EMCH 402
Experimental Stress Analysis
Technical Report
Determination of Principal Orientations and Stress Magnitudes through
Photoelastic Models and Birefringent Coatings
Co-Author A (Lead Author): Donaldson, Robert
Co-Author B: Stachnik, Michael
Co-Author C: Shaw, Erik
Lab Group No: Group 3, Team 4
4/2/14, 10:00 am
Submission Date: 4/29/14
INTRODUCTION:
This lab implements a transmission polariscope and a reflection polariscope to explore
polariscope techniques and analyze photoelasticity deformation measurements. These non-distructive,
whole field analytical techniques are powerful methods of stress and strain analysis. The observations and
measurements can be utilized to determine stress distribution in the material. These optical methods
provide visual representation of stress distribution allowing for analysis of discontinuities in a material.
These tools can be used to determine critical stress points and concentrations in irregular geometries to
gain an understanding of the model before having to implement numerical methods.
The transmission polariscope will be implemented in the plane configuration and the circular
configuration. The plane configuration consists of a polarizer, analyzer, and a light source. The circular
configuration consists of two additional quarter wave plates. Both configurations require photoelastic
model materials. The plane polariscope configuration is used to determine the principle orientations of the
model from recording isoclinic fringe patterns. The stress-fringe coefficient for the model will be
determined from the isochromatic fringe pattern produced from the circular polariscope configuration.
The reflection polariscope produces isochromatic fringe patterns from the birefingent coating on
the model being stressed. Birefringent materials refract light when under mechanical stress. Deformation
can be measured from the stress concentrations which can be related to strain via the stress optic law.
This lab will examine three different photoelasic model specimens. A C-shaped specimen and a
beam specimen will be observed with the transmission polariscope under loads in each polariscope
configuration. An aluminum specimen with a center hole and birefringent coating will be observed with
the reflection polariscope. The testing of these specimens will provide hands on interpretations of fringe
patterns and allow us to apply photoelasticity to analyze stress concentrations in the models as well as
calibrate the material to identify the material.
The completion of this lab shows the power of these polariscope techniques and their wide range
of applications from identifying materials to determining stress concentrations. On the contrary if these
analytic tools are not used properly large errors are produced as was experienced in the conclusion of this
lab.
1
THEORY:
The governing theories behind polariscope techniques are the wave theory of light,
retardation, intensity, and the stress optic law. The stress optic law being of most importance as it
related optical phenomena such as retardation and intensity to mechanical aspects.
In order to understand the underlying theory we must first start with the wave theory of
light. The wave equation can describe optical effects associated with photoelastisity by
evaluating the electrical and magnetic components of light. This leads us to the simplified
expression, equation 1, for the magnitude of light as a function of position along the axis of
propagation.
𝐸 = acos
2𝜋
(𝑧 + 𝛿 − 𝑐𝑡)
𝜆
(1)
The superposition property of light waves allows for two waves to have the same wavelength and
frequency but different amplitudes and phases. This is considered retardation. When considering the
phase, δ, and taking the difference between two waves, equation 2, we get retardation.
𝛿 = 𝛿2 − 𝛿1
(2)
Intensity is a function of retardation between waves of light as described in equation 3 where a is
amplitude and proportional to intensity.
𝐼 ∝ 𝑎2 = 𝑓(𝛿) (3)
Understanding these basic optical concepts we can apply them to the photoelastic material. When the
photoelastic material has light passed through it the wave enters the principal stress directions. This
causes the waves to travel at different velocities leading to phase retardation between the two
components. This gives us the relationship of the stress-optic law in equations 4.
∆=
2𝜋ℎ
𝐶(𝜎1 − 𝜎2 )
𝜆
(4)
Where ∆ is angular retardation, h is specimen thickness, C is the stress-optic coefficient and 𝜎1 and 𝜎2 are
the principal stresses. We define the fringe order in equation 5 due to interference from the polariscope
which reveals additional fringes that are dependent of relative retardation.
𝑁=
∆
2𝜋
(5)
This allows us to define the stress fringe coefficient or material fringe value as described in equation 6.
𝑓𝜎 =
𝜆
𝐶
(6)
Rearranging and substituting gives us the in-plane principal stress difference for 𝜎1 > 𝜎2 in equation 7.
2
𝜎1 − 𝜎2 =
𝑁𝜎
ℎ
(7)
Therefore we can arrive at the analogous strain relationship in equation 8.
𝜀1 − 𝜀2 =
𝑁𝑓𝜀
,
ℎ
1+𝜐
𝑤ℎ𝑒𝑟𝑒 𝑓𝜀 = (
) 𝑓𝜎
𝐸
(8)
MATERIALS AND METHODS:
Three different photoelastic model specimens were used in this lab. One was a C-shaped
specimen, another was a beam of an unknown material, and the third was an aluminum specimen with a
center hole and birefringent coating. The three different specimen can be viewed below as Figure 1.
Figure 1: (Left to Right) C-Shaped Specimen, Aluminum Specimen with hole and coating, Beam Specimen (didn’t
get a whole photo of it)
These three specimen were subjected to a load in order to view various fringes in both light and
dark fields. The fringes would then be used later for experimental strain calculation and compared to the
theoretical strain. In this lab, a photoelasticity deformation technique was used with both a transmission
and reflection polariscope. Isoclinic fringe pattern data in the plane polariscope dark fringe configuration
was collected during this experiment. This data was collected in order to determine principal orientation
of the photoelastic specimens. For the circular transmission polariscope, isochromatic fringe pattern data
was recorded in both the light and dark fields. For 2-D photoelastic analysis, a photoelastic model
material was calibrated in order to determine the stress-fringe coefficient 𝑓𝜎 of the beam specimen. Once
the 𝑓𝜎 is calculated, the beam’s material can be determined based on known values of 𝑓𝜎 . In the second
part of the lab, a reflection polariscope was used to measure deformation of the opaque aluminum
specimen with the birefringent coating. Whole and half-order isochromatic fringe patterns are employed
in order to determine the strain-field based on the stress-optic law. A compensator was used to determine
fractional fringe orders to completely analyze the effects of stress concentration.
3
In the first part of the lab a C-shaped specimen was loaded in the middle of the transmission
polariscope and it was observed in both light and dark fields with the circular polariscope and in dark
field with the plane polariscope. The setup of the transmission polariscope can be observed on the next
page in Figure 2.
Figure 2: Transmission plane polariscope. Wave Plates can be inserted for
circular transmission polariscope
The C-shaped specimen was loaded in a diametral compressive matter, but due to the shape of the
specimen, it underwent both compressive and tensile forces. A small amount of load was applied until
isoclinic fringes were observed in dark field. Once loaded to the desired amount, the load was recorded to
be 14psi. Symmetry of 0/90 degree isoclinics were observed which determined that the specimen was
loaded in the proper manner. The analyzer was rotated in 15 degree increments from 0 to 90 degrees in
order to view the locations where isoclinics of all parameters pass through. These points represent the
isotropic, or also known as singular points. An image was captured of the C-shaped specimen and can be
viewed below in Figure 3. Also displayed in the images are the dark fringe order for both compressive
and tensile fringes.
Figure 3:C-shaped specimen in dark field. Left, tensile fringe orders. Right, compressive fringe orders.
4
In the next part of the experiment a beam was assembled in the transmission polariscope in a four
point bending setup. Two people were needed to setup the beam in this manner because of the four blocks
were needed to be setup simultaneously at the perfect locations. Once the setup was completed, the
alignment was checked using the isoclinic fringes again like in the C-shaped specimen. Isochromatic
fringe data was observed and recorded in the circular polariscope arrangement. The beam was loaded
until the 4th fringe was just visible on the edge of the specimen in dark field. An image was captured using
a monochromatic filter and also using white diffused light. These images can be observed below in Figure
4 and Figure 5. The fringe orders are also displayed for both the light and dark field on the photographs.
Figure 4: Beam loaded in four point bending in light field with monochromatic filter.
Figure 5:Beam loaded in four point bending in dark field.
In the last part of the experiment, we used a reflective polariscope to load and observe an
aluminum specimen with a hole in it with a birefringent coating. The specimen was setup also with a four
point bending as well. A digital compensator was also used in this part of the experiment to record the
fractional fringe order. The setup can be observed below in Figure 6 on the next page.
5
Figure 6:Setup of the reflection polariscope with the digital compensator
After the specimen was setup in the proper location the room light was turned off and the light
source was turned on. Specimen alignment was checked using isoclinic fringes as before to make sure it
was perfectly aligned. The beam was then loaded in regular force increments until 2.03 fringes appeared
in the specimen, which was based on our calculations. The final load was recorded at 214psi once the max
number of fringe order was observed. We captured a digital image using a monochromatic filter and an
image under diffused white light but they were never uploaded. Finally, we recorded fractional fringe
orders using the digital compensator. 1/16th inch marks were made on the specimen along the path
between the edge of the hole to the outside of the specimen.
RESULTS:
Photoelastic materials and birefringent coatings were analyzed using the two techniques outlined
in the Materials and Methods section. The first was the transmission polariscope. We used the
transmission polariscope to analyze a beam in four point bending, and a C-shape undergoing compressive
force. We then used a reflection polariscope to analyze an aluminum specimen. The specimen had a
circular hole that acted as a stress concentrator, and also had a birefringent coating. This coating allowed
us to observe stresses in the aluminum despite the aluminum not being a photoelastic material.
The transmission polariscope allowed for the calculation of fσ, which was needed to analyze the
stresses in the material. The beam in four point bending was used for this calculation. The polariscope
was set up in a dark field configuration. The beam was photographed using a monochromatic filter to
make the fringes more identifiable. The fringes are shown in Error! Reference source not found..
6
Figure 7:Beam in four point bending, dark field
Using the image and specimen dimensions, a plot of y vs. N was made, where y is the distance
from the central axis and N is the fringe order as shown in Figure 8. The slope of this plot was used to
find the stress-fringe coefficient. This was used to approximate what material the beam was made out of,
as well as allowing analysis of stress states.
By combining the statics expression for a beam in bending with the stress optic law, Eq. (1) was
obtained with related the distance from the center axis (y) with the fringe order (N).
𝑤3
𝑦 = 𝑓𝜎 (6𝑃𝑑) 𝑁
(1)
Plotting the distance y from the neutral axis as a function of fringe order is show in Error!
Reference source not found.. Using a best fit line, an equation for the line could be determined. This is
shown as Eq. (2).
𝑦 = 0.188𝑥 = 0.188𝑁
(2)
The result of equating Eq. (1) to the equation of the line in Eq. (2) and rearranging is shown as
Eq. (3).
6𝑃𝑑
𝑓𝜎 = 0.188 ( 𝑤 3 ) = 0.188 (
6∗228𝑝𝑠𝑖∗0.94𝑖𝑛
)
(1.504𝑖𝑛)3
(3)
7
Figure 8:Fringe distance from central axis versus fringe order
Solving for fσ now yielded 127 lb/(in-fringe).
After the stress fringe coefficient was calculated, 2-D photoelasticity of a C-shaped photoelastic
model was done. Using the theoretical elasticity equations for polar stress of a thick half circular ring
under diametrical compression shown as Eq. (4) and Eq. (6), σθ could be calculated. This was our
theoretical stress magnitude.
2𝐴 =
2𝐴 =
𝑃
(4)
𝑏
𝑎 2 −𝑏2 +(𝑎2 +𝑏2 ) 𝑙𝑛( )
𝑎
14𝑝𝑠𝑖
1.279𝑖𝑛2 −2.227𝑖𝑛2 +(1.279𝑖𝑛2 +2.227𝑖𝑛2 ) 𝑙𝑛(
2.227𝑖𝑛
)
1.279𝑖𝑛
= 41.9
(5)
𝑎 2 𝑏2
𝑎 2 +𝑏2
) − 𝑟 ] 𝑠𝑖𝑛𝜃
𝑟3
(6)
8.11
6.595
) − 𝑟 ] 𝑠𝑖𝑛𝜃
𝑟3
(7)
𝜎𝜃 = 2𝐴 [3𝑟 − (
𝜎𝜃 = 41.9 [3𝑟 − (
To determine the experimental magnitude, we used the stress optic law, but did not account for
thickness. Therefore, our equation was as shown in Eq. (8).
𝜎𝜃 = 𝑁𝑓𝜎
(8)
Using these equations, plots of magnitude vs. position were generated. The disc was in
compression along the inner diameter and tension along the outer diameter.
The compression data is shown in Error! Reference source not found. and Figure 9.
8
Figure 9:Magnitude vs. position, compression side
The tension data is shown in Error! Reference source not found. and Figure 10.
Figure 10:Magnitude vs. position, tension side
The max stresses for the tension and compression sides are shown in Error! Reference source
not found.. The max fringe order is also shown. For the compression side, the max fringe order was
N=5, while for the tension side, the max fringe order was N=3.
9
Next, an aluminum specimen with a birefringent coating was studied with a reflection
polariscope. A Babinet-Soliel compensator was used to measure fractional fringe order. Measuring
fractional fringe order is necessary to obtain accurate strain readings of locations of interest that do not
coincide with a full fringe. Thus, the compensator was used to observe the strain state at incremental
distances away from a stress concentrator.
In order to determine NQ (fringe order), Eq. (9) was used. The results are tabulated in Appendix
A along with the experimental and theoretical strain differences.
𝑁𝑄 = 2 − (𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑓𝑟𝑖𝑛𝑔𝑒 𝑜𝑟𝑑𝑒𝑟 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡)
(9)
Using the fractional fringe orders, Figure 11 was made. This plot shows how the fringe order,
and therefore strain, decreases rapidly the farther away from the stress concentrator the measurement is
made.
Figure 11: Fringe order vs position measured by B-S compensator
When the compensator gave a negative reading, the fringe orders were bigger than 2; this is only
the case for the first measured fringe. To find the correction factor one must use the following Eq. (10).
ℎ 𝑐 𝐸 𝑐 (1+𝜈𝑠 )
𝐹𝐶𝑅 = [1 + ℎ𝑠 𝐸𝑠 (1+𝜈𝑐)]
( 10 )
Note c stands for coating and s stands for specimen.
0.119∗0.36(1+0.29)
𝐹𝐶𝑅 = [1 + 0.0625∗10(1+0.28) ] = 1.06908
( 11 )
10
Then by using the corrected reinforcement factor, we can find the experimental principal strain
difference from the following Eq. (12).
𝑠
𝑠
𝜀𝑥𝑥
− 𝜀𝑦𝑦
= 𝐹𝐶𝑅 (𝑁𝑄 ∗ 𝐹)
( 12 )
Where F was given to be 633µε. For each fringe order, the experimental strain differences were
calculated and all the values were assembled to form Figure 12.
Figure 12:Experimentally determined strain differences along the position between AB
on the specimen
Next, theoretical values for the strain difference were calculated by first deriving the principal
stresses along AB using the equations for stresses. The resultant equations are Eq. (13) and Eq. (14).
𝜎𝑟𝑟 =
𝜎0
[(1 −
2
𝑎2
3𝑎2
)
(1
+
{
2
𝑟
𝑟2
𝜎𝜃𝜃 =
𝜎0
[(1 +
2
𝑎2
3𝑎 4
) ({1 + 4 } 𝑐𝑜𝑠2𝜃)]
2
𝑟
𝑟
− 1} 𝑐𝑜𝑠2𝜃)]
( 13 )
( 14 )
Where a stands for the radius of the hole, and r stands for the location of the fringe.
These values can translate easily into x and y coordinates since theta is always zero since we want
to know the values along the x axis. So 𝜎𝜃𝜃 translates into 𝜎𝑦𝑦 and 𝜎𝑟𝑟 translates into 𝜎𝑥𝑥 . 𝜖𝑥𝑥 𝑎𝑛𝑑 𝜖𝑦𝑦
can be determined from the generalized Hooke’s law again given in Shukla and Dally as eq. 2.15.
1
𝜈
1
𝜈
𝜖𝑥𝑥 = 𝐸 𝜎𝑥𝑥 − 𝐸 (𝜎𝑦𝑦 + 𝜎𝑧𝑧 )
𝜖𝑦𝑦 = 𝐸 𝜎𝑦𝑦 − 𝐸 (𝜎𝑥𝑥 + 𝜎𝑧𝑧 )
( 15 )
( 16 )
The stresses are from the above calculated values, and the material properties from the specimen
are used in this case to calculate the strains. The specimens Young’s Modulus is 10Msi and the Poisson’s
ratio of the specimen is 0.29.
With these values the theoretical strain difference can be calculated from the following equation:
11
𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝑆𝑡𝑟𝑎𝑖𝑛 𝑑𝑖𝑓𝑓 = |𝜖𝑥𝑥 − 𝜖𝑦𝑦 |
( 17 )
From the calculated values for theoretical strain, the plot in Error! Reference source not found.
was formed below showing the theoretically determined principal strain difference vs. position along path
AB.
Figure 13: Theoretically calculated strain difference along the x axes from the edge of
the hole to the edge of the specimen
DISCUSSION:
Comparing the stress fringe coefficient obtained from the experimental data to literature values,
was unsuccessful. This could likely be due to inaccuracies with the measurement of y, which was done
using pixels from the photo. The stress fringe coefficient obtained was 127 lb/in-fringe. This would put
the material in range of a celluloid material. However, the material was definitely not cellulose based. It
was more likely polycarbonate. Polycarbonate has stress fringe coefficient of 40 lb/in-fringe respectively.
There is a large error associated with this, so the material is not conclusively polycarbonate. For the data
analysis, a stress fringe coefficient of 40 lb/in-fringe was used by the class to analyze the data.
The calculations for stress in the C-shaped specimen had high errors, ranging from 68 to 110
percent. Once again, measurement of the position r was done by counting pixels, which has inherent
error. While quantitatively our results had high error, qualitatively they indicated that the compression
stress increased towards the inner diameter, and the tension stress increased towards the outer diameter.
This is reasonable and can be supported by the fringes seen on the specimen.
Using the birefringent model, we were able to obtain the correct characteristic shape for the plot
of microstrain vs. position. However, there was once again high error in the experimental data. This was
most likely due to inexperience with the Babinet-Soliel compensation device.
12
CONCLUSIONS:
The results obtained from the experimental data largely did not agree with the theoretical
calculations. Qualitatively, plots were obtained that showed representative indicators of what stresses
were experienced by the specimens studied. Large sources of error were attributed to inexperience with
the technique as well as inaccuracies with measurement of experimental data.
The results showed that photoelasticity is powerful analytic tool that has a wide range of
applications. Fringe order can be used to determine a materials stress fringe coefficient. That in turn
could be used to identify an unknown material, and calculated stresses in the material using the stressoptic law. The Babinet-Soliel compensator could be used to measure fractional fringes. This is especially
useful when dealing with small numbers of fringes and obtaining more accurate information about stress
states at specific points. Adding a birefringent coating to a material that has no photoelastic properties,
like aluminum, is a useful way to study specimens like bridge trusses or support beams. Photoelasticity
has many useful applications, and both transmission and reflection polariscopes are powerful analytic
tools.
APPLICATIONS:
The separation by integration of equilibrium equations consists of an integration of the
equilibrium equations for elastic conditions. The equilibrium equations are the partial equations with the
absence of volume forces and applies to the case of plane stress. After integration of the equilibrium
equations and the use of the finite difference approximation, the equation can be written in this way:
𝑗
𝜎𝑥𝑗 = 𝜎𝑥0 − ∑𝑖=1
Δ𝜏𝑥𝑦𝑖
Δ𝑦
Δ𝑥
18
This equation is needed to establish the values of principal stresses and strains. This equation
allows 𝜎𝑥𝑗 to be found at the point of interest. Increases in shear stress Δ𝜏𝑥𝑦𝑖 can be obtained from the
values 𝜏𝑥𝑦𝑖 of the points above and below the lines in Figure A1 of the Appendices. In this figure, the x
and y axes are arbitrarily chosen and the separation is shown as being parallel to the x-axis. To solve the
equation, data on the fringe order and the angle of the principal directions (1, 2 in Figure A1) are needed
for the line of separation and the lines above and below it as well. The stress optic law is also needed to
solve for 𝜎𝑥0 once all of the fringe orders, mechanical properties, specimen specs, and angles of principal
directions are recorded. The stress optic law can be found in the appendices as equation B1.
Once 𝜎𝑥𝑗 is calculated at the point of interest, 𝜎𝑦𝑗 and 𝜏𝑥𝑦𝑗 can be calculated through the stress
optic law and the representation of Mohr’s circle. The isoclinic angle must be used in the evaluation of
the shear stress gradient. This measurement is less precise than the measurement of isochromatic fringe
order. To reduce errors, it was determined that equation 18 could be also expressed as a function of the
difference of principal stresses or principal strains instead of a function of shear stress. This equation can
be viewed in the appendices as equation B2.
13
The advantage of this technique was that it did not require any special instruments for stress
separation. Although, this technique did present a few disadvantages, one being propagation and
accumulation of a number of numerical errors. This rendered the assessment of 𝜎𝑥0 because of the
presence of residual stress as well as the evaluation of the gradient of shear stress. In addition to
numerical errors, another downfall was because it was a lengthy technique besides in the simplest of
cases. The technique was later modified when graphical techniques were used to reduce the time spent on
integration. The technique was also improved after computers were invented and a few algorithms were
developed for the calculations of the stress differences. The most recent improvement was when a wholefield separation technique was implemented, which eliminated the need for pixel on a free boundary for
every integration line.
REFERENCES:
[1]
Fernández, M. Solaguren-Beascoa, J. M Alegre Calderón, P. M Bravo Díez, and I. I Cuesta
Segura. "Stress-separation Techniques in Photoelasticity: A Review." The Journal of Strain
Analysis for Engineering Design 45.1 (2010): 1-17. Print.
14
APPENDIX:
Section A: Figures
Figure A1: A series of points used to analyze and apply the shear difference technique.
Section B: Equations
Stress Optic Law:
𝐸
𝑁 𝜆
0
𝜎𝑥0 = ± 1+𝜈 ( ℎ𝑘
) cos2 𝛼0
𝑗
Difference of Principal Stresses: 𝜎𝑥𝑗 = 𝜎𝑥0 − ∑𝑖=1
Δ(𝜎1𝑖 −𝜎2𝑖 )𝑠𝑖𝑛2𝛼𝑖
Δ𝑥
2Δ𝑦
B1
𝑗
− ∑𝑖=1
(𝜎1𝑖 −𝜎2𝑖 )𝑐𝑜𝑠2𝛼𝑖
Δ𝑦
Δαi Δ𝑥 B2
15
Section C: Sample Calculations
Calculating stress using stress optic law
Experimental:
𝜎𝜃,26.6 = 𝑁𝑓𝜎
𝜎𝜃,26.6 = −2 ∗ 40 = −80𝑝𝑠𝑖
Theoretical:
𝜎𝜃,26.6 = 41.9 [3 ∗ 1.52 − (
8.11
6.595
)
−
] sin(.464)
1.523
1.52
𝜎𝜃,26.6 = −47.0𝑝𝑠𝑖
Percent error:
(80 − 101.6)
𝑃. 𝐸. = 𝑎𝑏𝑠[
] × 100 = 21.2%
101.6
Calculating NQ using B-S compensator
𝑁𝑄 = 2 − 9 − 0.430 = 2.43
Tabulated data from B-S compensation technique:
Fringe
Position
0.5
0.5625
0.625
0.6875
0.75
0.8125
0.875
0.9375
1
Average Recorded Fractional
Fringe Order Measurement
-0.43
NQ
2.43
Experimental Strain
Difference (µε)
1644.447396
0.05
0.385
0.72
0.815
0.85
0.875
0.91
0.97
1.95
1.615
1.28
1.185
1.15
1.125
1.09
1.03
1319.618281
1092.914627
866.2109741
801.9218783
778.236422
761.3182389
737.6327826
697.0291432
Theoretical Strain
Difference (µε)
8.505668528
5.882078302
4.467520446
3.668963955
3.202849023
2.924549501
2.756320911
2.654518701
2.593739356
16
Calculating principle strain difference
𝑠
𝑠
𝜀𝑥𝑥
− 𝜀𝑦𝑦
= 1.06908(1.95 ∗ 633µε) = 1319.62µε
Calculating stress for fringe order 1.95
𝜎𝑟𝑟 =
214𝑝𝑠𝑖
0.5042
0.5042
[(1 −
+
∗
− 1} cos(2 ∗ 0))] = 50.82𝑝𝑠𝑖
)
(1
{3
2
0.56252
0.56252
𝜎𝜃𝜃 =
214𝑝𝑠𝑖
0.5042
0.5044
((1
+
∗
[(1 +
)
{3
}) cos(2 ∗ 0))] = 506.79𝑝𝑠𝑖
2
0.56252
0.56254
Example strain calculations are shown for fringe order 1.95
𝜖𝑥𝑥 =
1
0.29
𝑖𝑛
(0.50679𝑘𝑠𝑖 + 0) = −9.62𝐸 − 6
∗ 0.05082𝑘𝑠𝑖 −
10000𝑘𝑠𝑖
10000𝑘𝑠𝑖
𝑖𝑛
𝜖𝑦𝑦 =
1
0.29
𝑖𝑛
(0.05082𝑘𝑠𝑖 + 0) = 4.92𝐸 − 5
∗ 0.50679𝑘𝑠𝑖 −
10000𝑘𝑠𝑖
10000𝑘𝑠𝑖
𝑖𝑛
Calculation of theoretical strain difference for fringe order 1.95
𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝑆𝑡𝑟𝑎𝑖𝑛 𝑑𝑖𝑓𝑓 = |−9.62𝐸 − 6 − 4.92𝐸 − 5| = 5.88𝜇𝜖
17