Liftings and stresses for planar periodic frameworks

TL;DR

This paper extends Maxwell's theorem to periodic frameworks, linking stresses and liftings, and explores their deformation and rigidity properties with applications in crystallography and materials science.

Contribution

It introduces a periodic analog of Maxwell's theorem and applies it to analyze rigidity and auxetic properties of planar periodic frameworks.

Findings

Established a periodic Maxwell's theorem for frameworks.

Proved deformation and rigidity properties for periodic pseudo-triangulations.

Applied results to auxetic structures and ultrarigid frameworks.

Abstract

We formulate and prove a periodic analog of Maxwell's theorem relating stressed planar frameworks and their liftings to polyhedral surfaces with spherical topology. We use our lifting theorem to prove deformation and rigidity-theoretic properties for planar periodic pseudo-triangulations, generalizing features known for their finite counterparts. These properties are then applied to questions originating in mathematical crystallography and materials science, concerning planar periodic auxetic structures and ultrarigid periodic frameworks.