User:GeoZorby/Draft RUS

Introduction

Resonant ultrasound spectroscopy (RUS) is a laboratory technique whose use in

History

Interest in elastic properties made its entrance with

Theory

First you have to solve the problem of calculating the

).

Lagrangian minimization

All RUS measurements are performed on samples that are free vibrators. Because a complete

(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

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

Here, ${\displaystyle u_{i}}$ is the ith component of the

from harmonic time dependence, ${\displaystyle c_{i,j,k,l}}$ is a component of the elastic tensor, and ρ is the

Cartesian coordinate

directions.
To find the minimum of the Lagrangian, you must 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.

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

Rayleigh-Ritz Variational Method

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 (

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 a 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

The Inverse Problem

The

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

Measurements

The most common method for detecting the mechanical resonant spectrum is illustrated in Fig. 2, where a small parallelepiped-shaped sample is lightly held between two

${\displaystyle f_{n}}$ (from which the elastic constants are determined) and the is always on the order of a few parts per million.
Unlike in a conventional ultrasonic measure, in a method that resonates the sample a strong coupling between the transducer and the sample is not required, because the sample behaves as a natural amplifier.^[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. For variable-temperature measurements the sample is held between the ends of two buffer rods that link the sample to the transducers (Fig. 3) because the transducers must be kept at room temperature. In terms of pressure, on the contrary, there is a limit of only a few bars, because the application of higher pressures leads to dampening of the resonances of the sample.^[1]

Samples

RUS can be applied to a great range of samples sizes, with a minimum in the order or a few hundred

, 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 compressed 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).

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

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
Polycrystalline samples should ideally be fully dense, free of cracks and without preferential orientation of the

Transducers

Unlike all other ultrasonic techniques, RUS

Applications

As a versatile tool for charactering elastic properties of solid materials, RUS has found applications in a variety of fields.
In

.

References