Resonating-Orthotropic-Cube Method for Elastic Constants

Following studies by Demarest (1969) and by Ohno (1976), we describe measurements and analysis that yield, from a single cube-shape specimen, in a single measurement, the complete set of anisotropic elastic-stiffness constants, the C_ij. Experimentally, we place a cubic specimen between two piezoelectric transducers, which excite and detect the cube's macroscopic free-vibration (fundamental-mode) frequencies, up to 10 MHz. From the specimen's shape, size, and mass, and from the measured resonance-frequency spectrum, we analyze for the C_ij within a given tolerance ϵ_i: . (No sum on i.) Here ƛ_i relates to the measured resonance frequencies, and λ_i represents eigenvalues calculated by a Rayleigh-Ritz method using Legendre-polynomial approximating functions. Legendre-polynomial orthogonality ensures a diagonal mass matrix [m], which simplifies the resulting eigenvalue problem: . For materials with certain symmetries, the coefficient matrix [k] reduces to a block-diagonal matrix, which reduces computational effort and simplifies vibration-mode identification.