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 2024-

T6 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.

1

Table of Contents

1. Introduction & Background.............................................................................3

2. Equipment and Procedure................................................................................4

3. Data, Analysis & Calculations.........................................................................5

4. Results..............................................................................................................8

5. Conclusions......................................................................................................8

6. References........................................................................................................8

7. Raw Notes........................................................................................................9

2

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:

allongitudin

lateral

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

Rubber

Lead

Copper

Aluminum

Copper

Polystyrene

Brass

Ice

Polystyrene foam

Stainless Steel

Steel

Beryllium

Isotropic lower limit

0.5

0.48- ~0.5

0.44

0.37

0.35

0.34

0.34

0.33

0.33

0.3

0.30

0.29

0.08

-1

Table 1

3

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.

Clateral

allongitudin

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.

4

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.

5

Table 2

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

Equation 3

Equation 4

The initial calculation of Poisson's ratio can be found using these two strains.

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.

6

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.

Equation 6

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

Equation 7

7

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

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 2024-

T6, 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.

<http://silver.neep.wisc.edu/~lakes/PoissonIntro.html>

Elgun, Serdar Z. "Poisson's Ratio". 11 Sept. 2006.

<http://info.lu.farmingdale.edu/depts/met/met206/poissonsratio.html>

8

7. Raw Notes

Figure 3

9

Figure 4

10