Lab #2: Stress Measurements using Digital Image
Correlation (DIC)
Brighton Smith
Texas A&M University, Dallas, TX, 77843, United States
I.
Experimental Summary
Aerospace engineering relies on stress analysis to ensure the structural integrity of
components under varying loads. Traditional finite element analysis (FEA) cannot factor in
stress concentrations around features like holes or joints, which can lead to the underestimation
of failure points. One metric to quantify this effect is the stress concentration factor (K), which
represents the ratio between the maximum stress near a feature to the nominal stress (when that
feature is not present). However, accurately measuring K is challenging because of the complex
geometric irregularities that the features can introduce, especially with point-based techniques of
strain gauges.
This lab uses a different method for stress analysis called digital image correlation (DIC) to
calculate K and bypass many of the complications that arise when introducing various geometric
features. DIC is used to capture full-field strain data by comparing images of loaded and
unloaded samples, enabling precise strain measurements. This lab can be split into two parts:
tensile testing of a baseline aluminum dogbone sample and tensile testing of an aluminum
dogbone sample with a hole in its center. The goal was to extract information such as Young’s
modulus (E) and Poisson’s ratio (ν) after performing DIC on the baseline sample, and then to
calculate K by performing DIC on the sample with the hole and comparing it to the other data.
The first step of this lab was to measure the geometry features of the two samples such as
gauge length, sample width, sample thickness, and hole diameter. Several measurements were
taken of each feature to obtain an average and standard deviation. The second step was to use a
dark marker to apply a random surface pattern of dots to one side of each sample. This pattern
eventually provided the necessary contrast to perform DIC. The next step was to perform a
tensile test of the baseline sample by mounting the sample in an Instron tensile tester. A digital
camera was placed directly in front of the sample and a zero-force reference image was taken.
The sample was then progressively loaded in steps of 0.5 kN until failure. After each loading
step, an image was taken of the sample using the digital camera, and the load and deformation
gauge output from the Instron machine was recorded. Once the baseline sample broke, it was
replaced by the sample with the hole, and the loading process was repeated.
The data analysis portion of this lab began with performing DIC on the baseline sample using
Vic-2D software. By comparing the zero-force reference images to the subsequent ones, the
average horizontal (εxx) and vertical (εyy) strains were computed for each loading step along with
standard deviations. Stress was then calculated for each loading step using the Instron data, and
an engineering stress-strain curve was plotted for both the strain data calculated from the Instron
1
deformation gauge and the strain data generated from DIC. The Young’s modulus was calculated
for each data set by performing an uncertainty-weighted least-squares (UWLS) fit and finding
the slope. The DIC strain data was also plotted, and Poisson’s ratio was calculated by performing
a UWLS fit and taking the negative of the slope. DIC was then performed on the dogbone
sample with the hole by marking two reference points on the zero-force reference image and
using the subsequent images to generate the horizontal and vertical strain for each loading step at
the points. The vertical stress (σyy) at both points for each loading step was calculated using the
previously found Young’s modulus and Poisson’s ratio, the DIC strain data, and a constitutive
relationship. Finally, K was calculated for each reference point by taking the ratio of the vertical
stress at the highest loading step for the sample with the hole to the nominal stress at the highest
loading step. Uncertainty was propagated through each intermediate step and the standard
deviation for each of the major results such as Young’s modulus, Poisson’s ratio, and both K
values was calculated.
II.
Table 1
Data and Results
Geometric features for the Baseline Dogbone Sample.
Parameter
Gauge Length
(mm)
Sample Width
(mm)
Sample
Thickness (mm)
Cross-Sectional
Area (mm2)
Values
73.33
14.76
1.24
18.3
14.76
1.27
18.7
14.73
1.24
18.3
Average
—
14.75
1.25
18.5
Standard
Deviation
0.011
0.01
0.01
0.2
Table 2
Geometric Features for the Dogbone Sample with Hole.
1
The standard deviation for singular measurements was assumed to be equivalent to the measurement error, which
was ± 0.01 mm for the calipers.
2
Parameter
Gauge
Length (mm)
Sample
Width (mm)
Sample
Thickness
(mm)
Hole
Diameter
(mm)
Cross-Sectional
Area at
Symmetry Line
(mm2)
Values
73.51
14.68
1.22
2.84
14.4
14.86
1.24
14.9
14.63
1.22
14.4
Average
—
14.72
1.23
—
14.6
Standard
Deviation
0.01
0.01
0.01
0.01
0.2
Table 3
Instron Data from Tensile Testing for Both Samples.2
Baseline Dogbone Sample
Dogbone Sample with Hole
Nominal Load
Magnitude (kN)
Load
Magnitude
(kN)
Displacement
(mm)
Load
Magnitude
(kN)
Displacement
(mm)
0.5
0.4808
0.057
0.4881
0.051
1.0
1.002
0.132
0.9914
0.108
1.5
1.511
0.217
1.49
0.167
2.0
1.974
0.309
1.998
0.236
2
Uncertainty from the Instron data was determined by the half-range method, wherein the uncertainty is calculated
as half of the smallest division or increment of the instrument’s scale.
3
2.5
2.491
0.429
2.482
0.323
3.0
3.008
0.544
2.94
0.6393
3.5
3.478
0.668
—
—
4.0
3.926
0.877
—
—
Table 4
Axial Stress and Strain Values Generated from Instron Data.
Baseline Dogbone Sample
σyy (GPa)
0.0260 ± 3 × 10
Dogbone Sample with Hole
εyy
1 × 10
σyy (GPa)
± 2 × 10
0.0334 ± 5 × 10
εyy
1 × 10
± 2 × 10
0.0542 ± 6 × 10
1.8 × 10
± 2 × 10
0.068 ± 1 × 10
1.5 × 10
± 2 × 10
0.0818 ± 9 × 10
3.0 × 10
± 2 × 10
0.102 ± 1 × 10
2.2 × 10
± 2 × 10
0.107 ± 1 × 10
4.2 × 10
± 2 × 10
0.137 ± 2 × 10
3.2 × 10
± 2 × 10
0.135 ± 2 × 10
5.9 × 10
± 2 × 10
0.170 ± 3 × 10
4.4 × 10
± 2 × 10
0.163 ± 2 × 10
7.4 × 10
± 2 × 10
0.202 ± 3 × 10
8.7 × 10
± 2 × 10
0.188 ± 2 × 10
9.2 × 10
± 2 × 10
—
4
—
0.212 ± 2 × 10
Table 5
—
1.20 × 10
± 2 × 10
DIC Strain Data for Baseline Dogbone Sample.
Nominal Load Magnitude (kN)
Table 6
3
εxx
εyy
0.0
−5 × 10
0.5
2 × 10
± 5 × 10
5 × 10
1.0
1 × 10
± 5 × 10
1.0 × 10
± 2 × 10
1.5
−2 × 10
± 5 × 10
1.5 × 10
± 2 × 10
2.0
−2 × 10
± 5 × 10
2.0 × 10
± 2 × 10
2.5
−3 × 10
± 5 × 10
2.6 × 10
± 2 × 10
3.0
−4 × 10
± 5 × 10
3.2 × 10
± 2 × 10
3.5
−6 × 10
± 4 × 10
3.9 × 10
± 2 × 10
± 2 × 10
−5 × 10
± 3 × 10
± 2 × 10
DIC Strain Data and Calculated Axial Stress for Dogbone Sample with Hole.
Nominal Load
Magnitude (kN)
0.5
—
εxx
εyy
σyy (GPa)
Pt. A3
Pt. B
Pt. A
Pt. B
Pt. A
Pt. B
0.00564
-0.00178
0.00169
0.00009
0.177
-0.021
Pt. A was farther from the hole than Pt. B
5
1.0
-0.00964
0.00271
0.00193
0.00089
-0.032
0.089
1.5
-0.01312
0.00385
0.00302
0.00131
-0.022
0.130
2.0
0.00218
-0.00252
0.00201
0.00282
0.144
0.121
2.5
-0.00591
0.00010
0.00876
0.00374
0.404
0.211
2 × 10
2 × 10
2 × 10
2 × 10
0.006
0.003
Standard
Deviation
Table 6
Final Results.
Young’s Modulus
from Instron Data
(E)
Young’s
Modulus from
DIC Data (E)
Poisson’s Ratio
(ν)
—
—
—
Pt. A
Pt. B
19.2 ± 0.2
48.9 ± 0.6
0.26 ± 0.01
2.37 ± 0.04
1.24 ± 0.02
6
Stress Concentration Factor (K)
Figure 1
Figure 2
A Comparison of Stress and Strain for the Baseline Dogbone Sample.
A Comparison of Horizontal and Vertical Strain for the Baseline Dogbone
Sample.
7
III.
Discussion and Conclusion
From Figure 1, it's evident that the Young’s modulus obtained from two distinct datasets
differs significantly. This discrepancy likely arises because the Instron displacement gauge relies
on the apparatus's displacement rather than the sample's. During loading, it seemed the sample
wasn't tightly clamped, potentially leading to inaccurate displacement readings. In contrast, DIC
analyzes the sample’s strain field directly, yielding a Young’s modulus much closer to the
widely accepted value of 69 GPa. Notably, only data up to the nominal loading condition of 3.5
kN was used to calculate Young’s modulus, as the sample reached its yield point between 3.5 kN
and 4 kN. The choice of uncertainty-weighted least squares (UWLS) was deliberate; strain
uncertainty outweighed stress uncertainty, requiring UWLS. Conversely, uncertainty least
squares (ULS) focuses solely on the dependent variable's uncertainties. Strain was chosen as the
uncertain value for both fits in Figure 1 due to its larger uncertainties. Similarly, horizontal strain
(εxx) was selected for Figure 2, its uncertainties surpassing those of vertical strain (ε yy). Notably,
the fit slope in Figure 2 bears the opposite sign to Poisson’s ratio as it represents the negative
ratio of εxx to εyy.
As mentioned previously, an apparent error source was the inadequate clamping of the
sample during loading. Future labs can mitigate this by ensuring proper clamping before
proceeding. Additionally, smearing of the surface pattern on the baseline dogbone sample may
have occurred during placement in the Instron apparatus, potentially affecting DIC accuracy. In
hindsight, applying a new surface pattern on the opposite side of the sample could have
circumvented this issue entirely.
IV.
Appendix
Many intermediate calculations were used to arrive at the final stress concentration factors
(K) displayed in Table 6. Therefore, it was imperative to accurately propagate uncertainty
through each intermediate step. The following outlines the process used to determine the
standard deviation for the K values.
σ =
𝜕𝑦
𝜎
∂𝑥
+
𝜕𝑦
𝜎
∂𝑥
(1)
𝜕𝑦
𝜎
∂𝑥
+ ⋯+
is the general propagated uncertainty equation. To transform this into the specific case, we have
σ =
𝜕𝐾
𝜎
∂σ
+
𝜕𝐾
𝜎
∂σ
=
with
8
1
σ
𝜎
+ −
σ
σ
σ
(2)
σ
=
=
𝜎
𝜎
+
+
+
𝜎
( ( )
𝜎
+
𝜎
)
+
𝜎
(3)
𝜎
+
𝜎
and
𝜎
=
𝜎
+
𝜎
=
𝜎
+
𝜎
Where F is the applied load and A is the cross-sectional area at the symmetry line.
9
(4)