TY - JOUR

T1 - Mobility of symmetric block-and-hole polyhedra

AU - Guest, Simon

AU - Fowler, Patrick

AU - Schulze, Bernd

N1 - This is the author’s version of a work that was accepted for publication in International Journal of Solids and Structures. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in International Journal of Solids and Structures, 150, 2018 DOI: 10.1016/j.ijsolstr.2018.05.029

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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.

AB - 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.

KW - Symmetry

KW - Rigidity

KW - Mechanisms

KW - Block-and-hole polyhedra

KW - Bar-joint frameworks

KW - Panel-hinge structures

U2 - 10.1016/j.ijsolstr.2018.05.029

DO - 10.1016/j.ijsolstr.2018.05.029

M3 - Journal article

VL - 150

SP - 40

EP - 51

JO - International Journal of Solids and Structures

JF - International Journal of Solids and Structures

SN - 0020-7683

ER -