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Topological mechanics and phononics

Banff International Research Station for Mathematical Innovation and Discovery

This talk will explore elastic and mechanical properties and mode structures of model periodic lattices, such as the square, kagome, pyrochlore, and jammed packings with central-force springs, that are at or near the Maxwell limit mechanical stability with coordination number z equal to twice the spatial dimension d. It will discuss the origin and nature of zero modes and elasticity of these structures under both periodic (PBC) and free boundary conditions (FBC), and it will investigate lattices (a) whose zero modes under the two boundary conditions are essentially identical, (b) whose phonon modes in the bulk are gapped with no zero modes in the periodic spectrum (except at zero wavenumber) but include zero-frequency surface in the free spectrum, and (c) whose bulk phonon modes include isolated points or lines where their frequency is zero. In case (a), lattices are generally in a type of critical state that admits states of self-stress in which there can be tension in bars with zero force on any node. Distortions away from that state gap the spectrum and give rise to surface modes under free boundary conditions whose degree of penetration into the bulk diverges at the critical state. The gapped states have a topological characterization, similar to those of polyacetylene and topological insulators, that define the nature of zero-modes at the boundary between systems with different topology. Case (c) is closely analogous to Weyl semi-metals with isolated points in the Brillouin zone where valence and conduction bands meet. These critical lattices generally have macroscopic elastic distortions, called Guest Modes, that cost no energy.

Mathematics; Mechanics of deformable solids; Statistical mechanics, structure of matter; Mathematical physics

video/mp4

Author affiliation: University of Pennsylvania

2017-06-11

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61910

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/