Infinite Bar-Joint Frameworks, Crystals and Operator Theory

Abstract

A theory of flexibility and rigidity is developed for general infinite bar-joint frameworks (G,p). Determinations of nondeformability through vanishing flexibility are obtained as well as sufficient conditions for deformability. Forms of infinitesimal flexibility are defined in terms of the operator theory of the associated infinite rigidity matrix R(G,p). The matricial symbol function of an abstract crystal framework is introduced, being the matrix-valued function on the $d$-torus representing R(G,p) as a Hilbert space operator. The symbol function is related to infinitesimal flexibility, deformability and isostaticity. Various generic abstract crystal frameworks which are in Maxwellian equilibrium, such as certain 4-regular planar frameworks, are proven to be square-summably infinitesimally rigid as well as smoothly deformable in infinitely many ways. The symbol function of a three-dimensional crystal framework determines the infinitesimal wave flexes in models for the low energy vibrational modes (RUMs) in material crystals. For crystal frameworks with inversion symmetry it is shown that the RUMS appear in surfaces, generalising a result of F. Wegner for tetrahedral crystals.

Sign error corrected

References in corpus (5)

• Algorithms for 3D rigidity analysis and a first order percolation transition
• Rigid unit modes in tetrahedral crystals
• Crystal frameworks, symmetry and affinely periodic flexes
• Rigidity-based approach to the boson peak in amorphous solids: from sphere packing to amorphous silica
• Finite motions from periodic frameworks with added symmetry

Cited by in corpus (11)

• Crystal frameworks, symmetry and affinely periodic flexes
• Matroids of Gain Graphs in Applied Discrete Geometry
• Polynomials for Crystal Frameworks and the Rigid Unit Mode Spectrum
• The first-order flexibility of a crystal framework
• The rigidity of infinite graphs
• Inductive constructions for frameworks on a two-dimensional fixed torus
• Quotient graphs of symmetrically rigid frameworks
• The almost periodic rigidity of crystallographic bar-joint frameworks
• Flexible placements of periodic graphs in the plane
• The mechanical modes of a $2$-periodic triangulated surface
• String-node nets and meshes