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Hybrid lattice metamaterials with auxiliary resonators made of functionally graded materials

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Abstract

This paper aims to enhance and tune wave-propagation characteristics of periodic architected structures by adding auxiliary resonators made of functionally graded materials (FGMs). For this purpose, cantilever FGM beams are added to periodic metamaterials with square and hexagonal topologies, and the effects of the material distribution of the added resonators on their wave-attenuation performance are analyzed. More specifically, a combination of locally resonant and Bragg-type bandgaps is formed as a result of adding FGM resonators, while the conventional structures have no bandgap in the considered region. The studied low-frequency region is of high importance, and the appearance of wide bandgaps there opens horizons for new structural and acoustic applications. Further, these bandgaps depend on the material parameters of the resonators, and their location and width are changed, systematically as functions of the elastic modulus ratio, density ratio, and non-negative power-law exponent of the resonators. For the numerical analysis, a finite element formulation is developed for an FGM beam, and the wave propagation is studied using Bloch’s theorem. According to the results, with increasing the elastic modulus contrast of the auxiliary FGM resonators, the locations of the bandgaps move higher, while increasing the density ratio contrast moves them to lower frequencies. Additionally, the effect of adding auxiliary FGM resonators on the directionality of the wave propagation is studied using the iso-frequency contours of the first dispersion branches of each structure. The results of the present study can be a starting point for using FGM resonators to design tunable elastic/acoustic metamaterials with the ability to filter waves in predefined frequency ranges.

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Authors and Affiliations

  1. School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran

    Hamid Jafari, Soroush Sepehri, Mohammad Reza Hairi Yazdi, Mahmoud Mosavi Mashhadi & Mir Masoud Seyyed Fakhrabadi

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  1. Hamid Jafari
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Appendix

Appendix

The shape functions for the 6-degrees-of-freedom frame element with length l are presented in the current manuscript:

$$\begin{aligned} {\xi _1}= & {} \left( {1 - \frac{x}{l}} \right) , \end{aligned}$$
(A.1)
$$\begin{aligned} {\xi _2}= & {} \left( {\frac{1}{{{l^3}}}} \right) ( {2{x^3} - 3{x^2}l + {l^3}} ) ,\end{aligned}$$
(A.2)
$$\begin{aligned} {\xi _3}= & {} \left( {\frac{1}{{{l^3}}}} \right) ( {{x^3}l - 2{x^2}{l^2} + x{l^3}} ), \end{aligned}$$
(A.3)
$$\begin{aligned} {\xi _4}= & {} \frac{x}{l} ,\end{aligned}$$
(A.4)
$$\begin{aligned} {\xi _5}= & {} \left( {\frac{1}{{{l^3}}}} \right) ( { - 2{x^3} + 3{x^2}l} ) ,\end{aligned}$$
(A.5)
$$\begin{aligned} {\xi _6}= & {} \left( {\frac{1}{{{l^3}}}} \right) ( {{x^3}l - {x^2}{l^2}} ). \end{aligned}$$
(A.6)

where x denotes the local coordinate of the element.

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Jafari, H., Sepehri, S., Yazdi, M.R.H. et al. Hybrid lattice metamaterials with auxiliary resonators made of functionally graded materials. Acta Mech 231, 4835–4849 (2020). https://doi.org/10.1007/s00707-020-02799-0

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