On the effective shear speed in 2D phononic crystals

TL;DR

This paper introduces two new analytical estimates for the effective shear wave speed in 2D phononic crystals, compares them with numerical methods, and evaluates their accuracy across different lattice configurations.

Contribution

The paper develops two novel closed-form estimates for the effective shear speed in 2D phononic crystals using plane wave expansion and monodromy matrix approaches.

Findings

PWE estimate is most accurate for densely packed stiff inclusions.

MM estimate provides the best overall fit across various configurations.

An efficient numerical scheme for computing the effective speed is proposed.

Abstract

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.