Guest, S.D., Fowler, P.W. orcid.org/0000-0003-2106-1104 and Schulze, B. (2018) Mobility of symmetric block-and-hole polyhedra. International Journal of Solids and Structures, 150. pp. 40-51. ISSN 0020-7683
Abstract
Block-and-hole polyhedra can be derived from a bar-joint triangulation of a polyhedron by a stepwise construction: select a set of non-overlapping disks defined by edge-cycles of the triangulation of length at least 4; then modify the interior of each disk by an addition or deletion operation on vertices and edges so that it becomes either a rigid block or a hole. The construction has a body-hinge analogue. Models of many classical objects such as the Sarrus linkage can be modelled by block-and-hole polyhedra. Symmetry extensions of counting rules for mobility (the balance of mechanisms and states of self-stress) are obtained for the bar-joint and body-hinge models. The extended rules detect mechanisms in many cases where pure counting would predict an isostatic framework. Relations between structures where blocks and holes are swapped have a simple form. Examples illustrate the finer classification of isostatic and near-isostatic block-and-hole polyhedra achievable by using symmetry. The present approach also explains a puzzle in standard models of mobility. In the bar-joint model, a fully triangulated polyhedron is isostatic, but in a body-hinge version it is heavily overconstrained. When the bodies are panels with hinge lines intersecting at vertices, the overconstraints can be explained in local mechanical terms, with a direct symmetry description. A generalisation of the symmetry formula explains the extra states of self-stress in panel-hinge models of block-and-hole polyhedra.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | ©2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). |
Keywords: | Symmetry; Rigidity; Mechanisms; Block-and-hole polyhedra; Bar-joint frameworks; Panel-hinge structures |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > Department of Chemistry (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 22 Oct 2018 15:26 |
Last Modified: | 22 Oct 2018 15:26 |
Published Version: | https://doi.org/10.1016/j.ijsolstr.2018.05.029 |
Status: | Published |
Publisher: | Elsevier |
Refereed: | Yes |
Identification Number: | 10.1016/j.ijsolstr.2018.05.029 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:137441 |