Measuring Strain Using X-Ray Diffraction
James Belasco
Office of Science, SULI Program
Villanova University
Stanford Linear Accelerator Center
Menlo Park, CA
August 17, 2007
Prepared in partial fulfillment of the requirements of the Office of Science, U.S.
Department of Energy Science Undergraduate Laboratory Internship (SULI)
Program under the direction of Dr. Michael Woods at the Stanford Linear
Accelerator Center.
Participant: ________________________________________________
Signature
Research Advisor: ______________________________________________
Signature
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Table of Contents
Abstract ............................................................................................................................... 3
Introduction ......................................................................................................................... 3
Materials and Methods ........................................................................................................ 5
Results ................................................................................................................................. 7
Discussion ........................................................................................................................... 7
Acknowledgements ............................................................................................................. 9
References ......................................................................................................................... 10
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Abstract
Measuring Strain Using X-Ray Diffraction. JAMES BELASCO (Villanova University,
Villanova, PA, 19085) APURVA MEHTA (Stanford Linear Accelerator Center, Menlo Park,
CA, 94025)
Determining the strain in a material has often been a crucial component in determining the
mechanical behavior and integrity of a structural component. While continuum mechanics
provides a foundation for dealing with strain on the bulk scale, how a material responds to strain
at the very local level—the understanding of which is fundamental to the development of a
cohesive framework for the behavior of strained material—is still not well understood. One of
the critical components in determination of the behavior of materials under strain at a local scale
is an understanding how global average deformation, as a response to an externally applied load,
gets distributed locally. This is critical and very poorly understood for a polycrystalline
materials—the material of choice for a large variety of structural components. We studied this
problem for BCC iron using x-ray diffraction. By using a nanocrystalline iron sample and taking
x-ray diffraction patterns at different load levels and at different rotation angles, a complete 2nd
rank strain tensor was determined for the three sets of crystallites with three distinct
crystallographic orientations. The determination of the strain tensors subsequently allowed the
calculation of the elastic modulus along each crystallographic plane. When compared to
measured values from single crystal for the corresponding crystal orientations, the data from our
polycrystalline sample demonstrated a higher degree of correlation to the single crystal data than
expected. The crystallographic planes demonstrated a high degree of anisotropy, and therefore,
to maintain displacement continuity, there must be a secondary mode of strain accommodation in
a regime that is conventionally thought to be purely elastic.
Introduction
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Every successful major bridge builder
elastic, or reversible strain, produce two
for half a century has used continuum
different types of perturbations in the
mechanics to analyze the response of
lattice structure of a material. Elastic
materials to strain. However, while
strain causes a stretching in the lattice
continuum mechanics can model the
bonds while plastic strain results in
deformation of materials at large scales,
dislocation and motion of slip-bands.
at the very local scale materials
The ideal method for measuring such
invariably deform and fail, (even in large
changes is X-ray diffraction, which
structures failure is often initiated by a
produces a distinctly different pattern for
single crack). At the very local scale,
the two different types of strains. Elastic
materials are not the homogeneous and
strain, which is the primary focus of this
isotropic structures continuum
paper, alters the spacing of the
mechanics assumes them to be, but
crystalline lattice and therefore results in
instead are composed of an anisotropic
shifting of the diffraction peaks [2].
arrangement of discreet atoms and
Thus, as a sample is strained along a
molecules. Thus, a more complete
particular axis, X-ray diffraction patterns
understanding of mechanical behavior is
can be taken that will elucidate the
necessary to reconcile the discrepancy
deformation in the lattice structure.
between the two views of the behavior
Using BCC iron, a ring pattern can be
of materials as they are subjected to
obtained that can be mathematically
stress.
analyzed to generate the full second
The two types of strain, classified
as plastic, or irreversible strain, and
order strain tensor. These tensors thereby
indicate the particular deformation of the
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sample under strain as it responds along
the amount of out of plane strain in the
multiple axes. By calculating the strain
material
tensors for each of the three diffraction
Data collection Procedure
rings of iron, the (110), (200), and (211),
X-ray diffraction patterns (see Figure 1)
(each of which corresponds to a different
were taken with a set wavelength of
crystallographic orientation) and
0.975 Å as the prepared iron samples
comparing them to one another the
were rotated from 0 degrees to + 45
deformation of the crystallites with those
degrees with a diffraction pattern taken
crystallographic orientations can be
every 9 degrees. To study the elastic
determined. Ultimately, the elastic
deformation, the sample was strained to
modulus of each lattice plane can be
the onset of plastic deformation (the
calculated through an analysis of the
plastic strain limit taken to be that
change in the strain level for increasing
determined by traditional continuum
load levels.
mechanics sources) and the diffraction
Materials and Methods
patterns were taken at set intervals as the
Sample Preparation
sample was unloaded, to ensure that at
BCC iron samples with a
least under conventional mechanics
thickness of 13.5 microns were prepared
model the subsequent deformation is
for x-ray diffraction by cutting them into
purely elastic. Patterns were taken at five
dog bone shapes that were then inserted
separate load levels as well as at the
into a displacement controlled tensile
initial zero load level.
rig. The dog bone shape is used to
Pattern Analysis
facilitate purely tensile loading and limit
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An unstrained lattice produces a
Strain Tensor Determination
diffraction pattern with circular rings.
The change in ellipticity of a diffraction
When a sample is under elastic tensile
ring under load is seen as a noticeable
stress, the lattice spacing elongates in the
fluctuation (termed “wiggle”) in the
direction of the applied load and
normally straight line of the Q vs. chi
contracts in the orthogonal directions.
polar plot (see Figure 2). Using a
This results in an elliptical distortion of
program developed in MATLAB®, a
the diffraction rings. The eccentricity of
Gaussian fit was used to determine the
these ellipses directly corresponds to
location of the high intensity points of
how the sample has deformed under
the wiggle (Figure 3), the magnitude of
strain. In order to analyze the nature of
the intensity, and the width of the
this ellipse, the 2-D diffraction images
intensity curve. The resulting Q and chi
were converted to diffraction coordinates
coordinates of the intensity peaks were
(Q, chi) from pixel positions using
then transformed into the psi and phi
Fit2D. Fit2D itself was calibrated for
coordinates used in the standard sin2
specific measurements using LaB6
technique [1]. These calculated values
samples that have standardized
were then used to generate the
diffraction parameters. The eccentricity
coefficients for the strain tensors
of the ellipse and the resulting
equation according to:
magnitude of all the six independent
    11 cos2  sin 2    22 sin 2  sin 2    33 cos2 
components of the 2nd rank strain tensor
  12 sin 2 sin 2    23 sin  sin 2   13 cos sin 2
(1)
were determined from this converted
data by the procedure outlined below.
Where:   
d   d 0
d0
(2)
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And the change in d is the displacement
Figure 4. Assuming the elastic
of the sample as it stretches under strain.
deformation is linear with stress and
For each load level, the MATLAB®
strain, a linear fit of the strain data was
program takes data from all the
taken for each ring. The slope of this fit
diffraction patterns as output by Fit2D
provided the elastic modulus along each
and then uses the collective data points
crystallographic direction.
from each strain level in a least-squares
The theoretical values for the elastic
fit, weighted by widths provided by the
modulus along each of the
intensity curves, to solve for the second
crystallographic planes of the BCC iron
order strain tensors (εij) as well as
were calculated using:
generate the error values for each
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2
 s11  2[( s11  s12 )  12 s 44 ](l12 l 22  l 22 l32  l12 l31
)
E
calculated tensor. This process was then
(4)
repeated for each of the three diffraction
Where l1, l2, l3 are the direction cosines
rings produced by BCC iron.
for the particular lattice plane as given in
Results
Table 1, and the s values are the stiffness
The strain tensor data for each ring at
constants specific to a given material,
each load level was tabulated (see Table
shown in Table 2 for iron.
4,5,6) and for each ring the calculated
These values are compared to the single
strain tensor in the ε22 direction (the load
crystal values for the elastic modulus in
axis of the dog bone sample which will
the chosen crystallographic directions in
demonstrate the most dramatic stain)
Table 3.
was plotted in Origin against the stress at
each load level as demonstrated in
Discussion
7
The tabulated values for the strain tensor
(d0). The data was manually shifted
data demonstrate what is expected in
closer to the origin to compensate for the
terms of sample deformation along
inaccurate estimation of d0 values.
particular axes. As the strain increases in
Ideally the actual d0 value could be
the ε22 direction, the strain values in the
determined by analyzing calibration
ε33 become negative as would be
parameters. In any case, the shifted data
expected from a material elongating in
does not impact the calculation of the
one direction while contracting in the
elastic modulus for the different lattice
other direction.
directions.
The data as plotted in the ε22 graph
The measured values for the elastic
produces a pretty good linear fit,
modulus along the three crystallographic
providing credible elastic modulus
directions are surprisingly dissimilar
values along the different
from each other, and similar to the single
crystallographic directions. The error
crystal values. In fact, the measured
bars on the individual points, while
value for the (211) plane is exceptionally
present, are too small to be noticeable
close to the single crystal value. Rather
demonstrating a high confidence in the
than exhibiting the behavior expected by
method developed for obtaining strain
continuum mechanics—including the
from 2D diffraction pattern. However,
averaging out of strain values as the
the plotted ε22 data sets and fitted lines
material was stretched—such a
do not have intercepts at zero because
correlation indicates that in the
their location on the y-axis depends on
polycrystalline sample used the strain is
the particular unstrained lattice spacing
distributed along the different
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crystallographic directions in highly
distribute strain to accommodate the
anisotropic manner. Thus, although the
anisotropy observed here while still
sample is polycrystalline, the high
demonstrating elasticity.
anisotropy of the crystallographic
Acknowledgements
orientations of iron has a large effect as
This project was performed at the
the sample reacts to stress.
Stanford Linear Accelerator Center in
Such a conclusion is problematic
Palo Alto, California. I would like to
because this high anisotropy would seem
thank the U.S. Department of Energy for
to indicate that the crystalline structure
providing the funding and facilities for
would deform differently in the same
the Summer Undergraduate Laboratory
direction depending on crystallographic
Internship (SULI) program in which I
orientation resulting in discontinuities
was allowed to participate. I would also
between grains in the material, if they
like to thank my mentor, Apurva Mehta,
are not accommodated by a secondary
for his guidance, support, and expertise
mode of strain distribution. But the
as well as David Bronfenbrenner who
loading procedure utilized is supposed to
worked with me throughout this project.
result in purely elastic – single mode
deformation. Certainly continuum
mechanics is proven on a larger scale,
and the degree of anisotropy
demonstrated indicates that more
research must be done to determine by
what mechanism crystalline structures
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References
[1] I.C. Noyan, J.B Cohen. Residual
[3] Viktor Hauk. Structural and
Residual Stress Analysis by
Stress: Measurement by
Nondestructive Methods. New
Diffraction and Interpretation.
York: Elsevier, 1997. pp. 135-
New York: Springer-Verlag,
150
1987, pp. 117-120
[2] B.E. Warren. X-Ray Diffraction.
[4] Richard W. Hertzberg.
Deformation and Fracture
New York: Dover Publications,
Mechanics of Engineering
1969, pp. 102-115
Materials. New York: Wiley,
1995. pp.10-16.
Figure 1. Diffraction pattern for BCC iron highlighting the three analyzed rings.
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Figure 2. Results of Fit2D coordinate transformation of diffraction pattern to Q/chi space. a) the
transformed pattern of an unstrained sample b) the pattern from a strained sample
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Figure 4. Stress-strain plot for each crystallographic plane along the principle loading direction.
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