Lab 2 Poisson\'s Ratio PDF

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Ungraded Lab 2, mandatory group project assignment for ARCE 224 with Professor Peter Laursen...

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EXPERIMENT 2 | POISSON’S RATIO California Polytechnic State University San Luis Obispo ARCE 224-02

CGLT Structural Firm

Maerill Ceballos

Felipe Garcia

Chris Levy

4/18/17

Matthew Thurman

TABLE OF CONTENTS

COVER PAGE

1

TABLE OF CONTENTS

2

PURPOSE

3

PROCEDURE

3

BACKGROUND

4

DATA AND RESULTS

5

ERROR ANALYSIS

6

CONCLUSIONS

7

REFERENCES

9

APPENDIX A: RECORDED DATA

10

APPENDIX B: EQUIPMENT

11

APPENDIX C: SAMPLE CALCULATIONS

12

Page 2 of 12

PURPOSE To determine Poisson’s Ratio for different materials experimentally and compare these values with theoretical values.

PROCEDURE 1) Measure and record the initial dimensions of each specimen. 2) Calculate the theoretical yield strength (force) Fy for each specimen. 3) Apply strain gauges to specimen in lateral and axial direction. 4) Use Tinius Olsen testing machine to load each specimen up to 75% of their yield strength. 5) Tabulate axial and lateral strain, and Poisson’s Ratio for each load applied. 6) Average Poisson’s Ratio for each specimen and compare with theoretical Poisson’s Ratio for each material.

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BACKGROUND The ratio of strain in the lateral direction to the strain in the axial direction is known as Poisson’s ratio and is denoted by the Greek letter v (nu): ν = - ( lateral strain / axial strain ) = -εx/εy For a bar in tension, the axial strain  represents elongation (positive strain) and the lateral strain represents a decrease in width (negative strain). For compression the opposite situation occurs, with the bar becoming shorter (negative axial strain) and wider (positive lateral strain). Poisson’s Ratio is named after the famous French mathematician Simeon Denis Poisson (1781-1840), who attempted to calculate this ratio by a molecular theory of materials. Poisson’s Ratio has a positive value for most materials, and ranges from 0.25 to 0.35 for most metals. Materials with extremely low values of Poisson’s Ration include cork (practically zero) and concrete (approx. 0.15). The theoretical upper limit for Poisson’s ratio is 0.5, with rubber having a Poisson’s ratio close to 0.5. The theoretical upper limit of Poisson’s Ratio is determined to be 0.5 through the combination of the dilatation, the modulus of elasticity, and the bulk modulus equations. The upper limit of Poisson’s Ratio comes from the equation for dilatation, the change per unit volume: e = - ( uniform hydrostatic pressure / bulk modulus ) = -p/k where the bulk modulus is expressed as: k = modulus of elasticity /(3  × (1 - (2 × Poisson’s Ratio )) = E/(3 × (1-(2 × v))) Because it is assumed that Poisson’s Ratio is positive for engineering materials, it is determined that the range for Poisson’s Ratio must be from 0 to 0.5. The equation for the bulk constant directly limits Poisson’s ratio not to exceed 0.5. For this experiment strain will be recorded using a strain gauge, a rectangular metal piece fitted with a switchback pattern of wire. On the strain gauge, the wire runs only along the length of the metal piece. If the length of the metal piece is extended then the wire will experience an extending as well. The increased length of the wire will reflect an increase in the resistance of the system, which by Ohm’s Law, results in a drop of the voltage in the system. This drop in voltage could then be interpreted to obtain either the specimen’s axial or lateral strain. If the metal piece is fitted so the wire is parallel to the axis of a specimen that experiences axial elongation, the metal and the wire on it experience that same elongation and can determine the axial strain. Since no wire runs parallel to the lateral portion of the object, there is virtually no resistance generated by the lengthening of wire in the specimen’s lateral direction. This means that the voltage drop is only affected by the resistance generated due to lengthening in the axial direction, therefore it can be interpreted as axial strain. To record lateral strain, the strain gauge is simply oriented laterally along the object.

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DATA AND RESULTS Table 1. Dimensions of Aluminum 6061-T6 Thickness (in)

Width (in)

Area (in)

Length (in)

0.537

0.1249

0.0671

6.05

Table 2. Dimensions of Steel 1010 Thickness (in)

Width (in)

Area (in)

Length (in)

0.535

0.1253

0.0670

6.10

Table 3. Load, Lateral and Axial Strain, and Poisson’s Ratio for Aluminum 6061-T6 Load (lb)

Lateral Strain

Axial Strain

Poisson’s Ratio

Calculated Axial Strain

353

-213 x 10-6

554 x 10-6

0.384

520.1 x 10-6

700

-414 x 10-6

1145 x 10-6

0.366

1033 x 10-6

1052

-611 x 10-6

1704 x 10-6

0.359

1552 x 10-6

1416

-813 x 10-6

2295 x 10-6

0.354

2089 x 10-6

1749

-999 x 10-6

2845 x 10-6

0.351

2581 x 10-6

Average Poisson’s Ratio

0.363

Table 4. Load, Lateral and Axial Strain, and Poisson’s Ratio for Steel 1010 Load (lb)

Lateral Strain

Axial Strain

Poisson’s Ratio

Calculated Axial Strain

502.5

-90 x 10-6

285 x 10-6

0.316

258.6 x 10-6

1005.0

-182 x 10-6

593 x 10-6

0.307

517.2 x 10-6 

1507.5

-286 x 10-6

915 x 10-6

0.313

775.9 x 10-6 

2010.0

-392 x 10-6

1235 x 10-6

0.332

1034 x 10-6 

2512.5

-502 x 10-6

1568 x 10-6

0.320

1293 x 10-6 

Average Poisson’s Ratio

0.315

Page 5 of 12

ERROR ANALYSIS Table 5. Comparison of Experimental and Theoretical Poisson’s Ratio of Aluminum 6061-T6 and Steel 1010 Material

Theoretical Poisson’s Ratio

Experimental Poisson’s Ratio

Percent Error of Experimental Poisson’s Ratio (%)

Maximum Difference in Experimental Poisson’s Ratio

Aluminum 6061-T6

0.33

0.363

10.0

0.033

Steel 1010

0.30

0.315

5.00

0.025

Average Percent Error

7.50

Table 6. Experimental and Calculated Axial Strain of Aluminum 6061-T6 Axial Strain

Calculated Axial Strain

Percent Error of Measured Axial Strain (%)

554 x 10-6 

520.1 x 10-6 

6.52

1145 x 10-6 

1033 x 10-6 

10.8

1704 x 10-6 

1552 x 10-6 

9.80

2295 x 10-6 

2089 x 10-6 

9.86

2845 x 10-6 

2581 x 10-6 

10.2

Average Percent Error

9.44

Table 7. Experimental and Calculated Axial Strain of Steel 1010 Axial Strain

Calculated Axial Strain

Percent Error of Measured Axial Strain (%)

285 x 10-6 

258.6 x 10-6 

10.2

593 x 10-6 

517.2 x 10-6 

14.7

915 x 10-6 

775.9 x 10-6 

18.0

1235 x 10-6 

1034 x 10-6 

19.4

1568 x 10-6 

1293 x 10-6 

21.3

Average Percent Error

16.7

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CONCLUSIONS Poisson’s ratio: The experimentally-derived Poisson’s ratio for aluminum of 0.363, gave a 10% error based on its theoretical value of 0.33 and the experimentally derived Poisson’s ratio for steel of 0.315, gave a 5% error from its theoretical Poisson’s ratio of 0.30. Data with 10% error or less is typically considered accurate, therefore the experimental values are viable. Furthermore, several credible sources provide a value 0.35 as the theoretical Poisson’s ratio for aluminum, which is approximately 0.1 away from our experimentally derived Poisson’s ratio for aluminum and would yield a percent error of 4%. Among these sources the discrepancy in the theoretical Poisson’s ratio for aluminum is attributed to different testing methods or the change in standard over time. Regardless, among the different values, the experimentally derived Poisson’s ratio for aluminum and steel falls beneath the 10% error and is considered viable. (Tables 6 and 7) The experimental Poisson’s ratios for aluminum and steel gathered at each increment were relatively consistent. The maximum difference in the calculated Poisson’s ratios for both aluminum and steel were found to be 0.033 and 0.025, respectively. These small values are expected out of the experimental values, given that Poisson’s ratio is an ideal constant. The minor inconsistencies in Poisson’s ratios calculated at increments for steel and aluminum are attributed to inaccurate readings by the strain gauges. (Table 5) Axial strain:The average percent error between measured axial strain and calculated axial strain for aluminum is 9.44%. The average percent error between measured axial strain and calculated axial strain for steel is 16.7%. These high percent errors may have been caused by inaccurate readings of the strain gauges, since it is not likely that specimen areas were measured incorrectly, or that the testing machine gave inaccurate readings of applied load, or that the moduli of elasticity used for each material are incorrect (these are the elements used to calculate axial strain). It is possible, however, that the timing of the readings and the increments of the given load not exactly added at the 15% increases required may have also contributed to the error. The fact that the experimentally-derived Poisson’s Ratio for each material had reasonable percent errors, and the Poisson’s Ratios calculated at each increment for each material were reasonably consistent, makes it more likely that the high percent error between measured and calculated axial strain is due to inaccurate readings by the strain gauges. (Appendix C, Tables 6 and 7) The measured axial strain for aluminum started at 554 x 10-6  and ended at 2845 x 10-6  . The -6 measured axial strain for steel started at 285 x 10 and ended at 1568 x 10-6  . Both experienced loads approximately began at 15% of their respective yield strengths and ended close to 75% of their yield strengths. Because aluminum started with less axial strain than steel and ended with more, the data shows that aluminum experiences more axial strain than steel, and experiences increasing axial elongation at a greater rate than steel. (Tables 3 and 4) Lateral strain: The measured lateral strain for aluminum started at -90 x 10-6  and ended at -502 x 10-6  . The measured axial strain for steel started at 285 x 10-6  and ended at 1568 x 10-6  . Aluminum started with more negative lateral strain and ended with more negative lateral strain than steel, so the data does not show directly which material experiences increasing negative

Page 7 of 12

CONCLUSIONS - CONTINUED lateral strain at a greater rate, but it does show that aluminum overall experiences more negative lateral strain than steel. (Tables 3 and 4) Material comparison: The theoretical Poisson’s ratios for aluminum and steel is 0.33 and 0.30, respectively. These positive values indicate that aluminum and steel both hold materialistic properties that, when under loading, naturally expand in one direction and contract in the other. For this experiment, both specimens were stretched axially and shortened laterally when tension was applied at their axial ends. Given that steel has a lower Poisson’s ratio value, it can be concluded that it is a stiffer material compared to aluminum. (Table 5) In summary, steel elongates axially to a lesser magnitude and at a lesser rate than aluminum, and shrinks laterally to a lesser magnitude than aluminum from 15% to 75% of each material’s yield strength. Cork evaluation: Cork has a near-zero value for Poisson’s ratio, which makes it a good choice for a bottle stopper. A near-zero Poisson’s ratio for cork translates to this: if axial force is exerted on the cork to remove it, it does not experience lateral strain and shrink away from the neck of the bottle. If axial force is exerted on the cork to push it in, it does not experience lateral strain and expand, which would prevent it from being pushed into the mouth of the bottle. Under loads such as axial compression or tension most materials tend to expand, or contract in the lateral direction. Therefore, this minimal lateral deformation or near-zero Poisson’s ratio is what makes cork an effective bottle seal and stopper.

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REFERENCES [1]

“Aluminum Alloy 6061 - Composition, Properties, Temper and Applications of 6061 Aluminum,” http://www.azom.com/article.aspx?ArticleID=3328  (West Midlands, UK: AZo Materials, March 2006).

[2]

Beer, Johnston, DeWolf, and Mazurek, Mechanics of Materials , 7th Edition (New York: McGraw-Hill Education, 2015). Section 4. Appendix B.

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APPENDIX A: RECORDED DATA

Figure 1. Original Data

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APPENDIX B: EQUIPMENT

Figure 2. Aluminum 6061-T6 Test Specimen

Figure 3. Steel 1010 Test Specimen

Figure 4. Test Setup

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APPENDIX C: SAMPLE CALCULATIONS Axial strain εx= σ/E = (P/A)/E Maximum Difference Maximum Difference = highest value - lowest value Poisson’s ratio ν = - ( lateral strain / axial strain ) = -εx/εy Percent error percent error = | experimental value - theoretical value | / theoretical value ) × 100 Load Increment P = Py × (0.15 × x)

where x = number increment

Yield Strength Py = σy × A

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