Bulk longitudinal wave reflection/transmission in periodic piezoelectric structures with metallized interfaces

A theoretical study is performed of the bulk acoustic wave propagation in periodic piezoelectric structures with metallized interperiod boundaries. A crucial specific feature of such structures is that the bounded acoustic beam incident perpendicular to an interface can generate scattered (i.e. reflected and transmitted) waves over the whole area of the interface rather than only within the spot where this acoustic beam crosses the interface as it occurs in the absence of metallization. This extra generation is due to the electric potential which is induced by the incident wave on the whole metallized boundary rather than only on its part. The expressions are obtained for the reflection and transmission coefficients in the case where a longitudinal wave propagates along the 6-fold symmetry axis of a hexagonal piezoelectric. The periodicity is realized by inserting thin metallic layers (electrodes) into otherwise homogeneous material perpendicular to its 6-fold axis. The derived expressions allow the determination of the amplitude of waves arising both inside and outside the incident acoustic beam. The analysis of these expressions shows that the extra generation in question is able to significantly alter the distribution of the wave amplitudes as compared with the pattern which is obtained without taking into account the wave fields appearing outside the domain occupied by the incident acoustic beam.

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.

Shear surface waves in phononic crystals

The existence of shear horizontal (SH) surface waves in two-dimensional periodic phononic crystals with an asymmetric depth-dependent profile is theoretically reported. Examples of dispersion spectra with bandgaps for subsonic and supersonic SH surface waves are demonstrated. The link between the effective (quasistatic) speeds of the SH bulk and surface waves is established. Calculation and analysis is based on the integral form of a projector on the subspace of evanescent modes which means no need for their explicit finding. This method can be extended to the vector waves and the three-dimensional case.

Analytical formulation of three-dimensional dynamic homogenization for periodic elastic systems

Homogenization of the equations of motion for a three-dimensional periodic elastic system is considered. Expressions are obtained for the fully dynamic effective material parameters governing the spatially averaged fields by using the plane wave expansion method. The effective equations are of Willis form with coupling between momentum and stress and tensorial inertia. The formulation demonstrates that the Willis equations of elastodynamics are closed under homogenization. The effective material parameters are obtained for arbitrary frequency and wavenumber combinations, including but not restricted to Bloch wave branches for wave propagation in the periodic medium. Numerical examples for a one-dimensional system illustrate the frequency dependence of the parameters on Bloch wave branches and provide a comparison with an alternative dynamic effective medium theory, which also reduces to Willis form but with different effective moduli.

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

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.

Effective Willis constitutive equations for periodically stratified anisotropic elastic media

A method to derive homogeneous effective constitutive equations for periodically layered elastic media is proposed. The crucial and novel idea underlying the procedure is that the coefficients of the dynamic effective medium can be associated with the matrix logarithm of the propagator over a unit period. The effective homogeneous equations are shown to have the structure of a Willis material, characterized by anisotropic inertia and coupling between momentum and strain, in addition to effective elastic constants. Expressions are presented for the Willis material parameters which are formally valid at any frequency and horizontal wavenumber as long as the matrix logarithm is well defined. The general theory is exemplified for scalar SH motion. Low frequency, long wavelength expansions of the effective material parameters are also developed using a Magnus series, and explicit estimates are derived for the rate of convergence.