High-frequency homogenization in periodic media with imperfect interfaces

Assier, Raphael; Touboul, M. C.; Lombard, Bruno; Bellis, Cédric

doi:10.1098/rspa.2020.0402

Cited by 9 publications

(7 citation statements)

References 34 publications

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“…The cross effect is predicted by the T ij coefficients where one is of order one while the other is nearly null and vice versa, depending on which point of the Brillouin zone is considered, X(π/2, 0) or Γ(0, π/2). Some of the dispersion curves nearly touch in figure 10, in which case a refined asymptotic ansatz need be implemented, that can be found in [237,238], and also in a precursor study on long-wave high-frequency motion of elastic layers [239].…”

Section: Bloch Dispersion Curvesmentioning

confidence: 90%

Mechanical metamaterials

2023

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Mechanical metamaterials, also known as architected materials, are rationally designed composites, aiming at elastic behaviors and effective mechanical properties beyond (“meta”) those of their individual ingredients – qualitatively and/or quantitatively. Due to advances in computational science and manufacturing, this field has progressed considerably throughout the last decade. Here, we review its mathematical basis in the spirit of a tutorial, and summarize the conceptual as well as experimental state-of-the-art. This summary comprises disordered, periodic, quasi-periodic, and graded anisotropic functional architectures, in one, two, and three dimensions, covering length scales ranging from below one micrometer to tens of meters. Examples include extreme ordinary linear elastic behavior from artificial crystals, e.g., auxetics and pentamodes, “negative” effective properties, behavior beyond classical linear elasticity, e.g., arising from local resonances, chirality, beyond-nearest-neighbor interactions, quasi-crystalline mechanical metamaterials, topological band gaps, cloaking based on coordinate transformations and on scattering cancellation, seismic protection, nonlinear and programmable metamaterials, as well as space-time-periodic architectures.

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Section: Bloch Dispersion Curvesmentioning

confidence: 90%

Mechanical metamaterials

2023

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“…Many further examples can also be found in e.g. [21,33,[36][37][38]40] and references therein. Before we start solving the homogenized Schrödinger equation (2.1) for different gradient functions g(X), it is important to understand the range of values of the parameters T and α that are relevant.…”

Section: (C) Admissible Parameter Valuesmentioning

confidence: 99%

On the problem of comparing graded metamaterials

Davies,

Fehertoi-Nagy,

Putley

2023

Proc. R. Soc. A.

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We use a simple effective model, obtained through the application of high-frequency homogenization, to tackle the fundamental question of how the choice of gradient function affects the performance of a graded metamaterial. This approach provides a unified framework for comparing gradient functions efficiently and in a way that allows us to draw conclusions that apply to a range of different wave regimes. We consider the specific problem of single-frequency localization, for which the appropriate effective model is a one-dimensional Schrödinger equation. Our analytic results both corroborate those of existing studies (which use either expensive full-field wave simulations or black-box numerical optimization algorithms) and extend them to other metamaterial regimes. Based on our analysis, we are able to propose a design strategy for optimizing monotonically graded metamaterials and offer an explanation for the lack of a universal optimal gradient function.

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“…The mirror symmetry implies that Kj=KN−j+1 false(j=1,…,N) and similarly for Mj. This model describes imperfect transmission of elastic waves through glue layers or cracks [30]. Conservation of energy is proven in [31].…”

Section: Generalizationsmentioning

confidence: 99%

“…The usual case of perfect contact is recovered when Kjfalse→+normal∞ and Mj=0. The inner product is defined in appendix A of [30]. Since ∂ωβj=2 Mj ω≥0, the assumptions of theorem 5.1 are satisfied.…”

Section: Generalizationsmentioning

confidence: 99%

Surface impedance and topologically protected interface modes in one-dimensional phononic crystals

Coutant,

Lombard

2024

Proc. R. Soc. A.

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When semi-infinite phononic crystals (PCs) are in contact, localized modes may exist at their boundary. The central question is generally to predict their existence and to determine their stability. With the rapid expansion of the field of topological insulators, powerful tools have been developed to address these questions. In particular, when applied to one-dimensional systems with mirror symmetry, the bulk-boundary correspondence claims that the existence of interface modes is given by a topological invariant computed from the bulk properties of the PC, which ensures strong stability properties. This one-dimensional bulk-boundary correspondence has been proven in various works. Recent attempts have exploited the notion of surface impedance, relying on analytical calculations of the transfer matrix. In the present work, the monotonic evolution of surface impedance with frequency is proven for all one-dimensional PCs with mirror symmetry. This result allows us to establish a stronger version of the bulk-boundary correspondence that guarantees not only the existence but also the uniqueness of a topologically protected interface state. This correspondence is extended to a larger class of one-dimensional models that include imperfect interfaces, array of resonators, or dispersive media. Numerical simulations are proposed to illustrate the theoretical findings.

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High-frequency homogenization in periodic media with imperfect interfaces

Cited by 9 publications

References 34 publications

Mechanical metamaterials

Mechanical metamaterials

On the problem of comparing graded metamaterials

Surface impedance and topologically protected interface modes in one-dimensional phononic crystals

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