Willis elastodynamic homogenization theory revisited for periodic media |
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| Affiliation: | 1. Department of Civil, Environmental, and Geo- Engineering, University of Minnesota, Minneapolis, MN 55455, United States;2. Laboratoire de Mécanique des Solides, Ecole Polytechnique, F-91128 Palaiseau Cedex, France;1. School of Mechanical, Aerospace and Nuclear Engineering, Ulsan National Institute of Science and Technology, UNIST-gil 50, Eonyang-eup, Ulju-gun, Ulsan, 44919, South Korea;2. Center for Safety Measurement, Division of Industrial Metrology, Korea Research Institute of Standards and Science (KRISS), 267 Gajeong-ro, Yuseong-gu, Daejeon, 34113, South Korea;3. Department of Mechanical and Aerospace Engineering, Institute of Advanced Machines and Design, Seoul National University, 599 Gwanak-ro, Gwanak-gu, Seoul, 151-744, South Korea;1. Department of Mathematics, Imperial College London, London SW7 2AZ, UK;2. 80 Capital LLP, London W1S 4JJ, UK;3. Aix-Marseille Université, CNRS, Centrale Marseille, 13013 Marseille, France;1. The University of Texas at Austin, Applied Research Laboratories, 10000 Burnet Road, Austin 78758, USA;2. The University of Texas at Austin, Department of Electrical & Computer Engineering, 1616 Guadalupe Street, Austin 78701, USA;1. Université de Bordeaux - Institut de Mécanique et d’Ingénierie, CNRS UMR 5295 - Talence, France;2. Université de Bordeaux - Centre de recherche Paul Pascal, CNRS UPR 8641 - Pessac, France |
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| Abstract: | The theory of elastodynamic homogenization initiated by J.R. Willis is revisited for periodically inhomogeneous media through a careful scrutiny of the main aspects of that theory in the 3D continuum context and by applying it to the thorough treatment of a simple 1D discrete periodic system. The Bloch theorem appears to be central to appropriately defining and interpreting effective fields. Based on some physical arguments, three necessary conditions are derived for the transition from the microscopic description to the macroscopic description of periodic media. The parameters involved in the Willis effective constitutive relation are expressed in terms of two localization tensors and specified with the help of the corresponding Green function in the spirit of micromechanics. These results are illustrated and discussed for the 1D discrete periodic system considered. In particular, inspired by Brillouin's study, the dependency of the effective constitutive parameters on the frequency is physically interpreted in terms of oscillation modes of the underlying microstructure. |
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| Keywords: | Homogenization Bloch waves Dynamics Constitutive behavior Inhomogeneous material |
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