Rigidity of periodic and symmetric structures in nature and engineering

Guest, Simon D.; Fowler, Patrick W.; Power, S. C.

doi:10.1098/rsta.2013.0358

Cited by 9 publications

(10 citation statements)

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“…The analysis of the rigidity and flexibility of periodic infinite bond-node structures is an ongoing endeavour in materials science and pure mathematics (Connelly, Ivić and Whiteley [2], Guest, Fowler and Power [7]). In this connection, it is well known in crystallography and engineering that many crystals and periodic structures which are critically coordinated exhibit a rich set of localized zero energy crystal vibrations (phonons) (Giddy et al.…”

Section: Introductionmentioning

confidence: 99%

Crystal flex bases and the RUM spectrum

Kitson¹,

Power²,

Badri³

2021

Proceedings of the Edinburgh Mathematical Society

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A theory of infinite spanning sets and bases is developed for the first-order flex space of an infinite bar-joint framework, together with space group symmetric versions for a crystallographic bar-joint framework ${{\mathcal {C}}}$ . The existence of a crystal flex basis for ${{\mathcal {C}}}$ is shown to be closely related to the spectral analysis of the rigid unit mode (RUM) spectrum of ${{\mathcal {C}}}$ and an associated geometric flex spectrum. Additionally, infinite spanning sets and bases are computed for a range of fundamental crystallographic bar-joint frameworks, including the honeycomb (graphene) framework, the octahedron (perovskite) framework and the 2D and 3D kagome frameworks.

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Section: Introductionmentioning

confidence: 99%

Crystal flex bases and the RUM spectrum

Kitson¹,

Power²,

Badri³

2021

Proceedings of the Edinburgh Mathematical Society

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“…The topic appears in a wide range of fields from geometry to crystallography to engineering, and indeed the mechanical behaviours of many of these structures remain unexplored. Some examples of such behaviours include auxeticity in materials of negative Poisson’s ratio [ 4 – 9 ], origami-based folding and deployment [ 10 – 14 ], and deformability of hierarchically arranged structures [ 15 – 17 ], in addition to the fundamental mechanical properties of rigidity and flexibility [ 18 – 22 ].…”

Section: Introductionmentioning

confidence: 99%

“…Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. of hierarchically arranged structures [15][16][17], in addition to the fundamental mechanical properties of rigidity and flexibility [18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Transition mechanism for a periodic bar-and-joint framework with limited degrees of freedom controlled by uniaxial load and internal stiffness

2018

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A specific periodic bar-and-joint framework with limited degrees of freedom is shown to have a transition mechanism when subjected to an external force. The static nonlinear elasticity of this framework under a uniaxial load is modelled with the two angular variables specifying the rotation and distortion of the linked square components. Numerically exploring the equilibrium paths then reveals a transition state of the structure at a critical value of the internal stiffness. A simplified formulation of the model with weak nonlinear terms yields an exact solution of its transition state. Load–displacement behaviour and stability for the two systems with or without approximation are analysed and compared.

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“…periodic and aperiodic bond-node structures in proteins and materials) and in technology (eg. formation control for autonomous multi-agent systems, sensor network localization, robotics and CAD software) (see for example [2,3,9,10,18]).…”

Section: Introductionmentioning

confidence: 99%

“…Thus, for any framework (G, ϕ) with ϕ bounded, condition (a) of Corollary 4.4 is satisfied and so C(G, ϕ) : p (V ; X) → p (E; Y ) is not bounded below for all p ∈ [1, ∞). This applies, in particular, to the rigidity matrices of periodic bar-joint structures ( [10]). Proposition 4.6.…”

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confidence: 99%

Coboundary operators for infinite frameworks

Kastis¹,

Kitson²,

Power³

2019

Mathematical Proceedings of the Royal Irish Academy

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We consider, from the point of view of operator theory, a class of infinite matrices in which the matrix entries are determined by an underlying graph structure with accompanying geometric data. This class includes the rigidity matrices of infinite bar-joint frameworks as well as the incidence matrices of infinite directed graphs. We consider the following questions: When do these matrices give rise to bounded operators? Can we compute the operator norm? When are these operators compact? And when are they bounded below?

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Rigidity of periodic and symmetric structures in nature and engineering

Abstract: One contribution of 14 to a Theo Murphy Meeting Issue 'Rigidity of periodic and symmetric structures in nature and engineering' .

Cited by 9 publications

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Crystal flex bases and the RUM spectrum

Crystal flex bases and the RUM spectrum

Transition mechanism for a periodic bar-and-joint framework with limited degrees of freedom controlled by uniaxial load and internal stiffness

Coboundary operators for infinite frameworks

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