### Waves in quasicrystalline phononic structures: self-similar stop/pass band distribution and negative refraction

Fluids and Materials Seminar

13th December 2018, 2:00 pm – 3:00 pm

Main Maths Building, SM3

Elastic wave propagation in phononic structures composed of repeated elementary cells generated according to the Fibonacci substitution rule [1] is investigated. These systems are an example of one-dimensional quasicrystalline structures [2], and their dispersive properties are studied by imposing Floquet-Bloch conditions and applying the transfer matrix method. By means of this general approach, two different problems are addressed: i) harmonic axial wave propagation in quasicrystalline structured rods, and ii) oblique propagation of antiplane shear waves in Fibonacci laminates. Although they are two different physical phenomena, their governing equations and the corresponding dispersion relations possess the same form. The invariance properties of the transfer matrix trace, describing the whole stop/pass band structure associated with any arbitrary elementary cell, are studied. Nonlinear recursive maps relating the traces of the transfer matrices corresponding to consecutive cells of the Fibonacci sequence are obtained. We show that for both quasicrystalline structured rods and Fibonacci laminates, stop/pass bands within a defined range of frequencies for a given sequence are distributed following a self-similar pattern and this is repeated according to a scaling law for subsequent orders [3]. The self-similar distribution of the stop/pass band structure is governed by scaling factors which are derived analytically through the linearization of the iterative maps connecting the traces of the transfer matrix associated with the concecutive cells of the sequences.

The transmission of an antiplane wave obliquely incident at the interface between an elastic substrate and a Fibonacci laminate is also investigated [4]. The diffraction angles associated with the transmitted modes are estimated by means of the space averaging procedure of the Poynting vector. We show that, with respect to a periodic classical bilayer [5], on the one hand, beyond a certain frequency threshold, high order Fibonacci laminates can provide negative refraction for a wider range of angles of incidence, on the other, they allow negative wave refraction at lower frequencies. Numerical results illustrate that in order to obtain negative refraction for a selected range of frequencies, the properties of the Fibonacci laminates can be controlled using the scaling factors governing their Bloch-Floquet spectrum.

The detected self-similar structure of the stop and pass band distribution and the observed negative refraction phenomena represent two important features that can be exploited in order to design quasicrystalline phononic metamaterials.

References

[1] Gei M. Waves in quasiperiodic structures: stop/pass band distribution and prestress effects, Int. J. Solids Struct., 47 (2010) 3067-3075.

[2] Poddubny A.L. and Ivchenko E.L., Photonic quasicrystalline and aperiodic structures, Physica E, 42, (2010) 1871-1895.

[3] Morini, L. and Gei, M., Waves in one-dimensional quasicrystalline structures: dynamical trace mapping, scaling and self-similarity of the spectrum, J. Mech. Phys. Solids 119, (2018) 83-103.

[4] Morini L., Eyzat Y. and Gei M., Negative refraction in quasicrystalline multilayered metamaterials, J. Mech. Phys. Solids 124, (2019) 282-298.

[5] Willis, J., Negative refraction in a laminate, J. Mech. Phys. Solids 97, (2016) 10-18.

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