Estimation and Determination of Isotropic Elastic Constants in Resonant Ultrasound Spectroscopy

Abstract

Accurate measurement of elastic constants in resonant ultrasound spectroscopy (RUS) depends on a perfect matching in the calculated and measured mode frequencies of free vibration of the solid specimen under study. Calculation of these frequencies requires estimated values of the elastic constants of the material. The present work proposes and demonstrates a method to derive initial guess values of the essential parameters for an isotropic and homogeneous material from the acquired RUS spectra itself. Specimen samples are prepared in the shape of rectangular parallelepiped having nearly same cross-sectional dimension but with different lengths. For particular compression (shear) mode corresponding to length l, the frequency f is inversely proportional to l. The slope m of f versus 1/l plot equals half of the compression (shear) velocity and this in turn gives an estimate of $$ c_{11} $$ ($$ c_{44} $$). With these parameters as the input guess parameters, RUS fitting method is executed to find out the best fit results. $$ c_{44} $$ obtained in this way is a good estimate close to its actual value. Further refinement of the parameter values is achieved by identifying volume oscillation mode frequencies for various sample lengths from the best fit frequency table of RUS output. The slope m of f versus 1/l plot for these modes corresponds to the bulk modulus (K). $$ c_{11} $$ calculated from K and $$ c_{44} $$ is a better estimate and a second step fitting method with these parameters as the input values gives final output values with improved accuracy. Elastic constants of commercially available specimens of aluminum, copper, lead, steel and brass are measured in this method assuming macroscopic homogeneity. Results are compared and found to be in good agreement with available literature values.