Mechanics Of Materials - Poisson's Ratio

Mechanics of Materials Laboratory Lab #5 Poisson's Ratio

David Clark Group C 9/15/2006

Abstract The purpose of the following experiment is to determine Poisson's ratio for 2024T6 aluminum, as well as serve as an outline for the procedure to test various other materials. Poisson's ratio refers to a characteristic dimensionless number which accurately predicts the amount of strain experienced in non-parallel directions to an applied load. To find this value, a cantilever setup was combined with dual strain gages to record changes in lateral and longitudinal strain. Poisson's ratio for the aluminum specimen was found to be 0.307, which is less than 1% from the scientifically accepted 0.310.

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Table of Contents 1. Introduction & Background............................................................4 2. Equipment and Procedure............................................................5 3. Data, Analysis & Calculations.......................................................6 4. Results..........................................................................................9 5. Conclusions...................................................................................9 6. References....................................................................................9 7. Raw Notes...................................................................................10

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1. Introduction & Background When a material experiences deformation, it not only changes on the axis of the applied load, but also in the perpendicular direction as well. This is known as Poisson's effect and can be predicted by Poisson's ratio. For a specimen experiencing a simple uniaxial load, this ratio is expressed as:

υ=−

ε lateral ε longitudinal

Equation 1 More complex loading configurations utilize the same principles; however these setups were not used in the determination of Poisson's ratio in this exercise. These more advanced conditions are beyond the scope of this lab. Poisson's ratio is constant for all homogeneous, isotropic, linearly elastic materials. For most materials, this value is between 0.0 and 0.5. Poisson's ratio should not be confused with stiffness or hardness. Materials with similar Poisson's ratio may have completely different Young's Moduli. For example, diamond is very hard whereas cork can be deformed by hand without and special equipment; however they both have very low Poisson's ratios (Lakes.) Below is a table of common ratios for various materials. Material

Poisson's ratio

Isotropic upper limit

0.5

Rubber

0.48- ~0.5

Lead

0.44

Copper

0.37

Aluminum

0.35

Copper

0.34

Polystyrene

0.34

Brass

0.33

Ice

0.33

Polystyrene foam

0.3

Stainless Steel

0.30

Steel

0.29

Beryllium

0.08

Isotropic lower limit

-1

Table 1

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The following experiment determines Poisson's ratio for 2024-T6 aluminum. This process uses the change in strain from an unloaded to a loaded configuration. The change in strain for both lateral and longitudinal directions is easily found, however due to the nature of the instrumentation used, correction must be used. The following form is the final equation used in this experiment.

υ=−

ε longitudinal ε lateral

⋅C

Equation 2 where C is the correction factor. For more information and an example of determining this correction factor, see section 3.

2. Equipment and Procedure This experiment was conducted using the following equipment: 1. Cantilever flexure frame: A simple apparatus to hold a rectangular beam at one end while allowing flexing of the specimen upon the addition of a downward force. 2. Metal beam: In this experiment, 2024-T6 aluminum was tested. The beam should be fairly rectangular, thin, and long. Specific dimensions are dependant to the size of the cantilever flexure frame and available weights. 3. P-3500 strain indicator: Any equivalent device that accurately translates to the output of strain gages into units of strain. 4. Two strain gages: 5. Micrometers and calipers: The specimen should be secured in the flexure frame such that an applied force can be placed opposite of the securing end of the fixture. The strain gage on the top of the test material should run longitudinally (or parallel to the length) at the same length as a second strain gage running perpendicularly on the adjacent side.

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Figure 1 Whenever taking a reading from a strain gage, consult the strain gage measurement device for optimal setup. When this experiment was performed, the following setup was utilized: •

The independent lead to the P+

•

One dependent lead to the S-

•

One dependent lead to a dummy connection (D120)

The strain indicator should be initially set to read zero strain for the longitudinal gage. The indicator should then be set to read the initial strain for the lateral gage. A load should be placed such that an increase in strain is created. (Note: The load applied should be verified to be below the yield stress to minimize damage to the specimen and ensure the integrity of the results.) Next, the strain indicator should be reconfigured to read the longitudinal gage. The net change in stress for both gages should be recorded for later calculations.

3. Data, Analysis & Calculations The gage factor for the strain gage used was 2.085 and the transverse sensitivity was 1.0. Both these factors are dependant upon the strain gage used and are generally given by the manufacturer. The following table lists the initial and final strain gage measurements, along with the net strain.

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Longitudinal (με) Undeflected 0 Deflected 1645 Net Strain 1645

Lateral (με) 80 580 -500

Table 2 The net longitudinal and lateral strain were found by the following equations:

εˆlongitudinal = εˆlongitudinal ( deflected ) − εˆlongitudinal (undeflected ) = 1645 − 0 = 1645µε Equation 3

εˆlateral = εˆlateral ( deflected ) − εˆlateral (undeflected ) = 580 − 80 = 500µε Equation 4 The initial calculation of Poisson's ratio can be found using these two strains.

υ=−

ε longitudinal − 500 =− = 0.304µε ε lateral 1645 Equation 5

The correction factor that appears in equation 2 is found from visual inspection of the transverse sensitivity correction chart. Kt is supplied by the manufacturer of the strain gage. This value should be traced up to the 0.304 line on the chart. Since this line is not graphed, visual estimation must be used.

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Figure 2 The estimated correction factor in this case is 1.01. Inserting this value into equation 2, the new determination for Poisson's ratio is found in equation 6.

υ=−

( − 500) C

εˆlongitudinal

=−

( − 500)(1.01) 1645

= 0.307

Equation 6 The scientifically accepted value for Poisson's ratio of 2024-T6 aluminum is:

υ std = 0.31 Equation 7 8

The percent error in the calculated ratio can be found using equation 8. %error =

υ std − υ calculated 0.310 − 0.307 × 100 × 100 = 0.967% υ std 0.310 Equation 8

4. Results Poisson's ratio for the 2024-T6 beam was 0.307. This value has an error of 0.967% from the accepted 0.31. The main source of error may be in the correction factor for transverse sensitivity. Since this factor was found by visual inspection of a chart, the actual value is difficult to determine. Other sources of error include, but are not limited to, imperfections in the adhesive on the strain gage and resolution of the strain indicator.

5. Conclusions For most materials that are available in bar form, the following experiment provides acceptable results. This was reflected in the percent error using aluminum 2024T6, which was less than 1% from the accepted value.

6. References Gilbert, J. A and C. L. Carmen. "Chapter 7 – Poisson's Ratio Flexure Test." MAE/CE 370 – Mechanics of Materials Laboratory Manual. June 2000. Lakes, Ron. "Meaning of Poisson's Ratio." University of Wisconsin. 11 Sept. 2006. Elgun, Serdar Z. "Poisson's Ratio". 11 Sept. 2006.

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7. Raw Notes

Figure 3

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Figure 4

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