Theoretical Investigations into Sample Size Effects on Ultrasonic Measurements of Elastic Moduli

Abstract

The relation between ultrasonic velocity and Young's modulus depends on sample dimension transverse to the direction of ultrasound propagation. For a sufficiently small transverse dimension, the phase and group velocities are the same and are independent of transverse dimension and independent of Poisson's ratio. For infinitely large transverse dimensions, the phase and group velocities are the same and dependent on Poisson's ratio. At intermediate sizes, several modes of ultrasonic wave propagation combine; the phase velocity is sensitive to transverse sample dimension, due to resonance effects and is not equal to the group velocity. The usual determination of ultrasonic velocity from the measured transit time of an ultrasonic pulse yields the group velocity. Analytical solutions consider the propagation of pure sine waves and consequently yield the phase velocity. This work extends previous analyses by considering propagation of a relatively short ultrasonic pulse, typical of those used for measurement of wave speed from the pulse propagation time. The main conclusion is that the phase velocity and transverse variation of the sine waves that do propagate are frequency dependent, resulting in distortion of a propagated pulse that is composed of sine waves of many frequencies. In the case studied here, the distortion is severe enough that the apparent wavespeed depends considerably on sample length and on the area of the sample surface in contact with the transducer.