The Buzz-o-sonic testing system was used to measure the Young’s moduli of seven bone shaped tensile specimens supplied by Dr. Richard Vinci at Lehigh University, Bethlehem, PA, USA. Although equations relating the flexural resonant frequency to the Young’s modulus (frequency equations) are not available in the ASTM standards or in the literature in general for these shapes, several methods were devised to find approximate frequency equations. The coefficients of determination, R2, were in the range 0.9995-0.9999.
The standard bone-shaped tensile specimen has uniform thickness but not width as shown in figure 1.

Figure 1: widths and lengths of bone-shaped tensiles specimen

Note - the specimen has uniform thickness


The standard equation relating the natural flexural frequency to the Young’s modulus of a bar of uniform cross-section can be written1:

eq1

Where:

f = flexural resonant frequency
beta1 = eigenvalue for the first (fundamental) mode of vibration
k = radius of gyration of the cross-section
L = sample length
E = Young’s modulus
rho= bulk density

For a rectangular or square cross-section, k = t/√12 where t = thickness.

For a freely supported bar, beta1~4.730

We are interested in obtaining the Young’s modulus (E) from the flexural frequency (f) thus:

 

eq2

 

Equations (1) and (2) ignore shear and rotary inertia. When these are factored in, we add a correction factor T1 2:

 

eq3

and

eq4

Where:

μ = Poisson’s ratio
L = sample length

 

Now, equations (3) and (4) apply to bars of uniform cross-section. The bone shaped samples tested were not of uniform cross-section and so beta1 will need to be modified*. The work here addresses this problem.

Seven bone-shaped specimens of different metals were used to determine the relevant frequency equations. The dimensions and masses of the bone shaped (Figure 1) tensile specimens are shown in Table 1.

Table 1: Dimensions and masses of the bone shaped samples


Note the small variation in the dimensions (other than the thickness t). Thus, there was a small variation in the total area of each sample. The main differences between the samples then, were in the masses/bulk densities and thicknesses.

The fundamental flexural resonant frequencies (Table 2) were easily excited using the cork impulse tool** included in the Buzz-o-sonic Lab Kit.

Table 2: Flexural resonant frequencies of the bone-shaped samples


However, because the values of beta1 and T1 were not known precisely, the samples were cut into bars of uniform cross-section and their Young’s moduli were measured using equations (3) and (4) (Table 3).

Table 3: Elastic properties of uniform bars cut from bone shaped specimens


The important properties are highlighted in pink. We can then use these properties to estimate the values of beta1 for the bone-shaped samples (Table 4). NOTE: a Poisson’s ratio of 0.3 was used to calculate T1 if T1 was used.

Table 4: Calculated Young’s moduli of the bone-shaped specimens


The fits were very good in all cases. However, the simplest method and one of the easiest from a practical viewpoint*** is the Ecalct* method which was derived from the following equation:

eq5

Where:

f = flexural resonant frequency
rho = bulk density
A = constant
t = thickness
x = exponent

Note: A is approximately a constant assuming that the mean values of L and Atotal do not vary significantly (Table 1). T1 is ignored or partly taken into account through the exponent x.


Ecalct* also gives the best fit, especially for the thicker Zn sample (the maximum deviation occurred for this sample in most cases).

A and t were determined by plotting E/f2rho vs. t as follows:

Figure 2: determining A and x in equation (5)


A power law gave the best fit†. As can be seen, x is -1.96: close to the value of 2 expected from equation (2). Thus, the Young’s modulus of a dog bone sample can be given by††:

eq6

Where t is in m, rho is in kg.m-3, and E is in Pa (SI units)

Using the mean Atotal from Table 1, the density in kg.m-3 can be given by:

eq7

Where:

m = mass in kg
rho = bulk density
atotal= mean total area in m2 = 3.274 10-3
t = thickness in m

 

The results for all of the different methods are shown in Figure 3.

Figure 3: Comparison of calculated Young’s moduli to the measured values


It would seem reasonable then, to simply measure only the mass and thickness of each bone shaped sample and then use one of the * methods shown in Table 4 to calculate the Young’s modulus and bulk density. It appears that the Ecalct* method (Table 4/equation (6)) would be the best choice giving a maximum error of ~1% - i.e. within experimental error. This method is now included in our Elastic Constants Excel workbook (included with the Buzz-o-sonic testing system).

 

*T1 should be modified, but it was found that it could either be safely ignored or equation (4) could be used
**This worked better than the standard impulse tool (3-4mm ball bearing cemented to a plastic strip as per ASM E1876-09…)
***All of the * methods require only that the thickness, mass, and out-of-plane flexural frequency be measured on each sample
†More sophisticated analyses could be used to optimize 1 and T1, but the power fit was more than adequate
††Using SI units and a higher precision fit to the data
  1. Daniel J. Gorman, "Free Vibration Analysis of Beams and Shafts," Publ. John Wiley and Sons (1975, originally published 1930)
  2. A. Spinner, and W. E. Tefft, "A Method for Determining Mechanical Resonance Frequencies and for Calculating Elastic Moduli from these Frequencies," Proceedings ASTM, 61 1221-1239 (1961)