EGR 320L Lab Report Week 1 - Frank Jamison

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

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. A stress/strain graph was then used to identify the elastic

range of the material, and least squares linear regression was used to determine the slope of the line

that best fit the data. The slope of this best fit line was the experimental determination of the Young's

Modulus for each material.

Data

The following tables and graphs show the raw data collected for the experiments. The first four tables

show the measured dimensions of the samples used in the experiment. The additional tables and graphs

are the readings from the Admet equipment used in the tensile testing of each sample. Tables are

broken down into groups of metals, plastics, and wires, and each group of samples tested are plotted on

a single graph.

Aluminum and Steel Sample Dimensions Polycarbonate Plastic Sample Dimensions

Length: 80 mm Length: 80 mm

Width: 4.0 mm Width: 4.0 mm

Thickness: 0.0762 mm Thickness: 0.381 mm

Thick Wire Sample Dimensions Thin Wire Sample Dimensions

Length: 80 mm Length: 80 mm

Diameter: 2.0 mm Diameter: 1.5 mm

Aluminum Sample 1 Data

Elongation (mm) Force (N) Elongation (mm) Force (N)

0.00 0.00 0.70 50.29

0.26 5.22 0.80 51.50

0.29 8.89 0.84 52.56

0.30 17.48 0.90 52.83

0.39 30.95 0.95 53.37

0.41 34.94 1.02 53.69

0.45 37.61 1.07 53.99

0.50 41.00 1.14 54.10

0.56 45.93 1.19 54.28

0.62 47.61 1.26 54.35

Aluminum Sample 2 Data


0.00 0.00 0.47 48.66

0.03 5.85 0.60 50.75

0.09 18.50 0.70 51.46

0.11 19.57 0.85 52.26

0.15 28.46 0.97 52.45

0.17 32.06 1.04 52.66

0.20 33.79 1.21 52.84

0.24 39.29 1.32 52.77

0.30 43.56 1.56 52.72

0.36 45.09 1.65 52.82

Steel Sample Data


0.00 0.00 0.42 166.66

0.06 23.77 0.44 172.75

0.09 34.28 0.51 193.32

0.13 50.47 0.53 197.25

0.15 60.09 0.60 211.70

0.17 70.15 0.70 226.98

0.21 87.27 0.76 229.81

0.23 95.39 0.93 238.16

0.25 104.31 1.15 240.17

0.29 119.68 1.32 240.38

0.33 133.86 1.69 240.61

0.36 145.59 1.91 240.00

0.38 152.06

Polycarbonate Plastic Sample 1 Data


0.00 0.00 7.41 85.52

0.59 23.86 10.03 72.90

0.84 32.93 14.98 75.19

1.06 41.72 20.13 74.44

1.18 45.55 25.34 74.54

1.40 51.64 30.30 74.60

1.72 61.17 35.16 76.86

1.92 64.94 40.21 78.95

2.19 69.92 45.24 77.93

2.56 75.70 50.44 76.42

3.07 81.85 54.98 76.55

4.09 87.55 56.22 76.26

5.27 89.26

0.00

50.00

100.00

150.00

200.00

250.00

300.00

0.00 0.50 1.00 1.50 2.00 2.50

Tensile Force F (N)

Elongation δ (mm)

Aluminum and Steel Tensile Tests

Aluminum Sample 1

Aluminum Sample 2

Steel Sample

Polycarbonate Plastic Sample 2 Data


0.504 14.37 6.696 88.11

0.613 20.42 8.053 83.78

1.045 37.04 9.169 74.91

1.183 42.69 11.183 74.77

1.377 48.77 13.204 73.70

1.501 53.12 15.348 74.35

1.713 58.30 18.961 75.21

1.945 64.17 23.608 75.69

2.115 67.03 27.449 75.64

2.262 68.94 34.725 75.30

2.543 74.07 40.160 76.54

3.128 81.70 45.370 76.75

3.678 85.76 50.282 77.30

4.575 89.00 55.106 77.77

5.609 89.48 56.324 78.02

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

100.00

0.00 10.00 20.00 30.00 40.00 50.00 60.00

Tensile Force F (N)

Elongation δ (mm)

Polycarbonate Plastic Tensile Tests

Sample 1

Sample 2

Thick Wire Sample Data


0.00 0.00 1.34 103.31

0.44 23.31 1.63 132.93

0.53 32.85 1.78 147.10

0.62 39.56 1.87 149.99

0.75 52.88 2.04 174.16

0.85 61.84 2.22 184.70

0.95 69.13 2.38 200.66

1.04 80.33 2.54 215.88

1.15 89.47 2.69 220.66

1.27 101.42 2.82 241.21

Thin Wire Sample Data


0.00 0.00 1.33 60.45

0.47 13.90 1.42 62.80

0.55 16.11 1.48 67.68

0.63 20.53 1.72 79.02

0.71 25.70 1.93 86.25

0.83 33.13 2.03 91.91

0.94 39.61 2.12 95.11

1.04 44.74 2.20 99.55

1.12 47.78 2.37 106.29

1.19 49.84 2.63 116.94

0.00

50.00

100.00

150.00

200.00

250.00

300.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Tensile Force F (N)

Elongation δ (mm)

Headphone Wire Tensile Tests

Thick Wire

Thin Wire

Results & Discussion

The following tables contain the cross‐sectional area calculations of each type of sample along with the

dimensions they were derived from. The cross sectional areas of the metal and plastic samples are

calculated by multiplying width by thickness (A = wt), and the cross sectional area of the wires is

calculated by multiplying pi by the radius squared (A = πr2).

Aluminum and Steel Sample Dimensions Polycarbonate Plastic Sample Dimensions

Width: 4.0 x 10‐3 m Width: 4.0 x 10‐3 m

Thickness: 7.62 x 10‐5 m Thickness: 3.81 x 10‐4 m

Area: 3.05 x 10‐7 m2 Area: 1.52 x 10‐6 m2

Thick Wire Sample Dimensions Thin Wire Sample Dimensions

Radius: 1.0 x 10‐3 m Diameter: 0.75 x 10‐3 m

Area: 3.14 x 10‐6 m2 Area: 1.77 x 10‐6 m2

As mentioned above, stress is calculated by dividing the applied force by the cross‐sectional area (σ=F/A), and strain is calculated by dividing the elongation by the original length of the sample (ε=δ/L). 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. Aluminum Sample 1 Stress and Strain Calculations

Strain (mm/mm) Stress (MPa) Strain (mm/mm) Stress (MPa)

0.00 0.00 8.75 x 10‐3 165

3.25 x 10‐3 17.1 10.0 x 10‐3 169

3.63 x 10‐3 29.2 10.5 x 10‐3 172

3.75 x 10‐3 57.3 11.3 x 10‐3 173

4.88 x 10‐3 102 11.9 x 10‐3 175

5.13 x 10‐3 115 12.8 x 10‐3 176

5.63 x 10‐3 123 13.4 x 10‐3 177

6.25 x 10‐3 135 14.3 x 10‐3 177

7.00 x 10‐3 151 14.9 x 10‐3 178

7.75 x 10‐3 156 15.8 x 10‐3 178

Aluminum Sample 2 Stress and Strain Calculations


0.00 0.00 5.88 x 10‐3 160

0.38 x 10‐3 19.2 7.50 x 10‐3 167

1.13 x 10‐3 60.7 8.75 x 10‐3 169

1.38 x 10‐3 64.2 10.6 x 10‐3 171

1.88 x 10‐3 93.4 12.1x 10‐3 172

2.13 x 10‐3 105 13.0 x 10‐3 173

2.50 x 10‐3 111 15.1 x 10‐3 173

3.00 x 10‐3 129 16.5 x 10‐3 173

3.75 x 10‐3 143 19.5 x 10‐3 173

4.50 x 10‐3 148 20.6 x 10‐3 173

Steel Sample Stress and Strain Calculations


0.00 0.00 5.25 x 10‐3 547

0.75 x 10‐3 78.0 5.50 x 10‐3 567

1.13 x 10‐3 112 6.38 x 10‐3 634

1.63 x 10‐3 166 6.63 x 10‐3 647

1.88 x 10‐3 197 7.50 x 10‐3 695

2.13 x 10‐3 230 8.75 x 10‐3 745

2.63 x 10‐3 286 9.50 x 10‐3 754

2.88 x 10‐3 313 11.6 x 10‐3 781

3.13 x 10‐3 342 14.4 x 10‐3 788

3.63 x 10‐3 393 16.5 x 10‐3 789

4.13 x 10‐3 439 21.1 x 10‐3 789

4.50 x 10‐3 478 23.9 x 10‐3 787

4.75 x 10‐3 499

Looking at the stress/strain graph for the first sample of aluminum, 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. I believe this to be a gross error where the sample wasn't seated

properly in the machine. 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. Using least squares linear regression, the

slope of the line, and thus the Young's modulus, is determined to be 46 GPa. The coefficient of

determination for this line is 0.961, and so it's a pretty good fit. Comparing our derived value to the

theoretical modulus of elasticity for aluminum, which is 70GPa[3], we can note our calculation is off by

34%. This is due, in part, to what I believe is the gross error stated above, and possibly a systematic

calibration error along with a rounding error. As you will see in the later samples, our derived modulus'

are consistently about half of the theoretical modulus'.

Looking at the stress/strain graph for the second aluminum sample, we see that it appears to be seated

correctly in the machine as there is no large slope discrepancy at the first data point. 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. The coefficient of determination for this calculation is 0.967. Our experimental

modulus differs from the theoretical modulus by 44%, and as stated above, we assume calibration and

rounding errors.

Looking at the stress/strain graph for the steel sample, we see a definite linear relationship through the

first 18 data points selected. The experimental modulus of elasticity of the sample is calculated to be 97

0.0

100.0

200.0

300.0

400.0

500.0

600.0

700.0

800.0

900.0

0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300

Stress σ(M

Pa)

Strain ε (mm / mm)

Stress / Strain of Aluminum and Steel

Aluminum 1

Aluminum 2

Steel

GPa and the coefficient of determination for the calculation is 0.991. The theoretical modulus of

elasticity of steel is 200 GPa[4] and thus our calculation is off by 51%. As we can clearly see, steel is much

stronger than aluminum while being considerably less elastic in nature.

Since the formulas for calculating stress and strain on samples does not differ, only the stress/strain

graphs are shown for the polycarbonate plastic and headphone wire samples.

As we can see from the stress/strain graph above, the two polycarbonate plastic samples produced

nearly identical graphs. In the first sample, I used the first 10 data points to calculate a Young's Modulus

of 1.5 GPa. The second sample also required the first 10 data points to make the Young's Modulus

calculation, and our derived value there was 1.4 GPa. The coefficients of determination for each sample

are 0.969 and 0.951 respectively. The theoretical modulus of elasticity of polycarbonate plastic is 2.0 ‐

2.4 GPa[5] and thus our calculations are off by between 25% and 42%. As before, I suspect calibration

and rounding errors as the cause for this discrepancy, 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.

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800

Stress, σ

(MPa)

Strain, ε (mm/mm)

Polycarbonate Plastic Stress / Strain Chart

Plastic Sample 1

Plastic Sample 2

Looking at the stress/strain graphs for the headphone wires, we see that they are very similar even

though they have different diameters. 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. I think

variations of this type are cause by the sample not being completely vertical in the machine.

Since the initial slopes are slightly off from the rest of the graphs, the origin points were omitted from

the modulus calculations, which were 2.3 GPa and 2.2 GPa for the thick and thin wires respectively. The

coefficient of determination for the thick wire was .999 and the coefficient of determination for the thin

wire was 0.996. The fact that the stress/strain slopes remained constant through all the data points, the

hypothesis is that the wires remain elastic until they break.

It is noted that most electronics use copper wiring as the conductor for the electric signal. And the

Young's modulus for copper is between 110 GPa and 128 GPa[6]. It is also noted that the material around

the copper wiring in many cases is PVC[7]. The modulus of elasticity for PVC is 490,000 psi[8], or roughly

3.4 GPa. Given this, our experimental Young's modulus is off by roughly 35% from the theoretical value.

It appears that the modulus of elasticity, at least in the experiment, is determined by the wire's

insulation. I believe this to be because the PVC is stretching over the interior copper wiring, and so while

the outside of the wire may have an elongation of 2.5 mm, we have no way of knowing what the actual

elongation of the copper wire is. Also, the interior copper wiring has a smaller cross‐sectional area and

so the modulus of elasticity should be higher.

0

10

20

30

40

50

60

70

80

90

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040

Stress σ(M

Pa)

Strain ε (mm/mm)

Headphone Wire Stress / Strain Chart

Thick Wire

Thin Wire

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, the answer is simple. It took

241 N of force to break the thick wire and 117 N of force to break the thin wire. If we assume a 45

degree angle of the thin wire at the y junction, each thin wire bears half the force of the thick wire.

Taking the smaller of the force used to break the thick wire or twice the force used to break the thin

wire, we divide 234 N of force by the gravitational constant 9.81 m/s2 to get a weight of approximately

23.9 kilograms, or 52.7 lbf.

In reality, it's hard to say. the jack is connected directly to the interior copper wires, the jack is usually

reinforced where it connects to the insulation, the angle at which the thin wires rest on the head varies

from person to person, there are just too many unknown variables to know for sure.

From these tests we've seen that metals have relatively low breaking points compared to plastics, and

that they are considerably less elastic. As a matter of fact, the plastics and PVS wire insulation did not

break during our tests, either by reaching the maximum distance the machine could pull them, of having

the interior wire snap.

We also noted that our numbers were 25% ‐ 50% off from theoretical values. One of the things that was

not done prior to testing the samples was a calibration test. Some of the graphs indicated that samples

weren't entered into the machine precisely vertically. And there are the inherent rounding errors that

come from doing experimental calculations with limited measuring equipment.

To be really thorough, I would like to see the entire wire tested, with all the reinforcements intact,

rather than just a small sample of the wire gripped from outside the insulation. Finding out the diameter

of the inside copper wiring would also be key to determining how much weight could be sustained.

In short, we know that an unreinforced wire gripped from the outside can bear 52.7 lbf and nothing

more.

References

1 Introduction to Engineering Analysis, page 123. 2 Introduction to Engineering Analysis, page 124. 3 Wikipedia entry for aluminum (http://en.wikipedia.org/wiki/Aluminium). 4 Wikipedia entry for Young's Modulus (http://en.wikipedia.org/wiki/Young%27s_modulus). 5 Wikipedia entry for polycarbonate (http://en.wikipedia.org/wiki/Polycarbonate). 6 Wikipedia entry for copper (http://en.wikipedia.org/wiki/Copper). 7 Alibaba.com (http://www.alibaba.com/showroom/high‐class‐earphone‐cheap‐mini‐headphones‐pvc‐

cable‐earbuds.html). 8 Wikipedia entry for polyvinyl Chloride (http://en.wikipedia.org/wiki/Polyvinyl_chloride).

EGR 320L Lab Report Week 1 - Frank Jamison

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