Experimental Mechanics (2012) 52:1407–1421
DOI 10.1007/s11340-012-9617-1
Optimization of Digital Image Correlation
for High-Resolution Strain Mapping
of Ceramic Composites
V.P. Rajan · M.N. Rossol · F.W. Zok
Received: 15 August 2011 / Accepted: 2 April 2012 / Published online: 14 April 2012
© Society for Experimental Mechanics 2012
Abstract Digital image correlation (DIC) is assessed as
a tool for measuring strains with high spatial resolution
in woven-fiber ceramic matrix composites. Using results of mechanical tests on aluminum alloy specimens
in various geometric configurations, guidelines are provided for selecting DIC test parameters to maximize
the extent of correlation and to minimize errors in
displacements and strains. The latter error is shown
to be exacerbated by the presence of strain gradients.
In a case study, the resulting guidelines are applied to
the measurement of strain fields in a SiC/SiC composite comprising 2-D woven fiber. Sub-fiber tow resolution of strain and low strain error are achieved. The
fiber weave architecture is seen to exert a significant
influence over strain heterogeneity within the composite. Moreover, strain concentrations at tow crossovers
lead to the formation of macroscopic cracks in adjacent
longitudinal tows. Such cracks initially grow stably, subject to increasing app lied stress, but ultimately lead to
composite rupture. Once cracking is evident, the composite response is couched in terms of displacements,
since the computed strains lack physical meaning in the
vicinity of cracks. DIC is used to identify the locations
of these cracks (via displacement discontinuities) and
to measure the crack opening displacement profiles as
a function of applied stress.
V.P. Rajan · M.N. Rossol · F.W. Zok (B)
Materials Department, University of California,
Santa Barbara, CA 93106-5050, USA
e-mail: zok@engineering.ucsb.edu
Keywords 3-D digital image correlation · Woven-fiber
ceramic-matrix composites · Spatial resolution ·
Strain error · Crack opening displacement
Introduction
Digital image correlation is a non-contact optical technique used to measure surface displacements and strain
fields of a test specimen subjected to an external stimulus (load, temperature, etc.) [1]. Displacements are
obtained by imaging a speckle pattern on the specimen
surface during the test and subsequently correlating
each image of the deformed pattern to that in the undeformed state [2]. Strains are obtained by differentiating
displacement fields. The technique combines sub-pixel
displacement accuracy [3] with excellent spatial resolution, allowing strains to be measured to within 10−4
or less [4]. In principle, these attributes make DIC
eminently suitable for probing strain distributions in
woven ceramic fiber/ceramic matrix composites. Conventional measurement techniques (e.g. strain gages
and extensometers) are often inadequate for three
principal reasons.
First, strain distributions in woven fiber composites
can be highly heterogeneous and dependent on the
weave architecture [5]. The local strains in a fiber
tow are influenced by numerous factors, including:
the orientation of the tow with respect to the loading
axis; the amplitude of out-of-plane tow undulation;
the local ‘environment’ around the tow, characterized
by the arrangement of neighboring tows and matrix;
and the elastic/fracture properties of the composite
constituents. Furthermore, the strains are expected to
1408
vary spatially over a length scale comparable to the
tow dimensions. To formulate high-fidelity models of
composite damage and failure, it is necessary to find the
maximum strain values at the tow level. The implication
is that strains must be resolved spatially at a length scale
smaller than that of the tows.
Second, strain variations arise in the presence of
structural features such as holes or notches. In these
cases, the gauge length for strain measurement must
be less than the smallest characteristic dimension of
the structural feature. Strain gauges and extensometers
lack the requisite spatial resolution to meet these requirements in most cases of practical interest.
Third, the mechanical behavior of ceramic composites must be understood at temperatures representative of the targeted service conditions (1200 − 1500◦ C).
Although contacting extensometers can be used to
obtain global strains, only non-contact methods are
viable for measuring strain variations in this domain.
Indeed, imaging capabilities needed to perform DIC
measurements at elevated temperatures have recently
been established and their use demonstrated on woven
C/SiC composites [6]. While the analyses of displacement and strain errors described later in this paper remain relevant, additional experimental challenges and
measuring artifacts, including illumination, heat haze
and speckle pattern degradation, arise at elevated temperatures.
Numerous studies have utilized digital image correlation to understand the mechanical behavior of fiber
composites [7–13]. Two recent studies [14, 15] have incorporated DIC to examine the mechanical behavior of
a 2-D C-fiber polymer matrix composite. They confirm
the expected features: notably, the heterogeneity of
strains at intra-tow and inter-tow length scales as well
as the periodicity of strains and its intimate relationship
to the fiber architecture. They also demonstrate the
suitability of DIC for measuring local strains with high
accuracy and spatial resolution.
Two additional important issues have been raised
in these and other papers. First, there are inherent
trade-offs between spatial resolution and displacement
accuracy. Second, there is complex interplay between
displacement error, strain error, and the parameters
chosen for the speckle pattern, image correlation and
strain calculation [1, 3, 4, 16–20]. Although several
studies have attempted to quantify these relationships,
most have been limited to correlation errors arising
from rigid body translations [17, 21] or no motion at
all [4]. Simulated experiments in which displacements
are applied numerically to a computer-generated image
have also been conducted [3, 18, 19, 22]. The latter
results represent best-case scenarios since they neglect
Exp Mech (2012) 52:1407–1421
experimental errors caused by camera vibration, imperfect image contrast, etc.
The present study represents part of a broad research
activity focused on strain evolution, damage and rupture of woven ceramic composites. The overarching objective of the present study is to establish a framework
for design and analysis of DIC experiments that yield
high-fidelity strain measurements in these materials.
This is accomplished by: (i) identifying and analyzing
sources of displacement and strain error in mechanical
tests that produce uniform strains as well as strain gradients; (ii) formulating analytic relationships between
these errors and applied strain, speckle size, and DIC
algorithm parameters; and (iii) utilizing the results to
design experiments to probe strain variations in a ceramic composite with the requisite spatial resolution.
The outline of the paper is as follows. First, the
principles of digital image correlation are reviewed.
Next, test results on a baseline (homogeneous) material
(Al 6061-T6) are analyzed to determine the relationships between displacement and strain error and the
DIC algorithm parameters. Three specimen geometries
are considered: one with nominally uniform strain (uniaxial tension) and two with strain gradients (open-hole
and center-notched tension). Because the latter geometries yield different strain gradients as well as different
length scales for strain variation, they provide useful
insights and guidance on the feasibility of measuring
strain gradients and small-scale strain heterogeneities
in woven composites. This part of the study complements previous work focused on errors arising from
rigid body motions [4, 17, 21]. Finally, a case study illustrates how the preceding analyses can be used to design
tests on a SiC/SiC composite. In addition to revealing
the effects of fiber architecture on strain distributions,
the tests are used to probe the initiation and growth
of cracks. To this end, displacement fields found via
digital image correlation are analyzed to locate cracks
and measure their opening displacements. Such measurements are crucial for calibrating models of crack
bridging in fiber composites.
Error Analysis
Digital Image Correlation
In-plane surface displacements are measured by tracking the deformation of a speckle pattern through a
series of digital images acquired during a mechanical
test. If the out-of-plane displacements of the sample
are negligible, a single (stationary) camera can be used.
Otherwise, to measure out-of-plane displacements and
Exp Mech (2012) 52:1407–1421
1409
to correctly extract in-plane displacements, stereo images (from two cameras) must be employed. The correlation algorithm attempts to locate a subset of the image
of the undeformed pattern corresponding to a subset
in the image of the deformed pattern. In mathematical
terms, it seeks to find [2]:
arg min
p
(G(ξ(x, p)) − F(x))2
(1)
x ∈ Subset
where G and F are the grayscale intensity values of
the deformed and undeformed subsets, respectively; ξ
is the shape function that describes the deformation
between the subsets; and p comprises the parameters
of the subset shape function, chosen to minimize the
sum of squared differences (SSD) between grayscale intensity values in the two corresponding subsets [2]. The
DIC software used in this work (Vic-3D, Correlated Solutions [23]) provides the option of using more complex
correlation schemes; these include center-weighting of
the sum in equation (1) and accounting for changes in
intensity values between images (offsets and scaling)
[23]. The minimization process results in the assignment
of a displacement vector (u, v, w) to the subset center. Correlation is subsequently repeated throughout
the sample to generate a displacement field over a
rectangular array of subset centers, defined as nodal
points.
There are several critical parameters for correlation.
One is the subset size, hsub (in pixels). As Bornert
et al. [18] have convincingly demonstrated, the spatial
resolution of displacement measurement is governed
predominantly by the subset size; sinusoidal displacement fields of a wavelength less than hsub cannot be
measured . The lower bound on subset size is set by
the requirement that each subset contain unique information to distinguish it from neighboring subsets. The
rule of thumb adopted in the DIC literature to satisfy
this requirement is that hsub > 3 · hsp , where hsp is the
speckle size [2]. Furthermore, to avoid aliasing, each
speckle must contain several pixels. The corresponding
rule of thumb is hsp > 3 pixels [2]. Conversely, large
speckles necessitate large subset sizes, thereby reducing
spatial resolution.
The order of the subset shape function also plays
an important role. For an nth-order shape function,
displacement errors scale with the n + 1th derivative
of displacement [18]; the shape function employed by
Vic-3D is first order (affine) with an additional term
to account for camera perspective. The requirement
that second-order derivatives be small (to minimize displacement error) sets an upper bound on the allowable
subset size.
When displacement fields are numerically differentiated (in the simplest case, by finite differences), the
resulting strain fields are usually ‘noisy’ [24]. The strain
fields are therefore filtered by Gaussian-weighted averaging over a prescribed N × N array of nodal points.
The distance between nodal points is defined as the
step size, hst (in pixels), and the gage length over which
strain averaging is performed is defined as the filter
length, h f (in pixels) (clearly, h f = N · hst ). The effects
of hsub , hst , and h f on displacement and strain error
both in uniform strain fields and under large strain
gradients are addressed below.
Experimental Details
Sample preparation
In this part of the study, tests were performed on
1.5 mm thick sheets of aluminum 6061-T6. Three test
configurations were employed: uniaxial tension, openhole tension, and center-notched tension. Samples were
machined via electro-discharge machining. The uniaxial tension test sample had dog-bone geometry with a
gauge section of 250 mm long and 25 mm wide. Both
the open-hole tension and center-notched tension specimens were machined from rectangular strips, 300 mm
long and 50 mm wide. The open-hole diameter was
12 mm, and center-notch was 12 mm in length with a
0.6 mm tip radius. Since the length scale for strain decay
in the vicinity of such features is controlled by the root
radius, the strain gradient in the center-notched sample
is expected to persist over a length scale 1/10th of that
in the open-hole test.
Two speckling techniques were employed. In both,
samples were first coated with flat white spray paint.
Speckles were subsequently applied using either a spray
canister with flat black spray paint or a Paasche airbrush with black water-soluble paint.
Mechanical testing and DIC setup
All samples were tested at room temperature on a hydraulic testing machine (MTS 810, Minneapolis, MN) at
a nominal strain rate of 10−4 /s. Samples were clamped
with hydraulic grips. Strains on the back-face of the
uniaxial tension specimen were measured using a laser
extensometer (Electronic Instrument Research, Irwin,
PA) over a gauge length of 25 mm.
Images for DIC were taken with a pair of digital cameras (Point Grey Research Grasshopper), each
with a CCD resolution of 2448 × 2048 pixels and a
70–180 mm lens (Nikon ED AF Micro Nikkor). The
1410
Exp Mech (2012) 52:1407–1421
focal length of the lenses was 70 mm, the aperture
setting was F-16, and the angle between cameras was
24◦ . For all experiments described in this work, either the maximum or the minimum focal length was
employed to minimize errors arising from differences
in magnification between the two cameras. For the
three aluminum alloy specimens, images were taken at
similar magnifications: 41 pixels/mm for the open-hole
and center-notched tension tests and 36 pixels/mm for
the uniaxial tension test. The area of the open-hole and
center-notched specimens within the field of view was
50 mm × 50 mm, while that for the uniaxial tension
specimen was 50 mm × 25 mm. The area of interest for
image correlation was selected to exclude un-speckled
regions such as the hole or notch.
Speckle pattern characterization
Two approaches are employed for quantifying the
speckle size distribution. One is an autocorrelation approach (AC), described by Rubin [25] and used in previous DIC studies [2, 17, 18]. The average speckle size,
hsp , is the width of the autocorrelation function, calculated from the points satisfying the equation A(u) =
0.5, where A is the autocorrelation function [2]. The
other approach relies on particle analysis (PA) techniques [26]. These have been implemented in various
software packages; the present analysis is performed
using ImageJ [27]. Prior to analysis, the camera image
is thresholded and converted to a binary image. Contiguous features of any size and circularity are found
within the new image. The effective diameter of each
feature (speckle) is defined as deq = 4 · A/P, where A
is its area and P is its perimeter. Using this approach,
circular speckles yield an effective diameter equal to
the actual diameter; for elliptical speckles, the effective
diameter is the geometric mean of the major and minor
axes of the ellipse.
Computation of error
In DIC displacement analyses, three types of errors are
typically reported: the bias, v, the standard deviation,
vSD , and the root-mean square error, vRMS [17, 18]. The
bias is a measure of the systematic deviation of the
measured displacements, vm , from their true (imposed)
values, vi , given by
n
1
(vm − vi )
v =
n j=1
(2)
Because the displacement bias is typically negligible
compared to the random error, characterized by the
standard deviation [18], vSD ≈ vRMS , where
vRMS
1 n
(vm − vi )2
=
n j=1
(3)
Only the RMS error is considered in this work. Strain
errors are computed in an analogous manner, with displacements, v, replaced by ǫ yy (y denoting the nominal
loading direction). Here, again, bias is typically much
smaller than the corresponding standard deviation and
thus only the RMS strain error is considered.
The imposed displacement field in the uniaxial tensile test is unknown a priori. It is taken to be of the
form
v = ǫi y + C1 x + C2
(4)
where x is the transverse in-plane direction; ǫi is the
imposed axial strain (taken to be that measured by a
‘virtual extensometer’ over a gage length of 40 mm
on the sample surface); C1 represents the displacement
gradient ∂v/∂ x associated with rigid body rotations
(small but not negligible); and C2 is the rigid body
translation (a natural consequence of applying extension to only one end of the test specimen). C1 and C2
are determined by fitting equation (4) to the measured
(DIC) displacement data. In this scheme, the displacement bias is assumed to be zero, and therefore the
RMS displacement error computed using equation (3)
is solely a measure of the random error. The sum in
equation (3) is evaluated over a sufficiently large array
of nodal points to ensure statistically significant results.
The RMS strain error is computed in a similar manner.
In the open-hole and center-notched tension tests,
only strain error is determined. Axial normal strain
values (ǫ yy ) from the DIC software are taken along
a line emanating from the edge of the hole or the
notch at the sample mid-plane along the x-direction.
Strains are interpolated between nodal points using
cubic splines. The true spatial variation of axial strain
along this line was obtained from finite element analysis, which utilized a material constitutive law calibrated
with the uniaxial tensile data. (Although the law was
calibrated for both elastic and plastic deformation, the
subsequent measurements were restricted to the elastic
domain only). The strains obtained from DIC and FEA
along with equation (3) are used to calculate the RMS
error.
Exp Mech (2012) 52:1407–1421
1411
Speckle size distributions for both the airbrush and
spray paint patterns and the relevant statistical values
are summarized in Fig. 1. The figure indicates that
the speckles from the airbrush pattern are finer than
that from the spray paint pattern. Furthermore, both
speckle size distributions are highly skew, implying
the presence of speckles several times larger than the
median value. The skewness can be problematic for
correlation; indeed, correlation is difficult to attain in
regions of the pattern where the speckle size is large.
These observations suggest that the median speckle
size is not the premier characteristic of a speckle size
distribution, since it is relatively insensitive to the presence of very large speckles. Statistics which account
for skewness, such as the autocorrelation speckle size
(which is implicitly weighted by the number of pixels
within a speckle) or the 90th percentile speckle size
(from PA) should be utilized instead. Hereafter, hsp
is taken to be that obtained from the autocorrelation
method. (Numerically, the values obtained from AC
are almost the same as the 90th percentile values.)
Correlation
Full image correlation requires that each subset in the
deformed image contain sufficient unique information
so that it can be located in the undeformed image [18].
This requirement sets a lower bound on the allowable
subset size. Figure 2(a) illustrates the effects of subset
hsp (pixels)
AC
Probability
0.3
Airbrush
Spray paint
Airbrush
9.2
4.7
PA
Median 90 perc.
3.5
10.7
2.2
4.9
0.2
Spray paint
AC
0.1
0
0
4
8
12
16
20
Speckle size
Fig. 1 Speckle size distribution from particle analysis (PA, solid
lines) and autocorrelation (AC, dashed lines)
(a)
hsub =3hsp
3
Airbrush
2
Spray paint
1
0
0.02
Displacement error (pixels)
Speckle size
% Subsets uncorrelated
4
Results and Discussion
Incomplete correlation
Full correlation
(b)
Airbrush
0.01
Spray paint
0
0
20
40
60
80
100
Subset size (pixels)
Fig. 2 Variation in (a) extent of correlation and (b) RMS displacement error with subset size for uniaxial tension tests on Al.
The predicted subset for full correlation from the rule hsub >
3 · hsp is shown by dashed lines in (a)
size (below and above this bound) on the degree of
correlation (based on error criteria prescribed by the
correlation algorithm). The results are broadly consistent with the rules of thumb for achieving optimal correlation: hsp > 3pixels and hsub > 3 · hsp [2]. The latter
describes the qualitative trend of Fig. 2(a)—that larger
speckle sizes require larger subset sizes for full correlation. The minimum subset size predicted by the rule of
thumb and the subset size required for full correlation
also agree quantitatively. For the spray painted speckles, hsp = 9.3 pixels and hence the minimum recommended subset size is hsub = 28 pixels. By comparison,
the experimental results show that 99% correlation is
obtained at 17 pixels and full (100%) correlation at 29
pixels. For the airbrushed speckles, hsp = 4.5 pixels and
the predicted lower bound on subset size is 13 pixels.
99% correlation is attained at a subset size of 13 pixels
and full correlation at 25 pixels.
Displacement error
Figure 2(b) shows the effect of subset size on displacement error for the uniaxial tension test at an
1412
Strain error
The sources of strain error in digital image correlation
are twofold. First, as mentioned previously, numerical
differentiation of ‘noisy’ displacement data results in
‘noisy’ strain data [24]. Filtering (averaging) is therefore employed. However, filtering can also introduce
strain error if derivatives of strain within the filtering
gage length are not negligible. In the uniaxial tension
test, the strains are uniform and thus only the former
error source is relevant; for the open-hole and centernotched tension tests, both sources must be considered.
The relationship between strain error, displacement
error, and subset size for the uniaxial tension test at
an applied strain of 0.002 is depicted in Fig. 3. Here,
hst = 2 pixels and h f = 10 pixels: both small in comparison to the selected subset sizes. Because displace-
Displacement error (pixels)
0.15
(a)
h
sub
= 15 pixels
hsub = 29 pixels
hsub = 59 pixels
0.1
0.05
0
0.001
0.002
0.005
0.01
0.02
0.05
Applied strain
700
(b)
ε = 0.002
600
Strain error x 10-6
applied strain of 0.002 (about half of the yield strain
and thus well within the elastic domain). For small
subset sizes, below that needed for full correlation, the
error scales approximately inversely with subset size.
In contrast, for larger subsets, the error asymptotes to
a constant value. Similar trends have been reported
in previous studies [17, 21]. The displacement errors
obtained in the present study (0.01 − 0.02 pixels) also
agree with the results of other studies [17, 28]. The
subset size that provides the best compromise between
spatial resolution and displacement accuracy lies just
below that at the asymptote; larger subset sizes yield
no further reduction in displacement error. The nearoptimal value employed in the subsequent strain error
analysis is taken to be 40 pixels for the spray paint
pattern. Note that, in general, the optimal subset size
depends on a number of factors, including the speckle
size distribution and presence of strain gradients within
the subset.
For subset shape functions that are first-order, displacement error is expected to be insensitive to the
magnitudes of rigid body displacements and/or uniform
strains within the subset [18]. This hypothesis can be
assessed for the case of uniform strain; the results are
shown in Fig. 3. For strains below yielding (i.e. < 0.004)
displacement error is indeed constant. Furthermore,
the error for low strains is identical to that for nominally
zero applied strain. This quantity was computed by
taking the standard deviation of displacements between
two sequential images (before loading). Interestingly,
for strains in excess of the yield strain, the error increases dramatically. It is surmised that the latter trend
is a consequence of the inherent heterogeneity of crystal plasticity at the length scales being probed by the
present measurements (h p /10 ≈ 3 µm).
Exp Mech (2012) 52:1407–1421
εRMS = νRMS/hsub
500
400
Measured
300
200
100
0
0
20
40
60
80
100
Subset size (pixels)
Fig. 3 Effects of strain and subset size on displacement and strain
errors (hst = 2 pixels, h f = 10 pixels)
ment values within a subset are correlated to one another, displacement data exhibit sinusoidal variations
with a wavelength roughly equal to the subset size.
Hence, differentiation of the displacement field to obtain strains yields errors obeying the scaling relationship ǫRMS ∝ vRMS / hsub . From the data in Fig. 3, the
scaling constant is found to equal 1; the agreement between the prediction and the actual strain error is good,
demonstrating that, for affine deformations, strain error can be ascribed solely to displacement error.
Because displacement values are correlated over a
length scale roughly equal to the subset size, strain
error can be minimized by choosing a gage length for
strain calculation comparable to the subset size. In
practice, this is equivalent to choosing a large step size:
hst > hsub /2 [4]. An alternate approach is to select an
arbitrarily small step size and subsequently filter the
resulting strain data. The effect of filtering on the strain
error for the uniaxial tension test at an applied strain of
0.002 is illustrated in Fig. 4. For the smallest step size,
two regimes are evident. When h f < hsub , averaging has
almost no effect due to the aforementioned correlation
Exp Mech (2012) 52:1407–1421
2.0
hst = 1
250
hst = 2
200
ε = 0.002
150
1.5
hst = 4
100
εRMS = vRMS/hf
hst = 7
Strain x 10-3
Strain error x 10-6
1413
Virtual
extensometer
hf = 134
hf = 50
1.0
hsub
hf = 10
50
30
5
10
20
30 40 50 60 80 100
y
0.5
hst = 15
(a) Uniaxial tension
200
Filter length, hf (pixels)
0
0
100
200
300
400
500
y−position (pixels)
Fig. 4 Strain error vs. filter length for various step sizes (hsub =
40 pixels). Strain error decreases inversely (solid line) with increasing filter length for h f > hsub . Strain error also decreases
with increasing step size
(b) Open-hole tension
3
x
1
v(y) = v(y0 ) + v ′ (y0 )(y − y0 ) + v ′′ (y0 )(y − y0 )2
2
(5)
and the corresponding strain, computed using (firstorder) forward differences, is
1
ǫ yy (y) = v ′ (y0 ) + v ′′ (y0 )(y − y0 )
2
(6)
hst = 27
2
hst = 19
hst = 2
hf = 135
1
hf = 10
0
0
100
200
300
x−position from hole edge (pixels)
(c) Center-notched tension
3
x
Strain x 10-3
of displacements and strains over a subset. In contrast,
when h f > hsub , the errors follow the expected scaling
relationship [12]: ǫRMS ∝ vRMS / h f .
Line scans of axial strain for the uniaxial tension
test in Fig. 5(a) demonstrate the efficacy of filtering in
reducing strain noise. Note that, when h f < hsub , strain
fields oscillate with a wavelength comparable to the
subset size. For large step sizes, the magnitude of strain
error is diminished [4].
Differences between the DIC virtual extensometer
strain and the strain measured by the laser extensometer on the specimen back-face are ≤ 50 µstrain,
indicating that systematic errors (strain bias) are small
compared to random errors (strain standard deviation).
The random errors are hundreds of µstrain before
filtering. Other metrics of strain bias, e.g. the average
shear strain across the specimen, are similarly small.
The assumption made at the outset of the analysis—
that strain bias is negligible compared to strain standard
deviation—is therefore validated.
Strain gradients constitute an additional source of
error, both because a secant line approximation is employed to compute displacement derivatives and because filtering is utilized to reduce noise. A Taylor
series analysis proves to be insightful. Suppose that ǫ yy
is to be determined at a point y0 . The displacements in
the neighborhood of y = y0 can be expressed as
Strain x 10-3
FEA
2
hf = 10
1
hf = 50
0
0
50
100
FEA
150
200
x−position from notch tip (pixels)
Fig. 5 (a) Line scans of strain (ǫ yy ) for uniaxial tension test (hst =
2 pixels, hsub = 40 pixels) for various filter lengths. Strain oscillations occur over a wavelength comparable to the subset size.
(b) Line scans of strain (ǫ yy ) for open hole tension test (hst = 2
pixels, hsub = 40 pixels) for minimal filtering (h f = 10 pixels) and
for constant filter length (h f = 135 pixels). Also shown for comparison is the distribution calculated by FEA. (c) Corresponding
line scans for the center-notched specimen (hst = 2 pixels, hsub =
20 pixels)
1414
Exp Mech (2012) 52:1407–1421
ǫ yy (y0 ) =
1 ′′
v (y0 )hst
2
(7)
That is, strain error is proportional to both the strain
gradient and the step size. An equivalent statement is
that large step sizes lead to a loss in spatial resolution of
strain measurements [4]. Filtering has a similar effect.
A Taylor series analysis of a uniformly weighted filter
yields
ǫ yy (y0 ) =
1 ′′
v (y0 )h f
4
(8)
for filters that are centered on the edge of a specimen
(e.g. the edge of a hole or notch) and
ǫ yy (y0 ) =
1 ′′′
v (y0 )h2f
24
(9)
for filters lying away from a specimen edge. The deleterious effects of strain gradients and higher-order derivatives within a filter length are manifest in these relations.
The preceding relationships are confirmed by the
results of the open-hole tension test. Figure 5(b) illustrates the salient trends. Line scans of strain, ǫ yy , along
the sample mid-plane are compared to the FEA prediction at an applied stress of 100 MPa. For small filter
lengths and step sizes, numerical differentiation noise is
not attenuated and the strains exhibit large-amplitude
oscillations (identical to those seen in the uniaxial
tension test). Conversely, for large filter lengths and
step sizes, strain estimates deviate from those predicted, both with increasing step size (at a constant
filter length) and with increasing filter length (at a
constant step size). The magnitude of the strain errors
(several hundred µstrain) is large compared to the
expected strain bias. At the hole edge, for step sizes
large enough to reduce strain variability from numerical differentiation, the errors are well described by
ǫRMS = 0.5
dǫ yy
dǫ yy
hst + 0.07
hf
dx
dx
(10)
The strain derivative terms were calculated using
the FEA results. The first term on the right side of
equation (10) is the same as that in equation (7). The
coefficient (0.07) of the second term differs from that
(0.25) obtained from the Taylor series analysis. The discrepancy is due to the fact that the Taylor series analysis
employs uniformly-weighted filtering while the DIC
strain calculation uses Gaussian-weighted filtering. The
latter provides superior results in the presence of strain
gradients.
The previous results can be utilized to generate a
‘map’ of acceptable parameters (magnification, subset
size, filter size) for a DIC experiment on a specimen
with a strain concentrator. The assumptions used to
construct the map are as follows. First, the step size
is chosen to be a small fraction of the subset size:
hst = hsub /10. Second, speckle size is assumed to be
‘optimal’: that is, the speckle pattern is reasonably oversampled so that 4 < hsp < 9 pixels [2, 17]. With these
assumptions, the displacement error (in pixels) should
be insensitive to magnification and speckle size and
controlled mainly by subset size [17]. The functional
form of the relationship is obtaining by fitting the data
in Fig. 2(b). (Note that the magnitude of the displacement error should be confirmed by performing rigid
body experiments before mechanical testing.) Third,
the conservative assumption is made that strain errors
arising from strain gradients and displacement noise are
additive. Then, the minimum magnification needed to
resolve a known strain gradient to a prescribed error
tolerance can be calculated using equation (10) and
relations expressed by Fig. 4. An example is shown in
Fig. 6, using an error tolerance of 300 µstrain. For an
applied stress of 100 MPa, the resulting strain gradient
is 10−6 /µm, which is indicated by the dashed line (the
magnification in the open-hole tension experiment being 41 pixels/mm). Evidently, only a narrow range of
subset size and filter length combinations yields acceptable strain data. One such combination is illustrated in
Fig. 5(b).
Minimum mag. (pixels/μm)/strain gradient (μm-1)
Since the distance between adjacent nodal points is
hst , the difference between the computed strain (from
equation (6)) and the actual strain, v ′ (y0 ), (i.e. the strain
error) becomes
10
x 10 4
hf /hsub = 3
8
2.5
2
1.5
1
6
4
Experimental
2
0
0
20
40
60
80
100
Subset size (pixels)
Fig. 6 Effects of subset size and filter length on the minimum
magnification/strain gradient required to achieve a specified
strain error (300 µstrain) in a DIC experiment
Exp Mech (2012) 52:1407–1421
1415
In the center-notched geometry, the strain gradient is
about an order of magnitude larger. The map indicates
that, to within a reasonable strain error, DIC parameters cannot be chosen to comply with the concurrent requirements of capturing the strain gradient near
the notch and mitigating the numerical differentiation
noise. Line scans in Fig. 5(c) confirm this result. The
gradient can be captured only by employing a higher
magnification. This would also require use of smaller
speckles.
These results are a direct consequence of the relationships between the length scales associated with
the structural features and the DIC analysis (hsub , h f ).
Specifically, in the present case, the gage length for
strain computation must be significantly smaller than
the notch root radius, which governs the length scale
for strain decay. However, to attenuate numerical
differentiation noise, the strain computation must be
performed over an area large compared to the subset
size, which must encompass several speckles and many
more pixels. While parameters can be chosen for the
open-hole and uniaxial tension tests to satisfy these
opposing requirements, no such selection is possible for
the center-notched test.
To summarize, errors are minimized by choosing a
speckle pattern and DIC parameters in accordance with
the following guidelines.
(i) Deformations within a subset should be affine;
that is, the strain gradients must be small.
(ii) The subset size, speckle size, and camera magnification should be chosen to obtain full correlation
Fig. 7 (a) A representative
sample of the woven fabric
and (b) topographical map of
SiC/SiC composite (at the
same magnification).
Crossover points and flat
segments can be discerned
from the changes in height.
(c) Tensile stress-strain
response and the points
(A,B,C,D) corresponding
to the strain field in (d)
as well as the strain and
displacement fields in Fig. 8
Weft tow
Warp tow
boundaries boundaries
10
0
-10
-20
3 mm
-30
(a)
(b)
250
0.0020
D
200
0.0018
C
0.0016
Stress (MPa)
B
0.0014
150
0.0012
100
0.0010
A
0.0008
0.0006
50
0.0004
0
0.0002
0
0.001 0.002 0.003 0.004 0.005
Strain
(c)
0
(d)
1416
Exp Mech (2012) 52:1407–1421
as well as the requisite displacement accuracy.
The rules of thumb hsp > 3 pixels and hsub > 3 ·
hsp provide a useful start.
(iii) The subset size should be sufficiently small that
the desired spatial resolution is achieved.
(iv) Step and filter sizes should be selected to preserve real strain gradients arising from material
or structural features while simultaneously minimizing noise due to numerical differentiation.
In some cases, the parameter selection process is
over-constrained. This can occur if the error tolerances
are low, the area of interest is large, or the length scales
for strain variation are small. Error analysis provides
useful insights in selecting parameters that yield the
best compromise between these competing objectives.
Fig. 8 Strain (top) and
displacement fields (bottom)
at three stress levels. The
white lines demarcate the tow
boundaries, established in
Fig. 7. Circled features are
examples of cracks emanating
from tow crossovers. The
crack with the x-y coordinate
system is the one analyzed in
Fig. 10. The black lines in the
displacement fields are
displacement contours, which
are separated by 0.5 µm
σ = 168 MPa
Case Study: SiC/SiC Composite
Experimental Details
Strain variations associated with the fiber weave in
a ceramic composite under uniaxial tension were
probed using DIC measurements. Samples were lasermachined from a SiC/SiC composite comprising six layers of 2-D woven fiber (eight-harness satin weave). The
specimen were machined so that the tensile direction
was aligned with one set of fiber tows. The specimens
had dog-bone geometry with a gauge length of 25 mm
and gauge width of 8.5 mm. The airbrush technique was
used for speckling. Fiberglass tabs were bonded to the
sample ends to facilitate uniform load transfer from the
grips to the sample. The testing machine and the DIC
equipment were identical to those used for the test on
σ = 186 MPa
σ = 200 MPa
εyy
0.010
0.009
0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0
v (μm)
50
40
yy
yy
y
y
x
x
x
x
30
xx
20
10
0
-10
-20
-30
3 mm
Exp Mech (2012) 52:1407–1421
1417
the Al specimens. A laser extensometer was used to
measure strain on the back-face of the specimen over
a gauge length of 25 mm.
In the test orientations employed, surface fibers reside within either long, relatively-flat segments of longitudinal tows or short segments of transverse tows,
each passing over a single longitudinal tow (Fig. 7(a)).
Because the matrix conforms to the undulations of the
underlying tows, crossovers and flat segments can be
discerned from the topographical map generated by
DIC (Fig. 7(b)).
DIC parameters were chosen to satisfy the following requirements: (i) a spatial resolution at a sub-tow
length scale; (ii) full correlation across the entire sample; and (iii) a field of view that encompasses the entire
specimen width. The last requirement yields an upper
bound on magnification. The magnification selected for
both experiments, 138 pixels/mm (attained using a lens
focal length of 180 mm), was sufficiently high to achieve
adequate spatial resolution while remaining below the
upper bound. At this magnification, the area within the
field of view was 12 mm × 8.5 mm, and hsp ≈ 9.5 pixels. A compromise between the first two requirements
2
Strain x 10−3
Estimated
error
1.5
y
1
Longitudinal tows
(a)
x
0.5
−3
−2
−1
0
1
2
3
y−position (mm)
Strain x 10−3
2.0
Estimated
error
1.5
1.0
(b)
0.5
Surface height, z (μm)
Fig. 9 (a) Line scans of strain
(ǫ yy ) along nominally flat
segments of longitudinal
tows, at an applied strain of
1.4 × 10−3 (about 30% of the
tensile failure strain). The
strain is not statistically
different from the
extensometer strain (dashed
line). (b) Line scans of strain
and (c) z-profiles along
crossover points, at the same
applied strain. Strain
elevations of about 30% of
the applied strain are found
in the center of the transverse
tow. (d) Locations of line
scans (flat segments and
crossover points) within the
fiber architecture
was achieved by selecting the smallest subset size that
yielded full correlation (hsub = 51 pixels). This choice
implies a spatial resolution of 0.36 mm, or roughly 30%
of the tow width.
Due to the insensitivity of displacement error to
strain, a good estimate of this error for low levels of
applied strain (< 0.01) is that obtained at nominally
zero strain. As stated previously, this quantity is computed by analyzing two sequential images taken before
loading. The resulting estimate of displacement error
is 0.013 pixels and the (unfiltered) strain error is 255
µstrain (comparable to values obtained from tests on
the aluminum alloy). Since the strain noise was unacceptably large in comparison to the strain variations
being measured, strains were filtered over a gage length
larger than the subset size but significantly smaller than
the tow width (h f = 75 pixels < htow = 160 pixels, hst =
5 pixels). Based on the trends in Fig. 4, the resulting
strain error induced by numerical differentiation is estimated to be approximately 100 µstrain. (Note that,
since ǫRMS ≈ ǫSD , ± 2ǫRMS gives a 95% confidence interval). Because strain gradients are unknown a priori,
other error quantities cannot be estimated. The results
10
(d)
0
−10
Transverse tow
−20
(c)
−30
−1.5
−1.0
−0.5
0
0.5
y−position (mm)
1
1.5
1418
Exp Mech (2012) 52:1407–1421
presented below show that strain gradients can indeed
become unacceptably high for accurate strain determination, especially in the vicinity of matrix cracks.
Results and Discussion
Representative results from tension tests are shown
in Figs. 8, 9, and 10. At small strains (roughly half
of the tensile failure strain), the strain fields are reasonably uniform (a result expected on the basis of
the satin nature of the fiber weave), with one notable exception. Whereas the axial strains within the
long segments of the longitudinal tows are uniform
over most of their length, strain concentrations arise
in locations where the transverse tows cross over the
longitudinal tows. These features are illustrated by the
line scans plotted in Fig. 9. Strain elevations can be
rationalized on the basis of straightening of axial tows at
crossover points [29]; in contrast, within nominally flat
10
(a)
Displacement, v (μm)
5
168 MPa
175 MPa
0
COD
186 MPa
v=
−5
200 MPa
−10
−1
hsub
0
−0.5
0.5
1
y−position (mm)
(b)
200 MPa
15
10
186 MPa
5
175 MPa
168 MPa
0
−0.5
0
0.5
1
C1 (y − ycen ) + C2 ymin ≤ y ≤ ycen − 0.67hsub
C3 (y − ycen ) + C4 ycen + 0.67hsub ≤ y ≤ ymax
(11)
20
COD (μm)
tow segments, undulations in the thickness direction
are small and hence the strain variations are similarly
small.
At larger strains (>0.002), the strain concentrations subsequently ‘bleed’ into the adjacent longitudinal tows. Approaching the composite failure strain
(0.004), strains in excess of 0.02 are seemingly attained
in these tows. Upon closer examination of the displacement data, it becomes apparent that these regions are
actually cracks, each producing an axial displacement
discontinuity. The computed strains are therefore not
true material strains. Furthermore, since these strains
are essentially equal to the ratio of the local crack
opening displacement (COD) to the subset size, the
computed values are highly sensitive to the selection of
the DIC analysis parameters.
Therefore, displacements (not strains) should be
used to assess the evolution of damage within the
composite. Displacement maps corresponding to the
strain maps are shown in Fig. 8. In addition to identifying the crack locations, the displacement data are
used to compute the COD profiles of the cracks. This
is accomplished in two steps (Fig. 10(a)). First, the
crack center is identified by locating the point at which
the (apparent) displacement gradient along a line that
straddles the crack reaches its maximum. Then, displacement data along this line are fit to a piecewise
linear relationship of the form:
1.5
x−position (mm)
Fig. 10 (a) Demonstration of procedure used to ascertain crack
opening displacement (COD). (b) COD profiles for the crack
highlighted in Fig. 8 at several stress levels
where y and v are the displacement and position,
respectively, in the direction of crack opening, and
Ci are fitting constants. This equation assumes firstorder displacements on either side of the crack. Since
displacement data may be unreliable very close to the
crack plane, data residing within two-thirds of a subset size from the crack center are excluded from the
fitting procedure. This requirement is represented by
the inequalities on the right side of equation (11). From
these fits, the displacement step and hence the COD
becomes vCOD = |C4 − C2 |. A threshold of 0.5 µm is
used as a minimum value to constitute the presence
of a crack. The procedure is repeated along each line
passing perpendicular to the subject crack to produce
the entire COD profile. Iterating at varying stress levels
provides information on crack evolution. Note that
crack locations can be determined much less accurately
than crack opening displacements; the accuracy of the
former measurement is comparable to the displacement
spatial resolution (i.e. subset size), while that of the
Exp Mech (2012) 52:1407–1421
1419
Fig. 11 Post-mortem
micrography of the CMC
tensile sample. (a) Section
plane with overlaid
displacement field showing
crack locations, (b) Optical
and (c) Electron micrographs
of large central crack, and
(d) Electron micrograph of
small crack. Red arrows
denote fiber breaks. Blue
arrows denote cracks in the
paint
(b)
(c)
(a)
20 μm
0.1 mm
Sectioning
plane
(d)
1 mm
20 μm
latter is comparable to the displacement error. If strain
fields are instead used to detect cracks, the resolution
of the measurement is related to the strain spatial resolution (i.e. filter length). This resolution is significantly
lower than the displacement spatial resolution if a large
filter length is employed. Efforts to determine crack
locations with high fidelity are also complicated by
bridging of cracks by the paint used in the speckle
pattern.
COD profiles of one such crack (indicated on Fig. 8)
are plotted in Fig. 10(b). At stresses slightly above that
for crack nucleation, the central region of the crack
exhibits an approximately elliptical profile, consistent
with the prediction for a weak-bridging scenario of a
through crack. The profiles near the tips diverge from
the elliptical shape, presumably because of influences
of neighboring cracks as well as the underlying tow
architecture. It may also be a consequence of a more
complex crack front in the through-thickness direction.
Approaching the ultimate tensile strength, cracks begin
to link. This is manifested as non-zero crack opening
displacements at the outer boundaries of the data in
Fig. 10(b). Current efforts are focused on extracting
pertinent bridging traction laws [30] from the displacement profiles.
Post-mortem sectioning and micrography confirms
the locations of cracks detected by DIC analysis
(Fig. 11). That is, each displacement discontinuity
captured by the DIC (Fig. 11(a)) corresponds to a transverse crack (Fig. 11(b), (c), and (d)). Each crack comprises breaks through several fibers within the surface
0◦ tow. The DIC analysis also agrees with micrography on the relative opening displacements of different
cracks.
Concluding Remarks
Digital image correlation is capable of measuring
full-field displacements and strains with accuracy and
spatial resolution unparalleled by strain gages and extensometers. It is especially well-suited for measuring pre-cracking strain distributions and post-cracking
damage evolution in fiber composites. However, design of DIC experiments that yield high-fidelity results
can be complex, primarily because numerous camera,
correlation, and post-processing parameters must be
selected simultaneously. These parameters can be categorized into two groups: those that must be chosen
before an experiment commences (speckle pattern,
camera magnification) and those that can be selected
after the experiment concludes (subset size, step size,
filter length). The present study shows that the optimal
results are obtained when even the latter parameters
are judiciously chosen prior to the test, to ensure that
the requisite fidelity is indeed achievable.
1420
By analyzing the results of mechanical tests on an
aluminum alloy in various test configurations, analytic relations for displacement and strain error have
been developed and validated. Each implies a tradeoff.
Higher camera magnifications entail not only higher
spatial resolution of displacements but also a more
limited field of view. The speckle size must be small
enough so that each subset has sufficient unique information and large enough so that at least three pixels
exist within each speckle. Larger subset sizes reduce
displacement error at the expense of spatial resolution.
Larger step sizes and filter lengths sacrifice the ability
to resolve strain gradients for attenuation of strain
noise. Analytic relations that capture these qualitative
statements have also been provided.
With appropriate analysis parameters, digital image
correlation can capture the evolution of strain within
ceramic composites. The present study has illustrated
that strain concentrations exist at tow crossovers even
in composites with seemingly ‘flat’ woven fabrics. At
higher stresses, the strain concentrations lead to the
formation of cracks in adjacent longitudinal tows that
ultimately cause rupture. In these cases, the DIC results
are clearly more meaningful when presented in terms
of displacements: strains calculated near a crack being
non-physical in nature. A procedure to discern the
locations of cracks as well as their opening displacement profiles has been presented. DIC measurements
are expected to provide crucial data for calibration of
high-fidelity models of the crack bridging process and
composite failure.
Acknowledgements This work was supported by the Pratt &
Whitney Center of Excellence at the University of California,
Santa Barbara (monitored by Douglas Berczik), and the US
AFOSR (Ali Sayir) and NASA (Anthony Calomino) under
the National Hypersonics Science Center for Materials and
Structures (AFOSR Prime Contract No. FA9550-09-1-0477 to
Teledyne Scientific and Sub-contract No. B9U538772 to UCSB).
The authors gratefully acknowledge the assistance of Renaud
Rinaldi with the finite element analysis.
References
1. Sutton M, McNeill S, Helm J, Chao Y (2000) Advances
in two-dimensional and three-dimensional computer vision.
Photomechanics 77:323–372
2. Sutton MA, Orteu J-J, Schreier HW (2009) Image correlation
for shape, motion, and deformation measurements. Springer
3. Schreier HW, Sutton MA (2002) Systematic errors in digital
image correlation due to undermatched subset shape functions. Exp Mech 42(3):303
4. Ke X-D, Schreier HW, Sutton MA, Wang YQ (2011) Error
assessment in stereo-based deformation measurements, Part
II: experimental validation of uncertainty and bias estimates.
Exp Mech 51(4):423–441
Exp Mech (2012) 52:1407–1421
5. Cox BN, Flanagan G (1997) Handbook of analytical
methods for textile composites. NASA Contractor Report
4750
6. Novak MD, Zok FW (2011) High-temperature materials testing with full-field strain measurement: experimental design
and practice. Rev Sci Instrum 82(11):115101
7. Bisagni C, Walters C (2008) Experimental investigation of
the damage propagation in composite specimens under biaxial loading. Compos Struct 85(4):293–310
8. Kazemahvazi S, Kiele J, Zenkert D (2010) Tensile strength
of UD-composite laminates with multiple holes. Compos Sci
Tech 70(8):1280–1287
9. Lagattu F, Brillaud J, Lafarie-Frenot M-C (2004) High strain
gradient measurements by using digital image correlation
technique. Mater Charact 53(1):17–28
10. Fuchs PF, Major Z (2010) Experimental determination of
cohesive zone models for epoxy composites. Exp Mech
51(5):779–786
11. Ramault C, Makris A, Van Hemelrijck D, Lamkanfi E, Van
Paepegem WS (2010) Comparison of different techniques for
strain monitoring of a biaxially loaded cruciform specimen.
Strain 47(S2):210–217
12. Orteu J-J, Cutard T, Garcia D, Cailleux E, Robert L (2007)
Application of stereovision to the mechanical characterisation of ceramic refractories reinforced with metallic fibres.
Strain 43(2):96–108
13. Pankow M, Justusson B, Salvi A, Waas AM, Yen C-F,
Ghiorse S (2011) Shock response of 3D woven composites:
an experimental investigation. Compos Struct 93(5):1337–
1346
14. Daggumati S, Voet E, Van Paepegem W, Degrieck J, Xu J,
Lomov SV, Verpoest I (2011) Local strain in a 5-harness satin
weave composite under static tension: Part I — Experimental
analysis. Compos Sci Tech 71(8):1171–1179
15. Anzelotti G, Nicoletto G, Riva E (2008) Mesomechanic
strain analysis of twill-weave composite lamina under unidirectional in-plane tension. Compos Part A Appl Sci Manuf
39(8):1294–1301
16. Wang Y-Q, Sutton MA, Ke X-D, Schreier HW, Reu PL,
Miller TJ (2011) On error assessment in stereo-based deformation measurements, Part I: theoretical developments for
quantitative estimates. Exp Mech 51(4):405–422
17. Robert L, Nazaret F, Cutard T, Orteu JJ (2007) Use of 3D digital image correlation to characterize the mechanical
behavior of a fiber reinforced refractory castable. Exp Mech
47(6):761–773
18. Bornert M, Brémand F, Doumalin P, Dupré J-C, Fazzini M,
Grédiac M, Hild F, Mistou S, Molimard J, Orteu J-J, Robert
L, Surrel Y, Vacher P, Wattrisse B (2009) Assessment of
digital image correlation measurement errors: methodology
and results. Exp Mech 49(3):353–370
19. Avril S, Bonnet M, Bretelle A-S, Grédiac M, Hild F, Ienny
P, Latourte F, Lemosse D, Pagano S, Pagnacco E, Pierron
F (2008) Overview of identification methods of mechanical parameters based on full-field measurements. Exp Mech
48(4):381–402
20. Knauss WG, Huang Y, Chasiotis I (2003) Mechanical measurements at the micron and nanometer scales. Mech Mater
35(3–6):217–231
21. Haddadi H, Belhabib S (2008) Use of rigid-body motion for
the investigation and estimation of the measurement errors
related to digital image correlation technique. Opt Lasers
Eng 46(2):185–196
22. Schreier HW, Braasch JR, Sutton MA (2000) Systematic errors in digital image correlation caused by intensity interpolation. Opt Eng 39(11):2915–2921
Exp Mech (2012) 52:1407–1421
23. Vic-3D (2007) ®Software. Correlated Solutions Incorporated, Columbia, SC. http://www.correlatedsolutions.com
24. Nicoletto G, Anzelotti G, Riva E (2009) Mesoscopic strain
fields in woven composites: experiments vs. finite element
modeling. Opt Lasers Eng 47(3–4):352–359
25. Rubin DM (2004) A simple autocorrelation algorithm for
determining grain size from digital images of sediment. J
Sediment Res 74(1):160–165
26. Lecompte D, Smits A, Bossuyt S, Sol H, Vantomme J,
Van Hemelrijck D, Habraken AM (2006) Quality assessment
of speckle patterns for digital image correlation. Opt Lasers
Eng 44(11):1132–1145
1421
27. Rasband WS (1997–2011) ImageJ, US National Institutes of
Health, Bethesda, Maryland, USA. http://imagej.nih.gov/ij/
28. Sutton MA, Yan J, Deng X, Cheng C-S, Zavattieri P (2007)
Three-dimensional digital image correlation to quantify deformation and crack-opening displacement in ductile aluminum under mixed-mode I/III loading. Opt Eng 46(5):
1–16
29. González C, Llorca J (2005) Stiffness of a curved beam subjected to axial load and large displacements. Int J Solids
Struct 42(5–6):1537–1545
30. Bao G, Suo Z (1992) Remarks on crack-bridging concepts.
Appl Mech Rev 45(8):355-366