Measurements and uses of elastic constants for determining the technical constants and Debye temperature

Abstract

This dissertation deals with the subject of testing the methodology of the PUCOT (Piezoelectric Ultrasonic Composite Oscillator Technique) for measuring the elastic constants, providing an improved method of using sound velocity measurements to predict the technical constants and an improved computational procedure to determine the elastic Debye temperature and the technical constants from sound velocity measurements. It has been shown that free vibration conditions may be forbidden for the length to diameter ratios used by the PUCOT. More adequate differential equations suitable for the PUCOT have been solved for the required relationships between the measured frequencies and the elastic constants. Experimental results are provided to support the theoretical findings, where it has been shown that the compliances may be obtained using the PUCOT but inverting them to stiffnesses leads to error propagation. It is suggested therefore that the PUCOT should only be used to supplement and/or complement the results of wave transmission methods in measuring the elastic constants. As for using sound velocity measurements as an averaging scheme, it has been shown that averaging schemes based on dynamic deformations follow the trend of those averaging schemes based on static deformations. That is, averaging the reciprocal velocity produces an upper bound for the technical constants similar to that of Voigt. Averaging the reciprocal cubed velocity on the other hand, produces a lower bound similar to the that of the Reuss method. The proposed averaging scheme which averages the reciprocal squared velocity yields intermediate results in good agreement with observations and in excellent agreement with Hill's method. Lastly, it has been shown that the required numerical integration to compute the Debye temperature and the technical constants can be simplified substantially by solving analytically the secular equation for the three phase velocities and using the Simpson 1/3 rule to perform the integration in 1/4 of the spherical shell representing the Brillouin zone in the crystal lattice. It is suggested that summation over 121 crystallographic directions suffices for convergence of the integration. Numerical results concerning the Debye temperature for a large body of materials are obtained and show excellent agreement with those obtained by more laborious procedures.