Title: Group Representation Theory for Elastic (possibly Granular?) Metamaterials

Author (Talk): Tejas Dethe, Princeton University

Abstract:

Elastic Phononic Crystals 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 relations, described by band diagrams, of waves propagating through phononic crystals are affected not only by material properties but also by the symmetry properties of the crystal. It has been shown that buckling of compressed phononic crystals as well as deformation of soft granular metamaterials 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 that disallow propagation of waves with frequencies within the band gap. However, it was unclear if there are predictive principles that can help understand these changes systematically. My work addresses this using a group representation theory-based framework to explain the effect of primitive unit cell symmetries on degeneracies in the band diagram for undeformed elastic phononic crystals. Group representation theory is a generalized framework that can be applied to systems governed by partial differential equations. When working with eigenproblems (as in wave propagation), it can help classify the eigenfunctions (i.e. wavemodes) by the spatio-temporal symmetry properties of the eigenoperator. Moreover, group representation theory can be combined with bifurcation theory to understand how solutions break symmetry to give rise to post-bifurcated states (as in buckling) as a critical parameter is changed (i.e. strain). Extending the application of group theoretic tools to systems tackled by the granular materials community could help make predictions based on symmetry and topological considerations where contact forces play an important role.