where the X1 axis is parallel to the
rolling direction of the sheet and assuming an isotropic constitutive law, this equation is greatly
simplified. When considering tilting in the rolling direction (φ = 0°) the equation becomes
dψ − d o
do
=
1
1
S 2σ 11 sin 2 ψ + S1 (σ 11 + σ 22 ) + S 2σ 13 sin 2ψ
2
2
(1)
Copyright ©JCPDS-International Centre for Diffraction Data 2008 ISSN 1097-0002
133
1
1 +ν
S2 =
are
E
E
2
the effective X-ray elastic constants (XECs, with elastic constants ν and E), which are associated
with a chosen family of planes in a specific material. If tilting in the transverse direction (φ =
90°), the σ indices 1 and 2 in Eq. (1) are exchanged. It seems reasonable to assume the effective
XECs may differ for rolling (RD) and transverse (TD) directions, as a result of the orthotropic
nature of rolled sheet metal; therefore we will investigate calibration of the XECs in both of
these directions. The second term in Eq. (1) is not used here, since the XRD system combines the
data from the two detectors by resetting the intercept of the linear lattice strain versus sin2ψ to
zero for both detectors. The third term (“splitting” term) is expected to be zero, since the top and
bottom surfaces are stress free and the measurements are made after initial plastic yield
throughout the thickness. Potential causes and corrections for divergence from linear behavior,
attributable to texture or intergranular stresses, will be discussed in the results and analysis
section.
where σij are the components of stress in the surface system, and S1 = −
ν
and
MECHANICAL TESTING PROCEDURE
Two categories of experiments are performed: (1) the Marciniak ram test for biaxial strains with
in situ XRD measurements to determine the multiaxial stress-strain response, and (2) uniaxial
tensile tests of as-received and biaxially prestrained samples for XEC calibration and XRD
verification.
Equi-biaxial (EB) and plane-strain (PS) experiments are performed using the Marciniak tooling.
After initial clamping of the specimen (and reinforcing washer) in the binder, the general
procedure includes three steps: (1) the material is loaded by an increase in ram height at a quasistatic rate, (2) the ram position is held and the XRD system is focused on the surface of the sheet,
and (3) a scan of selected ψ angles (13 in this case) is taken each with a small (± 3°) oscillation.
This procedure is then repeated for the next data point. The entire procedure (1 through 3) takes
about 8 min with each peak profile taking about 15 s (because of the low power and absorption).
The approximate penetration depth is 40 µm, resulting in about 4500 grains diffracting at any
given time. Intensity variation with position may be used to approximate the number of
crystallites sampled at a given ψ [5]. A limited data set (not statistically significant) of these
intensities results in a conservative sampling estimate of >70 grains for all ψ used, and typically
>150 grains. The macroscopic strain during each hold time is calculated by averaging the output
of a biaxial extensometer for the entire scan time (standard deviations in the data were typically
<0.005% strain). Uniaxial (U) experiments are performed similarly, but use a portable uniaxial
testing frame with a single axis extensometer and ASTM-E8 sub-sized flat specimens. This
method of uniaxial testing requires less material and machining than the equivalent Marciniak
method. Separate samples are tested in both the rolling and transverse directions for each strain
state.
The same portable uniaxial testing frame and ASTM-E8 specimen shape are used for the XEC
calibration experiments. During deformation, load (from a calibrated load cell) and strain (from
an 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 initial uniaxial yield
stress) and reloaded elastically. During these elastic unload/reload times, the uniaxial loading is
Copyright ©JCPDS-International Centre for Diffraction Data 2008 ISSN 1097-0002
paused at multiple points and XRD is used to measure the current stress. These experiments are
performed on as-received samples and biaxially prestrained (by the Marciniak method) samples
with 5%, 10%, 15%, and 20% prestrain, in both RD and TD.
Figure 2. Calibration experiment results: example AR TD (a) sin2ψ behavior and (b) the associated XEC
determination, and (c) overall XEC results versus maximum stress.
RESULTS AND ANALYSIS
For each of the five prestrain levels (including AR) and in each direction (RD and TD), a
uniaxial XEC experiment is performed with 5 to 9 elastic unload/reload series at various strain
levels resulting in 70 independent measurements of XEC. Figure 2(a) plots the lattice spacing
versus sin2ψ behavior for one series, and the linear fits to these data. The slopes of these fits are
plotted in Figure 2(b) versus the measured true stresses, and this is also fit with a line where the
slope is ½S2 for this elastic unload/reload. The uncertainty is calculated from the propagation of
the lattice strain uncertainty for one standard deviation. Figure 2(c) plots the calculated ½S2
values versus the maximum stress in the as-received sample (with RD in red and TD in blue).
The large variations with stress are seen near the 0.02% yield offset stress of 94 MPa where the
calculated uncertainties are the largest. The overall average ½S2 values for RD and TD are (19.3
± 0.9) TPa−1 and (19.7 ± 0.7) TPa−1, respectively. In the figure, calculated ½S2 values for the
biaxially prestrained samples are also shown, and the differences between RD and TD (solid and
open symbols, respectively) values are even less distinct. The trend for all the points shows a
slight variation of XEC with the maximum stress during plastic deformation, irrespective of the
method of deformation (uniaxial or biaxial), although the higher stress levels are only achievable
with biaxial deformation.
There is some variation from the linear behavior in Figure 2(a), which is in part random and
systematic errors. These are present even when measuring Al stress free powder samples, which
may be used to bound these errors to ± 9 × 10–5 strain. Some variations seen in this study deviate
from linearity by values slightly greater then these bounds, and in some cases follow a pattern
(similar to a single period oscillation). This could be a result of other issues, such as texture or
intergranular stresses, so these are now briefly discussed.
Although texture is an important factor in non-linear sin2ψ behavior in some materials (e.g., iron
and steels), the non-linear sin2ψ behavior seen here cannot be explained by the effects of
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Copyright ©JCPDS-International Centre for Diffraction Data 2008 ISSN 1097-0002
preferred orientation. XEC-simulations using ideal orientations [6] show that both the measured
texture sharpness of our samples (maximum of 2.2 times random) and the weak elastic
anisotropy of aluminum produce a much weaker non-linear behavior than observed.
Figure 3. Example of intergranular stress data: (a) lattice strains versus stress at ψ = –34.7° and (b) intergranular
strain variation with ψ angle for 5% EB prestrained TD sample.
The other source of variations from linearity considered here are intergranular stresses, which are
difficult to measure. If we assume intergranular stresses add an offset strain (εint) to the average
micromechanical strain ( ε ), then the measured strain is ε = ε + ε int [7], where ε equals the right
hand side of Eq. (1). This formulation assumes the εint values are not stress level dependent, but
are added to the average strain (which depends on the average stress). The εint values depend on
the particular crystallites (and the difference in the yield stress between adjacent grains)
diffracting at the current ψ angle, therefore a variation of εint with ψ angle (and with plastic
deformation) is expected. Recall that at each sin2ψ scan the intercept of the linear fit was set to
zero, and this effectively assumes that a constant do exists for all the ψ angles (and allows for
combination of the data from the two detectors). Therefore, in the absence of intergranular
strains, the variation of measured strain with applied stress should intercept at zero (because of
random and systematic errors this does not occur exactly, as can be seen when measuring a
stress-free powder sample). Figure 3(a) shows lattice strain versus stress for ψ = −34.7° (of the
TD 5% biaxially prestrained sample). The non-zero intercept of the linear fit could be partially as
a result of intergranular stresses. Figure 3(b) is the distribution of these values with ψ angle for
the same sample (errorbars are the propagated uncertainty). There are five points slightly (but
distinctly) above the powder sample level (9 × 10−5), and are potentially the result of
intergranular stresses. Analyzing the data from the entire matrix of calibration experiments,
similar behavior is seen for the initial loading of the biaxially prestrained samples and not for the
as-received samples. After uniaxial plastic deformation, the intergranular strains reduce to below
the ± 9 × 10−5 bounds for almost all samples, but some distinctly negative strains seem to
develop at ψ = ± 15° in a few cases. Recalculation of the ½S2 values after subtraction of the
calculated intergranular strains results in values similar (less than 0.5% difference) to those
shown in Figure 2(c). Therefore intergranular stresses do not significantly affect our
measurements.
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Copyright ©JCPDS-International Centre for Diffraction Data 2008 ISSN 1097-0002
Stress-strain curves from standard uniaxial fixed-rate tests are shown in Figure 4(a) and 4(b) for
the RD and TD, respectively, along with data points measured by XRD using the average ½S2
values described above. The XRD data coincide well with the standard test results, with some
variation at lower strain levels (especially for RD). The results for the XRD PS and EB
experiments calculated using the same ½S2 values are plotted in Figure 4(c) and 4(d). The PS
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 sets performed
months apart by different system operators to further test the method’s repeatability. With the
exception of one RD EB data point (at about 2.7% strain), the repeat tests agree well.
Figure 4. XRD uniaxial stress-strain results in (a) RD and (b) TD with uniaxial curves from standard testing. Biaxial
XRD stress-strain results for (c) PS and (d) EB, with RD and TD plotted in closed and open circles, respectively.
A more complicated calibration using the variation of ½S2 with stress level [as shown in Figure
2(c)] could be used and is the topic of future work. Based on the trend, this calibration would
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Copyright ©JCPDS-International Centre for Diffraction Data 2008 ISSN 1097-0002
increase the measured stress values at lower stress levels and decrease the measured stresses at
the higher stress levels, which may correct the minor differences seen in Figure 4(a) and 4(b).
Overall the method produces useful biaxial stress-strain curves for sheet metal samples through
very high levels of plastic strain, previously unattainable.
REFERENCES
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[2] Marciniak, Z.; Kuczyński, K. Int. J. Mech. Sci. 1967, 9, 609–612.
[3] Foecke, T.; Iadicola, M. A.; Lin, A.; Banovic, S. W. Metall. Mater. Trans. A 2007, 38, 306–
313.
[4] Noyan, I. C.; Cohen, J. B. Residual Stress: Measurement by Diffraction and Interpretation;
Springer-Verlag: New York, 1987; p. 118.
[5] Gnaeupel-Herold, T.; Prask, H. J.; Fields, R. J.; Foecke, T. J.; Xia, Z. C.; Lienert, U. Mater.
Sci. Eng., A 2004, 366, 104–113.
[6] Hauk, V. Structural and Residual Stress Analysis by Nondestructive Methods; Elsevier:
Amsterdam, 1997; p. 193 ff.
[7] Behnken, H. Phys. Status Solidi A 2000, 177, 401–418.
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