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Digital Image Correlation (Old Experiment)

Objective

One object of this experiment is to explore the displacement and strain behavior of structures using the digital image correlation technique. In addition, you will explore the interesting response of polypropylene, a material that exhibits different moduli in tension and compression. The experiment consists of two tests: 1) a four point bending test of a polypropylene specimen from which you will estimate the elastic moduli of polypropylene; and 2) a tension test of a second polypropylene specimen with a hole cut in it, from which you can determine the strain and stress concentrations caused by the hole. The loading for both tests will be accomplished using an Instron load frame.

Background

Digital Image Correlation

In the context of structural testing, Digital Image Correlation (DIC) is a method for tracking the point-wise displacements of a structure (typically a surface of the structure) using a series of images of the structure undergoing deformation. DIC is a non-intrusive measurement technique since nothing has to be mounted to the specimen directly. Furthermore, DIC can measure real structural component geometries in real world conditions. The DIC measurements are primarily limited by image resolution, such that higher resolution images produce more accurate results. Alternatively, a higher-resolution displacement field can be captured by zooming the camera’s field of view to a smaller portion of the specimen of interest. To use DIC, it is usually necessary to prepare the specimen by painting a high-contrast speckle pattern on the surface so that subsequent pre- and post-deformation images can be analyzed to accurately determine the displacement field on the structural surface.

There are some requirements and rules-of-thumb for producing good speckle patterns. First, the pattern should be “random”; a highly organized and repeatable pattern would produce ambiguity in the displacement measurement. Second, the size of the speckles (or high contrast objects) should be roughly the same as (or slightly larger than) a 3x3 pixel region on the digital image for optimal tracking of the displacement. If the speckles are smaller than this, the image magnification can be changed to meet this criterion. Finally, the “density” of speckle features should be sufficient to have an average of 3-4 such features in a 10x10 pixel region. DIC does not rely on correlating a single speckle features, but rather multiple features to get an average displacement in a (multi-pixel) sub-region of the image.

DIC has its origin in speckle imaging approaches used in solid mechanics, and correlation-based analysis methods developed in the 1980’s for object tracking in image processing applications and particle-based velocimetry measurements in fluid mechanics. In fact, DIC is very similar to a common velocity-field measurement approach used in fluid mechanics, Particle Image Velocimetry (PIV). In both DIC and PIV, the individual displacements of many small subregions of an imaged area are obtained by comparing images before and after the displacement has occurred. For each subregion, the “before” (pre) and “after” (post) images are cross-correlated, sometimes using Fast Fourier Transform (FFT) algorithms. The displacement for that subregion is the one that provides the best correlation between the two images. In DIC, the displacement is the desired quantity. In PIV, this displacement is divided by the (short) time between the two images to obtain the local velocity.

This analysis process is typically performed after recording a sequence of images. In DIC, after the displacement field is calculated, the strain field can be determined. The two-dimensional surface displacement field is characterized as u(x,y) and v(x,y), where u and v are the displacements in the x and y directions for a point originally at location (x,y). With u and v determined, we can obtain the surface strain field using the strain-displacement relationships

\large \varepsilon_x=\frac{\partial u}{\partial x}\qquad\varepsilon_y=\frac{\partial v}{\partial y}\qquad \gamma_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}

(1)

Note that differentiating the displacement data amplifies the noise in the data; so advanced analysis software like the package used here $^1$ employ additional processing approaches such as sophisticated smoothing to find the strain field from the displacement data. $^2$

$^1$ In this lab, we will use the Aramis analysis software package.

$^2$ You will also export the Aramis DIC data so you can analyze the full-field data.

Figure 1. The 3D DIC imaging systems, with two cameras mounted in a stereoscopic configurations and two light sources.

DIC (and PIV) systems come in different flavors. For example, the measured displacements of a thin region can be two-dimensional (2D) or three-dimensional (3D). In this lab, we will use a 3D DIC system (shown in Figure 1) that enables us to capture displacements in all three coordinate directions, including out-of-plane deformations. A 3D system provides more information than the more common 2D systems by adding an additional measurement. The 2D system requires only one camera (or equivalently, only camera imaging view point). To capture out-of-plane deflections, the 3D DIC system uses two cameras in a stereoscopic configuration. It is important to reiterate that 3D DIC (and PIV) systems provide three components of displacement (or velocity) from a surface (or thin planar region). There are also volumetric DIC and PIV approaches that provide 3D results for each location within a three-dimensional volume.

Four-Point Flexure Testing

The four-point flexure or bending test is designed to test the flexural response of a slender beam. The goal of the four point bending test is to create a state of pure bending. Pure bending is a stress state where the bending moment is constant and the shear resultant is zero everywhere. The diagram on the left in Figure 2 illustrates a pure bending condition in which only opposing bending moments are applied to either end of the beam. Unfortunately, it is difficult to generate a pure bending moment in a real experiment. Instead, we will use opposing off-set point loads that generate a couple at either end of a beam. This configuration is shown on the right diagram in Figure 2. The advantage of this loading condition is that over the central span there are no shear loads and the beam is subject only to a bending moment.

Figure 2. Idealized pure bending load (left); four point bending test load (right).

Figure 3 illustrates the setup of the test apparatus for the four point bending test. The test specimen is placed on rollers which are placed below and above the specimen. The load frame applies a compressive load to the experimental apparatus which transmits point loads to the top and bottom of the beam. Rollers are used to ensure that simple support conditions are imposed.

Figure 3. Apparatus schematic for a four point bending test

Under ideal conditions in pure bending, the strain in the beam should be linear through the thickness:

(2)

Furthermore, if the material is linear elastic and isotropic, then Hooke's law applies and the bending moment can be calculated as follows:

(3)

where I is the second moment of area of the beam, which is given as

(4)

In Eq. (4), w is the width of the beam, and h is its depth. Therefore, if we impose M through a four point bending test, and can estimate from the digital image correlation results, we can infer the elastic modulus from

(5)

Polypropylene, however, exhibits different elastic moduli under tension and compression. As a result, the neutral surface is not at the geometric centroid of the cross-section and we have to use a composite beam analysis technique to find the elastic modulus under tension and compression. The measured strain will be offset from the geometric centroid, i.e., Eq. (2) has to be modified as follows:

(6)

where b is a strain offset.

(7)

Integrating Eq. (7) yields

(8)

(9)

Next, we know from equilibrium that there cannot be an internal axial load in the beam. Therefore, the axial resultant (N) must be zero:

(10)

(11)

(12)

Eq. (12) can be solved for the modulus under tension, giving

(13)

Inserting this into Eq. (11) provides an expression for the compressive modulus

(14)

Stress Concentrations

Stresses around defects and sudden changes in a structure can be significantly higher than the average stress in the structure. These sharp increases in stress are called stress concentrations. A good example of a stress concentration is the behavior of the stress near a circular hole in a structure subject to uniform tension or compression. For an infinite plate loaded in-plane, the tangential stress around the edge of the hole has an analytic solution

(15)

(16)

(17)

In this lab, you will measure displacements and strains in a polypropylene test specimen (Figure 6) with a circular cutout subject to tension. The DIC system will enable us to visualize the distribution of strains around the hole and observe the stress and strain concentration.

Figure 4. Photograph of test specimen with hole.

Procedure

Specimen Preparation

A critically important step for a DIC experiment is to prepare the specimen's surface and to apply a speckle pattern that the software can interpret and transform into a strain field. Various resources such as Trilion's FAQ and Correlated Solution's technical article detail how best to do this. For this lab, the specimens have already been prepared, so the steps below are for your enlightenment.

Printed label method

If your specimen has a flat surface and good adhesion, self-adhesive labels with pre-printed speckle patterns can be directly applied to the specimen. The following process was followed to generate speckles for this lab:

Download the Correlated Solutions Speckle Generator software.
Convert the PDF to a high-resolution image using Adobe Acrobat Online or similar. This file can also be found below this list.
Open Dymo Connect, set up the Extra Large shipping label in portrait, and import the image from the previous step, scaling it up to full size.
Print labels.
Remove any previous labels and/or any large residues or deposits on the specimen surface.
Degrease the specimen's surface.
Carefully apply the label in the region of interest without folding the edges, and applying firm pressure to aid its adhesion.
Use a sharp knife to trim away the excess label, taking care to not cut away any from the region of interest.

Paint-and-toothbrush method

For certain surfaces, such as those that aren't flat, speckling paint directly onto the specimen is the preferred approach. One such paint application method is using a toothbrush as follows:

Lightly sand and deburr the specimen to remove any manufacturing artifacts. Wipe clean the specimen to remove any oils/residues that may prevent paint adhesion.
Apply a matte white paint base layer to the specimen to remove any reflectivity it inherently has. Do this using spray paint, shaking the can well and applying in multiple thin layers with ample time for each layer to dry before applying the next.
Dip a toothbrush into black paint, tapping off any excess. Having determined the appropriate distance away from the specimen to get the desired particle size beforehand, flick the brush's bristles to create a distribution of speckles over the base layer that will give good results. Consult the DIC manual for recommended particle size and coverage.

Calibration

The 3D stereovision DIC system requires both of its cameras to be calibrated in order to project facets (i.e. speckles) observed in their own frames into real-world co-ordinates. To achieve this, an object with well-known geometry is placed at various positions and orientations in the field-of-view of the cameras, with images then taken for the software to perform a "stereo-triangulation".

Calibration can be a lengthy process, so please work quickly and diligently so as not to run out of time. If the cameras get knocked out of the position or get disorientation at any point during the labs, this calibration will need to be repeated before data can be acquired. Therefore, take great care to not bump or knock the cameras as you move around the Instron.

If you do need to recalibrate the cameras, the TAs will guide you through the process using the prompted procedure in the Aramis software.

DIC System Calibration Procedure

Open the Aramis software from the Desktop
1. Click "File" and "New Project"
2. Select and arbitrary file name for this calibration (the file will not be saved)
3. Select the "3D" radio button
4. Click "Next" 3 times
5. Ensure the Max. deviation is set to 0.300 and then click "Next"
6. Click "Finish"
Enter calibration mode by clicking the black square button (4th from the left near the top of the screen)
1. Select the "CQG1171/55x44" option and click "Next"
2. Ensure the Focal length is 50.00 mm and click "Next"
3. Click Finish. You should now see a screen with two images (left image and right image).
Calibrate the system
1. With the help of the TAs, run through the 13 calibration images with the 55x44 calibration block and the white tilt stand.
  1. It is best to rotate the Instron grips such that the front of the grips is facing the jackscrews when doing calibration
  2. Use the slabs of metal and the instron grip rotation to raise and lower the calibration cube as necessary
  3. For images taken without the tilt stand, ensure the image angles are within +/-0.3 degrees. For images taken with the tilt stand, ensure the images are within +/-5.5 degrees.
  4. You can undo an image but you can only undo the current image. Once you click snap, the previous calibration image is saved and you cannot change it!
  5. You CANNOT undo the last (13th) calibration image!!!!
2. This step may take some time! Be patient. You will have plenty of time to finish your lab but poor calibration will result in poor data!
3. Wait for the calibration result pop-up to appear. Ensure the deviations (both calibration and scale) are relatively small and then click OK.

DIC Calibration Verification Procedure

Delete any previously taken images.
Enter Measurement mode by clicking the blue camera icon (5th from the left on the top of the screen).
Make sure the laser is off and the lights are on.
Take an unloaded reference image by clicking the blue camera icon on the bottom of the screen. This image will save as Stage 0.
Take a second unloaded image by clicking the same blue camera icon on the bottom of the screen. This image will save as Stage 1.
Exit measurement mode by clicking the blue camera icon (5th from the left on the top of the screen).
Mask the area of interest
1. Click on "Stage 0" in the explorer. Define a mask by clicking the blue and white "Define Mask" button on the top of the screen (6th from the left). A "define mask" pop up will show up.
2. Click "Mask all" (4th from the left). The bottom left screen should turn blue.
3. Click "unmask rectangle" (3rd from the right).
4. On the Stage 0 left image (bottom left image on the screen), left click and select the area of interest.
  1. You should select a region such that you capture as much of the speckle pattern as possible and a small amount of dark space. Ask the TAs for guidance if necessary.
    Dogbone specimen: You should capture the hole, speckle region, and a small amount of dark space on the left and right of the specimen.
    4 pt. bending beam: You should capture the speckle region between the inner rollers, and a small amount of dark space above and below the specimen
  2. You will see a green rectangle encompassing the region you select.
5. Right click inside the rectangle and the blue tint should be removed.
6. Click OK in the "Define Mask" pop-up and the pop-up will disappear.
With Stage 0 selected, click "Add a start point" (8th button from the left on the top of the screen)
Hold Control and click somewhere inside the green rectangle near the hole but not in the hole (for the open hole tension specimen) or inside the green rectangle on the speckle pattern (for the 4 point bending specimen).
1. In the pop-up, ensure that the intersection deviation is very low. Ask a TA if you are unsure.
2. Click Next. Click Create. Click Close.
Click "Compute Project" (3rd button from the right on the top of the screen).
1. Click Close once the project is computed and you will see the strain field.
2. Right click on the color legend, and under scaling select 2 sigma.
Observe the resulting surface plot and evaluate the quality of the surface detection. If you are satisfied with your results, move on – if not, assist the TA in re-calibrating the setup and repeat verification until you are satisfied.

Four-Point Bending Test

Table 1. Four-point loading parameters to be used in experiment (defined in Figure 2)

Prepare the specimen:
1. Rotate the Instron grips such that it is facing forward if necessary. You will need 2 people to do this and make sure a TA is supervising this process!
2. Place the 4 point bending fixture into the instron grips. You will need 2 people to do this and make sure a TA is supervising this process!
  1. Grip the bending fixture such that the grip section on the fixture is flush to the back of the clamps. If done correctly, there should be a gap in the front.
  2. It should also be gripped ~3mm below where the grip starts to diverge. See the image below for reference.
3. Locate the beam specimen that has a self-adhesive speckle pattern label applied.
4. Measure the thickness and width of the beam specimen being careful not to touch the speckle pattern with your fingers
5. With the help of a TA, ensure that there is no folding, fading, or any other damage to the speckle pattern. If there is, apply a new speckle pattern sticker to the specimen.
6. With the help of a TA, position the beam in the test fixture using a 12-inch lower support length and a 4-inch upper support length.
  1. Place the rollers in locations to produce the four-point bending parameters listed in Table 1. There are arrows on the fixture itself to depict where the rollers must be placed
  2. Ensure the bottom rollers are on placed below the beam. These 2 rollers are labeled with a "B"
  3. Ensure the top rollers are placed above the beam. These 2 rollers are labeled with a "T"
7. TA Note: If this line starts to become faded, redraw it with the sharpie in the green toolbox at the end of the lab.
8. Have a TA verify the position of the beam by turning on the laser, and using a slip of paper placed flush to the front surface of the beam. If the laser is in the center of the image on the Aramis software, then the beam placement is accurate.
9. Make sure to turn the laser off.
10. Zero the load and displacement on the Instron software.
Calibrate the system:
1. If you have not yet calibrated the cameras, follow the calibration procedure above to calibrate the DIC camera system.
2. If you have already calibrated the system, you may skip this step.
Verify the camera calibration:
1. Follow the camera calibration verification procedure above.
Perform the experiment:
1. Delete any previously-taken images
2. Enter Measurement mode by clicking the blue camera icon (5th from the left on the top of the screen).
3. Make sure the laser is off and the lights are on.
4. Take an unloaded reference image by clicking the blue camera icon on the bottom of the screen. This image will save as Stage 0.
5. Jog the Instron down until you've applied a bending load of 150 lbf.
6. With the specimen under load, inspect the speckle label for any signs of peeling/buckling/distortion. If you observe any, you will need to unload the specimen and re-apply a label according to the earlier instructions before following all previous steps to get back to this step.
7. Take an image of the specimen under load. This image will save as Stage 1.
8. Immediately unload the specimen, noting that the beam will slowly spring back to its natural unloaded position meaning you will need to jog the Instron further than you initially realize.
9. Exit measurement mode by clicking the blue camera icon (5th from the left on the top of the screen).
10. Mask the area of interest
  1. Click on "Stage 0" in the explorer. Define a mask by clicking the blue and white "Define Mask" button on the top of the screen (6th from the left). A "define mask" pop up will show up.
  2. Click "Mask all" (4th from the left). The bottom left screen should turn blue.
  3. Click "unmask rectangle" (3rd from the right).
  4. On the Stage 0 left image (bottom left image on the screen), left click and select the area of interest.
    You should select a region such that you capture as much of the speckle pattern as possible and a small amount of dark space. Ask the TAs for guidance if necessary.
    4 pt. bending beam: You should capture the speckle region between the inner rollers, and a small amount of dark space above and below the specimen
    You will see a green rectangle encompassing the region you select.
  5. Right click inside the rectangle and the blue tint should be removed.
  6. Click OK in the "Define Mask" pop-up and the pop-up will disappear.
11. With Stage 0 selected, click "Add a start point" (8th button from the left on the top of the screen)
12. Hold Control and click somewhere inside the green rectangle on the speckle pattern.
  1. In the pop-up, ensure that the intersection deviation is very low. Ask a TA if you are unsure.
  2. Click Next. Click Create. Click Close.
13. Click "Compute Project" (3rd button from the right on the top of the screen).
  1. Click Close once the project is computed and you will see the strain field.
  2. Right click on the color legend, and under scaling select 2 sigma.
14. Inspect the results for what you would expect to see for this load case. If you do not attain a usable result, try to remedy the situation in this order:
  1. Re-process the existing images with a different start point
  2. Verify lighting, focus, shutter time, with the TAs and re-capture unloaded and loaded images before re-processing
  3. Re-apply a speckle label and re-process
  4. Re-calibrate the cameras and re-process
15. If you are satisfied with your results, remove the specimen from the Instron and export your data.
  1. Click Export > Export All Points > Select "AE3610_01082024_Spring2024_ExportConfig", select your filename and save location, then hit OK.
  2. Make sure the "Selected Points Only" check box is unchecked before exporting data and the proper config file is used!

Open-Hole Tension Test

Prepare the specimen:
1. Rotate the Instron grips such that it is facing sideways if necessary. You will need 2 people to do this and make sure a TA is supervising this process!
2. Locate the open-hole dog-bone specimen that has a self-adhesive speckle pattern label applied.
3. Measure the specimen's thickness, width, and hole diameter.
4. With the help of a TA, ensure that there is no folding, fading, or any other damage to the speckle pattern. If there is, apply a new speckle pattern sticker to the specimen.
5. With the help of a TA, position the specimen in the clamping test fixture.
6. Zero the load and displacement on the Instron software.
Calibrate the system:
1. If you have not yet calibrated the cameras, follow the calibration procedure above to calibrate the DIC camera system.
2. If you have already calibrated the system, you may skip this step.
Verify the camera calibration:
1. Follow the camera calibration verification procedure above.
Perform the experiment:
1. Delete any previously-taken images
2. Enter Measurement mode by clicking the blue camera icon (5th from the left on the top of the screen).
3. Make sure the laser is off and the lights are on.
4. Take an unloaded reference image by clicking the blue camera icon on the bottom of the screen. This image will save as Stage 0.
5. Jog the Instron up until you've applied a tensile load of 2.2 kN.
6. With the specimen under load, inspect the speckle label for any signs of peeling/buckling/distortion. If you observe any, you will need to unload the specimen and re-apply a label according to the earlier instructions before following all previous steps to get back to this step.
7. Take an image of the specimen under load. This image will save as Stage 1.
8. Immediately unload the specimen, noting that an unloaded specimen has a load reading of 0 kN. You may ungrip one side of the specimen to fully unload the specimen.
9. Exit measurement mode by clicking the blue camera icon (5th from the left on the top of the screen).
10. Mask the area of interest
  1. Click on "Stage 0" in the explorer. Define a mask by clicking the blue and white "Define Mask" button on the top of the screen (6th from the left). A "define mask" pop up will show up.
  2. Click "Mask all" (4th from the left). The bottom left screen should turn blue.
  3. Click "unmask rectangle" (3rd from the right).
  4. On the Stage 0 left image (bottom left image on the screen), left click and select the area of interest.
    You should select a region such that you capture as much of the speckle pattern as possible and a small amount of dark space. Ask the TAs for guidance if necessary.
    Dogbone specimen: You should capture the hole, speckle region, and a small amount of dark space on the left and right of the specimen.
    You will see a green rectangle encompassing the region you select.
  5. Right click inside the rectangle and the blue tint should be removed.
  6. Click OK in the "Define Mask" pop-up and the pop-up will disappear.
11. With Stage 0 selected, click "Add a start point" (8th button from the left on the top of the screen)
12. Hold Control and click somewhere inside the green rectangle near the hole but not in the hole.
  1. In the pop-up, ensure that the intersection deviation is very low. Ask a TA if you are unsure.
  2. Click Next. Click Create. Click Close.
13. Click "Compute Project" (3rd button from the right on the top of the screen).
  1. Click Close once the project is computed and you will see the strain field.
  2. Right click on the color legend, and under scaling select 2 sigma.
14. Inspect the results for what you would expect to see for this load case. If you do not attain a usable result, try to remedy the situation in this order:
  1. Re-process the existing images with a different start point
  2. Verify lighting, focus, shutter time, with the TAs and re-capture unloaded and loaded images before re-processing
  3. Re-apply a speckle label and re-process
  4. Re-calibrate the cameras and re-process
15. If you are satisfied with your results, remove the specimen from the Instron and export your data.
  1. Click Export > Export All Points > Select "AE3610_01082024_Spring2024_ExportConfig", select your filename and save location, then hit OK.
  2. Make sure the "Selected Points Only" check box is unchecked before exporting data and the proper config file is used!

Data to be Taken

Thickness and width of the beam, and load value used in the four-point bending test.
Displacement and strain data from the Aramis software for the four-point bending test.
Thickness and width of the specimen, hole diameter, and load value used in the open-hole tension test.
Displacement and strain data from the Aramis software for the open-hole tension test.

Data Reduction

For the open-hole tension test, compute the stresses in your specimen based on the measured strains and the appropriate measured modulus of elasticity for polypropylene.

Results Needed for Data Report

Table of beam dimensions and load value used in the four-point bending test.
Table of specimen dimensions, hole size and load value used in the open-hole tension test.

For the four-point bending test, plots of the axial and transverse normal strain fields as a function of y (with the y-axis as defined in Figure 5), at three axial locations between the two inner load points (i.e., within the constant moment region). One of these should be the center axial location.

Figure 5. Through-thickness stress-strain distribution for material with different moduli in compression and tension.

A single graph containing plots of the tangential stress along a line perpendicular to the length dimension of your specimen that passes through the center of the hole, for each loading.
A single graph containing plots for each of the loadings of the tangential stress (normalized by its value far from the hole) as a function of angle around the hole, close to the edge of hole.

Digital Image Correlation (Old Experiment)

Objective

Background

Digital Image Correlation

\large \varepsilon_x=\frac{\partial u}{\partial x}\qquad\varepsilon_y=\frac{\partial v}{\partial y}\qquad \gamma_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}

(1)

$^1$ In this lab, we will use the Aramis analysis software package.

$^2$ You will also export the Aramis DIC data so you can analyze the full-field data.

Figure 1. The 3D DIC imaging systems, with two cameras mounted in a stereoscopic configurations and two light sources.

Our 3D DIC system, which employs two 5-megapixel cameras, will measure the displacements within a test volume that is centered on the test specimen. While the 3D DIC system provides accurate 3D displacement fields, it requires additional calibration effort compared to a simple, single-camera 2D system. The cameras are calibrated by orienting a thermally balanced plate with calibrated markings on it within the test volume. $^3$

$^3$ The Aramis software will guide you through the calibration.

Four-Point Flexure Testing

Figure 2. Idealized pure bending load (left); four point bending test load (right).

Figure 3. Apparatus schematic for a four point bending test

Under ideal conditions in pure bending, the strain in the beam should be linear through the thickness:

\large \varepsilon_x=-\kappa y

(2)

Furthermore, if the material is linear elastic and isotropic, then Hooke's law applies and the bending moment can be calculated as follows:

\large M=-\int_A\sigma_x ydA=-\int_A E\varepsilon_xydA=\int_A E\kappa y^2dA=EI\kappa

(3)

where I is the second moment of area of the beam, which is given as

\large I=\frac{wh^3}{12}

(4)

E=\frac{M}{I\kappa}

(5)

\large \varepsilon_x=-\kappa y-b

(6)

where b is a strain offset.

The through-thickness stress distribution is shown in Figure 5. The y-location of the neutral surface can be found as $y_N=-b/\kappa$ . The bending moment can be found by a similar integration as that in Eq. (3), except using Eq. (6) for the strain and integrating separately on either side of the neutral surface, i.e.,

\large M=-\int_{-w/2}^{w/2}\int_{-h/2}^{y_N}-E_T(\kappa y^2+by)dydx-\int_{-w/2}^{w/2}\int_{y_N}^{h/2}-E_C(\kappa y^2+by)dydx

(7)

Integrating Eq. (7) yields

\large M=wE_T\left(\kappa\left(\frac{y_N^3}{3}+\frac{h^3}{24}\right)+\frac{b}{2}\left(y_N^2-\frac{h^2}{4}\right)\right)+wE_C\left(\kappa\left(\frac{h^3}{24}-\frac{y_N^3}{3}\right)+\frac{b}{2}\left(\frac{h^2}{4}-y_N^2\right)\right)

(8)

Substituting our expression for $y_N$ in (8) and simplifying gives

\large M=wE_T\kappa\left(\frac{h^3}{24}+\frac{b^3}{6\kappa^3}-\frac{bh^2}{8\kappa}\right)+wE_C\kappa\left(\frac{h^3}{24}-\frac{b^3}{6\kappa^3}+\frac{bh^2}{8\kappa}\right) \\ \space \\ \space \\ =\frac{w}{24}E_T\kappa\left(h-\frac{2b}{\kappa}^2\right)\left(h+\frac{b}{\kappa}\right)+\frac{w}{24}E_C\kappa\left(h+\frac{2b}{k}\right)^2\left(h-\frac{b}{\kappa}\right)

(9)

Next, we know from equilibrium that there cannot be an internal axial load in the beam. Therefore, the axial resultant (N) must be zero:

\large N=0 =\int\sigma_x dA \\ \space \\ =\int_{-w/2}^{w/2}\int_{-h/2}^{y_N}-E_T(\kappa y+b)dydx+\int_{-w/2}^{w/2}\int_{y_N}^{h/2}-E_C(\kappa y+b)dydx

(10)

Integrating Eq. (10) and replacing $y_N$ with $-b/\kappa$ yields

\large 0=E_T\left(\frac{2b}{\kappa}-h\right)^2-E_C\kappa\left(\frac{2b}{\kappa}+h\right)^2

(11)

Now combining Eq. (11) and Eq. (9) for the bending moment, eliminating __ $E_C$ and using our expression for the 2nd moment of area, Eq. (4), results in

\large M=\frac{wh}{12}E_T\kappa\left(h-\frac{2b}{\kappa}\right)^2=E_T\kappa I\left(1-\frac{2b}{h\kappa}\right)^2

(12)

Eq. (12) can be solved for the modulus under tension, giving

\large E_T=\frac{M}{I\kappa\left(1-\frac{2b}{\kappa h}\right)^2}

(13)

Inserting this into Eq. (11) provides an expression for the compressive modulus

\large E_C=\frac{M}{I\kappa\left(1+\frac{2b}{\kappa h}\right)^2}

(14)

Stress Concentrations

\large\sigma_{\theta\theta}=\sigma_{av}(1-2\cos2\theta)

(15)

where the coordinate θ runs circumferentially around the circular hole and $\sigma_{av}$ is the average stress in the specimen far away from the hole. The direction θ=0 is aligned with the loading direction, or the x-axis in our geometry. The maximum value of the stress (due to the hole) normalized by $\sigma_{av}$ is known as the stress concentration factor.

The stresses in planar cylindrical coordinates (i.e., $\sigma_{rr}$ , $\sigma_{\theta\theta}$ , and $\sigma_{r\theta}$ ) can be calculated from Cartesian stresses (i.e., $\sigma_{x}$ , $\sigma_{y}$ and $\gamma_{xy}$ ) using standard coordinate transformations, for example

\large \sigma_{rr}=\sigma_x\cos^2\theta+\sigma_y\sin^2\theta+\sigma_{xy}2\sin\theta\cos\theta

(16)

\large \sigma_{\theta\theta}=\sigma_x\sin^2\theta+\sigma_y\cos^2\theta-\sigma_{xy}2\sin\theta\cos\theta

(17)

Figure 4. Photograph of test specimen with hole.

Procedure

Specimen Preparation

Printed label method

Download the Correlated Solutions Speckle Generator software.
Apply the following settings before exporting to PDF (the geometry can change, but this is set up specifically for Dymo Extra Large shipping labels using a LabelWrite 4XL thermal printer):
Convert the PDF to a high-resolution image using Adobe Acrobat Online or similar. This file can also be found below this list.
Open Dymo Connect, set up the Extra Large shipping label in portrait, and import the image from the previous step, scaling it up to full size.
Print labels.
Remove any previous labels and/or any large residues or deposits on the specimen surface.
Degrease the specimen's surface.
Carefully apply the label in the region of interest without folding the edges, and applying firm pressure to aid its adhesion.
Use a sharp knife to trim away the excess label, taking care to not cut away any from the region of interest.

Paint-and-toothbrush method

For certain surfaces, such as those that aren't flat, speckling paint directly onto the specimen is the preferred approach. One such paint application method is using a toothbrush as follows:

Lightly sand and deburr the specimen to remove any manufacturing artifacts. Wipe clean the specimen to remove any oils/residues that may prevent paint adhesion.
Apply a matte white paint base layer to the specimen to remove any reflectivity it inherently has. Do this using spray paint, shaking the can well and applying in multiple thin layers with ample time for each layer to dry before applying the next.
Dip a toothbrush into black paint, tapping off any excess. Having determined the appropriate distance away from the specimen to get the desired particle size beforehand, flick the brush's bristles to create a distribution of speckles over the base layer that will give good results. Consult the DIC manual for recommended particle size and coverage.

Calibration

If you do need to recalibrate the cameras, the TAs will guide you through the process using the prompted procedure in the Aramis software.

DIC System Calibration Procedure

Open the Aramis software from the Desktop
1. Click "File" and "New Project"
2. Select and arbitrary file name for this calibration (the file will not be saved)
3. Select the "3D" radio button
4. Click "Next" 3 times
5. Ensure the Max. deviation is set to 0.300 and then click "Next"
6. Click "Finish"
Enter calibration mode by clicking the black square button (4th from the left near the top of the screen)
1. Select the "CQG1171/55x44" option and click "Next"
2. Ensure the Focal length is 50.00 mm and click "Next"
3. Click Finish. You should now see a screen with two images (left image and right image).
Calibrate the system
1. With the help of the TAs, run through the 13 calibration images with the 55x44 calibration block and the white tilt stand.
  1. It is best to rotate the Instron grips such that the front of the grips is facing the jackscrews when doing calibration
  2. Use the slabs of metal and the instron grip rotation to raise and lower the calibration cube as necessary
  3. For images taken without the tilt stand, ensure the image angles are within +/-0.3 degrees. For images taken with the tilt stand, ensure the images are within +/-5.5 degrees.
  4. You can undo an image but you can only undo the current image. Once you click snap, the previous calibration image is saved and you cannot change it!
  5. You CANNOT undo the last (13th) calibration image!!!!
2. This step may take some time! Be patient. You will have plenty of time to finish your lab but poor calibration will result in poor data!
3. Wait for the calibration result pop-up to appear. Ensure the deviations (both calibration and scale) are relatively small and then click OK.

DIC Calibration Verification Procedure

Delete any previously taken images.
Enter Measurement mode by clicking the blue camera icon (5th from the left on the top of the screen).
Make sure the laser is off and the lights are on.
Take an unloaded reference image by clicking the blue camera icon on the bottom of the screen. This image will save as Stage 0.
Take a second unloaded image by clicking the same blue camera icon on the bottom of the screen. This image will save as Stage 1.
Exit measurement mode by clicking the blue camera icon (5th from the left on the top of the screen).
Mask the area of interest
1. Click on "Stage 0" in the explorer. Define a mask by clicking the blue and white "Define Mask" button on the top of the screen (6th from the left). A "define mask" pop up will show up.
2. Click "Mask all" (4th from the left). The bottom left screen should turn blue.
3. Click "unmask rectangle" (3rd from the right).
4. On the Stage 0 left image (bottom left image on the screen), left click and select the area of interest.
  1. You should select a region such that you capture as much of the speckle pattern as possible and a small amount of dark space. Ask the TAs for guidance if necessary.
    Dogbone specimen: You should capture the hole, speckle region, and a small amount of dark space on the left and right of the specimen.
    4 pt. bending beam: You should capture the speckle region between the inner rollers, and a small amount of dark space above and below the specimen
  2. You will see a green rectangle encompassing the region you select.
5. Right click inside the rectangle and the blue tint should be removed.
6. Click OK in the "Define Mask" pop-up and the pop-up will disappear.
With Stage 0 selected, click "Add a start point" (8th button from the left on the top of the screen)
Hold Control and click somewhere inside the green rectangle near the hole but not in the hole (for the open hole tension specimen) or inside the green rectangle on the speckle pattern (for the 4 point bending specimen).
1. In the pop-up, ensure that the intersection deviation is very low. Ask a TA if you are unsure.
2. Click Next. Click Create. Click Close.
Click "Compute Project" (3rd button from the right on the top of the screen).
1. Click Close once the project is computed and you will see the strain field.
2. Right click on the color legend, and under scaling select 2 sigma.
Observe the resulting surface plot and evaluate the quality of the surface detection. If you are satisfied with your results, move on – if not, assist the TA in re-calibrating the setup and repeat verification until you are satisfied.

Four-Point Bending Test

Table 1. Four-point loading parameters to be used in experiment (defined in Figure 2)

Prepare the specimen:
1. Rotate the Instron grips such that it is facing forward if necessary. You will need 2 people to do this and make sure a TA is supervising this process!
2. Place the 4 point bending fixture into the instron grips. You will need 2 people to do this and make sure a TA is supervising this process!
  1. Grip the bending fixture such that the grip section on the fixture is flush to the back of the clamps. If done correctly, there should be a gap in the front.
  2. It should also be gripped ~3mm below where the grip starts to diverge. See the image below for reference.
  3. Ensure that the front of the fixture is facing forward--The side labeled "Front"
3. Locate the beam specimen that has a self-adhesive speckle pattern label applied.
4. Measure the thickness and width of the beam specimen being careful not to touch the speckle pattern with your fingers
5. With the help of a TA, ensure that there is no folding, fading, or any other damage to the speckle pattern. If there is, apply a new speckle pattern sticker to the specimen.
6. With the help of a TA, position the beam in the test fixture using a 12-inch lower support length and a 4-inch upper support length.
  1. Place the rollers in locations to produce the four-point bending parameters listed in Table 1. There are arrows on the fixture itself to depict where the rollers must be placed
  2. Ensure the bottom rollers are on placed below the beam. These 2 rollers are labeled with a "B"
  3. Ensure the top rollers are placed above the beam. These 2 rollers are labeled with a "T"
  4. Ensure the letter "F" on all 4 rollers is facing the cameras. Furthermore, ensure all 4 rollers are flush to the front surface of the fixture. See the image below
  5. Place the beam on the rollers such that the front face of the beam is resting just behind the line drawn on the rollers. Use the lines on the bottom rollers to do this. See the image below
7. TA Note: If this line starts to become faded, redraw it with the sharpie in the green toolbox at the end of the lab.
8. Have a TA verify the position of the beam by turning on the laser, and using a slip of paper placed flush to the front surface of the beam. If the laser is in the center of the image on the Aramis software, then the beam placement is accurate.
9. Make sure to turn the laser off.
10. Zero the load and displacement on the Instron software.
Calibrate the system:
1. If you have not yet calibrated the cameras, follow the calibration procedure above to calibrate the DIC camera system.
2. If you have already calibrated the system, you may skip this step.
Verify the camera calibration:
1. Follow the camera calibration verification procedure above.
Perform the experiment:
1. Delete any previously-taken images
2. Enter Measurement mode by clicking the blue camera icon (5th from the left on the top of the screen).
3. Make sure the laser is off and the lights are on.
4. Take an unloaded reference image by clicking the blue camera icon on the bottom of the screen. This image will save as Stage 0.
5. Jog the Instron down until you've applied a bending load of 150 lbf.
6. With the specimen under load, inspect the speckle label for any signs of peeling/buckling/distortion. If you observe any, you will need to unload the specimen and re-apply a label according to the earlier instructions before following all previous steps to get back to this step.
7. Take an image of the specimen under load. This image will save as Stage 1.
8. Immediately unload the specimen, noting that the beam will slowly spring back to its natural unloaded position meaning you will need to jog the Instron further than you initially realize.
9. Exit measurement mode by clicking the blue camera icon (5th from the left on the top of the screen).
10. Mask the area of interest
  1. Click on "Stage 0" in the explorer. Define a mask by clicking the blue and white "Define Mask" button on the top of the screen (6th from the left). A "define mask" pop up will show up.
  2. Click "Mask all" (4th from the left). The bottom left screen should turn blue.
  3. Click "unmask rectangle" (3rd from the right).
  4. On the Stage 0 left image (bottom left image on the screen), left click and select the area of interest.
    You should select a region such that you capture as much of the speckle pattern as possible and a small amount of dark space. Ask the TAs for guidance if necessary.
    4 pt. bending beam: You should capture the speckle region between the inner rollers, and a small amount of dark space above and below the specimen
    You will see a green rectangle encompassing the region you select.
  5. Right click inside the rectangle and the blue tint should be removed.
  6. Click OK in the "Define Mask" pop-up and the pop-up will disappear.
11. With Stage 0 selected, click "Add a start point" (8th button from the left on the top of the screen)
12. Hold Control and click somewhere inside the green rectangle on the speckle pattern.
  1. In the pop-up, ensure that the intersection deviation is very low. Ask a TA if you are unsure.
  2. Click Next. Click Create. Click Close.
13. Click "Compute Project" (3rd button from the right on the top of the screen).
  1. Click Close once the project is computed and you will see the strain field.
  2. Right click on the color legend, and under scaling select 2 sigma.
14. Inspect the results for what you would expect to see for this load case. If you do not attain a usable result, try to remedy the situation in this order:
  1. Re-process the existing images with a different start point
  2. Verify lighting, focus, shutter time, with the TAs and re-capture unloaded and loaded images before re-processing
  3. Re-apply a speckle label and re-process
  4. Re-calibrate the cameras and re-process
15. If you are satisfied with your results, remove the specimen from the Instron and export your data.
  1. Click Export > Export All Points > Select "AE3610_01082024_Spring2024_ExportConfig", select your filename and save location, then hit OK.
  2. Make sure the "Selected Points Only" check box is unchecked before exporting data and the proper config file is used!

Open-Hole Tension Test

Prepare the specimen:
1. Rotate the Instron grips such that it is facing sideways if necessary. You will need 2 people to do this and make sure a TA is supervising this process!
2. Locate the open-hole dog-bone specimen that has a self-adhesive speckle pattern label applied.
3. Measure the specimen's thickness, width, and hole diameter.
4. With the help of a TA, ensure that there is no folding, fading, or any other damage to the speckle pattern. If there is, apply a new speckle pattern sticker to the specimen.
5. With the help of a TA, position the specimen in the clamping test fixture.
6. Zero the load and displacement on the Instron software.
Calibrate the system:
1. If you have not yet calibrated the cameras, follow the calibration procedure above to calibrate the DIC camera system.
2. If you have already calibrated the system, you may skip this step.
Verify the camera calibration:
1. Follow the camera calibration verification procedure above.
Perform the experiment:
1. Delete any previously-taken images
2. Enter Measurement mode by clicking the blue camera icon (5th from the left on the top of the screen).
3. Make sure the laser is off and the lights are on.
4. Take an unloaded reference image by clicking the blue camera icon on the bottom of the screen. This image will save as Stage 0.
5. Jog the Instron up until you've applied a tensile load of 2.2 kN.
6. With the specimen under load, inspect the speckle label for any signs of peeling/buckling/distortion. If you observe any, you will need to unload the specimen and re-apply a label according to the earlier instructions before following all previous steps to get back to this step.
7. Take an image of the specimen under load. This image will save as Stage 1.
8. Immediately unload the specimen, noting that an unloaded specimen has a load reading of 0 kN. You may ungrip one side of the specimen to fully unload the specimen.
9. Exit measurement mode by clicking the blue camera icon (5th from the left on the top of the screen).
10. Mask the area of interest
  1. Click on "Stage 0" in the explorer. Define a mask by clicking the blue and white "Define Mask" button on the top of the screen (6th from the left). A "define mask" pop up will show up.
  2. Click "Mask all" (4th from the left). The bottom left screen should turn blue.
  3. Click "unmask rectangle" (3rd from the right).
  4. On the Stage 0 left image (bottom left image on the screen), left click and select the area of interest.
    You should select a region such that you capture as much of the speckle pattern as possible and a small amount of dark space. Ask the TAs for guidance if necessary.
    Dogbone specimen: You should capture the hole, speckle region, and a small amount of dark space on the left and right of the specimen.
    You will see a green rectangle encompassing the region you select.
  5. Right click inside the rectangle and the blue tint should be removed.
  6. Click OK in the "Define Mask" pop-up and the pop-up will disappear.
11. With Stage 0 selected, click "Add a start point" (8th button from the left on the top of the screen)
12. Hold Control and click somewhere inside the green rectangle near the hole but not in the hole.
  1. In the pop-up, ensure that the intersection deviation is very low. Ask a TA if you are unsure.
  2. Click Next. Click Create. Click Close.
13. Click "Compute Project" (3rd button from the right on the top of the screen).
  1. Click Close once the project is computed and you will see the strain field.
  2. Right click on the color legend, and under scaling select 2 sigma.
14. Inspect the results for what you would expect to see for this load case. If you do not attain a usable result, try to remedy the situation in this order:
  1. Re-process the existing images with a different start point
  2. Verify lighting, focus, shutter time, with the TAs and re-capture unloaded and loaded images before re-processing
  3. Re-apply a speckle label and re-process
  4. Re-calibrate the cameras and re-process
15. If you are satisfied with your results, remove the specimen from the Instron and export your data.
  1. Click Export > Export All Points > Select "AE3610_01082024_Spring2024_ExportConfig", select your filename and save location, then hit OK.
  2. Make sure the "Selected Points Only" check box is unchecked before exporting data and the proper config file is used!

Data to be Taken

Thickness and width of the beam, and load value used in the four-point bending test.
Displacement and strain data from the Aramis software for the four-point bending test.
Thickness and width of the specimen, hole diameter, and load value used in the open-hole tension test.
Displacement and strain data from the Aramis software for the open-hole tension test.

Data Reduction

Using a best linear fit for the appropriate region of your $\varepsilon_x(y)$ plots from the four-point bending test, compute the slope ( $\kappa$ ) and intercept (b) relative to a coordinate axis centered on your specimen.
From your experimental data, determine the tensile and compressive moduli ( $E_T$ and $E_C$ ) for polypropylene.
From your plots of transverse normal strain $\varepsilon_y$ divided by the axial strain $\varepsilon_x$ , determine a value for Poisson’s ratio ( $\nu$ ) for polypropylene.
For the open-hole tension test, compute the stresses in your specimen based on the measured strains and the appropriate measured modulus of elasticity for polypropylene.

Results Needed for Data Report

Table of beam dimensions and load value used in the four-point bending test.
Table of specimen dimensions, hole size and load value used in the open-hole tension test.
“Image graphs” or “full-field color plots” $^1$ of the axial (horizontal) and transverse (vertical) displacement field for the field of view analyzed by the Aramis software in the bending test. Choose your color scaling wisely to accentuate any important gradients. Be sure to also include color bars showing how your colors map to the values of displacement.

$^1$ Here an “image” means a false-color image, where each displacement (or strain) measurement location is like a pixel in the false-color image, and the color of the pixel corresponds to the value being shown (axial displacement in this case). You can do this, for example, with the image function in Matlab

Images of $\varepsilon_x$ , $\varepsilon_y$ and $\gamma_{xy}$ for the field of view imaged and analyzed by the Aramis software in the bending test, with color bars.
For the four-point bending test, plots of the axial and transverse normal strain fields as a function of y (with the y-axis as defined in Figure 5), at three axial locations between the two inner load points (i.e., within the constant moment region). One of these should be the center axial location.

Figure 5. Through-thickness stress-strain distribution for material with different moduli in compression and tension.

Plot of the transverse normal strain $\varepsilon_y$ divided by the axial strain $\varepsilon_x$ along the center line used for Result 5.
Table of measured values of $\kappa$ , b, and the polypropylene properties $E_T$ , $E_C$ , and $\nu$ . Include in your table the published values for the polypropylene properties (include your source for those values).
Images of the axial strain ( $\varepsilon_x$ ) and transverse strain ( $\varepsilon_y$ ) fields for the region analyzed by the Aramis software in the open-hole tension test, with color bars, for both loadings.
Images of the axial ( $\sigma_x$ ) and transverse stress ( $\sigma_y$ ) fields based on your measured strains and measured modulus of polypropylene, with color bars, for both loadings.
Images of the normal radial stress ( $\sigma_{rr}$ ) and normal tangential stress ( $\sigma_{\theta\theta}$ ) fields for the open-hole tension tests, with color bars, for both loadings.
A single graph containing plots of the tangential stress along a line perpendicular to the length dimension of your specimen that passes through the center of the hole, for each loading.
A single graph containing plots for each of the loadings of the tangential stress (normalized by its value far from the hole) as a function of angle around the hole, close to the edge of hole.