Equivariant Bifurcations for Tunable Wave Propagation in Elastic Phononic Crystals

Elastic Phononic Crystals (EPC) are soft metamaterials that have periodic modulations in material properties like shear modulus and density, whose microstructure is characterized by a repeatable region called the unit cell. The dispersion relation, described by band diagrams, of waves propagating through EPCs is affected not only by material properties but also by the symmetry properties of the crystal. It has been shown that bucking of EPCs can tune wave propagation properties – for example, by decreasing the number of intersections (degeneracies) in the band diagram, potentially opening band gaps. Since buckling changes the symmetries of the EPC, we have previously used a group representation theory-based framework to explain the effect of primitive unit cell symmetries on degeneracies in the band diagram for undeformed EPCs. Interpreting buckling as a bifurcation, we now apply equivariant bifurcation theory to EPCs to predict symmetries of the post-buckled structures and their associated modified wave propagation properties. Such interplay between elastic bifurcations and symmetries of EPCs can lead to a potential general approach to formulating rational design rules for material-agnostic deformation-tunable EPCs.