Journal
JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS
Volume 111, Issue -, Pages 414-433Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.jmps.2017.11.017
Keywords
Computational multiscale analysis; Homogenization; Micromorphic continua; Floquet-Bloch transform; Acoustic metamaterials; Phononic crystals
Funding
- European Research Council under the European Union's Seventh Framework Programme / ERC [339392]
- European Research Council (ERC) [339392] Funding Source: European Research Council (ERC)
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This paper presents a general multiscale framework towards the computation of the emergent 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 eigen-vectors 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. (C) 2017 The Authors. Published by Elsevier Ltd.
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