-- Main.thaox385 - 14 May 2014

Measuring Young’s Modulus in an Aluminum bar Using the Double Exposure Technique

Gina Thao and Cong Chen

Methods of Experimental Physic 2014

Abstract

We measure the Young’s Modulus of an aluminum bar by using holographic interferometry at different applied weights. Holographic interferometry can be used in the testing of stress and strain evaluation of a material [1]. This will allow us to measure the distance, Δy, created by the disturbance of the force applied to the aluminum bar. Our final results come to be about (621 ± 1.7) GPa for 2g and (1533±1.7) GPa for 100g compared to the accepted value, 69GPa. We are over by 8σ and 21σ respectively.

Introduction

<a name="Introduction"></a> Holography is the process by which the amplitude and phase variation across a wavefront recorded and subsequently reproduced. These images that are created by Holography are called holograms. Holograms are created by using the double-exposure interferometry. Double-exposure interferometry is the measurement of the same object twice, once with a reference point and then exposing it again after applying a stress to the object creating a deformation in the image. To create holograms we will be using a diode laser and shine it through a holographic plate. A holographic plate is a diffraction grating tool that separates different wavelengths of light to a high precision. The technique of holographic interferometry allows us to take precious measurement on the order of 100nm.

As mentioned, double-exposure interferometry allows us to see the diffractions created by the stress applied. After the initial exposure on to the plates, we expect to see a change in the images on the plate from the reference point to the change after the applied stress. This displacement between the two will result in fringes and can only be seen after the exposure. Without holography, we are not able to see this change this is the motivation for using such a technique.

Specifically, for our experiment we will be measuring the stiffness in an elastic isotope material. Elastic materials are materials that return to its original state after stress levels are applied which can be characterized by Young’s modulus. Young’s modulus is the quantity that describes a materials response internally to the amount of stress applied normal to opposite faces [2].

Theory

A fringe pattern will appear on the obtained hologram where the separate reconstructed wavefronts interfere. Light fringes will appear where the wavefronts interfered constructively; this occurs when the displacement of the waves is an integral number of wavelengths. Dark fringes will appear where the wavefronts interfered destructively; this occurs when the displacement of the waves is an integral number of half wavelengths [3].

The reconstructed fringe pattern on the holographic plate follows Bragg’s Law of diffraction

where n is an integer, λ is the wavelength of incident wave, d is the spacing between the planes, and θ is the angle between the incident ray and the scattering planes. In our experiment, double-exposure interferometry will be used to measure the vertical displacement of a metal beam due to applied mass.

This shows the two stages of exposure one at the reference point and the other at the bending point. The angle is the angle produced by the incident of light and angle is the angle produced by the reflected light from the beam [3]

From figure above, the optical path length when the beam is bent, is longer than that of the non-bent. Since is much smaller than the length of the beam we can take the assumption that the vertical displacement is always perpendicular to the bar and the change of angle and is negligible. The distance between the reference point and bending point. It is what we are looking for Δy to determine the Young's modulus. Δy can be represented as shown below.

The determination of the bending displacement comes from the property of the bent beam and the stress.

. The theoretical equation for a beam bending under stress shows the displacement as:

where E is Young’s modulus for the material in one orientation, I is the moment of area, is the weight of the applied mass, is the total weight of the bar, l is the length of the bar, x is the distance measured from the edge of the clamp to the fringe [3]. In our experiment, we are only interested in the displacement produced by the applied mass, therefore can be ignored (6) becomes

Finally the Young’s modulus of the material we want to measure is given

Apparatus

For our experimental set-up we place a laser about 35 to 40cm away from the holographic plate and the aluminum bar that is 31cm x 2.5cm x 0.5cm. The distance from the plate to the laser is to account for the whole bar under the plate to be fully illuminated by the laser light. The plate will be place on top of the two table clamps place next to the bar so that the bar and the plate won’t be in complete contact. Figure 2 shows the general schematic of the experiment. The diagram shows the plate being laid on top of the aluminum bar with a laser light beaming at the plate at a distance and angle. This kind of schematic is called contact copy. Contact copy gets rid of additional vibration if there is no movement between the bar and the plate [4]. The beam on the bar will exert an approximate length of 6cm with a width of 2.5cm.

This is a general schematic of our experimental set-up, with the laser at a fix angle on both images and the plate on top of the bar on the table clamps. The left image is the set-up with no weights added (initial point) and on the right has the added weights (bend point).

Results

These are the images that we obtained. The one on the left is for the 2g and the one on the right is what we obtained for 100g. Here the 2g's interferece patterens are more prominent than those of the 100g's.

From the two samples that we obtained, with weights 2g and 100g, we use imageJ to find the distances between each interference pattern. Which we need to find in order to solve for Δy in our (2). From the graph given from imageJ we can see that each peak represents the distance between each interference pattern. Using the list of data points imageJ provides and plotting the points into excel we can approximate where each peak is. This gives us a value of about 36 peaks for the 2g and 29 peaks for the 100g. The 2g sample gave us a clear image of the interference pattern while the 100g wasn’t that clear which only allowed us to get a certain amount of data from that sample. For the 2g we obtain about 1566 data points and for the 100g about 208 data points. In addition to the differences of the observable data collection we can clearly see that the peaks in the 2g are more pronounce vs the ones in the 100g as shown below.

The image on the left is the graph of the 2g produce from imageJ and the one on the right is the graph for the 100g. As shown above the 2g have more pronounced peaks vs the 100g.

Plotting a graph of our peaks and the distance between each peak allows us to see if we are missing a peak. After finding those peaks we find the values of the angle α and Δy. Angle β is negligible since it is the reflected light from the laser we set it as 0 in (5). Our value of Δy are (5.8503±0.0004) m for 2g and (4.7125±0.0003) m for 100g. From Δy we can use (8) to find E. Our E comes out to be (621 ± 1.7) GPa for 2g and (1533±1.7) GPa for 100g. Our values compare to the accepted values are over by 8σ and 21σ. From this we can conclude that when using smaller weights will deviate less from the accepted value than using bigger weights.

Conclusion

We had a lot of troubles creating the holograms. Once we were able to create the 2g, we were not able to recreate the image. It baffled us for a while but we couldn’t figure out what was wrong. We have tried varying some variables that we thought could contribute to destroying the interference patterns. First we tried to lessen the vibration by standing a couple of feet away from the table when the plate is expose to the laser and also tried not to breath too much near the experiment; since it is very sensitive to vibration. After a couple of failed trials after the variation we tried to vary the exposure time from 10s to 5s, 3s, 15s but that still didn’t work. By the luck of making a mistake when we were adding the 100g weight and letting it settle before we expose the light we then found out that, that was our mistake. We let it settle for too long the vibration from the air or little movements might have come back.

Other contributing factors could be that we were unable to collect more data and hence do not have an accurate value for Young’s modulus. For other MXP students that would like to do this experiment they might want to study the reason why smaller weights deviate less than larger weights.

Acknowledgements

We want to thank Kurt Wick and Joe Sobek without them we couldn’t have the project and have successful data.

References

• 1. Fein, Howard. "Holographic Interferometry : Nondestructive Tool." The Industrial Physicist., 1997. 37-39. The Industrial Physicist. American Institute of Physics. Web. 9 May 2014. < http://www.aip.org/tip/INPHFA/vol-3/iss-3/p37.pdf >.

  2. Glen Elbert.Web. 9 Mar 2014. < http://physics.info/elasticity/ >.

  3. http://mxp.physics.umn.edu/s07/projects/s07_holographicbeambending/theory.htm

  4. http://www.integraf.com/a-simple_holography.htm

  5. Wetter , Marc, and Ian Campbell. "Double Exposure Holographic Interferometry." mxp.physics.umn.edu. N.p., 9 May 2006. Web. 9 Mar 2014.< http://mxp.physics.umn.edu/s06/Projects/S06_Hologram/ >.