Efficient band-structure calculations of non-classically damped phononic materials by Bloch mode synthesis in state space

Bloch mode synthesis (BMS) techniques enable efficient band-structure calculations of periodic media by forming reduced-order models of the unit cell. Rooted in the framework of the Craig-Bampton component mode synthesis methodology, these techniques decompose the unit cell into interior and boundary degrees-of-freedom that are nominally described, respectively, by sets of normal modes and constraint modes. In this paper, we generalize the BMS approach by state-space transformation to extend its applicability to generally damped periodic materials that violate the Caughey-O’Kelly condition for classical damping. In non-classically damped periodic models, the fixed-interface eigenvalue problem may, in general, produce a mixture of underdamped and overdamped modes. We examine two mode-selection schemes for the reduced order model and demonstrate the underlying accuracy-efficiency trade-offs when qualitatively distinct mixtures of underdamped and overdamped modes are incorporated. The proposed approach provides a highly effective computational tool for analysis of large models of phononic crystals and acoustic/elastic metamaterials with complex damping properties. This investigation does not only extend the applicability of BMS techniques to the most generally damped models of periodic media, it also advances our understanding of the nature of damping modes and the non-trivial manner by which they contribute to the wave propagation properties.