801
X-Ray Stress A
28. X-Ray Stress Analysis
Jonathan D. Almer, Robert A. Winholtz
28.1 Relevant Properties of X-Rays ................
28.1.1 X-Ray Diffraction........................
28.1.2 X-Ray Attenuation......................
28.1.3 Fluorescence .............................
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28.2 Methodology ........................................
28.2.1 Measurement Geometry ..............
28.2.2 Biaxial Analysis ..........................
28.2.3 Triaxial Analysis .........................
28.2.4 Determination
of Diffraction Peak Positions ........
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28.3 Micromechanics of Multiphase Materials.
28.3.1 Macrostresses and Microstresses...
28.3.2 Equilibrium Conditions ...............
28.3.3 Diffraction Elastic Constants.........
28.4 Instrumentation ...................................
28.4.1 Conventional X-Ray
Diffractometers ..........................
28.4.2 Special-Purpose Stress
Diffractometers ..........................
28.4.3 X-Ray Detectors .........................
28.4.4 Synchrotron and Neutron Facilities
28.5 Experimental Uncertainties ...................
28.5.1 Random Errors ...........................
28.5.2 Systematic Errors ........................
28.5.3 Sample-Related Issues................
28.6 Case Studies .........................................
28.6.1 Biaxial Stress .............................
28.6.2 Triaxial Stress ............................
28.6.3 Oscillatory Data Not Applicable
to the Classic Model....................
28.6.4 Synchrotron Example:
Nondestructive, Depth-Resolved
Stress........................................
28.6.5 Emerging Techniques and Studies
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811
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28.7 Summary ............................................. 817
28.8 Further Reading ................................... 818
References .................................................. 818
X-rays are an important tool for measuring stresses,
particularly residual stresses, in crystalline materials.
x-ray stress measurements are used to help solve material failure problems, check quality control, verify
computational results, and for fundamental materials
research.
X-ray diffraction can be used to precisely determine the distance between planes of atoms in crystalline
materials through the measurement of peak positions.
These positions can be used to determine elastic strains
in each crystalline phase of the material. These strains
can then be converted to stresses using appropriate
elastic constants. Plastic deformation can be detected
through changes in diffraction peak widths rather than
peak shifts. Laboratory-based x-ray sources typically
penetrate only a few tens of microns into common materials, yielding stresses averaged over the near-surface
region. Deeper depths can be accessed using destructive
layer-removal methods, with appropriate corrections.
Alternatively, more highly penetrating neutrons or high-
Part C 28
X-ray diffraction is a powerful non-destructive
technique capable of measuring elastic strain in
all crystalline phases of a material, which can
be converted to stress using appropriate elastic
constants. With laboratory sources typical penetration depths are on the micron-level, and deeper
depths can be evaluated using (destructive) layer
removal methods or higher-energy x-rays (esp.
from synchrotron sources) or neutrons. Diffraction also provides complementary information on
crystallographic texture and plastic deformation.
Potential sources of errors in stress measurements are outlined. Finally, some case studies and
emerging techniques and studies in this field are
highlighted.
802
Part C
Noncontact Methods
energy x-rays [28.1, 2]. can be used to examine the
interior of components nondestructively.
The subject of stress measurement with x-ray
diffraction has been treated in great depth by Noyan
and Cohen [28.3] and more recently by Hauk [28.4].
In addition, related reports have been presented by
the Society for Automotive Engineers [28.5], the Cen-
tre Technique de Industries Méchaniques [28.6], the
American Society for Metals [28.7], and most recently as a United Kingdom good practice guide [28.8].
Here we provide an updated account of the subject, including case studies from both laboratory- and
synchrotron-based experiments, together with a summary of emerging studies in the field.
28.1 Relevant Properties of X-Rays
Part C 28.1
In this section, several properties of x-rays relevant to
diffraction stress analysis are summarized.
28.1.1 X-Ray Diffraction
X-rays provide a means of examining the atomic-scale
structure of materials because their wavelengths are
similar to the size of atoms. The measurement of lattice spacings in crystalline materials with x-rays utilizes
diffraction, the constructive interference of x-ray waves
scattered by the electrons in the material. If the crystal
is oriented properly with respect to the incident x-ray
beam, constructive interference can be established in
particular directions, giving rise to a diffracted x-ray
beam. Constructive interference arises when the path
length for successive planes in the crystal equals an
integral number of wavelengths. This condition is summarized with Bragg’s law
nλ = 2d sin θ .
(28.1)
Here, λ is the wavelength of the x-rays, d is the interplanar spacing in the crystal, θ is the diffraction angle, and
n is the number of wavelengths in the path difference
between successive planes. In practice, n is eliminated
from (28.1) by setting it equal to 1 and considering
the diffraction peaks arising with n > 1 to be occurring
from higher-order lattice planes, i. e., diffraction from
(100) planes with n = 4 are considered as arising from
the (400) planes with n = 1. The interplanar spacing
or d-spacing in the crystal is measured along a direction that bisects the incident and diffracted x-ray beams.
A vector in this direction with length 2 sin θ/λ is termed
the diffraction vector, as shown in Fig. 28.1.
Equation (28.1) can be used in two ways to determine d-spacings in materials. First, a nominally
monochromatic x-ray beam can be used and the diffraction angle θ precisely measured. Second, a white or
polychromatic x-ray beam can be incident on the specimen and the wavelengths scattered at a particular angle
measured. This is usually done with a solid-state detector that can precisely measure the energy of detected
photons using the wavelength–energy relationship for
electromagnetic radiation
hc
(28.2)
,
E
where h is Planck’s constant, c is the speed of light, and
E is the energy of the photons. Substituting numerical
values for the constants, if λ is in Angstroms and E is
in keV, the wavelength energy relation becomes
λ=
λ=
12.39842
.
E
(28.3)
28.1.2 X-Ray Attenuation
As x-rays pass through matter they are attenuated. This
is an important consideration in measuring stresses
with x-rays. For typically used reflection geometries,
the attenuation defines the depth being sampled in
a stress measurement while, for transmission geometries, it will define the thickness of a specimen that can
be examined.
An x-ray beam of intensity I0 passing through
a thickness t of a material will be attenuated to an intensity I given by
I = I0 exp(−μl t) ,
(28.4)
where μl is the linear attenuation coefficient for the
material. The linear attenuation coefficient can be computed for a sample having N elements using
μl = ρ
N
wi (μm )i ,
(28.5)
i=1
where ρ is the sample density and wi and μm,i are,
respectively, the weight fraction and mass attenuation
coefficient of element i. The μm are inherent properties
X-Ray Stress Analysis
a)
28.1 Relevant Properties of X-Rays
803
b)
X3
Ψ
X3'
σ
Diffraction
vector
X1
θ
Ψ
X2
c)
Ψ
Diffraction
vector
Fig. 28.1 (a) Coordinate systems used in diffraction stress measurement, showing the specimen coordinates (X i ) and
laboratory coordinates (X i ) and the manner in which the specimen is rotated to orient the diffraction vector along the
laboratory X 3 axis; definition of tilt angle ψ for (b) Ω-gonio-metry and (c) Ψ -goniometry, with the in-plane stress
direction indicated
of a given element and commonly tabulated. Between
absorption edges (which lead to fluorescence, see below) these values vary with wavelength λ and elemental
atomic number Z as
μm ∝ Z 4 λ3 ,
(28.6)
which when combined with (28.2) indicates that absorption decreases considerably with x-ray energy.
Table 28.1 lists μl values for selected materials and
common laboratory x-ray energies. Also tabulated is
the inverse value τ1/e , which represents the x-ray path
length at which I = I0 /e ≈ 0.37I0 , and which is seen
to be in the micron range. The actual penetration depth
into the sample depends on the diffraction geometry
used, and this is covered further in Sect. 28.2.1.
28.1.3 Fluorescence
When the incident x-ray energy is higher than the
(element-specific) energy required to remove an electron (e.g., K-edge electron) from its shell, the electron
will be ejected. This ejected electron is called a photoelectron and the emitted characteristic radiation is called
fluorescent radiation. For strain measurements, fluorescence is unwanted as it increases x-ray background
levels, thereby decreasing the signal-to-background
ratio and reducing accuracy in peak position determination.
The fluorescence signal can be eliminated by using x-ray energies below the fluorescence energies of
the elements comprising the sample, though this is not
always practical. For tube sources, containing both continuous (bremsstrahlung) radiation and characteristic
(Kβ and Kα ) radiation, the fraction of higher-energy
(fluorescence-forming) x-rays can be reduced relative
to the signal-forming (typically Kα ) x-rays through
the use of an incident-beam energy filter in the form
of a metal or oxide foil or monochromator. Alternatively, fluorescence signals can be reduced by the use
of a diffracted-beam energy filter (foil or monochromator) and/or through use of an energy-discriminating
(e.g., solid-state) detector.
Part C 28.1
σ
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Part C
Noncontact Methods
Table 28.1 X-ray penetration depths, common reflection/radiation combinations for stress measurements, and elastic properties
for selected materials
Material
(structure)
μ (1/mm) and (τ(1/e), μm)
Mo–Kα
Cu–Kα
λ = 0.71 Å
λ = 1.54 Å
Cr–Kα
λ = 2.29Å
14.3
132
397
(69.9)
(7.6)
(2.5)
Al (fcc)
Elastic
aniso- a
tropy
1.2
310
2547
904
(3.3)
(0.4)
(1.1)
Ferrite (bcc)
2.3
Part C 28.2
311
2633
934
(3.2)
(0.4)
(1.1)
Austenite (fcc)
3.3
hkl b
2θ
(deg)b
111
53.8
200
49.9
311 (Cr)
200
139.0
52.1
141.7
211 (Cr)
156.0
178.7
732 (Mo)
111
154.5
163.2
209.8
200
124.4
420 (Cu)
111
422
439
1291
(2.4)
(2.3)
(0.8)
Nickel (fcc)
2.5
107
919
2718
(9.4)
(1.1)
(0.4)
-
b
c
d
155.4
200
52.3
163.6
145.7
152.1
164.0
90.9
83.0
139.5
E/(1 + ν)
bulk
(GPa) d
142.3
219.4
144.2
213 (Cu)
a
147.0
200
420 (Cu)
002
Titanium (hcp)
E/(1 + ν)
Hill
(GPa) c
87.6
83.6
Anisotropy is calculated for cubic materials as 2c44 /(c11 − c12 ), where cij are the elastic constants from [28.9] and [28.10].
Suggested reflection/radiation combinations and corresponding Bragg angles for laboratory stress measurements are shown.
Calculated using cij from [28.9] and [28.10] and using the Hill–Neerfeld average (Sect. 28.3.3).
From [28.11]
28.2 Methodology
Stress measurement with diffraction involves measuring
lattice d-spacings in different directions in a specimen
utilizing Bragg’s law and then using them to compute
strains through
ε=
sin θ0
d − d0
=
−1 .
d0
sin θ
(28.7)
Here d0 and θ0 are the unstressed d-spacing and the corresponding Bragg angle. We wish to obtain the stress
and strain components in the specimen with respect to
the specimen coordinate system shown in Fig. 28.1. By
observing the diffraction with a diffraction vector oriented along the X 3 axis, we can determine the strain in
this direction with (28.7). The X 3 axis is oriented with
respect to the specimen coordinate system by the an-
gles φ and ψ. Applying tensor transformation rules to
the specimen coordinate system we may write the measured strain, εφψ , in terms of the strains in the specimen
coordinate system:
dφψ − d0
d0
= ε11 cos2 φ sin2 ψ + ε22 sin2 φ sin2 ψ
εφψ =
+ ε33 cos2 ψ + ε12 sin 2φ sin2 ψ
+ ε13 cos φ sin 2ψ + ε23 sin φ sin 2ψ . (28.8)
By measuring strains with diffraction in at least six
independent directions, the strains in the specimen coordinate system can be determined by a least-squares
procedure [28.12]. After the strains in the specimen co-
X-Ray Stress Analysis
ordinate system have been computed, the stresses in this
coordinate system can be determined with Hooke’s law
as
S1
1
εij − δij 1
εii ,
(28.9)
σij = 1
2 S2
2 S2 + 3S1
28.2.1 Measurement Geometry
Placing the diffraction vector along a particular X 3 axis
can be accomplished in an infinite number of ways by
rotating the diffraction plane (containing the incident
and diffracted beams) around the diffraction vector. In
practice this is accomplished in two ways, termed Ω
goniometry and Ψ goniometry, which are illustrated
in Fig. 28.1b,c. In Ω goniometry the specimen is rotated by an angle ψ about an axis perpendicular to the
diffraction plane, while for Ψ goniometry this rotation
is performed about an axis within the diffraction plane.
These rotations orient the diffraction vector to the specimen coordinate system by the angle ψ in Fig. 28.1a.
For both goniometry methods the orientation angle φ is
obtained by rotating the specimen about the X 3 axis.
The penetration depth into the sample, z, depends
on the x-ray path length into and out of the sample, and
thus on measurement geometry. This depth is given for
both types of goniometry as [28.13]
σ23 , and σ33 , which have a component perpendicular
to the surface. It is therefore common to assume a biaxial stress state and measure the stress in a particular
in-plane direction (given by the angle φ in Fig. 28.1a).
If we drop the σi3 components and define the stress in
the φ direction as
σφ = σ11 cos2 φ + σ12 sin 2φ + σ22 sin2 φ
(28.11)
and solve for dφψ we obtain
dφψ =
1
S2 d0 σφ sin2 ψ + S1 d0 (σ11 + σ22 ) + d0 .
2
(28.12)
This equation gives the measured d-spacing as a function of the tilt angle. For a stressed material, at ψ = 0◦
the strain is given by the Poisson effect of the stress. As
the specimen is tilted, the measured strain varies according to strain transformation rules, varying linearly with
sin2 ψ. If the measured d-spacing is plotted versus the
variable sin2 ψ, there should be a linear relationship. It
is good to check experimentally that this relationship is
indeed linear by measuring at multiple ψ tilts. Nonlinear relations can reveal a number of problems and can
invalidate the analysis. Case studies illustrating both linear and nonlinear behavior are presented below. If a line
is fitted to the d versus sin2 ψ data, the resulting slope
will be proportional to the stress σφ . The stress can be
obtained by multiplying the experimentally determined
slope by 12 S2 and d0 . The stress-free lattice spacing, d0 ,
is generally unknown. Since it is simply a multiplier
in (28.12), the intercept of the d versus sin2 ψ plot or
another approximation can be used with only a small
error.
28.2.3 Triaxial Analysis
ln II0 sin2 θ − sin2 ψ + cos2 θ sin2 ψ sin2 η
,
z=
2μl
sin θ cos ψ
(28.10)
where η is the sample rotation around the diffraction
vector and equal to 0◦ (90◦ ) for Ω (Ψ ) goniometry,
respectively, and other quantities are as previously defined. The maximum penetration depth is thus at ψ = 0◦
and θ ⇒ 90◦ such that z max ⇒ ln(I0 /I )/2μl .
28.2.2 Biaxial Analysis
Because conventional x-rays only penetrate a few tens
of microns into most engineering materials, the stresses
measured are usually biaxial, lacking components σ13 ,
805
When using high-energy x-rays or neutrons, the penetration depth is larger, so that all the components of the
stress tensor may be present. In addition, as we will see
in Sect. 28.3, triaxial analysis may be needed for conventional laboratory-based x-ray sources if multiphase
materials are investigated. To determine the complete
stress tensor the lattice spacing is measured in a number
of directions/angles, and strains in these directions are
calculated with the stress-free lattice spacing d0 . The triaxial strain components εij are computed using (28.8),
which can be rewritten as
εφψ =
6
k=1
ak f k (φ, ψ) ,
(28.13)
Part C 28.2
where δij is the Kronecker delta function and S1 =
(ν/E)hkl and S2 /2 = ((1 + ν)/E)hkl are the diffraction elastic constants (DEC), which are discussed in
Sect. 28.3.3. Equations (28.7)–(28.9) encapsulate the
basic model of stress measurement by diffraction. Practical implementation of this model will be discussed in
the following sections.
28.2 Methodology
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Part C
Noncontact Methods
Part C 28.2
where ak are the six strain components determined by
fitting the measured data, and the functions f k (φ, ψ)
are functions of the two variables φ and ψ. At least
six independent strain measurements are needed to
solve (28.13) for the strain components. Greater accuracy is obtained by increasing the number of strain
measurements and by using measurement directions
that have a wide angular distribution. It is good practice
to measure at a number of ψ values (both positive and
negative) for several different constant φ values. Plots
of the measured strain values for each of the φ values
versus sin2 ψ help reveal the quality of the measured
data. Examining (28.8), we see that if the shear strains
ε13 and ε23 are zero, the ε versus sin2 ψ plots will be linear with the positive and negative ψ data overlapping on
this line. If the shear strains ε13 and ε23 are present, the
positive and negative branches of the plot will take different paths following two sides of an ellipse. If the data
do not follow one of these shapes, it indicates a problem
and fitting the data to (28.8) is unlikely to give accurate stress values. Plots of ε versus sin2 ψ which can be
properly fitted are shown as a case study below.
An important issue for accurate triaxial stress
measurement is obtaining an accurate value of the
stress-free lattice spacing. In contrast to the biaxial
case where it is just a multiplier, the stress-free lattice
spacing is subtracted from the measured d-spacings to
obtain a strain. Consequently an accurate unstressed lattice spacing d0 is needed for triaxial analysis [28.14].
An error in the stress-free lattice spacing results in an
error in all the resulting computed strain values, which
yields an erroneous hydrostatic stress/strain component
appearing in the final stress/strain tensor determination.
If an accurate stress-free lattice spacing is unavailable,
the deviatoric components of the stress and strain tensors will still be accurate even though the hydrostatic
components are not [28.15].
28.2.4 Determination
of Diffraction Peak Positions
Lattice spacings are determined from diffraction by
careful determination of diffraction peak positions. As
a stressed specimen is tilted and diffraction peaks are
recorded, the position of the peak will shift as a different
component of strain is resolved. Strains in crystalline
materials are small (usually 10−4 or less) and, thus,
peak shifts will be small, often less than half the width
of the peak. Therefore, the diffraction peak must be
carefully recorded and the peak position precisely determined. The position of the peak can be defined as the
2θ value of the centroid of the diffraction intensity or the
position of maximum diffracted intensity. For symmetric diffraction peaks these quantities will be the same,
but in general, they will be different. The same definition should be used throughout a stress measurement.
Theoretically, the peak centroid is more appropriate, but
is not widely used in practice.
In order to determine the peak position, the intensity
versus 2θ data is usually fitted to a mathematical function representing the peak profile [28.3,16]. A Gaussian
peak with a linear background is given by
2θ − 2θp 2
+ 2θm + b ,
I = I0 exp −4 ln 2
W
(28.14)
where I0 is the diffracted intensity over the background
level, 2θp is the peak position, W is the peak full-width
at half-maximum, m is the slope of the background,
and b is the background intercept. A Gaussian often
fails to fit the tails of the diffraction peak well. Adding
a Lorentzian component to the peak function gives
a pseudo-Voigt function,
⎧
⎪
⎨
2θ − 2θp 2
I = I0 η exp −4 ln 2
⎪
W
⎩
⎫
⎪
⎬
1
+ (1 − η)
2 ⎪ + 2θm + b ,
2θ−2θp
⎭
1+
W
which will usually better fit the tails. Here η is the fraction of the Gaussian component in the peak and (1 − η)
is the fraction of the Lorentzian component.
With conventional x-ray tubes the characteristic
x-rays usually contain two closely spaced wavelengths,
so the diffraction peaks are actually a doublet of two
closely spaced peaks. For a Kα1 –Kα2 doublet, the
higher-angle peak will have approximately half the
height of the lower-angle peak due to the intensity ratio of the respective wavelengths in the incident beam.
Constraining the peak height ratio to 0.5 can improve
the fitting of the doublet. With sufficient broadening, the
two peaks in the doublet can be treated as a single peak
and fitted with one of the above equations. If the two
peaks do not sufficiently overlap (i. e., they are distinct),
they should be fitted with two peaks separated in 2θ by
δ. Note that adding the second peak to the fit function
does not require the addition of any new fit parameters.
The separation between the two peaks δ is computed
X-Ray Stress Analysis
from the known x-ray wavelengths and Bragg’s law
−1 λ2 sin(2θ1 /2)
− 2θ1 .
δ = 2 sin
(28.15)
λ1
Intensity (Counts)
450
I0 = 203.44 ± 2.67
2θ = 144.071 ± 0.00275
W = 0.463429 ± 0.00546
m = – 8.7897 ± 2.03
b = 146.46 ± 293
400
350
300
250
200
150
143.5
144
144.5
145
2θ (Degrees)
Fig. 28.2 211 diffraction peak from the β phase of a 60–40 brass
along with fit to a Gaussian doublet. The best fit parameters from
(28.14) are given in the inset box. The value of δ was calculated
prior to fitting to have a value of 0.60409
advantage is that the degree of doublet overlap can
vary with ψ, causing shifts in the location of maximum intensity due to varying doublet overlap, which
will erroneously be recorded as due to residual stress.
Fitting a doublet peak function to the data automatically accounts for this potential problem. With linear or
area detectors, the whole peak is automatically recorded
and there is little need to fit just the top portion of the
peak.
28.3 Micromechanics of Multiphase Materials
In Sect. 28.2, methods for measuring stress states with
diffraction were outlined. In order to properly interpret these stresses, we now distinguish between types
of stresses, outline key equilibrium conditions, and provide information on diffraction elastic constants needed
to compute stresses from measured strains.
28.3.1 Macrostresses and Microstresses
Macrostresses are the stresses that appear in a homogeneous material. They vary slowly on the scale
of the microstructure and are the stresses that are revealed by dissection techniques. Residual stresses, as
they have been traditionally treated, are macrostresses
and arise due to nonuniform deformations on a macro-
807
scopic scale. Microstresses, in contrast, arise from
the microstructural inhomogeneities in the material
and nonuniform deformations on this scale. While
microstresses can often be an impediment to determining the true residual macrostresses in a material,
they are increasingly becoming of interest in their own
right, with the recognition that they can also influence material behavior [28.18]. Residual stresses have
also been distinguished in the literature as types I,
II, and III, as illustrated in Fig. 28.3. Type I stresses
are macrostresses that are constant over many grains
in the material. Type II stresses are microstresses and
can vary between phases, as well as between grains
within a given phase. Typical diffraction data provide
the grain-averaged type II stress value within a given
Part C 28.3
Figure 28.2 shows a 211 diffraction peak from the
β phase of a 60–40 brass, measured with Cr Kα1 –Kα2
radiation and fitted with a Gaussian peak profile, including a second peak to account for the doublet.
With an area or linear detector, the background away
from the peak position is automatically recorded and
should be included in the fitting. A simple line is usually
sufficient, though a higher-order polynomial or other
forms can also be used to represent the background accurately. If the diffraction peak is recorded by point
counting, the background may not be sufficiently well
recorded away from the peak to accurately fit. Recording the background can be time consuming and may
not be worthwhile, particularly if the background is not
changing as the specimen is tilted. In these instances
the background should be assumed constant in 2θ and
only this constant included in the fit function. If a background function with a slope is used and intensity data
away from the peak that would allow an accurate determination of the slope is absent, erroneous peak positions
can result from the fitting, due to correlations between
the slope and peak position.
Fitting the top 15% of the diffraction peak to
a parabola has also been utilized for x-ray stress measurements [28.17]. For point counting this method has
the advantage that time is not spent collecting the whole
diffraction peak, speeding up the measurements. A dis-
28.3 Micromechanics of Multiphase Materials
808
Part C
Noncontact Methods
phase. Type III stresses are also microstresses but represent stress variations within single grains. Residual
stresses of types I and II will result in diffraction peak
shifts, as they change the mean lattice spacing, while
type III stresses result in diffraction peak broadening
rather than peak shifts. Measurement of type III stresses
is outside the current scope and will not be further discussed. For a two-phase material, the measured stress
from peak shifts can be written as
t α I
σij = σij + II σijα ,
t β I
β
σij = σij + II σij ,
(28.16)
Part C 28.3
where the superscripts t, I, and II stand for the total
stress in a phase (α or β), the type I macrostresses,
and the type II microstresses, respectively. The total
stresses and microstresses are represented as averages
(denoted by carets) as they can vary significantly within
the diffracting volume, whereas the macrostress does
not. Note also that the macrostress component is common to both phases and thus does not have a phase
designation. These equations hold for each component
of the stress tensor. Using (28.16), the microstresses in
the α and β phases must balance to zero when weighted
by their volume fractions, thus
β
(28.17)
(1 − f ) II σijα + f II σij = 0 .
Here f is the volume fraction of the β phase. The total stress components in the α and β phases can be
σ
α
σ
β
σ II
I
σ II
β
σ II
measured with diffraction measurements as outlined
in Sect. 28.2.3. If the volume fractions of the phases
are known, the components of the macrostress tensor
and microstress tensors can be then determined using (28.16) and (28.17).
28.3.2 Equilibrium Conditions
As measured with conventional (low-energy) x-rays,
residual macrostresses with a component perpendicular
to the surface, σ13 , σ23 , and σ33 (with the x3 axis normal to the surface) are typically negligible. This can be
seen via the following equilibrium relations for residual
macrostresses:
∂σ11 ∂σ12 ∂σ13
+
+
=0,
∂x1
∂x2
∂x3
∂σ21 ∂σ22 ∂σ23
+
+
=0,
∂x1
∂x2
∂x3
∂σ31 ∂σ32 ∂σ33
+
+
=0.
(28.18)
∂x1
∂x2
∂x3
These relationships show that the stress components σi3
can only change from their value of zero at the free surface if there are gradients in the other components of
stress within the surface. These components of stress
cannot normally have high enough gradients within the
surface to cause the components σi3 to reach measurable levels within the penetration depth of conventional
x-rays (the biaxial assumption, see Table 28.1).
Another equilibrium relation that proves useful is
that the stress integrated over a cross section of a solid
body must be zero
σij dA = 0 .
(28.19)
A
This relation can be used as a check on experimental
results and can also be used to determine the stress-free
lattice spacing.
σ III
28.3.3 Diffraction Elastic Constants
Fig. 28.3 Definition of macrostress (σ I ) and microstresses
in a two-phase material. The measured peak shift in a given
phase is proportional
to the total mean elastic stress in that
phase (σ I + σ II ). Variations in stress within a grain (σ III )
cause peak broadening rather than peak shifts
Elastic constants are needed to compute stresses from
measured diffraction strains in polycrystalline materials. Due to the selective nature of the diffraction
process and presence of crystalline elastic anisotropy,
the so-called diffraction elastic constants (DEC) differ,
in general, from the bulk elastic constants. Here, two
methods for attaining DEC are described:
1. computing them using single-crystal elastic constants and a given micromechanical (grain-averaging)
model, and
X-Ray Stress Analysis
2. measuring the constants in situ using known elastic
loads.
•
•
The higher the crystalline anisotropy, the larger the
deviation between DEC and bulk values for a given
diffraction peak. Thus, use of bulk values will give
a corresponding error in absolute stress value for
a given diffraction measured strain (28.12).
These deviations are especially important for lowmultiplicity peaks (e.g., 111/200). For highermultiplicity peaks (e.g., 732 in ferrite) the DEC
approach the bulk values. Such peaks have the
added benefit of being less sensitive to texture, as
shown by Hauk in textured steel [28.25].
A second method is to measure the DEC in situ using known loads. This represents the inverse problem to
stress analysis described throughout the rest of the chapter, where stresses are unknown and DEC are assumed
known. Loads should be applied using either uniaxial
tension or four-point bending, since the applied stress
distributions in such cases can be determined accurately from elasticity theory, and the diffraction strains
are recorded for reflections of interest. The American
Society for Testing and Materials has developed a standard test method for the measurement of the diffraction
elastic constant S2 /2 for a biaxial stress measurement,
which can give some practical guidance [28.26]. Such
methods are particularly important when elastic constants of the material are unknown or in question. For
example, alloying can modify DEC considerably from
elemental values, as found by Dawson et al. [28.27]
on aluminum alloys. Tables of measured DEC for various materials (including common alloys) can be found
in [28.3, 7]. Taking this method one step further, Daymond [28.28] found that, by using strain from multiple
diffraction peaks and a weighted Rietveld method, it
was possible to calculate bulk elastic properties E and
ν in polycrystals. He found good matches to bulk properties for both textured and untextured cubic (steel) and
hexagonal (titanium) crystal symmetries.
We note that the use of bulk elastic constants (E, v)
in place of hkl-dependent DEC is common. While this
can cause an error in the absolute stress values, relative
changes in stress (e.g., between samples of the same
composition, or positions in a given sample) may still
be valid.
Finally, we note that the grain-interaction models described above are appropriate for bulk samples,
but van Leeuwen et al. [28.29] found that in the case
of thin films (e.g., vapor-deposited nickel) the Vook–
Witt [28.30] model is more appropriate to describe the
measured strains. The latter model takes the special geometry of thin films into account for specifying the
strain and stress state, and can be used to calculate
nonlinear d–sin2 ψ behavior even in the absence of
crystallographic texture. For additional information on
strain measurements in thin films two reviews on the
subject are noted [28.31, 32].
28.4 Instrumentation
A variety of equipment can be used for diffraction stress
measurements. Conventional x-ray diffractometers are
often used, as well as special-purpose stress diffractometers which may be laboratory-based or portable.
At synchrotron x-ray and neutron facilities the working area is typically larger than with laboratory-based
diffraction systems, permitting larger samples and/or
ancillary equipment. Associated instrumentation is generally specialized and outside the scope of this chapter.
28.4.1 Conventional X-Ray Diffractometers
Conventional x-ray diffractometers are widely available
and used for a variety of applications. These can of-
809
Part C 28.4
The most widely used micromechanical models
are those from Voigt [28.19], Reuss [28.20], Neerfeld–
Hill [28.21,22], and Eshelby–Kroner [28.23]. The Voigt
and Reuss models assume that all grains in a polycrystalline aggregate have, respectively, the same strain and
stress, and represent, respectively, the upper and lower
bounds of the elastic constants. It is generally found that
the Neerfeld–Hill (which is the arithmetic average of the
Voigt and Reuss values), and Eshelby–Kroner models
best match experimentally determined DEC. Equations
for determining DEC for various crystal symmetries can
be found in [28.3, 4], and stress-analysis software packages may provide computational tools (e.g., [28.24]).
In Table 28.1, Hill model values of the inverse DEC,
2/S2 = (E/(1 + ν))hkl , are compared with bulk values
(i. e., those expected for an isotropic body) for various
materials and hkl peaks. We note the following trends
from the table:
28.4 Instrumentation
810
Part C
Noncontact Methods
ten be utilized for stress measurement, though software
to control the data acquisition and analysis will often
be deficient for this task, requiring a greater degree of
operator expertise. Conventional x-ray diffractometers
may have several limitations, making stress measurement difficult or impossible. They often have limited
space for specimens, have limited or no ability to
achieve ψ-tilts, have incompatible beam defining slits,
and provide poor access to the raw data for analysis. If
Ψ -goniometry is to be used with a diffractometer having a χ-circle, the x-ray tube needs to be oriented for
point focus to limit the length of the beam along the
diffractometer axis.
Part C 28.5
28.4.2 Special-Purpose Stress
Diffractometers
Various stress diffractometers have been specifically designed to perform these measurements. They have the
ability to handle larger and heavier specimens and have
software to automate the data collection and analysis.
Most modern stress diffractometers use linear or area
detectors to increase the measurement speed, and utilize
the Ψ -goniometry because it does not defocus the x-ray
optics. More information and a photograph of a stress
diffractometer are given in a case study in Sect. 28.6.1.
Laboratory-based units will be more flexible diffraction
instruments, usually having texture measurement capabilities and the ability to do general powder diffraction.
Portable stress diffractometers can be taken into the
field to make stress measurements on very large objects
without having to move them.
28.4.3 X-Ray Detectors
X-ray detectors can be broadly classified into point,
line, or area detectors. Point detectors include proportional, scintillation, and solid-state detectors. For
each the x-ray energy is proportional to the measured pulse height, so that energy discrimination can
be accomplished through electronic filtering. The
energy resolution is highest for solid-state detectors (ΔE/E ∼ 3–5%) and lowest for scintillation
counters (ΔE/E ∼ 30–40%). Thus, solid-state detectors generally have lower background levels and
are the standard choice for energy-dispersive measurements. If necessary, secondary monochromators
and/or foils can be used in conjunction with scintillation or proportional detectors to reduce background
levels.
As their name implies, linear and area detectors measure, respectively, one- and two-dimensional
angular sections of x-ray diffraction cones. As this information is collected simultaneously, these detectors
allow for significantly faster strain measurements than
point counters, which require scanning over the 2θ region of interest. Area detectors have the added benefit
of providing simultaneous grain size and texture information through intensity variations around a diffraction
cone. These detectors generally have limited energy resolution, and incident-beam filters can be employed to
limit background levels.
28.4.4 Synchrotron and Neutron Facilities
Both synchrotron and neutron facilities provide unique
capabilities for strain and stress analysis. Synchrotrons
offer x-ray fluxes several orders of magnitude higher
than x-ray tubes, and as such can be used for timeresolved experiments. In addition, these sources have
much lower divergence and source sizes, so focal
spots from mm down to tens of nm can be used for
spatially resolved measurements. Furthermore, thirdgeneration facilities [the Advanced Photon Source
(APS), USA; the European Synchrotron Radiation Facility, France; and SPRing-8, Japan] are excellent
sources of high-energy x-rays, with concomitant high
penetration depths (over 1 mm in most materials). This,
in turn, enables many in situ and bulk studies that are
either not practical or possible with laboratory-based
sources. Even higher penetration depths are provided by
neutrons (up to the cm level), with similar in situ capabilities. Access to synchrotron and neutron facilities
is typically granted (for free) based on peer-reviewed
proposals, but proprietary (fee-based) work may be arranged at certain facilities.
28.5 Experimental Uncertainties
With proper care, diffraction stress measurements can
readily be made with sufficient accuracy and precision
to solve many problems. There are, however, ran-
dom and systematic errors in the measurement process
that should be carefully considered when evaluating
diffraction stress measurement results. The best way
X-Ray Stress Analysis
28.5.1 Random Errors
Random errors in x-ray measurements arise from the
statistical nature of x-ray counting and the random
component of proper alignment of the specimen and
instrument. The statistical errors will typically be the
dominant error in measurements with laboratory xrays or neutrons. These errors can be driven arbitrarily
low by collecting data for longer periods of time,
but the improvement varies as the square root of the
time.
When a diffraction peak is recorded and fitted to
find its position, there will be an uncertainty associated with this position due to the statistical nature of
recording x-ray intensities. Fitting programs usually
give an estimate of the uncertainty in the fitted parameters, including the peak position. This uncertainty can
be propagated through the analysis to estimate the uncertainty in the stresses. The uncertainty in a d-spacing
is given by differentiating Bragg’s law [28.3]
STD(dψ ) = dψ cot θ
STD(2θp )
2
π
180
.
(28.20)
Here, STD(dψ ) represents an estimated standard deviation in the value of dψ and STD(2θp ) is an estimated
uncertainty in the peak position arising from the fitting
of the diffraction peak. The π/180 factor is a conversion
of degrees to radians, assuming that the uncertainties in
peak position are in degrees. Similarly, the uncertainty
in a strain is given by
STD(εφψ ) = cot θ
STD (2θ)
2
π
180
811
.
(28.21)
These uncertainties can then be used in the fitting of the
d-spacing or strains εφψ to the biaxial or triaxial models ((28.12) and (28.13)). A proper fitting package will
propagate the uncertainties in the input data to the fitted
parameters.
A goodness-of-fit parameter can be computed to
help determine if the measured data suitably fits the
model used [28.36]. The goodness of fit is the probability that the observed deviations from the model fit
arise from the estimated errors. A low goodness of fit
can indicate that the errors are underestimated or that
the data do not fit the model and the resulting stresses
should not be considered valid. It has been recommended that goodness-of-fit parameters should be better
than 10−3 –10−5 before considering the model fit to be
valid [28.36]. Poor goodness-of-fit parameters may indicate the presence of systematic errors and/or samplerelated issues, both of which are described below.
Measurements made under computer control can
take some quick preliminary measurements of the x-ray
intensities and control the measurement time to give
a specified statistical error in the final measurements.
28.5.2 Systematic Errors
Systematic errors in diffraction stress measurement
consist primarily of imperfect optics and misalignments
in the diffractometer, both of which cause shifts in
the diffraction peak position from its true position. For
conventional x-ray stress measurement using reflection
geometries, a number of researchers have given equations to estimate the uncertainties in stress due to these
sources, and these are summarized in Table 28.2. The
size of these systematic peak shifts generally increases
Table 28.2 References for systematic instrumental errors
in diffraction stress measurements
Source of error
Refs.
Specimen displacement (ω-goniometry)
Specimen displacement (Ψ -goniometry)
Horizontal beam divergence
Vertical beam divergence
Missetting of diffractometer angles
ψ-axis displacement (ω-goniometry)
ψ-axis displacement (Ψ -goniometry)
Curvature of specimen surface
[28.3, 6, 34]
[28.3, 6]
[28.3]
[28.3, 34]
[28.35]
[28.3, 6, 34]
[28.3, 6]
[28.3, 6]
Part C 28.5
to evaluate the errors present is to repeatedly measure
a stress-free standard, such as an annealed powder from
the material of interest. The average stress measured
should give a measure of the systematic errors present,
while the variation in results should give a measure of
the random errors present. The American Society for
Testing and Materials ASTM outlines a standard test
method for verifying the alignment of an x-ray stress
diffractometer [28.33] and specifies that an annealed
powder measured according to accepted procedures
should give a value of stress less than 14 MPa (2 ksi).
Measuring an annealed powder is a time-consuming
process and may not be practical. It is desirable to obtain good estimates of the random and systematic errors
in a measurement based on just that single measurement. It is also desirable to be able to estimate the errors
so that the measurements can be made to a specified
accuracy. Finally, it can be a good practice to repeat
measurements to check reproducibility.
28.5 Experimental Uncertainties
812
Part C
Noncontact Methods
Part C 28.5
with the tilt angle ψ and with decreasing diffraction
peak position 2θ. Care should be taken if extreme values
of ψ are to be used (greater than 60◦ ) or with diffraction peaks below 2θ of 130◦ . For transmission methods
with high-energy x-rays or neutrons, successful measurements can be successfully made with low-2θ peaks,
in part because they often have limited (or no) requisite
sample movement.
Generally the largest instrumental error is that due
to specimen displacement, i. e., not having the sample
surface at the center of the diffractometer axis. This results in a diffraction peak shift that depends on ψ and
thus introduces an error in the measured stresses. This
error can be minimized by using parallel beam geometry in which vertically aligned Soller slits are used to
define the diffraction angles. This geometry only works
for point detectors that are scanned across the diffraction peak, which is much slower than using linear or
area detectors. This component of error can change as
specimens are moved for measurements at different locations or as specimens are changed. This error will
also change for a specimen from the value recorded on
a stress-free powder. Placement of the specimen surface on the diffractometer axis is typically achieved with
special alignment gages for dedicated instruments. Additionally, one can measure the specimen displacement
for cubic materials with diffraction [28.3]. The lattice
parameter a is determined from a series of diffraction
peaks and plotted versus cos2 θ/ sin θ. The slope of this
plot gives the specimen displacement Δx according to
the relation
Δx cos2 θ
ahkl − a0
=
,
a0
RG sin θ
(28.22)
where RG is the goniometer radius and a0 is the true
lattice parameter, which can be approximated with the
intercept of the plot.
For triaxial analysis an accurate value of the stressfree lattice spacing must be used. This should be
measured on the same diffractometer used for the stress
measurements. Inaccuracies in the stress-free lattice
spacing lead to an error in the hydrostatic component
of the stress tensor [28.15]. If a single stress-free lattice spacing is used for a series of measurements, this
error will be systematic and changes in the hydrostatic
component between measurements will be meaningful.
In some situations, notably welds, the stress-free lattice
spacing can vary from point to point in the specimen
due to composition gradients. If the full tensor is then
desired, stress-free reference values must be measured
for each point sampled, which can be accomplished by
sample sectioning.
28.5.3 Sample-Related Issues
The error analysis described above applies to ideally
behaved specimens, and thus represents a lower limit
on the error in stress values. Here we describe potential sample-related issues that can modify strain data
relative to this ideal behavior.
Texture Effects
Crystallographic texture is common in materials and is
due to directional processing methods, such as rolling
in bulk materials or directional deposition in thin films
and coatings. The presence of texture causes systematic variations in the peak intensity as a function of
tilt. This can be distinguished from random (stochastic) variations in intensity, which indicate improper
grain averaging (addressed below). If strong texture is
known or suspected to be present, it can be beneficial
to measure a pole figure on the peak intended for strain
analysis, which both provides quantitative texture information and aids determination of the most appropriate
orientations to measure. Pole figure measurements are
outside the scope of this chapter; the reader is referred
to [28.37] for details.
The presence of texture, coupled with elastic
anisotropy, can lead to oscillations in d versus sin2 ψ
plots. In such cases the diffraction elastic constants
vary on a macroscopic scale with orientation, so grains
with different DEC are sampled at each (φ and ψ)
tilt setting. Methods to account for texture have been
developed by various researchers [28.4, 38, 39]. One
particularly straightforward method is to use highmultiplicity planes, such as the (732) reflection in steel,
which Hauk [28.25] showed provides more linear d versus sin2 ψ behavior than lower-multiplicity peaks (e.g.,
(211)). Drawbacks to the use of these peaks are that
they are less intense, and thus subject to higher statistical error, than higher multiplicity peaks, and require
higher-energy x-ray sources (e.g., Mo tubes) than those
commonly used and available.
Grain Averaging
As illustrated in Fig. 28.3, the measured strain can vary
substantially from grain to grain in multiphase materials, and even within a single phase due to elastic and/or
plastic anisotropy. Thus, in (typical) cases where the
macrostress is the desired quantity, it is desirable to
sample many grains such that these intergrain variations
X-Ray Stress Analysis
Stress Gradients
The fact that the penetration depth changes with tilting (28.10) can be important if strain/stress gradients
exist within the penetration depth, as different (depthaveraged) strain values will be sampled at different
tilts. This effect can cause curvature in d versus sin2 ψ
plots, which can be analyzed by assuming a particu-
Stress error (MPa)
200
180
160
Error
Envelope
140
120
100
80
60
40
20
0
400
900
1400
1900
Number of grains sampled in x-ray spot
Fig. 28.4 Difference between measured (using x-ray
diffraction) and known stresses as a function of the number of sampled grains (after [28.40]). Each point represents
a single diffraction measurement, taken across a bent
austenitic stainless-steel specimen containing a gradient
both in residual stress and grain size (variation from
62–248 grains/mm2 ). Note the inverse relationship between sampled grains and stress error
lar depth dependence of the stress [28.3]. This issue
can be of particular concern for coatings where microstructural gradients on the μm level are common,
and measurement schemes have been developed for
such cases [28.41]. If the stress variations over larger
ranges than the penetration depth are of interest, these
can be measured by destructive layer-removal methods (Sect. 28.6.2) or by nondestructive profiling with
higher-energy x-ray or neutron sources (Sect. 28.6.4).
28.6 Case Studies
In this section a series of case studies are presented.
Laboratory x-ray diffraction is used for biaxial and
triaxial stress analysis, for both well-behaved and oscillatory d versus sin2 ψ data. In addition, synchrotron
high-energy x-ray diffraction is used for nondestructive
stress profiling.
28.6.1 Biaxial Stress
Biaxial residual stress measurements with the sin2 ψ
method are the most commonly performed diffraction
813
stress measurements. In this example, biaxial x-ray
stress measurements were used to help optimize the
processing of coil springs for improved fatigue resistance. Large coil springs were heat treated to form
tempered martensite, shot peened, and then preset. Figure 28.5 shows the stress diffractometer used to make
biaxial residual stress measurements on the springs.
Cr-Kα x-rays were used and the tempered martensite 211 diffraction peak was recorded with a linear
position-sensitive detector. Specimen tilting was performed with the Ω-goniometry. The positions of the
Part C 28.6
are averaged out. The need for averaging is especially
important in the case of large grain sizes and/or small
x-ray beam sizes, and is exacerbated by the fact that
only a small fraction of the irradiated grains diffract.
Common methods to enhance averaging are to oscillate
(in φ and/or ψ) and/or translate the sample during data
collection.
The importance of grain averaging was illustrated
by Prime et al. [28.40], who examined a bent austenitic
steel sample using both neutron and x-ray diffraction.
The bending provided a gradient in both residual stress
and grain size (determined from optical microscopy),
and laboratory x-ray stress measurements were taken
along the gradient direction. These x-ray stresses were
compared with the calculated macrostresses (which
agreed well with bulk neutron results), and the error
in x-ray stresses (difference from calculations) is plotted versus number of sampled grains in Fig. 28.4. While
there is considerable scatter in the data, there is clearly
an inverse trend between stress error and the number of
grains sampled.
Finally it is noted that the other extreme to polycrystalline averaging is to measure strain in single grains.
Such measurements provide fundamental insights into
polycrystalline deformation behavior, as well as unique
input to deformation models. New methods are being
developed to perform such measurements in individual
grains within polycrystals (Sect. 28.6.5).
28.6 Case Studies
814
Part C
Noncontact Methods
Part C 28.6
These measurements were used to study processing
methods for spring manufacturing. Figure 28.7 shows
the residual stresses with depth below the shot-peened
surface for two different heat treatment methods. The
stresses below the surface were measured after layer
removal and then later corrected for the stress relaxation that occurs with layer removal. One heat treatment
method leads to decarburization of the surface, which
diminishes the compressive residual stresses the material will support. It should be noted that, with
decarburization, the stress-free lattice parameter will
vary with depth because of its dependence on the carbon content. Because the stresses are biaxial in nature,
accurate stress-free lattice parameters are not needed,
simplifying the analysis.
28.6.2 Triaxial Stress
Fig. 28.5 Stress diffractometer making measurements on
a segment of a coil spring (photo by Yunan Prowato of
NHK International Corporation)
diffraction peaks were determined from the average of
the two half-maximum points on the peak. The d versus sin2 ψ plot is shown in Fig. 28.6 and is seen to be
linear. The goodness-of-fit is 0.57, indicating that the
estimated errors from the diffraction peak position determinations account for the observed error to a high
level of certainty. The slope and intercept from the
d versus sin2 ψ plot were used, along with the DEC
E/(1 + ν) = 176 GPa, to compute the residual stress as
−329 MPa.
The data plotted in Fig. 28.8 were obtained from the
surface of a ground steel bar [28.36]. The data were
collected with a conventional x-ray diffractometer utilizing Cr-Kα radiation. The grinding direction is aligned
along the φ = 0◦ direction. Along the grinding direction we see ψ-splitting, indicating the presence of shear
stresses, σ13 . Perpendicular to the grinding direction,
φ = 90◦ , ψ-splitting is absent, indicating that the shear
stresses σ23 are negligible. Along φ = 45◦ , there is moderate ψ-splitting, indicating a component of shear stress
perpendicular to the surface along this direction. From
the discussion in Sect. 28.3, we know that these stresses
arise as microstresses balanced by stresses of opposite
sign in another phase. Here, the shear microstresses
d-spacing (Å)
1.1706
1.1704
sin2 ψ
2θ
d-spacing
(Å)
0.041819
0.189426
0.325714
0.479062
156.296
156.439
156.598
156.742
1.170401
1.170096
1.169758
1.169455
Slope
Intercept
Stress
–0.0021916
1.1704953
–328.71 MPa
1.1702
1.17
1.1698
1.1696
1.1694
GOF = 0.573
1.1692
0
0.1
0.2
0.3
0.4
0.5
0.6
sin2 ψ
Fig. 28.6 Plot of d versus sin2 ψ for a shot-peened steel coil spring, with data and linear fit parameters included (data by
Yunan Prowoto of NHK International Corporation)
X-Ray Stress Analysis
are balanced by stresses in the carbide phases of the
steel. They arise from the shear deformations imposed
on the material from the grinding process. Low volume
fraction, low crystal symmetry, and considerable peak
broadening from the large deformations make measurement of the stresses in the carbide phase extremely
difficult and they were not measured here.
Fitting this data along with the known d0 and DEC
to (28.8)–(28.9) gives the stress tensor [28.12]:
⎛
⎞ ⎛
⎞
527 −8 −40
7 5 1
⎜
⎟ ⎜
⎟
σij = ⎝ −8 592 5 ⎠ ± ⎝5 7 1⎠ MPa .
−40 5 102
1 1 3
28.6 Case Studies
815
Residual stress (MPa)
0
–100
–200
–300
– 400
Decarburized
–500
–600
–700
(28.23)
Here we confirm the presence of the shear stress σ13 and
the negligible σ23 .
28.6.3 Oscillatory Data
Not Applicable to the Classic Model
Frequently, the strains measured in specimens will not
behave with orientation as predicted by (28.8). This
equation assumes that the strain measured in different directions transforms as a tensor quantity. Because
the volume of material diffracting in each orientation is different, this does not have to hold true. If
the stresses and strains partition themselves inhomogeneously on average, the measured strains will not
behave according to (28.8) and numerically forcing
the fit to compute stresses is questionable. Materials with texture due to deformation or growth, such
as cold-rolled sheet or thin films [28.42], will frequently show this behavior. Figure 28.9 shows a set
of biaxial data on a specimen of ground, cold-rolled
steel [28.36]. The data clearly does not show the linear behavior as a function of sin2 ψ, which is partly
attributed to the presence of sample texture (not shown).
A goodness-of-fit value of 6.2 × 10−49 was computed,
indicating an extremely low probability that the deviations from the model fit can be accounted for by the
estimated errors on the data points [28.36]. It should
be noted that, even if the data do not adequately fit
the models, the computed stresses may prove useful as a quality control tool or nondestructive test,
even though the interpretation of the results as stresses
may not be warranted. Advanced analysis methods, described in Sect. 28.6.5, may in certain cases be used
to more accurately determine mean stresses from such
data.
–900
Not decarburized
0
0.05
0.1
0.15
0.2
Depth (mm)
Fig. 28.7 Residual stresses measured with x-ray diffraction on the
outside of two coil springs as a function of depth. One spring has
experienced decarburization during heat treatment while the other
did not. Both raw data (solid lines) and layer-removal-corrected data
(dashed lines) are shown
28.6.4 Synchrotron Example:
Nondestructive, Depth-Resolved
Stress
Here we illustrate the use of high-energy x-rays from
a synchrotron source to nondestructively measure strain
and stress versus depth [28.43]. A schematic of the
experiments is shown in Fig. 28.10a. Cylindrical steel
specimens of 9 mm diameter were heat treated to
form a tempered martensite matrix and nanosize M2 C
strengthening precipitates. After heat treatment, specimens were laser peened and then subjected to rolling
contact fatigue (RCF). Measurements were performed
at the 1-ID beamline at the APS, Argonne National
Laboratory. An x-ray energy of 76 keV and a conical
slit [28.44] were used to create a diffraction volume of
≈ 20 × 20 × 150 μm3 . The high penetration power at this
x-ray energy (τ1/e ≈ 3 mm) allowed for transmission
measurements, wherein an area detector was placed after the conical slit to collect diffraction over a plane
encompassing (nearly) the axial (ε11 ) and normal (ε33 )
strain directions in a single exposure. Thus, sample tilting was not required to evaluate two principal strain
components, as opposed to the reflection-geometry
methods cited above. Specimens were translated along
the vertical (x3 ) direction, in 20 μm increments, relative to the fixed probe volume, to obtain strain and
stress information versus depth, and translated in the
Part C 28.6
–800
816
Part C
Noncontact Methods
d-spacing (Å)
1.1705
1.17
1.1695
1.169
Part C 28.6
θ = 0°,
θ = 0°,
θ = 90°,
θ = 90°,
θ = 45°,
θ = 45°,
1.1685
1.168
0
0.1
0.2
0.3
0.4
ψ (+)
ψ (–)
ψ (+)
ψ (–)
ψ (+)
ψ (–)
0.5
sin2 ψ
Fig. 28.8 Plot of d versus sin2 ψ for a ground steel spec-
imen showing the presence of ψ-splitting, indicating the
presence of shear stresses with a component normal to the
surface
horizontal (x1 ) direction to evaluate different RCF wear
tracks.
The axial stress σ11 was determined from the strains
from the martensite (211) reflection, using DEC for
martensite and assuming an equibiaxial strain state
(ε11 = ε22 ) (Fig. 28.10b). Significant compressive residd-spacing (Å)
1.1714
1.17135
ual stresses were observed near the surface after heat
treatment, and these stresses were further increased after peening, with a maximum value near the surface.
Furthermore, the RCF was found to change the residual
stress profile, with a subsurface maximum (≈ 100 μm
deep) observed under a wear track.
It should be noted that both white-beam (energydispersive) x-ray [28.45] and neutron [28.46] techniques can also be used for such deeply penetrating,
nondestructive measurements, with some limitations.
As in the case study shown, three-dimensional volumes are defined by the incident- and diffracted-beam
slits, with neutrons generally having larger probe volumes ( 1 mm3 ) due to flux limitations. White-beam
measurements can suffer from high background levels,
which can limit the measurable signal due to the limited
dynamic range of typical solid-state detectors.
28.6.5 Emerging Techniques and Studies
Both neutrons and synchrotron x-ray sources carry the
obvious disadvantages of not being located on-site and
having limited beamtime availability. However, the results of these measurements can be useful in a general
sense, both to check against conventional x-ray stress
measurements [28.40,47] and to provide critical tests of,
and input to, deformation models [28.48]. Furthermore,
they possess unique aspects such as being well suited
for in situ measurements, especially under mechanical
and/or thermal loading, and permitting nondestructive
strain mapping such as in the case study described
above. Here, we note emerging studies and techniques
in the field of diffraction strain analysis, largely made
possible by these sources.
1.1713
1.17125
1.1712
1.17115
1.1711
1.17105
1.171
GOF = 6.2 ×10– 49
1.17095
1.1709
0
0.1
0.2
0.3
0.4
0.5
sin2 ψ
Fig. 28.9 Plot of d versus sin2 ψ for a ground steel spec-
imen, illustrating data that does not fit the biaxial stress
model, with a correspondingly low goodness-of-fit (GOF)
value
Single-Grain Studies
In these studies individual grains within polycrystalline
aggregates are evaluated (e.g., [28.49, 50]). The basic
concept is to use beam sizes on the order of the grain
size (typically 1–100 μm), spatially locate diffracting
grains using single-crystal orientation methods, and determine strain using peak shifts. As these techniques
rely on small beam sizes, they have so far been done at
synchrotron sources. These techniques can also be used
for structural characterization, including grain boundary
mapping [28.51] and evaluating dislocations quantitatively [28.52] as well as dynamically [28.53]. Finally,
these techniques can be combined with spectroscopy
techniques such as extended x-ray absorption fine structure (EXAFS) and/or fluorescence for both materials
and environmental science applications [28.54].
X-Ray Stress Analysis
Combined Strain and Imaging Studies
For these studies, typically a relatively large (mm-sized)
x-ray beam and an area detector are used to image heterogeneities/defects, and then a smaller beam is used for
localized diffraction/strain measurements. For x-rays,
the typical imaging contrast mechanism is absorption
(radiography), but the coherence of synchrotron sources
also allows for more sensitive phase-contrast imaging [28.55]. Examples of work in this area includes
deformation studies on composite systems [28.56] and
evaluation of creep damage [28.57].
Advanced Quantitative Analysis
There have been several recent efforts to extend some
fundamental concepts of quantitative texture analysis
to describe orientation-dependent elastic strains/stresses
in polycrystalline materials [28.39, 65, 66]. Strain
measurements from multiple sample orientations are
represented on so-called strain pole figures, which are
analogous to pole figures used in texture analysis,
and which may be inverted to obtain the underlying
strain/stress distribution [28.39, 65, 66]. Using these
Top view, specimen
x1
Peened
surface
x2
9 mm
Conical
slit
RCF wear tracks
20 µm
APS
x-rays
E = 76 keV
x3
2θ ~ 8°
x2
Beam stop
100 µm
9 mm
Side view, experiment
Area
detector
b) Axial residual stress σ11 (MPa)
0
–200
– 400
–600
–800
–1000
–1200
Unpeened, outside wear track
Peened, outside wear track
Peened, under wear track
–1400
–1600
–1800
0
0.5
1
1.5
Depth into sample x2 (mm)
Fig. 28.10 (a) Setup for three-dimensional spatially resolved strain
measurements using high-energy synchrotron x-rays, a conical slit,
and area detector, with specimen photo shown in inset. (b) Measured
residual stresses in the axial (x1 ) direction for three cases
techniques, it is possible to examine the micromechanical states of specific crystallite populations within the
aggregate in greater detail than with standard methods.
Such techniques also provide the modeling community
with a powerful motivation and validation tool.
28.7 Summary
•
X-rays are capable of measuring elastic strain
in all crystalline phases present, using shifts in
diffraction peak positions. These strains can be
converted to elastic stresses using elastic constants. Diffraction also provides information on
peak intensities and widths that can be used to
analyze texture and plastic deformation, respectively.
817
•
•
With laboratory x-rays the probed depth is typically
microns. Deeper depths can be sampled either by
(destructive) layer-removal methods or (nondestructively) using higher-energy x-rays or neutrons.
Errors in strain values can arise from many sources,
including counting statistics, instrument misalignment, and sample-related sources. Methods to
assess experimental error include measuring an
Part C 28.7
Strain in Nontraditional Materials
While diffraction has traditionally only been used
to measure strain in crystalline materials, Windle
et al. [28.58] showed that strain can be extracted from
amorphous materials (polymers), by measuring changes
in peak positions and/or radial distribution functions
with applied load. This work has recently been extended with synchrotron studies [28.59, 60] on bulk
metallic glasses. Another emerging area is biomaterials
studies, including bone [28.61, 62], teeth [28.63], and
synthetic coatings on implants [28.64]. These materials
(e.g., bone) often contain additional ordering on longer
length scales than typically evaluated by wide-angle
scattering (up to the μm level [28.62]), and deformation on these levels can be evaluated by small-angle
scattering [28.61] (28.1).
a)
28.7 Summary
818
Part C
Noncontact Methods
annealed sample and/or repeating a given measurement. In addition, the use of incorrect diffraction
elastic constants to convert strain to stress will give
a proportional error in absolute stress values.
•
With the maturation and anticipated growth of both
synchrotron and neutron facilities, their use for advanced strain and stress analysis is expected to
continue well into the future.
28.8 Further Reading
General information on diffraction stress measurements
can be found in [28.3–8].
Part C 28
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