Representation Theory for Wave Propagation through Buckled Phononic Crystals

Elastic Phononic Crystals (EPC) are soft deformable metamaterials that have periodic modulations in material properties such as shear modulus, bulk 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 phononic crystals is affected not only by material properties but also by symmetry properties of the crystal. It has been shown that buckling of compressed phononic crystals could tune wave propagation properties – for example, by decreasing the number of intersections (degeneracies) in the band diagram, potentially leading to the opening of band gaps. In our previous work, we used a group representation theory (GRT)-based framework to explain the effect of primitive unit cell symmetries on degeneracies in the band diagram for undeformed EPCs. In this work, we extend our GRT framework to include the effect of deformation- and buckling-induced symmetry-breaking on wave propagation. We find that degeneracies in the post-buckling band diagrams can be explained by the symmetries of the stress distribution that characterize the post-buckling crystal and the effect of compression on non-primitive unstressed unit cells that are used to capture post-buckling behavior. We then have the potential to predict how different loading paths affect symmetry and hence wave propagation properties, which would be helpful for the rational design of deformation tunable EPCs.