Effects of crystalline anisotropy on resonant acoustic loss of torsional quartz viscometers.

Vibrational modes of unrestrained elastic cylinders of trigonal crystals are studied using Ritz-based polynomial approximations for displacements formulated in rectangular Cartesian coordinates. The selected orientation of the threefold trigonal axis is perpendicular to the cylinder axis, corresponding to the configuration employed in torsional quartz viscometry (TQV) for characterizing Newtonian fluids. A revised working equation for TQV is derived, incorporating effects of crystalline anisotropy, and Ritz results are used to numerically quantify effects of acoustic radiation from surface-normal displacements and viscous loss from nontorsional surface-parallel displacements of resonant modes corresponding to the purely torsional modes of isotropic cylinders traditionally employed as an approximation in TQV analysis. For a cylinder typical of TQV, with 3 mm diameter and 50 mm length, the anisotropy-related correction to the extracted fluid viscosity is a positive shift of 36 ppm relative to the isotropic approximation, if radiative losses are neglected. This contribution is independent of fluid properties. Radiative losses depend on the properties of the fluid and reduce the extracted viscosity. The total magnitude of corrections varies between several tens of parts per million for low density gases to values on the order of 0.01% for normal liquids near atmospheric pressure and 0.06% for superfluid helium.

Continuum-based free vibration of circular trigonal and isotropic plates.

The free vibration behavior of completely unrestrained elastic circular plates with trigonal and isotropic material symmetry is studied with an approach involving approximate continuum solutions to the three-dimensional theory of linear elasticity. Of primary interest are (1) the influence of trigonal material symmetry on the modes of free vibration and (2) the accuracy of thin plate theory relative to the more exact three-dimensional theory. Resonant frequencies are calculated from the weak form of the equations of motion for the plate through the use of approximation functions and the Ritz method formulated in cylindrical coordinates. This approach enables the resulting eigenvalue problem to be split through group-theoretical symmetry analysis. Representative examples are given and quantitative limits are discussed.

Vibrational modes of free nanoparticles: from atomic to continuum scales.

Vibration analysis of free standing silicon nanoparticles, with sizes ranging from 0.732 to 4.223 nm, are calculated using two different methods: molecular mechanics and classical continuum elasticity. Three different geometries are studied: cubes, spheres, and tetrahedrons. Continuum mechanics methods provide good estimates of the lowest natural frequency of particles having at least 836 (R>1.5 nm) and 800 (R>1.28 nm) atoms for cube- and tetrahedron-shaped nanostructures, respectively. Equations for vibrational frequencies of smaller particles as a function of size are proposed. The vibrational modes obtained from both methods were practically the same for the sphere- and tetrahedron-shaped particles with a large number of atoms. However, for the cube geometry only the shape of the modes corresponding to the lowest couple of frequencies occur in the same order. In general, vibrational modes shapes obtained using both methods are the same although the order in which they appear may be shifted.

Vibrational modes of nanolines.

Brillouin-light-scattering spectra previously have been shown to provide information on acoustic modes of polymeric lines fabricated by nanoimprint lithography. Finite-element methods for modeling such modes are presented here. These methods provide a theoretical framework for determining elastic constants and dimensions of nanolines from measured spectra in the low gigahertz range. To make the calculations feasible for future incorporation in inversion algorithms, two approximations of the boundary conditions are employed in the calculations: the rigidity of the nanoline/substrate interface and sinusoidal variation of displacements along the nanoline length. The accuracy of these approximations is evaluated as a function of wavenumber and frequency. The great advantage of finite-element methods over other methods previously employed for nanolines is the ability to model any cross-sectional geometry. Dispersion curves and displacement patterns are calculated for modes of polymethyl methacrylate nanolines with cross-sectional dimensions of 65 nm × 140 nm and rectangular or semicircular tops. The vibrational displacements and dispersion curves are qualitatively similar for the two geometries and include a series of flexural, Rayleigh-like, and Sezawa-like modes.

Elastic constants of layers in isotropic laminates.

The individual laminae elastic constants in multilayer laminates composed of dissimilar isotropic layers were determined using ultrasonic-resonance spectroscopy and the linear theory of elasticity. Ultrasonic resonance allows one to measure the free-vibration response spectrum of a traction-free solid under periodic vibration. These frequencies depend on pointwise density, laminate dimensions, layer thickness, and layer elastic constants. Given a material with known mass but unknown constitution, this method allows one to extract the elastic constants and density of the constituent layers. This is accomplished by measuring the frequencies and then minimizing the differences between these and those calculated using the theory of elasticity for layered media to select the constants that best replicate the frequency-response spectrum. This approach is applied to a three-layer, unsymmetric laminate of WpCu, and very good agreement is found with the elastic constants of the two constituent materials.

Traction-free vibrations of finite trigonal elastic cylinders.

The unrestrained, traction-free vibrations of finite elastic cylinders with trigonal material symmetry are studied using two approaches, based on the Ritz method, which formulate the weak form of the equations of motion in cylindrical and rectangular coordinates. Elements of group theory are used to divide approximation functions into orthogonal subsets, thus reducing the size of the computational problem and classifying the general symmetries of the vibrational modes. Results for the special case of an isotropic cylinder are presented and compared with values published by other researchers. For the isotropic case, the relative accuracy of the formulations in cylindrical and rectangular coordinates can be evaluated, because exact analytical solutions are known for the torsional modes. The calculation in cylindrical coordinates is found to be more accurate for a given number of terms in the series approximation functions. For a representative trigonal material, langatate, calculations of the resonant frequencies and the sensitivity of the frequencies on each of the elastic constants are presented. The dependence on geometry (ratio of length to diameter) is briefly explored. The special case of a transversely isotropic cylinder (with the elastic stiffness C14 equal to zero) is also considered.

Symmetrization of Ritz approximation functions for vibrational analysis of trigonal cylinders.

In the Ritz method of calculating vibrational normal modes, a set of finite series approximation functions provides a matrix eigenvalue equation for the coefficients in the series and the resonant frequency. The matrix problem usually can be block-diagonalized by grouping the functions into subsets according to their properties under the symmetry operations that are common to the specimen geometry and crystal class. This task is addressed, in this study, for the case of cylindrical specimens of crystals belonging to one of the higher trigonal crystal classes. The existence of doubly degenerate resonant modes significantly complicates the analysis. Group-theoretical projection operators are employed to extract, from series approximation functions in cylindrical coordinates, the terms that transform according to each irreducible representation of the point group. This provides a complete symmetry-based block diagonalization and categorization of the modal symmetries. Off-diagonal projection operators are used to provide relations between the displacement patterns of degenerate modes. The method of analysis is presented in detail to assist in its application to other geometries, crystal structures, and/or forms of Ritz approximation functions.