A Floquet-Bloch decomposition of the elastodynamical equations for the computation of wave’s dispersion in damped mechanical systems

International audienceIn the context of mid-frequency elastodynamical analysis of periodic structures, the Floquet-Bloch theorem has been recently applied. The latter allows the use of limited-size modeling of a representative cell to characterize waves dispersion properties of the assembled structure. The theorem provides a rigorous and well posed spectral problem representing waves dispersion in viscolelastic media. In particular, the wave-FEM (WFE) method and associated techniques are based on the Floquet-Bloch approach. Most of the current applications are limited either to undamped or to slightly damped systems. Through the mesh of the k-space (k being the wave number vector) inside or at the boundary of the first Brillouin zone is performed. For each wave headings a linear spectral problem is solved in order to obtain the corresponding eigenfrequency and the corresponding Floquet vector. Such process is more easily implemented for undamped structures but face some drawbacks when damping is considered. In this paper, the Floquet-Bloch theorem is used to set up an alternative technique in order to estimate the dispersion characteristics of a periodic structure. The approach is based on the same basic assumption, but the generalized eigenvalue problem which has to be solved differs from the one usually considered in WFE. Instead of discretising the k-space, the harmonic frequency and the wave heading are used to scan the k-space. Then the formulated eigenvalue problem is solved and the dispersion characteristics are obtained, including spatial attenuation terms. Some fundamentals properties of the eigensolutions are discussed, and the methodology is finally applied on a 2D waveguide application which can be found in the literature for an undamped case. The same example is considered with various damping levels, in order to illustrate the performances and specificities the efficiency of proposed approach. The proposed approach finds application in the analysis of wave propagation in the presence of damping materials or shunted piezoelectric patches, as well as in actively controlled systems, where equivalent damping terms are associated with the considered control scheme