Analyse dynamique des métamatériaux élastiques imparfaits

This thesis investigates the impact of periodicity defects on wave propagation in non-periodic structures. Initially motivated by the existence of manufacturing imperfections, this is a critical issue as periodicity defects can significantly influence a material's expected performance linked to its periodic characteristic, for example by potentially narrowing useful bandgaps and introducing unintended defect modes within passbands. In addition, strategically adding defects could provide a promising means of designing new metamaterials with significantly enhanced wave control capabilities. The main objective of this thesis is to develop theoretical and numerical methods for accurately and efficiently assessing the impact of periodicity defects on dispersion curves and mode shapes in non-periodic structures. A major challenge in this field is that the classical Floquet-Bloch theory, designed for periodic structures, is no longer applicable when there are periodicity defects.To address this challenge, the methodology adopted in this work includes two main components: numerical experiments to illustrate the influence of defects, and an extension of Floquet-Bloch analysis to model slightly disordered periodic media. First, a series of numerical experiments quantitatively characterizing the influence of defects on the dispersion relation and associated eigenmodes of imperfect periodic structures is presented in the case of elastic waves. These numerical experiments introduce localized defects into an initially perfectly periodic structure on a much larger scale than the period of the original structure, enabling the Floquet-Bloch theory to be effectively employed at this larger scale. These numerical results reveal, in line with theoretical results from the literature, that these defects can induce locally-supported (close to the defect) non-propagating modes at frequencies within the bandgaps of the unperturbed structure. The monotonic evolution of the eigenfrequency of such a defect mode with the defect amplitude is monitored numerically. The results of these experiments are confirmed by time-domain finite element simulations over a bounded layer of periodic material, offering useful insights into practical applications of metamaterials. Second, an asymptotic analysis of the acoustic wave equation in a medium weakly perturbed away from periodicity is developed. A new method is proposed to predict the dispersion curves and associated eigenmodes. The non-periodic material properties of the original acoustic equation are modeled as the composition of periodic material fields and a non-periodic mapping, assumed asymptotically close to the identity. At the leading order in the amplitude of the disorder, the medium is driven by an acoustic equation with periodic coefficients, which can be analyzed by using the classical Floquet-Bloch theory. At first order, with simple eigenvalues, post-processing of the periodic eigenvalues and eigenfunctions of the leading order allows to account for the non-periodic character of the original coefficients. In the case of repeated eigenvalues, a residual is introduced, whose minimization makes it possible to recover the opening of the bandgaps caused by the loss of periodicity. Eventually, the only problem to be numerically approximated to obtain the dispersion curves and the mode shapes of the non-periodic medium is the periodic eigenvalue problem, which is defined on the unit cell of the undisturbed medium, generally much smaller than the domain over which defects are located.