Crystal frameworks, symmetry and affinely periodic flexes

2011/03/09 by S. C. Power, Power, Stephen · 2 citations
Engineering · Materials Science · #52C25 #Advanced Materials and Mechanics #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Silicone and Siloxane Chemistry #Structural Analysis and Optimization

paper · pdf · doi:10.48550/arxiv.1103.1914

openalex publication_date 2011/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Symmetry equations are obtained for the rigidity matrices associated with various forms of infinitesimal flexibility for an idealised bond-node crystal framework \C in \bR^d. These equations are used to derive symmetry-adapted Maxwell-Calladine counting formulae for periodic self-stresses and affinely periodic infinitesimal mechanisms. The symmetry equations also lead to general Fowler-Guest formulae connecting the character lists of subrepresentations of the crystallographic space and point groups which are associated with bonds, nodes, stresses, flexes and rigid motions. A new derivation is also given for the Borcea-Streinu rigidity matrix and the correspondence between its nullspace and the space of affinely periodic infinitesimal flexes.

Cited by

• Infinite Bar-Joint Frameworks, Crystals and Operator Theory
• Inductive constructions for frameworks on a two-dimensional fixed torus
• Periodic body-and-bar frameworks

Related

• The first-order flexibility of a crystal framework
• Crystal Frameworks, Matrix-valued Functions and Rigidity Operators
• Infinite Bar-Joint Frameworks, Crystals and Operator Theory
• The almost periodic rigidity of crystallographic bar-joint frameworks
• Frameworks with crystallographic symmetry