Experiment #1 EGR 320L Frank Jamison September 11, 2013 Modulus of Elasticity of Materials Derived Through Tensile Testing and the Maximum Weight That Can Be Supported by iPod Earbud Wires Introduction The purpose of this experiment is to see what maximum weight can be applied to a set of ipod earbuds befoe they break and to compare it to three metal samples (two aluminum and one steel) and two plastic samples (both polycarbonate plastic) to see how a composite of the two materials might stand up. By applying a measured amount of force to a sample of known dimensions, we can calculate the stress (σ=F/A)[1] which is the applied force divided by the cross‐sectional area of the sample, and the strain (ε=δ/L)[2] which is the elongation of the sample divided by its original length. From there we can use Hooke's Law to determine the modulus of elasticity of the material. Hooke's Law, stated in terms of stress and strain, states that the stress on a material is equal to its modulus of elasticity multiplied by the strain on the material while that material is not deformed beyond its ability to resume its original shape (σ = Eε)[2]. Since stress is proportional to strain, we know that the elastic range will be represented by a straight line on the graph. The slope of the line created in the elastic range by plotting the stress as a function of the strain will be our experimental determination of the modulus of elasticity of the material (E=Δσ/Δε). Experimental Procedure The samples of aluminum, steel, polycarbonate plastic, and the thick and thin wire portions of a pair of ipod earbuds was placed into an Admet tensile testing machine at National University. For the metal tests, a dog bone shaped sample was used in order to concentrate the tensile force in a measurable area measuring 80 mm x 4 mm x 0.762 mm. The samples were then pulled at a constant rate while the applied force and elongation of the materials were measured. The plastic samples used were of a similar shape, but were about five times thicker. The measurements of the test area of the dog bone shaped samples were 80 mm x 4 mm x 0.381 mm. For the wire samples an 80 mm length of each part of the earbud set, both the thick wire section form the ipod jack to the y junction and the thin wire section from the y junction to the actual earbud, were placed into the Admet machine with a clamp holding each end. The samples were subjected to an increasing tensile force until they broke. As the samples were being pulled apart, the elongation and the force being applied were measured. Data pairs of force and 39 0.26 0.02 1. The first four tables show the measured dimensions of the samples used in the experiment.99 54.50 52.26 Force (N) 50. and each group of samples tested are plotted on a single graph.22 8.48 30.89 17.35 .30 0.37 53.62 Force (N) 0.61 41.0 mm Thin Wire Sample Dimensions Length: 80 mm Diameter: 1. Data The following tables and graphs show the raw data collected for the experiments. The additional tables and graphs are the readings from the Admet equipment used in the tensile testing of each sample.80 0.29 0.28 54.0 mm Thickness: 0. plastics.00 0.95 34. and least squares linear regression was used to determine the slope of the line that best fit the data. Tables are broken down into groups of metals.61 Elongation (mm) 0. Aluminum and Steel Sample Dimensions Length: 80 mm Width: 4. and wires. The slope of this best fit line was the experimental determination of the Young's Modulus for each material.94 37.14 1.00 45.56 0.56 52.90 0.0 mm Thickness: 0.5 mm Aluminum Sample 1 Data Elongation (mm) 0.70 0.00 5.83 53.45 0.69 53.10 54.07 1.elongation (25 pairs for steel and plastic and 20 pairs for aluminum and the wires) were then used to calculate the stress and strain on the samples.93 47.41 0.29 51.0762 mm Polycarbonate Plastic Sample Dimensions Length: 80 mm Width: 4.95 1.19 1. A stress/strain graph was then used to identify the elastic range of the material.50 0.381 mm Thick Wire Sample Dimensions Length: 80 mm Diameter: 2.84 0. 20 0.06 Elongation (mm) 0.82 Force (N) 0.00 .38 240.47 60.28 50.26 52.33 0.39 104.15 0.70 226.53 0.09 0.09 0.75 51.70 0.38 Force (N) 0.36 Steel Sample Data Elongation (mm) 0.29 43.81 238.57 28.85 18.60 0.31 119.66 52.06 0.70 0.56 45.46 32.04 1.60 0.25 0.15 1.21 0.00 5.23 0.76 0.46 52.66 50.15 0.17 0.97 1.59 152.50 19.77 52.68 133.30 0.42 0.45 52.32 1.17 240.69 1.85 0.00 0.84 52.93 1.29 0.11 0.56 1.32 197.32 1.13 0.25 211.44 0. Aluminum Sample 2 Data Elongation (mm) 0.24 0.06 33.09 70.00 23.09 Elongation (mm) 0.17 0.86 145.47 0.65 Force (N) 48.72 52.15 87.98 229.79 39.75 193.61 240.51 0.00 0.66 172.27 95.36 0.91 Force (N) 166.21 1.77 34.03 0.16 240. 30 35.00 0.09 87.19 74.60 76.18 45.00 0.00 0.03 14.55 76.98 20.93 76.16 40.06 41.72 61.92 64.19 69.52 72.17 1.00 Tensile Force F (N) 250.64 1.Aluminum and Steel Tensile Tests 300.22 Force (N) 85.26 .50 2.94 2.50 Elongation δ (mm) Polycarbonate Plastic Sample 1 Data Elongation (mm) Force (N) 0.93 1.72 1.44 74.54 74.95 77.98 56.00 1.07 81.00 Aluminum Sample 2 100.40 51.56 75.86 0.26 Elongation (mm) 7.44 54.34 30.27 89.50 1.55 5.00 2.00 0.24 50.00 Steel Sample 50.00 0.70 3.85 4.13 25.90 75.55 1.84 32.00 Aluminum Sample 1 150.42 76.41 10.21 45.92 2.59 23.86 78.00 200. 169 11.17 2.262 68.282 55.160 45.543 74.575 89.00 0.613 20.77 78.21 75.69 75.70 74.00 50.053 9.94 2.11 83.00 10.696 8.449 34.00 5.54 76.30 76.324 Force (N) 88.961 23.00 10.77 1.00 Elongation δ (mm) 40.00 Sample 2 30.78 74.128 81.91 74.370 50.07 3.00 60.Polycarbonate Plastic Sample 2 Data Elongation (mm) Force (N) 0.945 64.75 77.609 89.12 1.70 3.00 .00 20.77 73.48 Elongation (mm) 6.115 67.183 13.348 18.30 77.04 1.204 15.678 85.608 27.00 90.501 53.00 Tensile Force F (N) 70.02 Polycarbonate Plastic Tensile Tests 100.377 48.725 40.64 75.00 Sample 1 40.42 1.00 50.713 58.045 37.504 14.183 42.76 4.00 20.00 80.37 0.69 1.00 30.35 75.106 56.03 2.00 0.30 1.00 60. 83 0.00 200.93 2.20 2.00 13.90 16.38 2.85 39.53 25.61 44.55 106.63 1.15 1.42 Elongation (mm) 1.04 2.84 Elongation (mm) 1.31 132.27 Force (N) 0.02 86.33 1.95 1.84 69.44 0.91 95.80 67.00 Tensile Force F (N) 250.11 99.54 2.66 215.75 0.37 2.00 Thin Wire 50.88 61.00 Thick Wire 100.50 1.03 2.13 39.13 80.88 220.00 0.00 23.74 47.63 Force (N) 60.22 2.71 0.25 91.04 1.31 32.85 0.82 Force (N) 103.94 1.00 1.62 0.63 0.69 2.11 20.00 0.45 62.04 1.12 2.55 0.00 150.87 2.50 Elongation δ (mm) 2.68 79.48 1.21 Thin Wire Sample Data Elongation (mm) 0.66 241.53 0.33 89.72 1.10 149.56 52.Thick Wire Sample Data Elongation (mm) 0.29 116.16 184.99 174.00 0.42 1.78 1.00 2.50 3.93 147.00 0.19 Force (N) 0.94 Headphone Wire Tensile Tests 300.47 0.70 200.12 1.00 0.47 101.70 33.78 49.34 1.00 . 00 8.77 x 10‐6 m2 As mentioned above.1 10.00 0.2 10. Aluminum Sample 1 Stress and Strain Calculations Strain (mm/mm) Stress (MPa) Strain (mm/mm) Stress (MPa) 0.75 x 10‐3 165 ‐3 ‐3 3. stress is calculated by dividing the applied force by the cross‐sectional area (σ=F/A).3 11.3 x 10 173 4.52 x 10‐6 m2 Thick Wire Sample Dimensions Radius: 1.0 x 10 169 3. and the cross sectional area of the wires is calculated by multiplying pi by the radius squared (A = πr2).63 x 10‐3 29.88 x 10‐3 102 11.05 x 10‐7 m2 Polycarbonate Plastic Sample Dimensions Width: 4.3 x 10 177 7.25 x 10 135 14.9 x 10‐3 175 ‐3 ‐3 5.9 x 10‐3 178 ‐3 ‐3 7. The cross sectional areas of the metal and plastic samples are calculated by multiplying width by thickness (A = wt).81 x 10‐4 m Area: 1.0 x 10‐3 m Area: 3.63 x 10‐3 123 13.Results & Discussion The following tables contain the cross‐sectional area calculations of each type of sample along with the dimensions they were derived from.13 x 10 115 12.0 x 10‐3 m Thickness: 7. Aluminum and Steel Sample Dimensions Width: 4. and strain is calculated by dividing the elongation by the original length of the sample (ε=δ/L).4 x 10‐3 177 ‐3 ‐3 6. The following section of data tables will show the calculated stress and strain of the metal samples at each data point and then a graph of all three metal samples is plotted on a single stress/strain graph.14 x 10‐6 m2 Thin Wire Sample Dimensions Diameter: 0.75 x 10 57.8 x 10 178 .75 x 10 156 15.8 x 10 176 5.75 x 10‐3 m Area: 1.0 x 10‐3 m Thickness: 3.25 x 10 17.00 x 10‐3 151 14.5 x 10‐3 172 ‐3 ‐3 3.62 x 10‐5 m Area: 3. 75 x 10‐3 10.2 1.75 x 10‐3 9.50 x 10‐3 6.88 x 10 313 3.25 x 10‐3 5.13 x 10‐3 439 ‐3 4.13 x 10 230 2.63 x 10‐3 286 ‐3 2.00 0.75 x 10‐3 143 ‐3 4.50 x 10‐3 111 ‐3 3.88 x 10‐3 7.63 x 10 166 1.50 x 10‐3 8.4 x 10‐3 16.6 x 10‐3 12.0 x 10‐3 15.00 0.13 x 10‐3 60.Aluminum Sample 2 Stress and Strain Calculations Strain (mm/mm) Stress (MPa) 0.50 x 10‐3 11.38 x 10‐3 6.75 x 10‐3 499 Strain (mm/mm) 5.00 0.6 x 10‐3 Stress (MPa) 160 167 169 171 172 173 173 173 173 173 Steel Sample Stress and Strain Calculations Strain (mm/mm) Stress (MPa) 0.0 ‐3 1.5 x 10‐3 21.88 x 10‐3 197 ‐3 2.1x 10‐3 13.63 x 10 393 4.88 x 10‐3 93.50 x 10 148 Strain (mm/mm) 5.4 ‐3 2.5 x 10‐3 19.2 1.7 ‐3 1.50 x 10 478 4.63 x 10‐3 7.13 x 10 112 ‐3 1.13 x 10‐3 342 ‐3 3.75 x 10‐3 78.1 x 10‐3 16.5 x 10‐3 20.9 x 10‐3 Stress (MPa) 547 567 634 647 695 745 754 781 788 789 789 787 .6 x 10‐3 14.38 x 10 64.38 x 10 19.00 ‐3 0.13 x 10 105 2.50 x 10‐3 8.1 x 10‐3 23.00 x 10 129 3. 0 Aluminum 2 300.0 0. and thus the Young's modulus. and so it's a pretty good fit.0 Aluminum 1 400. This is due. we assume calibration and rounding errors. is determined to be 46 GPa. in part.0050 0. As you will see in the later samples.961.0250 0. Data points 1 through 6 form a relatively straight line and these were the data points used to calculate Young's Modulus for this sample. we see a definite linear relationship through the first 18 data points selected. The coefficient of determination for this calculation is 0. Comparing our derived value to the theoretical modulus of elasticity for aluminum. Looking at the stress/strain graph for the steel sample. the slope of the line.0 500. our derived modulus' are consistently about half of the theoretical modulus'. The experimental modulus of elasticity of the sample is calculated to be 97 .0200 0. Our experimental modulus differs from the theoretical modulus by 44%. and as stated above. The first 9 data points are used to calculate our least squares line and the slope indicates that the Young's Modulus for this sample is 39 GPa.0150 0.967. The coefficient of determination for this line is 0. and possibly a systematic calibration error along with a rounding error.0100 0. Looking at the stress/strain graph for the second aluminum sample.0300 Strain ε (mm / mm) Looking at the stress/strain graph for the first sample of aluminum. Using least squares linear regression. we see that there is a large difference between the slope of thee line from the zero point to the first data point and the slope of the line from the first data point forward. which is 70GPa[3].0 Steel 200.0 800.Stress / Strain of Aluminum and Steel 900. we can note our calculation is off by 34%.0 Stress σ (MPa) 700.0 0.0 600.0000 0.0 100. I believe this to be a gross error where the sample wasn't seated properly in the machine. to what I believe is the gross error stated above. we see that it appears to be seated correctly in the machine as there is no large slope discrepancy at the first data point. 0 10. Since the formulas for calculating stress and strain on samples does not differ.700 0.4 GPa.100 0. Polycarbonate Plastic Stress / Strain Chart 70. along with the fact that these samples are for educational use and may not be as purer as the samples used to derive the theoretical data. The theoretical modulus of elasticity of polycarbonate plastic is 2.0 0. and our derived value there was 1.400 0. The coefficients of determination for each sample are 0.951 respectively.300 0. only the stress/strain graphs are shown for the polycarbonate plastic and headphone wire samples. The theoretical modulus of elasticity of steel is 200 GPa[4] and thus our calculation is off by 51%.500 0.600 0.0 Plastic Sample 2 20.200 0. ε (mm/mm) As we can see from the stress/strain graph above. .0 0. steel is much stronger than aluminum while being considerably less elastic in nature. I suspect calibration and rounding errors as the cause for this discrepancy.0 Stress. The second sample also required the first 10 data points to make the Young's Modulus calculation. σ (MPa) 50. As before.800 Strain.GPa and the coefficient of determination for the calculation is 0. I used the first 10 data points to calculate a Young's Modulus of 1.5 GPa.991.0 40.4 GPa[5] and thus our calculations are off by between 25% and 42%. the two polycarbonate plastic samples produced nearly identical graphs. As we can clearly see.969 and 0.0 60.0 Plastic Sample 1 30. In the first sample.0 ‐ 2.000 0. 4 GPa.5 mm.025 0.2 GPa for the thick and thin wires respectively. The coefficient of determination for the thick wire was . and so while the outside of the wire may have an elongation of 2.015 0. the hypothesis is that the wires remain elastic until they break. our experimental Young's modulus is off by roughly 35% from the theoretical value. The modulus of elasticity for PVC is 490.Headphone Wire Stress / Strain Chart 90 80 Stress σ (MPa) 70 60 50 Thick Wire 40 Thin Wire 30 20 10 0 0.040 Strain ε (mm/mm) Looking at the stress/strain graphs for the headphone wires. I believe this to be because the PVC is stretching over the interior copper wiring. It appears that the modulus of elasticity. There is a slight variation in the slope of the graphs between the zero point and the first data point than from the slope of the rest of the graph for each sample. And the Young's modulus for copper is between 110 GPa and 128 GPa[6]. at least in the experiment.999 and the coefficient of determination for the thin wire was 0.3 GPa and 2. which were 2. Also. Given this. Since the initial slopes are slightly off from the rest of the graphs. The fact that the stress/strain slopes remained constant through all the data points. . I think variations of this type are cause by the sample not being completely vertical in the machine. the origin points were omitted from the modulus calculations.035 0. It is noted that most electronics use copper wiring as the conductor for the electric signal. we have no way of knowing what the actual elongation of the copper wire is. we see that they are very similar even though they have different diameters.000 psi[8].020 0. the interior copper wiring has a smaller cross‐sectional area and so the modulus of elasticity should be higher.000 0.996.030 0.005 0. It is also noted that the material around the copper wiring in many cases is PVC[7]. or roughly 3.010 0. is determined by the wire's insulation. 5 Wikipedia entry for polycarbonate (http://en. page 123.wikipedia. the jack is usually reinforced where it connects to the insulation. We also noted that our numbers were 25% ‐ 50% off from theoretical values.7 lbf and nothing more.wikipedia. page 124. 7 Alibaba. we know that an unreinforced wire gripped from the outside can bear 52. we divide 234 N of force by the gravitational constant 9.org/wiki/Polyvinyl_chloride).81 m/s2 to get a weight of approximately 23.alibaba.com (http://www. Introduction to Engineering Analysis. rather than just a small sample of the wire gripped from outside the insulation.org/wiki/Aluminium).Conclusion So what's the answer to our initial question? How much weight can be supported be the earbuds of your average ipod? if the force were being applied from the outside of the wire. of having the interior wire snap. 3 Wikipedia entry for aluminum (http://en. the answer is simple. either by reaching the maximum distance the machine could pull them. Some of the graphs indicated that samples weren't entered into the machine precisely vertically. each thin wire bears half the force of the thick wire. it's hard to say. the plastics and PVS wire insulation did not break during our tests. It took 241 N of force to break the thick wire and 117 N of force to break the thin wire.org/wiki/Young%27s_modulus).com/showroom/high‐class‐earphone‐cheap‐mini‐headphones‐pvc‐ cable‐earbuds. I would like to see the entire wire tested. In reality. One of the things that was not done prior to testing the samples was a calibration test.org/wiki/Polycarbonate).wikipedia. 2 .html). there are just too many unknown variables to know for sure. the jack is connected directly to the interior copper wires.7 lbf. If we assume a 45 degree angle of the thin wire at the y junction. and that they are considerably less elastic. 4 Wikipedia entry for Young's Modulus (http://en. the angle at which the thin wires rest on the head varies from person to person. 8 Wikipedia entry for polyvinyl Chloride (http://en. 6 Wikipedia entry for copper (http://en. Finding out the diameter of the inside copper wiring would also be key to determining how much weight could be sustained. or 52. To be really thorough. From these tests we've seen that metals have relatively low breaking points compared to plastics. As a matter of fact.wikipedia. References 1 Introduction to Engineering Analysis. In short.wikipedia.org/wiki/Copper). with all the reinforcements intact.9 kilograms. Taking the smaller of the force used to break the thick wire or twice the force used to break the thin wire. And there are the inherent rounding errors that come from doing experimental calculations with limited measuring equipment.
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