Equilibrium stresses and rigidity for infinite tensegrities and frameworks

TL;DR

This paper extends the concepts of equilibrium stresses and rigidity from finite to infinite tensegrities and frameworks, providing new characterisations and conditions for continuous rigidity.

Contribution

It introduces asymptotic equilibrium stresses and generalises rigidity criteria for infinite frameworks, offering new theoretical insights and simplified proofs.

Findings

Asymptotic equilibrium stresses are defined for infinite tensegrities.

Generalised criteria for prestress stability and second order rigidity are established.

Finite framework proofs are adapted to infinite cases, ensuring continuous rigidity.

Abstract

Asymptotic equilibrium stresses are defined for countably infinite tensegrities and generalisations of the Roth-Whiteley characterisation of first-order rigidity are obtained. Generalisations of prestress stability and second order rigidity are given for countably infinite bar-joint frameworks and are shown to give sufficient conditions for continuous rigidity relative to certain prescribed motions. The proofs are based on a new short proof for finite frameworks that prestress stability ensures continuous rigidity.