### Abstract

Original language | English |
---|---|

Pages (from-to) | 414-433 |

Journal | Journal of the Mechanics and Physics of Solids |

Volume | 111 |

DOIs | |

Publication status | Published - 2018 |

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### Keywords

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

### Cite this

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**A general multiscale framework for the emergent effective elastodynamics of metamaterials.** / Sridhar, A.; Kouznetsova, V.; Geers, M.G.D.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - A general multiscale framework for the emergent effective elastodynamics of metamaterials

AU - Sridhar, A.

AU - Kouznetsova, V.

AU - Geers, M.G.D.

PY - 2018

Y1 - 2018

N2 - 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.

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.

KW - Computational multiscale analysis

KW - Homogenization

KW - Micromorphic continua

KW - Floquet–Bloch transform

KW - Acoustic metamaterials

KW - Phononic crystals

U2 - 10.1016/j.jmps.2017.11.017

DO - 10.1016/j.jmps.2017.11.017

M3 - Article

VL - 111

SP - 414

EP - 433

JO - Journal of the Mechanics and Physics of Solids

JF - Journal of the Mechanics and Physics of Solids

SN - 0022-5096

ER -