Generalized inverse problems in resonant ultrasound spectroscopy

Abstract

Determination of the elastic constants by RUS is an inverse problem because experimentally obtained resonant frequencies cannot be directly recalculated into the elastic constants. Instead, an approximate spectrum is calculated from the dimensions and crystallographic orientation of the sample, its mass, and a set of 'guessed' elastic constants, and the difference between this approximate spectrum and the experiment is iteratively minimized. RUS has been used for the determination of either the elastic constants, or crystallographic orientations of the material in the past, but the recent advancements in RUS methodology, in particular, the employment of the scanning laser vibrometry for identification of the vibrational modes, enable inverse determination of most of the input parameters simultaneously. We propose an extension of the classical RUS inversion procedure that allows us to precisely identify the crystallographic orientation and dimensions of the sample in addition to the elastic coefficients. The proposed algorithm was applied to generally oriented iron single crystals. After the shape and orientation optimization, we achieved an unprecedented match between calculated and measured spectrum, including a very high number of utilized resonant modes (>300). We show that the highest modes are extremely sensitive to the crystallographic orientation.