EPSRC Reference: 
EP/P01108X/1 
Title: 
Infinite bondnode frameworks 
Principal Investigator: 
Power, Professor S 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics and Statistics 
Organisation: 
Lancaster University 
Scheme: 
Standard Research 
Starts: 
03 July 2017 
Ends: 
02 July 2019 
Value (£): 
243,203

EPSRC Research Topic Classifications: 
Logic & Combinatorics 
Mathematical Analysis 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
The analysis of the rigidity and flexibility of bondnode structures and skeletal frameworks may be traced back to the 18th century mathematicians AugustinLouis Cauchy and Leonhard Euler and their considerations of polyhedra with hinged faces. They showed in particular that convex triangulated structures, such as geodesic domes, which have rigid bars connected together at their endpoints, by universal joints, are inherently rigid. In more recent times bondnode frameworks have played a vital role in mathematical models for crystals and materials, with framework bars representing strong bonds between particular atoms or between rigid polyhedral units. In particular, material zeolites provide diverse periodic networks of cornerlinked regular tetrahedra, with striking geometric and topological structure. It has been found, moreover, that the low energy excitation modes of a crystal, the socalled rigid unit modes (RUMs), or zero modes, are discernible from an infinitesimal rigidity analysis of the corresponding infinite bar and joint framework.
The main aims of the project are to develop a deeper understanding of the rigidity and flexibility of
(1) triangulated surface structures in three dimensions, including infinite triangulations and surfaces of higher genus,
(2) periodic and aperiodic bondnode frameworks of various categories.
In the first "classical" topic it is currently an open problem to determine which partial triangulations of a classical compact surface of a particular genus yield graphs with generically rigid realisations in space. A first step in this direction has been the recent characterisation by Cruickshank, Kitson and Power of generic minimal rigidity in the case of a torus with a superficial hole, or porthole. Moreover, the consideration of infinite triangulations leads to infinitely faceted structures, which are remarkably diverse even for a spherical surface, and to the completely new topic of determining the generic rigidity of (space embeddings) of compact and locally compact graphs.
The second "modern" topic aims to extend the theory of the rigid unit mode spectrum to the bondnode frameworks of bordered crystals, bicrystals and aperiodic crystals, including quasicrystals. In particular three new notions, namely the geometric spectrum, crystal flex complexity, and the mean flexibility dimension (generalising the RUM dimension), provide promising new invariants and signatures for a deeper understanding of these structures.
Additionally the project will consider mathematical limits of 3periodic bondnode structures which lie in a new category of constraint systems introduced recently by Power and Schulze, namely "stringnode meshes". These are bondnode networks which are "ultranano" in the sense that the set of nodes is dense in space. For these abstract fibred structures both strainfree rigidity phenomena and finite motion phenomena are possible, even in critically coordinated cases, and the project will tackle the analysis of the prediction and quantification of such phenomena.
In these varied new directions for bondnode structures there is great potential for the enrichment of the mathematical models used in Material Science. At the same time the computation of the new invariants will benefit from the methodology of simulation and computation familiar to applied scientists.

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