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Propagation of a longitudinal wave through the periodic structure composed of alternating elastic and piezoelectric layers is considered. The faces of each piezoelectric layer are electroded and connected via a circuit with the capacity $C$. It is shown that if $C<0$ then the Floquet-Bloch spectrum $ \omega(K)$ in a certain range of negative $C$ may possess a quasistatic absolute stopband starting at $\omega =0$. Other unusual features of the spectrum occurring at certain fixed values of $C<0$ are the infinite group velocity of the first branch at the origin point $\omega =0$, $K=0$ and the flat bands $\omega={\rm const}$.

The quasistatic limit of the antiplane shear-wave speed ('effective speed') $c$ in 2D periodic lattices is studied. Two new closed-form estimates of $c$ are derived by employing two different analytical approaches. The first proceeds from a standard background of the plane wave expansion (PWE). The second is a new approach, which resides in $\mathbf{x}$-space and centers on the monodromy matrix (MM) introduced in the 2D case as the multiplicative integral, taken in one coordinate, of a matrix with components being the operators with respect to the other coordinate. On the numerical side, an efficient PWE-based scheme for computing $c$ is proposed and implemented. The analytical and numerical findings are applied to several examples of 2D square lattices with two and three high-contrast components, for which the new PWE and MM estimates are compared with the numerical data and with some known approximations. It is demonstrated that the PWE estimate is most efficient in the case of densely packed stiff inclusions, especially when they form a symmetric lattice, while in general it is the MM estimate that provides the best overall fitting accuracy., Comment: 13 pages, 9 figures

The paper provides a rigorous analysis of the dispersion spectrum of SH (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 $\omega $ (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 $\omega (K,k)$ may be characterized through its cuts at constant values of $K, k$ and $\omega $ 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 $\mathbf{M}$. The pivotal object is the Lyapunov function $\Delta (\omega ^{2},k^{2}) \equiv 1/2\mathrm{trace}\mathbf{M}=\cos K$ 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 analyze the cuts of the surface $\omega (K,k).$ The derivatives of the functions $\omega (k)$ at fixed $K$ and $\omega (K)$ at fixed $k$ and of the function $K(k)$ at fixed $\omega $ are described in detail. The curves $\omega (k)$ 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., Comment: 35 pages, 4 figures

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 illustrated using the example of scalar SH motion. Low frequency, long wavelength expansions of the effective material parameters are also developed using a Magnus series and explicit estimates for the rate of convergence are derived., Comment: 20 pages, 3 figures