### Abstract

We examine the flexibility of periodic planar networks built from rigid corner-connected equilateral triangles. Such systems are locally isostatic, since for each triangle the total number of degrees of freedom equals the total number of constraints. These nets are two-dimensional analogues of zeolite frameworks, which are periodic assemblies of corner-sharing tetrahedra. If the corner connections are permitted to rotate, as if pin-jointed, there is always at least one collapse mechanism in two dimensions (and at least three mechanisms in three dimensions). We present a number of examples of such collapse modes for different topologies of triangular net. We show that the number of collapse mechanisms grows with the size of unit cell. The collapsible mechanisms that preserve higher symmetry of the network tend to exhibit the widest range of densities without sterical overlap.

Original language | English (US) |
---|---|

Pages (from-to) | 3517-3530 |

Number of pages | 14 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 465 |

Issue number | 2111 |

DOIs | |

State | Published - Nov 8 2009 |

### Fingerprint

### Keywords

- Flexibility
- Locally isostatic networks
- Zeolites

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)
- Physics and Astronomy(all)

### Cite this

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*,

*465*(2111), 3517-3530. https://doi.org/10.1098/rspa.2009.0307

**On the collapse of locally isostatic networks.** / Kapko, V.; Treacy, Michael; Thorpe, Michael; Guest, S. D.

Research output: Contribution to journal › Article

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, vol. 465, no. 2111, pp. 3517-3530. https://doi.org/10.1098/rspa.2009.0307

}

TY - JOUR

T1 - On the collapse of locally isostatic networks

AU - Kapko, V.

AU - Treacy, Michael

AU - Thorpe, Michael

AU - Guest, S. D.

PY - 2009/11/8

Y1 - 2009/11/8

N2 - We examine the flexibility of periodic planar networks built from rigid corner-connected equilateral triangles. Such systems are locally isostatic, since for each triangle the total number of degrees of freedom equals the total number of constraints. These nets are two-dimensional analogues of zeolite frameworks, which are periodic assemblies of corner-sharing tetrahedra. If the corner connections are permitted to rotate, as if pin-jointed, there is always at least one collapse mechanism in two dimensions (and at least three mechanisms in three dimensions). We present a number of examples of such collapse modes for different topologies of triangular net. We show that the number of collapse mechanisms grows with the size of unit cell. The collapsible mechanisms that preserve higher symmetry of the network tend to exhibit the widest range of densities without sterical overlap.

AB - We examine the flexibility of periodic planar networks built from rigid corner-connected equilateral triangles. Such systems are locally isostatic, since for each triangle the total number of degrees of freedom equals the total number of constraints. These nets are two-dimensional analogues of zeolite frameworks, which are periodic assemblies of corner-sharing tetrahedra. If the corner connections are permitted to rotate, as if pin-jointed, there is always at least one collapse mechanism in two dimensions (and at least three mechanisms in three dimensions). We present a number of examples of such collapse modes for different topologies of triangular net. We show that the number of collapse mechanisms grows with the size of unit cell. The collapsible mechanisms that preserve higher symmetry of the network tend to exhibit the widest range of densities without sterical overlap.

KW - Flexibility

KW - Locally isostatic networks

KW - Zeolites

UR - http://www.scopus.com/inward/record.url?scp=73149085473&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=73149085473&partnerID=8YFLogxK

U2 - 10.1098/rspa.2009.0307

DO - 10.1098/rspa.2009.0307

M3 - Article

AN - SCOPUS:73149085473

VL - 465

SP - 3517

EP - 3530

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 0962-8444

IS - 2111

ER -