On asymptotic elastodynamic homogenization approaches for periodic media |
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| Affiliation: | 1. Dipartimento di Matematica, Università di Bologna, Italy;2. Dipartimento di Matematica, Università di Salerno, Italy;1. Department of Mechanical Engineering, Sapienza University of Rome, Rome, Italy;2. School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA;3. The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA;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. Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA;2. Department of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands |
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| Abstract: | A fairly large family of asymptotic elastodynamic homogenization methods is shown to be derivable from Willis exact elastodynamic homogenization theory for periodic media under appropriate approximation assumptions about, for example, frequencies, wavelengths and phase contrast. In light of this result, two long-wavelength and low-frequency asymptotic elastodynamic approaches are carefully analyzed and compared in connection with higher-order strain-gradient media. In particular, these approaches are proved to be unable to capture, at least in the one-dimensional setting, the optical branches of the dispersion curve. As an example, a two-phase string is thoroughly studied so as to illustrate the main results of the present work. |
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| Keywords: | Homogenization Elastodynamics Asymptotics Effective motion equation |
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