Resonant ultrasound spectroscopy

Resonant ultrasound spectroscopy (RUS) is a laboratory technique used in geology and material science to measure fundamental material properties involving elasticity. This technique relies on the fact that solid objects have natural frequencies at which they vibrate when mechanically excited. The natural frequency depends on the elasticity, size, and shape of the object—RUS exploits this property of solids to determine the elastic tensor of the material. The great advantage of this technique is that the entire elastic tensor is obtained from a single crystal sample in a single rapid measurement. At lower or more general frequencies, this method is known as acoustic resonance spectroscopy. Resonant ultrasound spectroscopy (RUS) is a laboratory technique used in geology and material science to measure fundamental material properties involving elasticity. This technique relies on the fact that solid objects have natural frequencies at which they vibrate when mechanically excited. The natural frequency depends on the elasticity, size, and shape of the object—RUS exploits this property of solids to determine the elastic tensor of the material. The great advantage of this technique is that the entire elastic tensor is obtained from a single crystal sample in a single rapid measurement. At lower or more general frequencies, this method is known as acoustic resonance spectroscopy. Interest in elastic properties made its entrance with 17th century philosophers, but the real theory of elasticity, indicating that the elastic constants of a material could be obtained by measuring sound velocities in that material, was summarized only two hundred of years later. In 1964, D. B. Frasier and R. C. LeCraw used the solution calculated in 1880 by G. Lamè and H. Lamb to solve the forward problem, and then inverted it graphically, in what may be the first RUS measurement. Nevertheless, we had to wait the participation of geophysics community, interested in determining the earth's interior structure, to solve also the inverse problem: in 1970 three geophysicists improved the previous method and introduced the term resonant sphere technique (RST). Excited by the encouraging results achieved with lunar samples, one of them gave one of his students the task of extending the method for use with cube shaped samples. This method, now known as the rectangular parallelepiped resonance (RPR) method, was further extended by I. Ohno in 1976. Finally, at the end of eighties, A. Migliori and J. Maynard expanded the limits of the technique in terms of loading and low-level electronic measurements, and with W. Visscher brought the computer algorithms to their current state, introducing the final term resonant ultrasound spectroscopy (RUS). Firstly solve the problem of calculating the natural frequencies in terms of sample dimensions, mass, and a set of hypothetical elastic constants (the forward problem). Then apply a nonlinear inversion algorithm to find the elastic constants from the measured natural frequencies (the inverse problem). All RUS measurements are performed on samples that are free vibrators. Because a complete analytical solution for the free vibrations of solids does not exist, one must rely on approximations. Finite element methods base on balancing the forces on a differential volume element and calculating its response. Energy minimization methods, on the other hand, determine the minimum energy, and thus the equilibrium configuration for the object. Among the energy minimization techniques, the Lagrangian minimization is the most used in the RUS analyses because of its advantage in speed (an order of magnitude smaller than the finite-element methods). The procedure begins with an object of volume V, bounded by its free surface S. The Lagrangian is given by L = ∫ V ( K E − P E ) d V ( 1 ) {displaystyle L=int _{V}(KE-PE)dV(1)} where KE is the kinetic energy density K E = 1 2 ∑ i ρ ω 2 u i 2 ( 2 ) {displaystyle KE={frac {1}{2}}sum _{i} ho omega ^{2}u_{i}^{2}(2)} and PE is the potential energy density