Resonant ultrasound spectroscopy

This technique relies on the fact that solid objects have natural frequencies at which they vibrate when mechanically excited.

In 1964, D. B. Frasier and R. C. LeCraw used the solution calculated in 1880 by G. Lamé 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 for the participation of geophysics community, interested in determining the Earth's interior structure, to solve the inverse problem: in 1970 three geophysicists improved the previous method and introduced the term resonant sphere technique (RST).

Finally, at the end of the 1980s, 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).

Because a complete analytical solution for the free vibrations of solids does not exist, one must rely upon approximations.

is the ith component of the displacement vector, ω is the angular frequency from harmonic time dependence,

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:

[3] Setting the first term equal to zero yields the elastic wave equation.

that satisfies the previously mentioned conditions are those displacements that correspond to ω being a normal mode frequency of the system.

[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".

in a set of basis functions appropriate to the geometry of the body, substituting that expression into Eq.

The stationary points of the Lagrangian are found by solving the eigenvalue problem resulting from Eq.

[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.

(n=1,2,...) is calculated using estimated values for the elastic constants and the known sample dimensions and density.

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] The most common method for detecting the mechanical resonant spectrum is illustrated in Fig.

One transducer is used to generate an elastic wave of constant amplitude and varying frequency, whereas the other is used to detect the sample's resonance.

[2] Rather, keeping at minimum the couple between them, you get a good approximation to free surface boundary conditions and tend to preserve the Q, too.

The sample, either a fully dense polycrystalline aggregate or a single crystal, is machined in to a regular shape.

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.

For the higher symmetries, it is convenient to have different lengths edges to prevent a redundant resonance.

[4] Polycrystalline samples should ideally be fully dense, free of cracks and without preferential orientation of the grains.

Once prepared, the density must be measured accurately as it scales the entire set of elastic moduli.

[1] 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.

Corners are used because they provide elastically weak coupling, reducing loading, and because they are never vibrational node points.

[4] As a versatile tool for characterizing elastic properties of solid materials, RUS has found applications in a variety of fields.

In geosciences one of the most important applications is related to the determination of seismic velocities in the Earth's interior.

In a recent work,[7] for example, the elastic constants of hydrous forsterite were measured up to 14.1 GPa at ambient temperature.

This study showed that aggregate bulk and shear moduli of hydrous forsterite increase with pressure at a greater rate than those of the corresponding anhydrous phase.