A micromechanical model of rate and state friction: 1. Static and dynamic sliding

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In this paper we analyze the influence of shear and normal stress changes on frictional properties. This problem is fundamental as, for instance, sudden stress changes are naturally induced on active faults by nearby earthquakes. As any stress changes can be seen as resulting from a succession of infinitesimal stress steps, the role of sudden stress changes is crucial to our understanding of fault dynamics. Laboratory experiments carried out by Linker and Dieterich (1992) and Nagata et al. (2012), considering steps in normal and shear stress, respectively, show an instantaneous response of the state variable (a proxy for the evolution of contact surface in our model) to a sudden stress change. We interpret this response as being due to an (instantaneous) elastic response of the plastic and elastic contacts. We assume that the anelastic response of the plastic contacts is frozen during sudden stress changes. The contacts, which were driven by plasticity before the stress change, are elastically accommodated during the sudden variation of the load. On the contrary, when the loading is slowly varying, elastic deformation of plastic contacts can be neglected. Our model is able to explain the evolution law for the state variable reported by Linker and Dieterich (1992).

“…Consequently, the elastic load w e and plastic load w p per contact are given by

w_{e} = \frac{α_{2} (1 + α_{3})}{\overset{α}{true}} \frac{W}{N}

w_{p} = \frac{1 + α_{3}}{\overset{α}{true}} \frac{W}{N}

with

α_{3} = \frac{f_{e}}{1 - f_{e}}, f_{e} = \frac{α_{3}}{1 + α_{3}}

and

\overset{α}{true} = 1 + α_{2} α_{3}

We have also [ Perfettini and Molinari , ]

μ = \frac{\overset{α}{true} - 1}{\overset{α}{true}} μ_{e} + \frac{τ_{p} Σ_{r}^{p}}{σ Σ_{0}} = \frac{\overset{α}{true} - 1}{\overset{α}{true}} μ_{e} + \frac{μ_{p}}{\overset{α}{true}}

where μ e is the friction coefficient of elastic contacts considered constant and

μ_{p} = \frac{τ_{p}}{σ_{p}} = \frac{F_{p}}{W_{p}}

…”

Section: Main Results From the Micromechanical Model Of Frictionmentioning

confidence: 99%

τ_{p} = τ_{p}^{*} ([], 1 + β_{τ} \log ((), \frac{V}{V_{*}})) with β_{τ} > 0

Similarly, it is assumed that the steady state value of the plastic contact area is given by

Σ_{r (ss)}^{p} = \frac{σ}{σ_{*}} {Σ_{r}^{p}}^{*} ([], 1 - β_{Σ} \log ((), \frac{V}{V_{*}})) with β_{Σ} > 0

for fixed values of σ and V . Note that away from steady state, the plastic contact area

Σ_{r}^{p}

can depend on time, slip rate, and normal stress.…”

Section: Main Results From the Micromechanical Model Of Frictionmentioning

confidence: 99%

“…Perfettini and Molinari [] proposed the following generalized evolution law

\overset{θ}{true} = \overset{F}{true} (θ, V) - \frac{a}{b} {\overset{c}{true}}_{V} θ \frac{\overset{V}{true}}{V} - \frac{{\overset{c}{true}}_{σ}}{b} θ \frac{\overset{σ}{true}}{σ}

with

\overset{F}{true} (θ, V) = (1 - {\overset{c}{true}}_{V}) F (θ, V)

where

{\overset{c}{true}}_{V}

is a new frictional parameter. As discussed in section 5.3, equation reduces to the Linker‐Dieterich law for

{\overset{c}{true}}_{V} = 0

and F = F A or F = F S .…”

Section: Most Popular Evolution Laws For the State Variablementioning

confidence: 99%

“…Alternatively, it has been proposed that the term

σ ([], μ_{*} + b \log ((), \frac{θ}{θ_{*}}))

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A micromechanical model of rate and state friction: 2. Effect of shear and normal stress changes

Molinari

Perfettini

2017

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A Physics‐Based Rock Friction Constitutive Law: Steady State Friction

Aharonov

Scholz

2018

Experiments measuring friction over a wide range of sliding velocities find that the value of the friction coefficient varies widely: friction is high and behaves according to the rate and state constitutive law during slow sliding, yet markedly weakens as the sliding velocity approaches seismic slip speeds. We introduce a physics‐based theory to explain this behavior. Using conventional microphysics of creep, we calculate the velocity and temperature dependence of contact stresses during sliding, including the thermal effects of shear heating. Contacts are assumed to reach a coupled thermal and mechanical steady state, and friction is calculated for steady sliding. Results from theory provide good quantitative agreement with reported experimental results for quartz and granite friction over 11 orders of magnitude in velocity. The new model elucidates the physics of friction and predicts the connection between friction laws to independently determined material parameters. It predicts four frictional regimes as function of slip rate: at slow velocity friction is either velocity strengthening or weakening, depending on material parameters, and follows the rate and state friction law. Differences between surface and volume activation energies are the main control on velocity dependence. At intermediate velocity, for some material parameters, a distinct velocity strengthening regime emerges. At fast sliding, shear heating produces thermal softening of friction. At the fastest sliding, melting causes further weakening. This theory, with its four frictional regimes, fits well previously published experimental results under low temperature and normal stress.

A microscopic model of rate and state friction evolution

Rubin

2017

Whether rate‐ and state‐dependent friction evolution is primarily slip dependent or time dependent is not well resolved. Although slide‐hold‐slide experiments are traditionally interpreted as supporting the aging law, implying time‐dependent evolution, recent studies show that this evidence is equivocal. In contrast, the slip law yields extremely good fits to velocity step experiments, although a clear physical picture for slip‐dependent friction evolution is lacking. We propose a new microscopic model for rate and state friction evolution in which each asperity has a heterogeneous strength, with individual portions recording the velocity at which they became part of the contact. Assuming an exponential distribution of asperity sizes on the surface, the model produces results essentially similar to the slip law, yielding very good fits to velocity step experiments but not improving much the fits to slide‐hold‐slide experiments. A numerical kernel for the model is developed, and an analytical expression is obtained for perfect velocity steps, which differs from the slip law expression by a slow‐decaying factor. By changing the quantity that determines the intrinsic strength, we use the same model structure to investigate aging‐law‐like time‐dependent evolution. Assuming strength to increase logarithmically with contact age, for two different definitions of age we obtain results for velocity step increases significantly different from the aging law. Interestingly, a solution very close to the aging law is obtained if we apply a third definition of age that we consider to be nonphysical. This suggests that under the current aging law, the state variable is not synonymous with contact age.