ABSTRACT
A way to achieve negative refraction of elastic anti-plane shear waves is a transmission across an interface between a homogeneous substrate and a periodic transverse laminate. To achieve pure negative refraction, the frequency of the source should be lower than the upper limit of the second transition zone (TZ) of the harmonic spectrum of the laminate. An effective way to control the location of TZ is to consider a canonical configuration for the laminate, a concept that originates from the properties of quasi-crystalline sequences among which the Fibonacci one is a particular case. Based on the universal structure of frequency spectrum, we provide a method based on the reduced torus to study the effect of a change in canonical ratio on the limits of the TZ. A further contribution consists in the analytical estimate of the angle of refraction for a linear relationship between frequency and longitudinal wavenumber. This is achieved by determining the components of the in-plane Poynting vector. The outcome provides a tool for the selection of a suitable laminate-substrate combination to accomplish a particular angle of the refracted wave. Finally, it is shown that for some particular configurations, the transmitted energy displays a peak that can be exploited to maximize the amount of energy travelling across the laminate. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)'.
ABSTRACT
The dynamical properties of periodic two-component phononic rods, whose elementary cells are generated adopting the Fibonacci substitution rules, are studied through the recently introduced method of the toroidal manifold. The method allows all band gaps and pass bands featuring the frequency spectrum to be represented in a compact form with a frequency-dependent flow line on the surface describing their ordered sequence. The flow lines on the torus can be either closed or open: in the former case, (i) the frequency spectrum is periodic and the elementary cell corresponds to a canonical configuration, (ii) the band gap density depends on the lengths of the two phases; in the latter, the flow lines cover ergodically the torus and the band gap density is independent of those lengths. It is then shown how the proposed compact description of the spectrum can be exploited (i) to find the widest band gap for a given configuration and (ii) to optimize the layout of the elementary cell in order to maximize the low-frequency band gap. The scaling property of the frequency spectrum, that is a distinctive feature of quasicrystalline-generated phononic media, is also confirmed by inspecting band-gap/pass-band regions on the torus for the elementary cells of different Fibonacci orders. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 2)'.
ABSTRACT
A nonlinear interface constitutive law is formulated for modeling the mechanical behavior of the periodontal ligament. This gives an accurate interpolation of the few available experimental results and provides a reasonably simple model for mechanical applications. The model is analyzed from the viewpoints of both mathematical consistency and effectiveness in numerical calculations. In order to demonstrate the latter, suitable two- and three-dimensional nonlinear interface finite elements have been implemented.
