Propagation properties of elastic waves in semi-infinite phononic crystals and related waveguides

The transmission and reflection coefficients of two-dimensional semi-infinite solid-solid phononic crystal systems and fluid-fluid phononic waveguide structures have been investigated. The numerical results show that the transmission spectra for longitudinally and transversally polarized incident waves are different, and the spectra of the transmission and reflection coefficients of the semi-infinite system agree well with the band structure. The numerical results show that when a guided wave incident, localized modes are excited, and different polarities have different coupling efficiencies with the incident guided wave. At the same time, far from the cutoff frequency, the guided wave couples out of semi-infinite waveguide highly efficiently.     

Propagation of elastic waves in hexagonal piezoelastic crystals

The propagation of elastic waves in an arbitrary direction through a hexagonal piezoelectric crystal is investigated. Orientational relations are obtained in analytic form for the effective elastic moduli and the piezoelectric and dielectric constants, and also for the basic parameters of the elastic waves: the phase velocity, electromechanical coefficient, and the divergence between the displacement vector and the energy flux. Expressions are given for the directions in which the electromechanical coefficient takes an extremal value, and the basic parameters are calculated for cadmium sulfide.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 101–105, May, 1977.     

Conical refraction of elastic waves in absorbing crystals

The absorption-induced acoustic-axis splitting in a viscoelastic crystal with an arbitrary anisotropy is considered. It is shown that after “switching on” absorption, the linear vector polarization field in the vicinity of the initial degeneracy point having an orientation singularity with the Poincaré index n = ±1/2, transforms to a planar distribution of ellipses with two singularities n = ±1/4 corresponding to new axes. The local geometry of the slowness surface of elastic waves is studied in the vicinity of new degeneracy points and a self-intersection line connecting them. The absorption-induced transformation of the classical picture of conical refraction is studied. The ellipticity of waves at the edge of the self-intersection wedge in a narrow interval of propagation directions drastically changes from circular at the wedge ends to linear in the middle of the wedge. For the wave normal directed to an arbitrary point of this wedge, during movement of the displacement vector over the corresponding polarization ellipse, the wave ray velocity s runs over the same cone describing refraction in a crystal without absorption. In this case, the end of the vector moves along a universal ellipse whose plane is orthogonal to the acoustic axis for zero absorption. The areal velocity of this movement differs from the angular velocity of the displacement vector on the polarization ellipse only by a constant factor, being delayed by π/2 in phase. When the wave normal is localized at the edge of the wedge in its central region, the movement of vector s along the universal ellipse becomes drastically nonuniform and the refraction transforms from conical to wedge-like.     

Anharmonic interactions of elastic and orientational waves in one-dimensional crystals

Planar oscillations of a chain of dumbbell-shaped particles possessing three degrees of freedom are studied. This system models the dynamics of quasi-one-dimensional crystals consisting of elongated anisotropic molecules. A system of nonlinear differential equations describing the anharmonic interaction of the elastic and orientational waves in the lattice, corresponding to different degrees of freedom of the particles, is constructed assuming a cubic interparticle interaction potential. It is shown that in the low-frequency approximation the system obtained is equivalent to the equations of the moment theory of elasticity, widely employed for describing nonlinear and dispersion properties of layered crystals and phase transformations in alloys. Some types of three-wave collinear interactions are investigated, suggesting the possibility of exciting orientational waves in organic crystals because of their nonlinear interaction with acoustic waves. Fiz. Tverd. Tela (St. Petersburg) 39, 137–144 (January 1997)     

Wave velocities in shock-compressed cubic and hexagonal single crystals above the elastic limit

When solids are subjected to high-pressure shock-wave loading, multiple stress waves propagate with velocities dependent upon the elastic and inelastic compressibilities of the solid. The present paper shows that the inelastic or plastic waves in cubic and hexagonal single crystals do not necessarily propagate with the bulk sound speed as they do in isotropic elastic-plastic solids. This result is a consequence of anisotropy in the plastic deformation which depends on the slip plane orientation in the crystal and has important consequences with regard to the determination of compressibilities from shock-wave data. In particular, for wave propagation in the <110> directions of cubic crystals the departure from the bulk velocity can be significant (5–25 per cent). For wave propagation normal to the c-axis in hexagonal crystals, the plastic wave velocity also differs from the bulk sound speed (10–25 per cent). Plastic wave velocities are tabulated for a number of cubic crystals on the basis of the various slip systems common to these materials. The calculated velocities are then compared with experimental data on shock-loaded single-crystal aluminum and sodium chloride.     

Experimental evidence of rotational elastic waves in granular phononic crystals

A generalized theory of elasticity, taking into account the rotational degrees of freedom of point bodies constituting a continuum, was proposed at the beginning of the twentieth century by the Cosserat brothers. We report the experimental observation of coupled rotational-translational modes in a noncohesive granular phononic crystal. While absent in the classical theory of elasticity, these elastic wave modes are predicted by the Cosserat theory. However the Cosserat theory fails to predict correctly the dispersion of the elastic modes in granular crystals even in the long-wavelength limit.     

Theory of elastic waves in real crystals. II.

Starting from the canonical equations of an elastic field formulated in the first part of this paper, conservation laws for an elastic field are obtained which are analogous to the energy and momentum conservation laws for an electromagnetic field. Expressions are found for the group velocity and the damping coefficient of elastic waves in a medium with space-time dispersion. A model of a continuous medium containing mobile dislocations is examined as a specific example. The question of the possibility of extending the theory to nonlinear media is discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 13–20, November, 1978.     

Propagation of guided elastic waves in 2D phononic crystals

The phononic band structure of two-dimensional phononic guides is numerically studied. A plane wave expansion method is used to calculate the dispersion relations of guided elastic waves in these periodic media, including 2D phononic plates and thin layered periodic arrangements. We show that, for any guided elastic wave, Lamb or generalised Lamb modes, stop bands appear in the dispersion curves, displaying a phononic band structure in both cases.     

Collimation and enhancement of elastic transverse waves in two-dimensional solid phononic crystals

In this Letter, we numerically investigate the propagation characteristics of elastic transverse waves emitted by line sources embedded inside two-dimensional (2D) solid phononic crystals (PCs). The results show that collimation and enhancement of elastic transverse waves can be achieved at the band edge frequencies. We find that the collimation effect originates from the flat equifrequency contours (EFCs) at the band edge of appropriately designed 2D solid PCs. It is shown that, in addition to geometric symmetry, appropriate constituent material combination is essential to obtain flat EFCs at the band edge. A highly directional and enhanced elastic transverse wave source with a half power angular width of only 5.6° and an enhancement factor of 530 is realized simply by utilizing a finite-size 2D solid PC structure.     

Localisation of elastic waves in two-dimensional randomly disordered solid phononic crystals

Combined with the supercell technique, the plane wave expansion method is used to calculate the band structures of the two-dimensional solid–solid phononic crystals with the random disorders in either radius or location of the scatterers. Phononic systems with plumbum scatterers embedded in an epoxy matrix are calculated in detail. The influences of the disorder degree on the band structures for both anti-plane and in-plane wave modes are investigated. It is found that, with increase of the disorder degree, the band gaps become narrower with more flat bands appearing in the gaps. Both displacement distribution and response spectra show that at the flat bands, elastic waves are localised due to the presence of the disorder. Wave localisation is more pronounced at the flat bands near the lower/upper edge for the radius/location disorder. Wave propagation and localisation in a randomly disordered system with a point defect is also studied. The influence of the disorder on the point-defect state is discussed. The results show that the disorder can tune the frequencies of the defect states. It is particularly noticed that the double degenerate mode appearing within the gap of the mixed in-plane waves is split up into two separated ones when the random disorder is introduced into the system. Generally, the influence of the disorder is more pronounced for the mixed in-plane modes than the anti-plane modes. The analysis of this paper is relevant to the assessment of the influences of manufacture errors on wave behaviours in phononic crystals as well as the possible control of wave propagation by intentionally introducing disorders into periodic systems.     

Investigation of the phase velocities of guided acoustic waves in soft porous layers

A new experimental method for measuring the phase velocities of guided acoustic waves in soft poroelastic or poroviscoelastic plates is proposed. The method is based on the generation of standing waves in the material and on the spatial Fourier transform of the displacement profile of the upper surface. The plate is glued on a rigid substrate so that it has a free upper surface and a nonmoving lower surface. The displacement is measured with a laser Doppler vibrometer along a line corresponding to the direction of propagation of plane surface waves. A continuous sine with varying frequencies was chosen as excitation signal to maximize the precision of the measurements. The spatial Fourier transform provides the wave numbers, and the phase velocities are obtained from the relationship between wave number and frequency. The phase velocities of several guided modes could be measured in a highly porous foam saturated by air. The modes were also studied theoretically and, from the theoretical results, the experimental results, and a fitting procedure, it was possible to determine the frequency behavior of the complex shear modulus and of the complex Poisson ratio from 200 Hz to 1.4 kHz, in a frequency range higher than the traditional methods.     

The static and elastic properties of mixed diatomic crystals

An interionic potential model has been proposed to study the static and elastic properties of mixed diatomic crystals. The interaction system of this potential consists of the long-range Coulomb and three-body interaction and the short-range overlap repulsion. This potential has been used to calculate the cohesive energy, phase-transition pressure and volume, third-order elastic constants and pressure derivatives of the effective second-order elastic constants for NaCl-NaBr mixed crystals. These results agree reasonably well with the available experimental results on the host crystals and allow us to draw some meaningful conclusions for the mixed alkali halide crystals.     

Combined spectral estimator for phase velocities of multimode Lamb waves in multilayer plates

A novel combined spectral estimate (CSE) method for differentiation and estimation the phase velocities of multimode Lamb waves whose wave numbers are much close or overlap one another in multiplayer plates is presented in this paper, which based on auto-regressive (AR) model and 2-D FFT. Simulated signals in brass plate were processed by 2-D FFT and CSE. And experiments are performed by using two conventional angle probes as emitter and receiver on the same surface of three-layered aluminum/xpoxy/aluminum plates, which include symmetrical and unsymmetrical plates. The multimode Lamb waves are excited in these laminates, and the received signal is processed by 2-D FFT and CSE, respectively. The results showed that the phase velocities of multimode signals whose wave numbers are much closed cannot be differentiated by 2-D FFT, but CSE has strong spatial resolution. Compared the measured phase velocities with the theoretical values, the error is smaller than 2% on the whole. It promises to be a useful method in experimental signals processing of multimode Lamb waves.     

Cohesive and anharmonic elastic properties of mixed fluorite crystals

Summary The cohesive and anharmonic elastic properties of four mixed fluorite crystals (Ca _x Sr_1−x F₂, Sr _x Ba_1−x F₂ Ba _x Ca_1−x F₂ and Cd _x Pb_1−x F₂) have been investigated by means of a three-body potential (TBP) model which consists of the long-range Coulomb and three-body interactions and the short-range van der Waals attraction and overlap repulsion effective up to the second-neighbour ions. Due to the lack of measured data on cohesive energy, third-order elastic constants and pressure derivatives of the secondorder elastic constants of mixed fluorites, the accuracy of the present results has been tested by comparing them with the so-called experimental results generated by the application of Vegard’s law to their corresponding experimental values for the host fluorites.

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Riassunto Sono state ricercate le proprietà elastiche coesive e anarmoniche di quattro cristalli misti di fluorite (Ca _x Sr_1∮x F₂, Sr _x Ba_1−x F₂, Ba _x Ca_1−x F₂ e Cd _x Pb_1−x F₂) per mezzo di un modello del potenziale a tre corpi (TBP) che consiste di interazioni di Coulomb a lungo raggio e a tre corpi, di attrazione di van der Waals a corto raggio e di repulsione di sovrapposizione efficace fino agli ioni del secondo vicino. A causa della mancanza di dati misurati sull’energia coesiva, di costanti elastiche di terz’ordine e di derivate di pressione delle costanti elastiche di secondo ordine, l’accuratezza dei risultati attuali è stata provocata confrontandoli con i cosidetti risultati sperimentali generati dalla applicazione della legge di Vegard ai loro valori sperimentali corrispondenti per i fluoriti ospiti.

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Резюме С помощью трех-частичной потенциальной модели исследуются когезионные и ангармоничные упрыгие свойства четырех смешанных кристаллов флюорита (Ca _x Sr_1−x F₂, Sr _x Ba_1−x F₂, Ba _x Ca_1−x F₂ и Cd _x Pb_1−x F₂). Используемая модель состои т из длиннодейтсвующего кулоновского и трех-частичного веаимодействий и короткодействующего притяжения Ван дер Ваальса и перекрывающегося отталкивания, эффективного вплоть до вторых соседних ионов. Из-за недостатка зкспериментальных данных для когезионной энергии упругих постоянных третьего порядка и производных по давлехию упрутих постоянных второго поядка для смешанных флюоритов, проверяется точность полученных результатов посредством сравнения их с так называемыми экспериментальными результатами, образованными с помощью применения закона Вегарда к соответствующим экспериментальным величинам для первичных кристаллов флюоритов.

     

Observation of a non-adiabatic geometric phase for elastic waves

We report the experimental observation of a geometric phase for elastic waves in a waveguide with helical shape. The setup reproduces the experiment by Tomita and Chiao [A. Tomita, R.Y. Chiao, Phys. Rev. Lett. 57 (1986) 937–940, 2471] that showed first evidence of a Berry phase, a geometric phase for adiabatic time evolution, in optics. Experimental evidence of a non-adiabatic geometric phase has been reported in quantum mechanics. We have performed an experiment to observe the polarization transport of classical elastic waves. In a waveguide, these waves are polarized and dispersive. Whereas the wavelength is of the same order of magnitude as the helix’s radius, no frequency dependent correction is necessary to account for the theoretical prediction. This shows that in this regime, the geometric phase results directly from geometry and not from a correction to an adiabatic phase.     

Cohesive and elastic properties of alkaline earth chalcogenide crystals

In the present study an analysis of the cohesive and elastic properties has been performed in fourteen alkali chalcogenide crystals using the composite potential forms based on the Drude oscillator model. The short-range overlap repulsion has been considered up to the second nearest neighbour interactions. The van der Waals coefficients have been estimated from the Kirkwood-Muller formulae. Cohesive energies, bulk-modulus and its pressure derivatives calculated for the entire family of crystals under study are found to present a good agreement with available experimental data, better than those obtained by previous workers using the traditional Born potential model.On leave from theDepartment of Physics, G. B. Pant University, Pantnagar-263 145, India.Authors are thankful to Mr. Saleh O. Harraga for the computational help.     

Pseudomagnetic fields and Landau levels for out-of-plane elastic waves in gradient snowflake-shaped crystals

The study of pseudomagnetic fields (PMF) in classical waves draws a growing attention due to the strength of PMF far higher than real magnetic fields. Here, we show that a giant PMF for out-of-plane elastic waves can be created in the snowflake-shaped crystals by introducing a gradient angle modulation along one direction. In particular, we demonstrate that the Landau energy levels for out-of-plane elastic waves can be formed near the Dirac cone region, as a hallmark of high-field physics induced by PMF. Moreover, the sublattice polarized bulk states similar to the behavior of graphene electron in magnetic fields are achieved in our elastic systems. Furthermore, the magnetic-induced edge state propagation along bend pathway is also demonstrated. Our study provides a new paradigm for manipulating the elastic waves in aspects of information processing and energy transport.     

Propagation of thickness-twist waves in elastic plates with periodically varying thickness and phononic crystals

We study the propagation of thickness-twist (TT) waves in a crystal plate of AT-cut quartz with periodically varying, piecewise constant thickness. The scalar differential equation by Tiersten and Smythe is employed. The problem is found to be mathematically equivalent to the motion of an electron in a periodic potential field governed by Schrodinger’s equation. An analytical solution is obtained. Numerical results show that the eigenvalue (frequency) spectrum of the waves has a band structure with allowed and forbidden bands. Therefore, for TT waves, plates with periodically varying thickness can be considered as phononic crystals. The effects of various parameters on the frequency spectrum are examined.     

The explicit secular equation for surface acoustic waves in monoclinic elastic crystals

The secular equation for surface acoustic waves propagating on a monoclinic elastic half-space is derived in a direct manner, using the method of first integrals. Although the motion is at first assumed to correspond to generalized plane strain, the analysis shows that only two components of the mechanical displacement and of the tractions on planes parallel to the free surface are nonzero. Using the Stroh formalism, a system of two second order differential equations is found for the remaining tractions. The secular equation is then obtained as a quartic for the squared wave speed. This explicit equation is consistent with that found in the orthorhombic case. The speed of subsonic surface waves is then computed for 12 specific monoclinic crystals.