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## Abstract

The physics of dry friction is often modelled by assuming that static and kinetic frictional forces can be represented by a pair of coefficients usually referred to as μs and μk, respectively. In this paper we re-examine this discontinuous dichotomy and relate it quantitatively to the more general, and smooth, framework of rate-and-state friction. This is important because it enables us to link the ideas behind the widely used static and dynamic coefficients to the more complex concepts that lie behind the rate-and-state framework. Further, we introduce a generic framework for rate-and-state friction that unifies different approaches found in the literature.

We consider specific dynamical models for the motion of a rigid block sliding on an inclined surface. In the Coulomb model with constant dynamic friction coefficient, sliding at constant velocity is not possible. In the rate-and-state formalism steady sliding states exist, and analysing their existence and stability enables us to show that the static friction coefficient μs should be interpreted as the local maximum at very small slip rates of the steady state rate-and-state friction law.

Next, we revisit the often-cited experiments of Rabinowicz (J. Appl. Phys., 22:1373–1379, 1951). Rabinowicz further developed the idea of static and kinetic friction by proposing that the friction coefficient maintains its higher and static value μs over a persistence length before dropping to the value μk. We show that there is a natural identification of the persistence length with the distance that the block slips as measured along the stable manifold of the saddle point equilibrium in the phase space of the rate-and-state dynamics. This enables us explicitly to define μs in terms of the rate-and-state variables and hence link Rabinowicz's ideas to rate-and-state friction laws.

This stable manifold naturally separates two basins of attraction in the phase space: initial conditions in the first one lead to the block eventually stopping, while in the second basin of attraction the sliding motion continues indefinitely. We show that a second definition of μs is possible, compatible with the first one, as the weighted average of the rate-and-state friction coefficient over the time the block is in motion.

We consider specific dynamical models for the motion of a rigid block sliding on an inclined surface. In the Coulomb model with constant dynamic friction coefficient, sliding at constant velocity is not possible. In the rate-and-state formalism steady sliding states exist, and analysing their existence and stability enables us to show that the static friction coefficient μs should be interpreted as the local maximum at very small slip rates of the steady state rate-and-state friction law.

Next, we revisit the often-cited experiments of Rabinowicz (J. Appl. Phys., 22:1373–1379, 1951). Rabinowicz further developed the idea of static and kinetic friction by proposing that the friction coefficient maintains its higher and static value μs over a persistence length before dropping to the value μk. We show that there is a natural identification of the persistence length with the distance that the block slips as measured along the stable manifold of the saddle point equilibrium in the phase space of the rate-and-state dynamics. This enables us explicitly to define μs in terms of the rate-and-state variables and hence link Rabinowicz's ideas to rate-and-state friction laws.

This stable manifold naturally separates two basins of attraction in the phase space: initial conditions in the first one lead to the block eventually stopping, while in the second basin of attraction the sliding motion continues indefinitely. We show that a second definition of μs is possible, compatible with the first one, as the weighted average of the rate-and-state friction coefficient over the time the block is in motion.

Original language | English |
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Pages (from-to) | 70-93 |

Number of pages | 24 |

Journal | Journal of the Mechanics and Physics of Solids |

Volume | 78 |

Early online date | 2 Feb 2015 |

DOIs | |

Publication status | Published - May 2015 |

## Keywords

- stiction loss
- stick-slip
- friction angle
- slope stability
- landslide
- bistability

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