Dispersion relations of generalized one-dimensional phononic crystals

TL;DR

This work addresses the challenge of predicting dispersion in 1D phononic crystals with arbitrary scatterers by introducing a generalized 1D model that combines a homogeneous host (described by a state vector $\mathbf{u}(x)$ and evolution $d\mathbf{u}/dx = \mathbf{A}\mathbf{u} + \mathbf{q}(x)$) with a set of scatterers and Bloch periodicity for Bloch waves with wave number $k$. It develops an exact dispersion framework based on plane-wave expansion and a Green-function construction $\mathbf{G}(k,\xi)$, reducing to a finite eigenproblem after truncation and a null-space condition $[\mathbf{I} - \hat{\mathbf{G}}(k)\hat{\mathbf{K}}]\hat{\bm{\Psi}} = 0$; it is complemented by a weak-scattering iterative scheme that yields analytical first- and second-order corrections in terms of the structure factor $S(q)$ and the Green kernel, with convergence controlled by the spectral radius $\rho(\mathbf{J})$. The framework is validated on diverse 1D configurations, including an Euler-Bernoulli beam with point resonators and a Timoshenko beam with continuous resonators, demonstrating accurate dispersion predictions for both propagating and evanescent modes outside bandgaps and providing a computationally efficient path for metamaterial design. By unifying discrete and continuous scattering mechanisms under a single plane-wave-based approach, the method offers a versatile tool for engineering elastic wave control in 1D structures and sets the stage for extensions to higher dimensions and stronger scattering regimes.

Abstract

We present a comprehensive method for determining {both exact and approximate} dispersion {relations} for one-dimensional {resonant phononic} crystals, applicable to a wide range of structures, regardless of their specific characteristics. This general framework employs a unified mathematical model, referred to as generalized {one-dimensional (1D) phononic crystal}, in which {different} types of {waves} and scatterers{/resonators} {can be} {considered} by adjusting certain parameters. The generalized {1D phononic crystal} consists of both a host {one-dimensional} homogeneous elastic {material} with physical properties represented in matrix form and an arbitrary set of scatterers {within the unit cell,} including resonators (discrete and continuous), small material inclusions, or variations in cross-sectional area. Based on general assumptions, {and imposing the periodicity and Bloch solutions} we develop a matrix-based algorithm utilizing the plane wave expansion method to derive the solution. Additionally, we propose an iterative procedure that provides analytical expressions for the first- and second-order terms, particularly useful in the context of weak scattering. The convergence conditions of the method are rigorously defined. The {efficiency of the} approach is demonstrated through several numerical examples, highlighting its versatility in different waveguide configurations and scattering scenarios.

Figures (7)

• Figure 1: Generalized 1D phononic crystal formed by a elastic host medium with an arbitrary number $N$ of scatterers distributed in the unit cell of length $L$. The 1D structure has properties arranged in matrix $\mathbf{A}$, see Table \ref{['tab01']}. The scatterers considered can be of two types: (1) linear resonators attached pointwise to the waveguide and (2) small-width inclusions with other material or changes in cross-sectional dimensions. The properties of scatterers are introduced in matrices $\mathbf{K}_\alpha$, see Table \ref{['tab02']}
• Figure 2: Dispersion relations of the 1D elastic homogeneous medium. The different branches correspond to expressions of the form $k_jL - q_\nu$, $q_\nu = 2\pi \nu / L, \ , \ \nu = 0, \pm 1, \pm 2,\ldots$. Within the Brillouin zone, they constitute the same curve as folding the branch $q_0=0$.
• Figure 3: Unit cell for the 1D phononic crystal considered in Example 1. An Euler-Bernouilli beam with point-resonators formed by one resonance (simple spring-mass system).
• Figure 4: 1D phononic crystal formed by a beam (flexural waves) with $N=5$ discrete resonators at positions $\xi_\alpha/L = \{0.224, \ 0.471, \ 0.525, \ 0.774, \ 0.953\}$. (a) and (b) Imaginary and real part of dispersion relation in Example 1, respectively. Exact result (blue), Iterative approach with the first-order iteration (red markers) and unperturbed homogeneous solution (dashed-black). (c) Plot of the spectral radius of the Jacobian matrix $\mathbf{J}$, criterium of convergence is $\rho(\mathbf{J})<1$
• Figure 5: (a) and (b) Imaginary and real part of dispersion relation in Example 1, respectively. Exact result (blue), Iterative approach with the second-order iteration (red markers) and unperturbed homogeneous solution (dashed-black). (c) Plot of the spectral radius of the Jacobian matrix $\mathbf{J}$, criterium of convergence is $\rho(\mathbf{J})<1$