A general multiscale framework for the emergent effective elastodynamics of metamaterials

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Abstract

This paper presents a general multiscale framework towards the computation of the emer- gent effective elastodynamics of heterogeneous materials, to be applied for the analysis of acoustic metamaterials and phononic crystals. The generality of the framework is exemplified by two key characteristics. First, the underlying formalism relies on the Floquet–Bloch theorem to derive a robust definition of scales and scale separation. Second, unlike most homogenization approaches that rely on a classical volume average, a generalized homogenization operator is defined with respect to a family of particular projection functions. This yields a generalized macro-scale continuum, instead of the classical Cauchy continuum. This enables (in a micromorphic sense) to homogenize the rich dispersive behavior resulting from both Bragg scattering and local resonance. For an arbitrary unit cell, the homogenization projection functions are constructed using the Floquet–Bloch eigenvectors obtained in the desired frequency regime at select high symmetry points, which effectively resolves the emergent phenomena dominating that regime. Furthermore, a generalized Hill–Mandel condition is proposed that ensures power consistency between the homogenized and full-scale model. A high-order spatio-temporal gradient expansion is used to localize the multiscale problem leading to a series of recursive unit cell problems giving the appropriate micro-mechanical corrections. The developed multiscale method is validated against standard numerical Bloch analysis of the dispersion spectra of example unit cells encompassing multiple high-order branches generated by local resonance and/or Bragg scattering.
Original languageEnglish
Pages (from-to)414-433
JournalJournal of the Mechanics and Physics of Solids
Volume111
DOIs
Publication statusPublished - 2018

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elastodynamics
Metamaterials
homogenizing
Scattering
projection
cells
Eigenvalues and eigenfunctions
continuums
Macros
Numerical analysis
Acoustics
scale models
scattering
Crystals
eigenvectors
theorems
formalism
operators
gradients
expansion

Keywords

  • Computational multiscale analysis
  • Homogenization
  • Micromorphic continua
  • Floquet–Bloch transform
  • Acoustic metamaterials
  • Phononic crystals

Cite this

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title = "A general multiscale framework for the emergent effective elastodynamics of metamaterials",
abstract = "This paper presents a general multiscale framework towards the computation of the emer- gent effective elastodynamics of heterogeneous materials, to be applied for the analysis of acoustic metamaterials and phononic crystals. The generality of the framework is exemplified by two key characteristics. First, the underlying formalism relies on the Floquet–Bloch theorem to derive a robust definition of scales and scale separation. Second, unlike most homogenization approaches that rely on a classical volume average, a generalized homogenization operator is defined with respect to a family of particular projection functions. This yields a generalized macro-scale continuum, instead of the classical Cauchy continuum. This enables (in a micromorphic sense) to homogenize the rich dispersive behavior resulting from both Bragg scattering and local resonance. For an arbitrary unit cell, the homogenization projection functions are constructed using the Floquet–Bloch eigenvectors obtained in the desired frequency regime at select high symmetry points, which effectively resolves the emergent phenomena dominating that regime. Furthermore, a generalized Hill–Mandel condition is proposed that ensures power consistency between the homogenized and full-scale model. A high-order spatio-temporal gradient expansion is used to localize the multiscale problem leading to a series of recursive unit cell problems giving the appropriate micro-mechanical corrections. The developed multiscale method is validated against standard numerical Bloch analysis of the dispersion spectra of example unit cells encompassing multiple high-order branches generated by local resonance and/or Bragg scattering.",
keywords = "Computational multiscale analysis, Homogenization, Micromorphic continua, Floquet–Bloch transform, Acoustic metamaterials, Phononic crystals",
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A general multiscale framework for the emergent effective elastodynamics of metamaterials. / Sridhar, A.; Kouznetsova, V.; Geers, M.G.D.

In: Journal of the Mechanics and Physics of Solids, Vol. 111, 2018, p. 414-433.

Research output: Contribution to journalArticleAcademicpeer-review

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AB - This paper presents a general multiscale framework towards the computation of the emer- gent effective elastodynamics of heterogeneous materials, to be applied for the analysis of acoustic metamaterials and phononic crystals. The generality of the framework is exemplified by two key characteristics. First, the underlying formalism relies on the Floquet–Bloch theorem to derive a robust definition of scales and scale separation. Second, unlike most homogenization approaches that rely on a classical volume average, a generalized homogenization operator is defined with respect to a family of particular projection functions. This yields a generalized macro-scale continuum, instead of the classical Cauchy continuum. This enables (in a micromorphic sense) to homogenize the rich dispersive behavior resulting from both Bragg scattering and local resonance. For an arbitrary unit cell, the homogenization projection functions are constructed using the Floquet–Bloch eigenvectors obtained in the desired frequency regime at select high symmetry points, which effectively resolves the emergent phenomena dominating that regime. Furthermore, a generalized Hill–Mandel condition is proposed that ensures power consistency between the homogenized and full-scale model. A high-order spatio-temporal gradient expansion is used to localize the multiscale problem leading to a series of recursive unit cell problems giving the appropriate micro-mechanical corrections. The developed multiscale method is validated against standard numerical Bloch analysis of the dispersion spectra of example unit cells encompassing multiple high-order branches generated by local resonance and/or Bragg scattering.

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