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Archive of Applied Mechanics
Published in: Archive of Applied Mechanics 1/2023

18-05-2022 | Original

On the fractional homogenization of one-dimensional elastic metamaterials with viscoelastic foundation

Authors: Wei Ding, John P. Hollkamp, Sansit Patnaik, Fabio Semperlotti

Published in: Archive of Applied Mechanics | Issue 1/2023

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Abstract

This work investigates the application of space–time fractional-order operators to the simulation of linear elastic waves propagating in 1D periodic structures resting on a viscoelastic foundation. More specifically, this study focuses on the possible application of fractional-order mathematics as the foundation to develop efficient reduced-order models capable of capturing the wave dynamics in periodic, viscoelastic one-dimensional metamaterials. By leveraging a space–time fractional formulation of the wave equation, we develop a homogenized model capable of capturing either material or geometric inhomogeneity and viscoelastic behavior. First, we derive the dispersion relation for a 1D infinite periodic bar resting on a longitudinal viscoelastic foundation using integer order formulation, which serves as a reference point in this work. Then, we obtain the dispersion relationships associated with two different fractional formulations. The first formulation relies on the use of time-fractional derivatives and focuses on capturing the dissipation induced by the viscoelastic foundation. The second formulation relies on the use of space–time fractional derivatives in order to lead to a homogenized one-dimensional model of the periodic bar. In order to achieve real-valued fractional orders, a matching approach between the dispersion relations of the fractional- and integer-order differential equations is used. Numerical simulations show that the space–time fractional wave equation serves as an effective homogenized model that well represents the wave propagation in a 1D periodic bar on a viscoelastic foundation. The results also illustrate that the use of space-fractional derivatives allows modeling the dynamics within (low order) frequency band gaps, a result typically not achievable with classical homogenization techniques.
previous article Band gaps in a periodic electro-elastic composite beam structure incorporating microstructure and flexoelectric effects
next article Formation of quasi-static stop band in a new one-dimensional metamaterial

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Appendix
Available only for authorised users
Footnotes
1
Recall that for a 1D viscoelastic element the stress and strain (also strain rate) are related as: \(\sigma (x) = E \epsilon (x) + E^\prime {\dot{\epsilon }}(x) \equiv E {\mathrm {d}u}/{\mathrm {d}x} + E^\prime {\mathrm {d} {\dot{u}}}/{\mathrm {d}x}\) [13]
 
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Metadata
Title
On the fractional homogenization of one-dimensional elastic metamaterials with viscoelastic foundation
Authors
Wei Ding
John P. Hollkamp
Sansit Patnaik
Fabio Semperlotti
Publication date
18-05-2022
Publisher
Springer Berlin Heidelberg
Published in
Archive of Applied Mechanics / Issue 1/2023
Print ISSN: 0939-1533
Electronic ISSN: 1432-0681
DOI
https://doi.org/10.1007/s00419-022-02170-w

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