1. Introduction

Elastic moduli of solids are of interest both as fundamental data needed in the calculation of many solid state properties and as engineering data [1].' Modern sound velocity and resonance techniques permit their determination to high accuracy provided that the mass and demensions of specimens can be accurately determined and provided that care is taken to eliminate or allow for certain loading effects. One popular technique is to determine the frequencies of the fundamental normal modes of vibration by the Forster method [2] in which a slender rod or bar is suspended horizontally on two fine wires or threads passing around the bar near the nodes of vibration. One wire is vibrated by a driver energized by a variable frequency oscillator. The other wire is attached to a detector. The driving frequency is varied until the desired mode of resonance (usually flexural or torsional) is excited and the frequency at resonance is measured. Procedures to identify the mode which is excited and accurate equations for calculating elastic moduli from these frequencies have been described [3].

The simplicity of the Forster method and its adaptability to measurements at high temperature have led to its widespread use. The accuracy with which fre

Figures in brackets indicate the literature references at the end of this paper.

quency can be measured with a crystal-controlled counter and the reproducibility of a given resonance frequency in a particular experimental arrangement sometimes obscures the fact that serious systematic errors can be present. The two most gross errors are the improper identification of the vibrational mode of the specimen which is excited or the exciting of a vibration of some part of the apparatus other than the specimen. Such errors may appear unlikely to pass undetected, but experience in consulting with operators just beginning to use the method has shown that such errors do occur. More subtle errors which are usually smaller and may pass unnoticed even by experienced operators include loading effects caused by the suspension or the surrounding atmosphere and temperature errors. The loading effects cause the specimen to resonate at a frequency slightly different from that of free vibration. The free vibration required for the specimen also makes it impossible to contact the specimens with a thermocouple unless the latter is used as one of the suspensions. This is usually undesirable because Ordinarily a thermocouple placed very near the speci of the mechanical characteristics of thermocouple wire. men is used to measure the temperature. The temperature difference between this couple and the specimen may be appreciable in high temperature measurements. It is, therefore, very useful for an operator to have a standard specimen with accurately known resonance frequencies and to have the temperature de

pendence of these frequencies in order to estimate the magnitude of these errors in his apparatus. The calculation of elastic moduli from resonance frequencies is sufficiently complicated that an example is useful for this purpose also. The complexity in these calculations arises because for a complete determination of the two independent moduli an iteration process is required to solve the simultaneous equations.

The foregoing are the primary reasons leading to the development of a dynamic elasticity standard. A secondary motivation is the possibility of using the same bars also as a standard for static measurement of Young's modulus in flexure. The static (isothermal) moduli should be less than the dynamic (adiabatic) moduli by a small amount calculable from an equation given, for example, by Nye [4]. For alumina the difference for Young's modulus is only 0.2 percent. Static flexural measurements, when carefully made and correctly analyzed show good agreement with dynamic measurements when this small difference is allowed for. Sources of error (up to several percent) exist in static measurement techniques which are difficult to locate and eliminate. Thus, dynamic standards might be useful for debugging static measurements.

The present elasticity standards were chosen to meet the following criteria: (1) They should be in a size range normally used for dynamic elasticity measurements. (2) They should permit very accurate values of resonance frequencies at 25 °C to be certified for use in checking room temperature measurement procedure. (3) Their size and shape should permit accurate values of elastic moduli at 25 °C to be calculated and certified. (4) The material used for the standard should be stable over the temperature range 25 to 1000 °C to permit certification of the ratio of frequency at elevated temperature to frequency at 25 °C over this temperature range (from this information the frequencies themselves and the elastic moduli can be calculated over this temperature range).

2. Material

A standard elasticity bar should certainly be stable under normal use. The most likely sources of trouble with these bars are mechanical abrasion and thermal effects. A standard elasticity bar should always be handled with care so that damage from abrasions is not likely, but it is desirable that the material of the bar be sufficiently hard so that it would not be scratched by contact with metals. The bar should withstand heating in air to 1000 °C without change. These requirements suggest the use of a hard, dense oxide.

Fused silica was considered for this use and would apparently be quite suitable at 25 °C. It possesses the advantages of elastic isotropy and of a high degree of homogeneity. However, the density changes slowly with time at 903 °C [5] so that the temperature range for use would be limited to somewhat below 900 °C. Furthermore, devitrification at the surface occurs slowly at temperatures near 1000 °C; its rate may depend on atmosphere and surface condition. Fused

silica was therefore not chosen despite its homogeneity and ready availability because of the uncertainty about the temperature limit which would certainly have to be below 900 °C.

Polycrystalline alumina is elastically isotropic i formed by a process which does not produce a preferred orientation. It is even harder than fused silica so that it is very abrasion resistant. Polycrystalline alumina is normally porous; this reduces the elastic moduli but is not objectional unless the amount of porosity varies sufficiently with position to cause an unacceptable degree of inhomogeneity. A process for the production of nearly pore-free polycrystalline alumina has been developed and thin pieces are available commercially [6-9]. This process requires the presence of a small amount of MgO to reduce the rate of grain growth. The amount of MgO used (typically 0.25 wt %) is chosen to be within the range of solid solubility at the sintering temperature (typically 1850 °C). The extent of solid solubility depends on atmosphere and grain size; more importantly, the limit of solid solubility decreases with decreasing temperature below 1730 °C [10]. Alumina produced by this process will apparently be in a metastable state at temperatures sufficiently below the sintering temperature. No work on the rate of precipitation of excess MgO at 1000 °C has been located, but the strong temperature dependence of diffusion rates in alumina suggests that at temperatures below 1200 °C precipitation would be too slow to be significant in a period of 1000 hours. Polycrystalline alumina was therefore chosen and is believed to be sufficiently stable at 1000 °C for the present purposes although its behavior over extremely long times is not known. After consultation with R. C. Anderson [11] and P. J. Jorgensen [12] it was decided to use 0.1 percent MgO and to isostatically cold press and sinter a single block of nominal dimensions 2 by 25% by 47% in from which 40 specimen bars would be machined. The single-block approach offered the advantages of chemical homogeneity of the starting powder and of lower cost. Regions near the center of an isostatically cold pressed and sintered block are typically more porous than those near the surface, and it was recognized that the large size chosen might result in appreciable total porosity and some variation in porosity throughout the block. It was believed, however, that these effects would be small enough to be acceptable, and the single block was manufactured. Sintering was carried out under the direction of P. J. Jorgensen [12].

The sintered block was diamond ground to produce flat surfaces with opposite surfaces parallel and adjacent surfaces perpendicular. The density was determined from the mass (corrected for air bouyancy) and the dimensions (measured after overnight equilibration in a constant temperature room at 23.9 °C). The result was 3.933±0.002 gm/cm3 where the uncertainty is the gross standard deviation calculated from the individual standard deviations of the mass and length determinations. The density of pure single crystal alumina is 3.986 gm/cm3 as determined by hydrostatic weighing [13] or 3.987 gm/cm3 from x-ray parameter measure

and Tefft [3], who also describe the electronic components required. The following specific items of equipment were used in the present work but many similar components exist and would presumably be equally satisfactory. The driver was an Astatic model M 41-8 phonograph record recording head. It was driven by a Hewlett-Packard 463-A precision amplifier driven in turn by a General Radio frequency synthesizer Model 1163-AR7C with automatic sweep unit. The frequency was monitored with a Beckman timer Model 6147 which in turn was standardized with the NBS 100 kHz frequency so that the driving fre quency measurement should be accurate to 1 part in 10. The pickup was an Astatic L-121 crystal phonograph cartridge for use in air or an Astatic 51-3 ceramic phonograph cartridge for use at reduced pressures. The signal was amplified by an Applied Cybernetics System Model 1A260V battery-powered lownoise amplifier. The signal was continuously monitored on a Dumont Model 304-AR oscilloscope which was supplemented for measurements of response to automatic frequency sweep by a Tektronix type 564 storage oscilloscope with a type 3A72 dual-trace plug-in amplifier and a type 2B67 plug-in time base. The use of a frequency synthesizer having a sweep unit is very desirable for this work which requires the combined capabilities of searching over wide frequency ranges (to find the resonances) and of very precise tuning (to determine the frequency of a resonance response as little as 0.1 Hz broad). The automatic feature of the sweep unit used allowed smooth frequency change (typically 1 Hz frequency change in 10 seconds) in locating the exact maximum of the vibrational amplitude corresponding to resonance. Even with very low sweep rates the slow response of the specimens caused a shift of the maximum amplitude of response from the resonance frequency, but by sweeping in both directions, recording the amplitudes on the storage oscilloscope, and noting how the maximum responses approached each other as the sweep rate was lowered this effect could be overcome.

D13 were lightly inscribed on one end with a diamond point scriber. The block was then diamond sawed into bars in accordance with this pattern. The four large bars A3, B3, C9, and D9 were reserved for possible machining of cylinders and the remaining 40 bars were prepared as elasticity standards. Each bar was given a fine diamond grind to produce flat surfaces with opposite sides parallel and adjacent sides perpendicular. The mass (corrected for air bouyancy) and the dimensions of each bar are given in table 1.

Bars A3 and B3 were examined by x-ray techniques. The prime concern was the possible existence of a preferred orientation of the crystallites which would cause the specimens to be anisotropic. Back-reflection x-ray patterns showed no such preferred orientation [15]. They also yielded the information that the average crystallite size (determined from the number of spots for a given reflection, the beam size, and the adsorp tion) was 24 μm. High-resolution high-intensity diffractometer patterns showed almost exclusively the a-alumina phase as expected. A very weak pattern, identified as due to the presence of MgAl2O, (spinel), was also present; the intensity, however, indicated that the proportion of MgAl2O, was much less than one percent.

3. Experimental Procedure and Frequency Corrections

The general procedure for exciting a resonance and probing the vibrational pattern with a pickup to identify the vibrational mode has been described by Spinner

Each bar was suspended horizontally using fine silk thread and the loop method of suspension [16] in which both the thread from the driver needle and the thread from the pickup needle are cemented to their respective needles at two points. Each thread then forms a continuous loop and supports the specimen without a knot at any point. Approximate values of the flexural and torsional resonance frequency were determined with the thread placed symmetrically some distance from the nodes. The vibrational pattern was probed to verify the identification of the nodes as previously described [3]. The measured internal friction of lowloss specimens is strongly affected by the suspension position [17]; the resonance frequency is affected to an extent which is much smaller but which is still significant for the present accurate frequency determinations. As in the previous work [17], a specimen was suspended with the driver and pickup threads positioned symmetrically. To obtain a correction for the effect of the suspension one of the specimens was suspended at several pairs of symmetrically located

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was next determined for two bars. The elastic moduli of alumina are known to vary smoothly with temperature and to be approximately linear near 25 °C [18, 19]. The temperature coefficients for the flexural and torsional resonance frequencies for these two bars were determined to be 0.117 Hz/°C and -0.710 Hz/°C, respectively.

The bars are so nearly identical that the same correction factors for suspension effect and temperature can be used for all bars. As can be seen the correction factors for these suspensions and for minor deviations of the test temperature from 25 °C are small (small enough to be neglected in most elastic moduli measurements), but they are not negligible for the present purpose. Accordingly, the following procedure was adopted. A careful measurement of the flexural and torsional resonance frequencies for each bar was made by suspending the bar within a heavy-walled, acoustically insulated box and allowing it to come to temperature equilibrium. The temperature was read to 0.1 °C and each measured resonance frequency was taken as the average of the value corresponding to the maximum amplitude of vibration for a slow increasing-frequency sweep and a decreasing-frequency sweep. Values were determined to 0.01 Hz. These values, corrected as above for temperature and suspension effect are the free resonance frequencies in air at 25 °C.

The atmosphere is known to affect both the resonance frequency and damping and it is sometimes suggested, especially for damping, that the viscous drag on the moving specimen is responsible. While viscous drag may be important for certain types of internal friction measurements (e.g., the torsion of thin cylinders)

FIGURE 3. Torsional resonance frequency as a function of pressure of argon.

Single point is for air at one atmosphere pressure.

it is apparently not the predominant atmospheric effect on the frequencies of specimens of the present size and shape. To determine the cause and size of the effect of atmosphere on frequency, specimens were suspended in a large vacuum chamber which also contained the driver and pickup. The resonance frequency was first measured in air. The chamber was then evacuated and filled in stages with argon. Figures 2 and 3 show the dependence of the flexural and torsional resonance frequencies upon pressure of argon. The straight lines were drawn by a visual fit. Figures 2 and 3 show that the pressure dependence of the resonance frequencies is essentially linear over the entire range of measurement. This differs markedly from the bahavior of the viscosity which changes little down to pressures of about 1 mm of Hg and then decreases rapidly with futher lowering of the pressure. Such a linear dependence upon pressure is, however, to be expected from the variation of the acoustic impedance with pressure. For small acoustical loading the frequency should change in proportion to the acoustic impedance. The latter is proportional to pc which for an ideal gas is proportional to (pp)12 or to y2M2p where p is density, c is velocity of sound, p is pressure, y=C/C, and M is mean molecular weight [20]. The molecular weights of argon and air are 39.95 and 29.12, respectively [21]; their y values are 1.668 and 1.403 [20]. Thus the ratio of the shift of resonance frequency in going from vacuum to 1 atm pressure of argon to the same shift for air is predicted to be (yMon/(YM) 1.277. The measured frequency shifts are 1.24 Hz and 0.99 Hz for flexure giving a ratio of 1.25; the corresponding values for torsion are 2.90 Hz and 2.25 Hz giving a ratio of 1.29. The

argon

air

=