A. L. Shuvalov

Tunable effective constants of the one-dimensional piezoelectric phononic crystal with internal connected electrodes

One-dimensional propagation of a longitudinal wave through an infinite piezoelectric periodically layered structure is considered. The unit cell consists, in general, of piezoelectric multilayers separated by thin electrodes which are connected through a capacitor with capacity Cj that plays the role of the external electric control providing tunability of the mechanical properties. The main focus of the present study is on the effective properties characterizing the homogenized medium. Due to the electric boundary conditions, the 4×4 transfer matrix M through the period T can be reduced to 2 × 2 dimension governing the mechanical fields only. As a result, the homogenized medium is pure elastic though its effective material parameters depend on the piezoelectric and dielectric coefficients of the actual piezoelectric medium, as well as on Cj. The effective parameters and the impedance in the low-frequency limit ω→0 and at finite frequency are obtained and analyzed.

Elastodynamics of radially inhomogeneous spherically anisotropic elastic materials in the Stroh formalism

A method for solving elastodynamic problems in radially inhomogeneous elastic materials with spherical anisotropy is presented, i.e. materials having cijkl=cijkl(r) in a spherical coordinate system {r,θ,φ}. The time-harmonic displacement field u(r,θ,φ) is expanded in a separation of variables form with dependence on θ,φ described by vector spherical harmonics with r-dependent amplitudes. It is proved that such separation of variables solution is generally possible only if the spherical anisotropy is restricted to transverse isotropy (TI) with the principal axis in the radial direction, in which case the amplitudes are determined by a first-order ordinary differential system. Restricted forms of the displacement field, such as u(r,θ), admit this type of separation of variables solution for certain lower material symmetries. These results extend the Stroh formalism of elastodynamics in rectangular and cylindrical systems to spherical coordinates.

Evaluation of the effective speed of sound in phononic crystals by the monodromy matrix method (L)

A new formula for the effective quasistatic speed of sound c in 2D and 3D periodic materials is reported. The approach uses a monodromy-matrix operator to enable direct integration in one of the coordinates and exponentially fast convergence in others. As a result, the solution for c has a more closed form than previous formulas. It significantly improves the efficiency and accuracy of evaluating c for high-contrast composites as demonstrated by a 2D example with extreme behavior.A scheme for evaluating the effective quasistatic speed of sound c in two- and three-dimensional periodic materials is reported. The approach uses a monodromy-matrix operator to enable direct integration in one of the coordinates and exponentially fast convergence in others. As a result, the solution for c has a more closed form than previous formulas. It significantly improves the efficiency and accuracy of evaluating c for high-contrast composites as demonstrated by a two-dimensional scalar-wave example with extreme behavior.

Spectral properties of a 2D scalar wave equation with 1D periodic coefficients: Application to shear horizontal elastic waves

The paper provides a rigorous analysis of the dispersion spectrum of shear horizontal elastic waves in periodically stratified solids. The problem consists of an ordinary differential wave equation with periodic coefficients, which involves two free parameters ω (the frequency) and k (the wavenumber in the direction orthogonal to the axis of periodicity). Solutions of this equation satisfy a quasi-periodic boundary condition which yields the Floquet parameter K. The resulting dispersion surface ω(K, k) may be characterized through its cuts at constant values of K, k and ω that define the passband (real K) and stopband areas, the Floquet branches and the isofrequency curves, respectively. The paper combines complementary approaches based on eigenvalue problems and on the monodromy matrix M. The pivotal object is the Lyapunov function Δ ( ω 2 , k 2 ) ≡ 1 2 trace M = cos K which is generalized as a function of two variables. Its analytical properties, asymptotics and bounds are examined and an explicit form of its derivatives obtained. Attention is given to the special case of a zero-width stopband. These ingredients are used to analyse the cuts of the surface ω(K, k). The derivatives of the functions ω(k) at fixed K and ω(K) at fixed k and of the function K(k) at fixed ω are described in detail. The curves ω(k) at fixed K are shown to be monotonic for real K while they may be looped for complex K (i.e. in the stopband areas). The convexity of the closed (first) real isofrequency curve K(k) is proved thus ruling out low-frequency caustics of group velocity. The results are relevant to the broad area of applicability of ordinary differential equation for scalar waves in 1D phononic (solid or fluid) and photonic crystals.

Experimental and theoretical study of acoustic waves generated by a laser line pulse in an optically absorptive isotropic cylinder

The generation of acoustic waves by a line-focused laser pulse in an optically absorptive cylinder is studied experimentally and theoretically. Experiments are performed on a 5 mm diameter NG5 colored glass rod using Nd:yttrium aluminum garnet laser, which delivers 5 ns pulses. The numerical simulation is based on the semi-analytical model of a radially distributed thermoelastic source, which takes into account penetration of laser energy into the bulk of the sample. Good agreement between the experimental and calculated wave forms is observed. Comparison of these wave forms with an auxiliary simulation, which assumes the model of a dipole source located at the cylinder surface, reveals the effect of optical penetration on the shape of the wave form and also on the relative amplitude of bulk and surface waves.

Plane Inhomogeneous Elastic Waves in Transversely Isotropic Media: Explicit Analysis of Different Types of Degeneracy

An intrinsic trait of the plane inhomogeneous waves in elastic media is a particular parametrization duality, stemming from two components characterizing the complex wave vector (bivector) k with respect to the reference frame of orthonormal vectors. Correspondingly, the characteristic condition associated with the equation of motion involves two parameters p and λ, where p is the eigenvalue of the Stroh matrix N (λ) and λ is the eigenvalue of the complex Christoffel matrix λ (p). One of the consequences is a rather elaborate layout of the complex-valued degeneracy types implying repeated p, or λ, or both p and λ. Different types of degeneracy are characterized by dissimilar analytical, algebraic and topological features. The general theory, which has been established for an arbitrary anisotropic medium, is developed in the present paper into an explicit form for a transversely isotropic continuum. By reducing the number of geometrical parameters and enabling factorization of the characteristic polynomial, transverse isotropy fosters explicit resolving of the equations which define different degeneracy types. A particular thrust of the paper is concerned with the algebraic analysis of the conditions, under which the degenerate inhomogeneous modes, having linearly varying or/and circularly polarized amplitude, can be excited on a traction-free boundary by means of reflection or within the Rayleigh wave for, respectively, two- and one-parameter manifolds in the space of orientations of the free surface and propagation direction. Various settings are identified which ensure reflected and surface-localized wave packets comprising a degenerate inhomogeneous mode.

Acoustic waves generated by a laser point pulse in a micrometric fiber

Having emerged in the 1980s, the laser ultrasonics technique with its non‐contact generation and detection process overpasses the difficulties of coupling piezoelectric transducers with curved surfaces. To date, the authors [1] have been interested in acoustic generation for cylinders opaque at a given laser wavelength and for the acoustic source located at the cylinder surface. In this presentation, assuming point focusing of the laser pulses, we propose a three‐dimensional (3D) semi‐analytical model for acoustic waves generation and propagation in a partly transparent isotropic cylinder. First, the radial displacement at any position on the free surface is derived, in a 3D Fourier domain, for an inner point source. The response to a volume‐source distribution along a radius is obtained as a convolution of the above Green function with the corresponding source distribution caused by optical absorption. Three inverse transforms are then applied to obtain the radial displacement at the cylinder surface. Picosecond ultrasonics experiments are performed on different micrometric fibers and compared with calculated waveforms for different optical absorptive properties. References [1] Y.D. Pan, C. Rossignol and B. Audoin, Appl. Phys. Lett. 82, 4379 (2003).

Trapping of shear acoustic waves by a near-surface distribution of cavities.

For a halfspace containing random and uniform distribution of empty cylindrical cavities within finite depth beneath the surface, the dispersion spectrum of coherent shear horizontal waves is calculated and analyzed based on the effective-medium approach. The scattering-induced dispersion and attenuation are coupled with the effect of a surface waveguide filled with scatterers. As a result, the obtained spectrum bears certain essential particularities in comparison with the standard Love-wave pattern. Simple analytical estimates enable a direct evaluation of the concentration of scatterers from the dispersion data.

Coherent elastic wave propagation through nonuniform spatial distributions of cracks

Models for multiple scattering of elastic waves usually concern uniform distributions of scatterers. The aim of this work is to predict the propagation of SH coherent waves through nonuniform distribution of parallel closed cracks. The spatial variation of the distribution properties is taken into account via replacing heterogeneous media by stacks of effective homogeneous layers. Propagation in each layer is then governed by the effective acoustic impedance and the effective wave number, derived in the framework of Waterman and Truell approach. On this basis, the reflection and transmission coefficients of the nonuniform distributions are calculated by using the transfer matrix method. We focus especially on distributions of crack size and concentration. Impact of the distribution profiles is also investigated. The semianalytical predictions are compared with numerical results obtained by using a finite‐difference code.

On the use of the Willis Constitutive Laws of Elastodynamics of Stratified Media in reflection/transmission problems

The generalized constitutive relationships derived by Willis1,2 for elastodynamics of composites have incited an increasing interest in the community of effective media, and particularly in that of metamaterials. The structures for which homogeneous and dispersive equivalent media are sought include laminates,3,4 phononic crystals,5 as well as locally resonant materials.6 Those constitutive laws propose an extended vision of the notion of an effective medium since they fully generalize the linear relationship between the couple momentum/stress and the kinematic one particle velocity/strain through the tensors of anisotropic mass density (order 2), of elasticity (order 4) and of inertial coupling (order 3). Following general results obtained in [3] that provide the complete set of dispersive effective parameters describing exactly the “macroscopic” propagation in stratified media, this communication aims at exemplifying the use of the Willis model in different types of problems with interfaces coupling sev...