### Abstract

Sliding frictional interfaces at a range of length scales are observed to generate travelling waves; these are considered relevant, for example, to both earthquake ground surface movements and the performance of mechanical brakes and dampers. We propose an explanation of the origins of these waves through the study of an idealized mechanical model: a thin elastic plate subject to uniform shear stress held in frictional contact with a rigid flat surface. We construct a nonlinear wave equation for the deformation of the plate, and couple it to a spinodal rate-and-state friction law which leads to a mathematically well-posed problem that is capable of capturing many effects not accessible in a Coulomb friction model. Our model sustains a rich variety of solutions, including periodic stick-slip wave trains, isolated slip and stick pulses, and detachment and attachment fronts. Analytical and numerical bifurcation analysis is used to show how these states are organized in a two-parameter state diagram. We discuss briefly the possible physical interpretation of each of these states, and remark also that our spinodal friction law, though more complicated than other classical rate-and-state laws, is required in order to capture the full richness of wave types.

Original language | English |
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Article number | 0606 |

Pages (from-to) | 1 - 27 |

Number of pages | 27 |

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

Volume | 473 |

Issue number | 2203 |

Early online date | 5 Jul 2017 |

DOIs | |

Publication status | Published - 31 Jul 2017 |

### Keywords

- Detachment front
- Friction
- Global bifurcation
- Rate-and-state
- Self-healing slip pulse

### ASJC Scopus subject areas

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

## Fingerprint Dive into the research topics of 'A phase-plane analysis of localized frictional waves'. Together they form a unique fingerprint.

## Profiles

### Jonathan Dawes

- Department of Mathematical Sciences - Professor
- Centre for Networks and Collective Behaviour
- EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
- Water Innovation and Research Centre (WIRC)
- Institute for Mathematical Innovation (IMI)
- Institute for Policy Research (IPR)
- Water Informatics: Science and Engineering Centre for Doctoral Training (WISE)
- Centre for Mathematical Biology

Person: Research & Teaching

## Cite this

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

*473*(2203), 1 - 27. [0606]. https://doi.org/10.1098/rspa.2016.0606