Shape transformers for phononic band gaps tuning in two-dimensional Bloch-periodic lattice structures

Abstract

In this paper, shape transformers, a set of non-dimensional geometrical parameters that define beam cross-section efficiencies based on their size and shape, are employed in the wave propagation analysis of two-dimensional periodic lattice structures. Here, shape transformers of lattice unit-cell beams are used as design variables to tailor the frequency band gaps of the lattice structure, in the form of their mid frequencies and aspect ratios, by means of numerical Bloch-periodic simulation. Case studies are presented for lattices with square and hexagon Bravais symmetries where a total of six different unit-cell topologies are studied, considering 12 classes of different cross-section shapes. It is observed that a significant shift of the waveband branches towards lower frequencies are obtained with the variation of shape transformers from a void to a filled envelope configuration. Moreover, while changing the cross-section geometric variables, it is possible to determine associated values of maximum aspect ratios of band gap frequencies, thus identifying cross-section shapes that have better performance with respect to lattice dynamic characteristic. It is also found that the “H” and “I” cross-section shapes, in comparison to other shape families presented, resulted in the maximization of the band gap aspect ratio while maintaining a low frequency range, hence being ideal candidates for designing lattice metamaterials with lower frequency ranges of elastic wave attenuation. Furthermore, among all lattice topologies investigated, it is noted that the employment of diamond cross-section and its derivatives resulted in the greatest sensitivity of the mid-frequency shift of the band gap with respect to variations in cross-section geometric efficiency. It is also observed, as a general behavior for all unit-cells studied, that when scaling up the in-plane height of the cross-section, the aspect ratio of the band gap was maximized. Finally, using the Modal Assurance Criterion, we demonstrated that the variation in the shape transformers conserves the consistency between the eigenwaves of unit-cells at different characteristic points of the Irreducible Brillouin Zone. The presented study paves the way for the development and the systematic implementation of a novel wave attenuation tuning technique.