USING ARCHIMEDES' PRINCIPLE TO DETERMINE CROSS4ECTlONAL AREAS

~ ~ ~ ~ i l ~ Specimen

Aluminum Core

xperimentalists are often required to determine ma- terial properties such as Young's modulus of elastic- ity for use in structural analyses. Most of the time E this is a straightforward process; specimens are fab-

ricated and tested according to an ASTM or local standard. However, problems arise when the material of interest comes in a shape that is not amenable to standard specimen fab- rication and testing methods. For example, the Young's mod- ulus of a soft rubber coating on an aluminum roller used in a laser printer recently needed to be determined. The thick- ness of the thin rubber coating varied somewhat over length of the roller, making it nearly impossible to obtain a tensile specimen of uniform thickness or width. In addition to the difficulty of creating a uniform specimen, a further compli-

cut surfaces of the specimens were not completely flat, sug- gesting the use of eqs (1) and (2) would be in error.

USE OF ARCHIMEDES' PRINCIPLE Figure 3 illustrates how the Archimedes principle was used to measure the average cross-sectional area of the specimens. A specimen is weighed in air and in water and the volume (V) of the en- tire specimen can be calcu- lated using:

V = (3) W a r - Wwater

Figure 2 shows the measurement of the specimen thickness with a dial caliper. Each specimen was measured four times at the center, always with the caliper face away from sight to prevent measurement shopping. These measurements were averaged and areas were calculated using eqs (1) and (2). The results yielded slightly different cross-sectional ar- eas for each specimen. However, a correlation between these numerical results and visual observations of the specimens could not be made. At this point two things became appar- ent. First, using the nominal dimensions for R and t with eqs (1) and (2) shows that a change of .005 in. in the mea- surement of t causes about an 8 percent change in the cross- sectional area. In other words, an inconsistent squeeze on the caliper could make a significant imDact on the

cation came in measuring the width and thickness of the compliant specimen.

SPECIMEN PREPARATION AND ANALYSIS Three tensile specimens were sliced from the coated roller with a razor blade. Specimen sizes were visibly different. A sketch of the cross section of the roller is shown in Figure 1. The cross section of one tensile specimen cut from the roller is also shown. The outer radius (R ) of the coated roller was 0.306 in. at the center with a nominal coat- ing thickness of 0.109 in. From Fig. 1 it can be shown that the angle 0 is:

0 = cos-1 (R) R - t

and the cross-sectional area is:

A = (0 - cos0 sinO)R2 (2)

Using nominal values for R and cover thickness ( t ) yields a nominal cross-sectional area of 0.0355 in2. The biggest prob- lem with this approach in calculating the cross sectional area of the tensile specimen is that is provides the area at one point along the specimen. The thickness varied some- what with position along the roller. Hence, the calculation of numerous cross sectional areas along the length of the spec- imen would have been required.

S. Harris is a Graduate Student, and E.M. Odem and B.W. Tew are Associate Professors at the Department ofMechanica1 Engineering at the Uniuersity ofldaho,

. Moscow, ID. ' Fig. 2 Thickness measurement of rubber tensile specimen

Septernber/October 1998 EXPERIMENTAL TECHNIQUES I7

Table I -Rubber Specimen Measurements and Average Cross Sectional Area

SAMPLE

I

2

SUBMERGED SAMPLE AVERAGE CROSS WEIGHT WEIGHT VOLUME DENSITY LENGTH SECTION AREA

0.00633 Ib. 0.000882 Ib. 0.151 in1 0.0419 Iblin’ 5.00 in 0.0302 in2

0.00598 Ib. 0.000844 Ib. 0. I43 in1 0.0420 Iblin’ 4.95 in 0.0288 in2

3

Fig. 3 Archimedes’ Principle applied to rubber tensile specimens

0.00633 Ib. 0.000869 Ib. 0.152 in’ 0.0418 Iblin’ 4.98 in 0.0304 in2

v = 85.47’~ + 0.02

I I I I

0.00 0.02 0.04 0.06 0.08 0.10

True Strain

. Fig. 4 Modulus of elasticity of rubber samples yields an average cross-sectional area for the specimen. The ,

measured data and the resulting cross-sectional areas are . shown in Table 1. Weights for each sDecimen were obtained . CONCLUSION using a Denver Instrukent Co. modk XL-300 scale with 1- milligram resolution. The average cross-sectional areas de- termined using Archimedes’ principle were about 15 percent smaller than those obtained from dial caliper measurements averaged over the length of the specimens.

TENSILE TESTING Each sample was tested in a SATEC model T-5000 test ma- chine to obtain load versus cross-head displacement data. The load was divided by the appropriate cross-sectional area and the displacement data was divided by the gage length (Table 1). The resulting stresses and strains were converted to true stress and true strain and are shown in Fig. 4. The results are very repeatable from sample to sample. A linear regression of the data yielded a Young‘s modulus of 85.5 psi.

* The use of Archimedes’ principle in determining the average ’ cross sectional area for odd shaped or flexible specimens is

inexpensive, simple, and effective. In the application dis- : cussed in this paper, the Young’s modulus of the rubber coat- . ing was used in an ANSYS finite element analysis to predict . the nip width of the rubber coated roller, i.e., the width of . the contact patch between the rubber coated roller and the

heating element of a laser printer. These numerical predic- * tions were verified experimentally. Analytical predictions * and experimental data were almost identical. If nominal or * dial caliper measurements had been used to determine the ‘ average cross-sectional area of the specimens we would have *

had to explain about a fifteen percent discrepancy in our : results. We thought that such a simple but accurate ap- . proach might be useful for other experimentalists as well.

18 EXPERIMENTAL TECHNIQUES SeptemberlOctober 1998