Dynamic continualization of mechanical metamaterials with quasi-periodic microstructure.

This article focuses on characterizing a class of quasi-periodic metamaterials created through the repeated arrangement of an elementary cell in a fixed direction. The elementary cell consists of two building blocks made of elastic materials and arranged according to the generalized Fibonacci sequence, giving rise to a quasi-periodic finite microstructure, also called Fibonacci generation. By exploiting the transfer matrix method, the frequency band structure of selected periodic approximants associated with the Fibonacci superlattice, i.e. the layered quasi-periodic metamaterial, is determined. The self-similarity of the frequency band structure is analysed by means of the invariants of the symplectic transfer matrix as well as the transmission coefficients of the finite clusters of Fibonacci generations. A high-frequency continualization scheme is then proposed to identify integral-type or gradient-type non-local continua. The frequency band structures obtained from the continualization scheme are compared with those derived from the Floquet-Bloch theory to validate the proposed scheme. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1).'

On dispersion curve coloring for mechanical metafilters.

This paper formalizes smooth curve coloring (i.e., curve identification) in the presence of curve intersections as an optimization problem, and investigates theoretically properties of its optimal solution. Moreover, it presents a novel automatic technique for solving such a problem. Formally, the proposed algorithm aims at minimizing the summation of the total variations over a given interval of the first derivatives of all the labeled curves, written as functions of a scalar parameter. The algorithm is based on a first-order finite difference approximation of the curves and a sequence of prediction/correction steps. At each step, the predicted points are attributed to the subsequently observed points of the curves by solving an Euclidean bipartite matching subproblem. A comparison with a more computationally expensive dynamic programming technique is presented. The proposed algorithm is applied with success to elastic periodic metamaterials for the realization of high-performance mechanical metafilters. Its output is shown to be in excellent agreement with desirable smoothness and periodicity properties of the metafilter dispersion curves. Possible developments, including those based on machine-learning techniques, are pointed out.