Resonant ultrasound spectroscopy

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 upon approximations. Finite element methods are based on balancing the forces applied to a differential volume element, then 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

${\displaystyle L=\int _{V}(KE-PE)dV(1)}$

where KE is the kinetic energy density

${\displaystyle KE={\frac {1}{2}}\sum _{i}\rho \omega ^{2}u_{i}^{2}(2)}$

and PE is the potential energy density

${\displaystyle PE={\frac {1}{2}}\sum _{i,j,k,l}c_{i,j,k,l}{\frac {du_{i}}{dx_{j}}}{\frac {du_{k}}{dx_{l}}}(3)}$

Fig. 1: Computer-generated illustrations of some normal modes of vibrations for a rectangular parallelepiped sample.

Here, ${\displaystyle u_{i}}$ is the ith component of the displacement vector, ω is the angular frequency from harmonic time dependence, ${\displaystyle c_{i,j,k,l}}$ is a component of the elastic stiffness tensor, and ρ is the density. Subscripts i, j, etc., refer to Cartesian coordinate directions.

To find the minimum of the Lagrangian, calculate the differential of L as a function of u, the arbitrary variation of u in V and on S. This gives:

${\displaystyle \delta L=\int _{V}{\Bigl \{}\sum _{i}{\Bigl [}\rho \omega ^{2}u_{i}-\sum _{j,k,l}c_{i,j,k,l}{\frac {\delta ^{2}u_{k}}{\delta x_{j}\delta x_{l}}}{\Bigr ]}\delta u_{i}{\Bigr \}}dV-\int _{S}{\Bigl \{}\sum _{i}{\Bigl [}\sum _{j,k,l}{\vec {n}}c_{i,j,k,l}{\frac {\delta u_{k}}{\delta x_{l}}}{\Bigr ]}du_{i}{\Bigr \}}dS(4)}$

Because ${\displaystyle u_{i}}$ is arbitrary in V and on S, both terms in square brackets must be zero.[3] Setting the first term equal to zero yields the elastic wave equation. The second square bracketed term is an expression of free surface boundary conditions; ${\displaystyle {\vec {n}}}$ is the unit vector normal to S. For a free body (as we assume it), the latter term sums to zero and can be ignored.

Thus the set of ${\displaystyle u_{i}}$ that satisfies the previously mentioned conditions are those displacements that correspond to ω being a normal mode frequency of the system. This suggests that the normal vibrations of an object (Fig. 1) may be calculated by applying a variational method (in our case the Rayleigh-Ritz variational method, explained in the next paragraph) to determine both the normal mode frequencies and the description of the physical oscillations.[4] To quote Visscher, getting both equations from the basic Lagrangian is "a mathematical fortuity that may have occurred during a lapse in Murphy's vigilance".[5]

The actuation of this approach requires the expansion of the ${\displaystyle u_{i}}$ in a set of basis functions appropriate to the geometry of the body, substituting that expression into Eq. (1) and reducing the problem to that of diagonalizing a N×N matrix (eigenvalue problem). The stationary points of the Lagrangian are found by solving the eigenvalue problem resulting from Eq. (4), that is,

${\displaystyle \omega ^{2}Ea=\Gamma a(5)}$

where an are the approximations to the motion expanded in a complete basis set, E comes from the kinetic energy term, and Γ comes from the elastic energy term. The order of the matrices is ~10^3 for good approximations.

Equation (5) determines the resonance frequencies from the elastic moduli.[3]

The inverse problem of deducing the elastic constants from a measured spectrum of mechanical resonances has no analytical solution, so it needs to be solved by computational methods. For the indirect method, a starting resonant frequency spectrum, ${\displaystyle f_{n}^{cal}}$ (n=1,2,...) is calculated using estimated values for the elastic constants and the known sample dimensions and density. The difference between the calculated and measured resonance frequency spectrum, ${\displaystyle f_{n}^{mea}}$ (n=1,2,...) is quantified by a figure of merit function,

${\displaystyle F=\sum _{n}w_{n}(f_{n}^{cal}-f_{n}^{mea})^{2}(6)}$

where ${\displaystyle w_{n}}$ (n=1,2,...) are weight coefficients reflecting the confidence on individual resonance measurements. Then, a minimization of the function F is sought by regressing the values of all the elastic constants using computer software developed for this process.[6]

RUS can be applied to a great range of samples sizes, with a minimum in the order or a few hundred micrometers, but for the measurement of mineral elasticity it is used on samples typically between 1 mm and 1 cm in size.

The sample, either a fully dense polycrystalline aggregate or a single crystal, is machined in to a regular shape.[1] Theoretically any sample shape can be used, but you obtain a substantial saving in computational time using rectangular parallelepiped resonators (RPR), spherical or cylindrical ones (less time savings with cylinders).

Fig. 3: The sample assembly for a RUS variable-temperature measurement.

Since the accuracy of the measure depends strictly on the accuracy in the sample preparation, several precautions are taken: RPRs are prepared with the edges parallel to crystallographic directions; for cylinders only the axis can be matched to sample symmetry. RUS is rarely used for samples of lower symmetry, and for isotropic samples, alignment is irrelevant. For the higher symmetries, it is convenient to have different lengths edges to prevent a redundant resonance.

Measurements on single crystals require orientation of the sample crystallographic axes with the edges of the RPR, to neglect the orientation computation and deal only with elastic moduli.[4]

Polycrystalline samples should ideally be fully dense, free of cracks and without preferential orientation of the grains. Single crystal samples must be free of internal defects such as twin walls. The surfaces of all samples must be polished flat and opposite faces should be parallel. Once prepared, the density must be measured accurately as it scales the entire set of elastic moduli.[1]

Unlike all other ultrasonic techniques, RUS ultrasonic transducers are designed to make dry point contact with the sample. This is due to the requirement for free surface boundary conditions for the computation of elastic moduli from frequencies. For RPRs this requires a very light touch between the sample's corners and the transducers. Corners are used because they provide elastically weak coupling, reducing loading, and because they are never vibrational node points. Sufficiently weak contact ensures no transduced correction is required.[4]