How collective asperity detachments nucleate slip at frictional interfaces

de Geus, Tom W. J.; Popović, Marko; Ji, Wencheng; Rosso, Alberto; Wyart, Matthieu

doi:10.1073/pnas.1906551116

How collective asperity detachments nucleate slip at frictional interfaces

Tom W. J. de Geus https://orcid.org/0000-0002-1694-3375 [email protected], Marko Popović https://orcid.org/0000-0003-2360-3982, Wencheng Ji, +1 , Alberto Rosso, and Matthieu Wyart [email protected]Authors Info & Affiliations

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved October 14, 2019 (received for review April 16, 2019)

November 7, 2019

116 (48) 23977-23983

https://doi.org/10.1073/pnas.1906551116

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Significance

Understanding how slip at a frictional interface initiates is important for a range of problems including earthquake prediction and precision engineering. The force needed to start sliding a solid object over a flat surface is classically described by a “static friction coefficient”: a constant established by measurements. It was recently questioned whether such constant exists, as it was shown to be poorly reproducible. We provide a model supporting that it is stochastic even for very large system sizes: Sliding is nucleated when, by chance, an avalanche of microscopic detachments reaches a critical radius, beyond which slip becomes unstable and propagates along the interface. It leads to testable predictions on key observables characterizing the stability of the interface.

Abstract

Sliding at a quasi-statically loaded frictional interface can occur via macroscopic slip events, which nucleate locally before propagating as rupture fronts very similar to fracture. We introduce a microscopic model of a frictional interface that includes asperity-level disorder, elastic interaction between local slip events, and inertia. For a perfectly flat and homogeneously loaded interface, we find that slip is nucleated by avalanches of asperity detachments of extension larger than a critical radius

A_{c}

governed by a Griffith criterion. We find that after slip, the density of asperities at a local distance to yielding

x_{σ}

presents a pseudogap

P (x_{σ}) \sim {(x_{σ})}^{θ}

, where

θ

is a nonuniversal exponent that depends on the statistics of the disorder. This result makes a link between friction and the plasticity of amorphous materials where a pseudogap is also present. For friction, we find that a consequence is that stick–slip is an extremely slowly decaying finite-size effect, while the slip nucleation radius

A_{c}

diverges as a

θ

-dependent power law of the system size. We discuss how these predictions can be tested experimentally.

The sliding of a block that rests on a flat surface starts when the applied tangential force passes some threshold

F_{S}

, which is proportional to the normal force

F_{N}

. Their ratio defines the friction coefficient

μ \equiv F_{S} / F_{N}

, which typically decreases with increasing sliding velocity when the latter is small (1–5). This phenomenology leads to stick–slip, whereby driving a system quasi-statically results in periods of loading that are punctuated by sudden macroscopic slip events. Experimental observations support that these events proceed by “fracture” (6–9): After a nucleation phase in which slip appears locally and evolves slowly (10, 11), a well-defined rupture front appears that travels ballistically across the frictional interface, unzipping it. This front is accompanied by a stress field in the elastic bulk that is well described by that of a propagating crack (12, 13). By contrast, the nucleation phase is much less understood. It is observed that (i) its spatial extension

A_{c}

decreases with increasing shear stress (10), (ii) there is a considerable variability in the tangential force magnitude at which macroscopic slip nucleates (10, 14, 15), and (iii) acoustic emission (16, 17) supports that nucleation occurs by bursts of spatially resolvable (16) detachments of micrometer-sized asperities (18–20). Explaining these facts is relevant for earthquake predictions (21) as well as to forecast the variability of the measured friction coefficient (10, 14, 15), of importance for precision engineering (22).

At a continuum level, rate-and-state models (23–26) are powerful phenomenological descriptions of frictional interfaces, in which friction depends on the sliding velocity as well as some history-dependent state

ϕ

of the interface. Several length scales appear in these approaches (11, 25), including a Griffith length beyond which sustained slip pulses can propagate, as well as a larger length at which these pulses can nucleate fracture, reminiscent of a first-order phase transition* (27). Yet these descriptions are coarse grained and phenomenological, and connecting them with asperity-level phenomena where disorder that arises from surface roughness is preponderant remains a challenge. Likewise, the precise meaning of the state variable

ϕ

, often thought as capturing the aging of contacts, remains to be clarified. One microscopic view is that a sliding frictional interface shares similarities with the plastic flow of amorphous materials (28). Interestingly, it was recently observed that the “state” of bulk amorphous materials (29, 30) can be quantified by the density of soft spots about to yield locally, which scales as

P (x_{σ}) \sim {(x_{σ})}^{θ}

, where

x_{σ}

is the stress increment required for a given spot to yield.^† The existence of a nontrivial exponent

θ > 0

was shown to be a necessary consequence of the long-range elastic interactions and of the nonmonotonic (varying in sign) stress redistribution triggered by plastic events (32–35). This parallel raises the intriguing possibility that frictional interfaces are characterized by some exponent

θ

as well.

Our goal is to propose a description of frictional interfaces that captures disorder at the asperity level, long-range elastic interactions between local slip events, and inertia. When the latter is absent, the physics are well understood and fall into the universality class of the depinning transition^‡ with monotonic interactions (36–38), for which a continuous transition at a unique, well-defined, macroscopic critical force

F_{c}

(39) separates a flowing and an arrested phase. There exists no macroscopic stick–slip, and at

F_{c}

motion corresponds to power-law avalanches in which many local slip events act in concert. However, what happens to this scenario when inertia matters (as it does at a frictional interface) is a matter of debate. The popular view is that inertia destroys criticality: The transition becomes first order, stick–slip appears, and for significant inertia macroscopic slip events are nucleated by a few asperities acting together (40, 41). However, another scenario has been proposed by Schwarz and Fisher (42) based on a simplified cellular automaton model describing short-range elasticity, in which stick–slip is a slowly decaying finite-size effect that vanishes in the thermodynamic limit, for which the transition is continuous. Nevertheless, Maimon and Schwarz (43) later argued that for physical systems the scenario developed in ref. 40 was presumably correct and that the conclusions of ref. 42 on the absence of hysteresis in infinite systems were nongeneric and valid only for a finely tuned model.

In this work we introduce a numerical model of flat, homogeneously loaded, frictional interfaces where inertia is properly treated by discretizing the bodies in contact by finite elements. Asperities at the interface are described by elements endowed with a random potential that represents the presence of surface roughness, allowing for sudden local slip events when a local (random) threshold stress is reached.^§ Our model thereby differs from existing ones as no microscopic constitutive friction model (such as slip-weakening, velocity-weakening, or rate-and-state; see, e.g., the spring–block models of refs. 48–51 and references therein) is presumed. In contrast, slip-weakening emerges as a consequence of mechanical noise (elastic waves generated by inertia) emitted when asperities detach, which can cause the nucleation of macroscopic slip.

Our main findings are that (a) surprisingly, the scenario developed in ref. 42 is correct: Stick–slip is a finite-size effect, although we find its power-law decay with system size to be so slow that it is significant even in very large systems. In that regime, the friction coefficient is intrinsically stochastic, consistent with observation (ii) above. (b) Despite the interaction being monotonic, due to inertia the interface presents a nontrivial exponent

θ

characterizing a pseudogap in the density of asperities about to yield. We argue that this conclusion will hold more generally to depinning problems with inertia and long-range elasticity. (c) Nucleation is triggered when avalanches get bigger than a critical radius, governed by a Griffith criterion, that diverges as a power law of the system size. Experimentally, the presence of avalanches is consistent with the measured distribution of acoustic emission (iii), while Griffith’s criterion is supported by a decreasing nucleation length with increasing stress (i). We relate the exponents associated with properties (a) and (c) to

θ

and those characterizing avalanches and confirm our predictions numerically. Finally, we propose experiments to test our results and measure

θ

.

Model

The geometry of our setup is illustrated in Fig. 1. It comprises a frictional interface (in red) embedded between two identical isotropic linear elastic materials (in blue), all discretized using finite elements and interacting in the same manner. The elements along the frictional interface (referred to as “blocks”) are plastic: They respond elastically (with the same elastic constants as the elastic bodies) up to a local yield strain (see below). To mimic energy leakage at the boundaries (by the transmission of elastic waves), we consider a viscous damping in the bulk whose magnitude is such that waves travel on the order of the system size before decaying (Methods). The system is periodic in the horizontal direction, while the top and bottom boundaries are used to impose an event-driven quasi-static simple shear. In this protocol, the strain is increased up to the next plastic event (the response to this increase is purely elastic and in mechanical equilibrium), after which an infinitesimal strain increment is applied, triggering (an avalanche of) plasticity. Once motion stops, this sequence is repeated.

Fig. 1.

Finite-element model of a frictional interface. The two bulk solids display linear elasticity, corresponding to a purely quadratic potential $V (ε)$ in each blue element (where the shear strain $ε$ is the norm of the deviatoric part of the strain tensor). The weak middle layer represents the detachment and reattachment of asperities. The potential in each such element (referred to as “block,” in red) is random and made of sequences of parabola separated by cusps, leading to sharp plastic events. The system is subjected to quasi-static simple shear by keeping the bottom boundary fixed while displacing the top boundary using infinitesimal strain increments after which energy is minimized.

The frictional interface consists of

N

“elasto-plastic” blocks (finite elements) of linear size

h

, each representing one or a few asperities.^¶ Similar blocks are used in models of plasticity of amorphous materials (46, 47). Each block is characterized by a random potential

V (ε)

function of the equivalent shear strain

ε

(the norm of the deviatoric part of the strain tensor) (Fig. 1, Bottom-Right Inset).

V (ε)

is constructed from a sequence of quadratic potentials of identical curvature, whose intersections define the yield strains. Disorder is introduced by randomly drawing the yield strains

Δ ε_{y}

from some distribution, independently for each block. We chose a Weibull distribution

P (Δ ε_{y}) = k {(Δ ε_{y})}^{k - 1} \exp [- {(Δ ε_{y})}^{k}]

with

k = 2

. To acquire statistics we consider an ensemble of independent realizations and focus on the two-dimensional (2D) case where larger systems can be reached (Methods and SI Appendix, section II).

A Single Plastic Event

Under shear loading, a block responds linearly up to reaching the local yield strain, corresponding to a cusp in

V (ε)

. Past that point, the block releases some of its elastic energy and settles in a new equilibrium position determined by the potential energy of the element and the interaction with its surroundings. Such plastic shear strain leads to a permanent redistribution of shear stress in the system that decays as a force dipole

1 / r^{d}

(ref. 53 and SI Appendix, section III), with

r

being the distance from the block and

d

the dimension of the space (here

d = 2

). Along the weak layer the kick in shear stress is strictly positive (and decays in space as

1 / r^{d}

), corresponding to a monotonic interaction. This effect alone can destabilize other blocks, leading to an avalanche of yielding events.

In addition, each yielding event emits elastic waves, causing a transient stress, whose amplitudes decay in space as a force monopole

1 / r^{d - 1}

. This effect can trigger yielding of blocks that would have remained stable otherwise, causing a dynamical weakening effect: Plastic activity leads to more inertial mechanical noise, which in turn creates more plastic activity.

Avalanches as Precursors of Macroscopic Slips

A typical stress–strain response is shown in Fig. 2A. The system first responds elastically, followed by a steady-state stick–slip behavior (highlighted in gray). The stick–slip phase consists of loading intervals punctuated by macroscopic slip during which all blocks yield many times, on average causing the stress to drop from

σ_{n}

to

σ_{c}

(Fig. 2A). Such macroscopic slips are fracture-like, as supported by the time evolution that presents a ballistic propagation front that travels at a supershear velocity (SI Appendix, section III), consistent with recent experiments (7, 8, 12, 13).

Fig. 2.

(A) Example of the stress–strain, $σ$ – $ε$ , response of a system of size $N = 3^{6}$ . The considered steady state is highlighted in gray. The ensemble average stresses before and after macroscopic slip (a system-spanning event), $σ_{n}$ and $σ_{c}$ , are indicated with dashed lines. (B) Schematic of a stick–slip cycle. The stress $Δ_{σ} \equiv σ - σ_{c}$ at which events are manually triggered to gather statistics is also indicated. Note that the apparent absence of small stress drops due to avalanches in panel A is because these avalanches are rare, as detailed in the text. (C) Distribution of slip size, $P (\tilde{S})$ , where $\tilde{S}$ is defined as the total number of times the blocks yielded during a single event. Note that we present all data, in panel C and in Figs. 3–5, in terms of the largest system ( $N = 3^{6} \times 2$ ) and validate their robustness in terms of system size in *SI Appendix*, section V.

As in experiments (17), we observe microscopic activity during the loading phases. It corresponds to events that failed to nucleate macroscopic slip and as such are important to analyze. The distribution of slip sizes

\tilde{S}

, defined as the total number of times that blocks yield during an event, shows a clear separation in two types of events: macroscopic slips at

\tilde{S} ≫ N

(indicating that blocks have yielded many times) and avalanches that occur during the loading phase^∥; see Fig. 2C and the sketch in Fig. 2B. However, the occurrence of avalanches is too rare to be insightful.

To gain more information about the avalanches, we manually trigger events at different stresses

Δ_{σ} \equiv σ - σ_{c}

, by locally applying a shear displacement perturbation to a randomly selected block along the weak layer (SI Appendix, section II). If all blocks were elastic, the displacement would simply snap back to the original equilibrium configuration. But for the elasto-plastic blocks an avalanche can be triggered, leading to a new equilibrium state.

We first focus on

Δ_{σ} = 0

. The distribution of avalanches sizes,

P (S)

, is obtained by eliminating events that result in macroscopic slip (defined as an event in which all

N

blocks yielded at least once). Strikingly, we find a power-law distribution of avalanches at

σ = σ_{c}

,

P (S) \sim S^{- τ}

[1]

with the exponent

τ ≃ 1.5

(Fig. 3A) and a fractal dimension

d_{f} ≃ 1.7

. The latter relates spatial extension

A

(the number of sites that yielded at least once) of an avalanche to its size:

S \sim A^{d_{f}}

[2]

(Fig. 3C). These results imply

P (A) \sim A^{- d_{f} (τ - 1) - 1}

[3]

as confirmed in Fig. 3B. We conclude that the stress

σ_{c}

, after macroscopic slip, is a critical point at which the distribution of avalanche sizes is scale-free.

Fig. 3.

(A) Distribution of avalanche sizes, $P (S)$ , at different stresses $σ$ at or above the critical stress $σ_{c}$ . Note that macroscopic slips, whereby the avalanche grew unstable and spanned the entire system, have been filtered from this distribution. The shaded regions correspond to the cutoff size, $S_{c}$ (see text below Eq. 4, for measurement), for each of the shown stresses. (B) Same as panel A for the area, $A$ , of avalanches. (C) Measurement of the fractal dimension, $d_{f}$ : the relationship between the area, $A$ , and the avalanche size, $S$ , at $σ = σ_{c}$ . The expected scaling and measured exponents are shown in *Insets*. In each panel the dashed line marks the power-law scaling with the indicated exponent.

Mechanism for Fracture Nucleation

Our central observation in Fig. 3 A and B is that increasing

σ

above the critical point

σ_{c}

leads to smaller and smaller cutoffs

A_{c}

and

S_{c}

for the distributions

P (A)

and

P (S)

. At first glance this is surprising, since at large stresses one may expect avalanches to be bigger. In fact, these cutoffs signify that large avalanches run away and lead to macroscopic slip (not included in these distributions). Thus the cutoffs

A_{c}

and

S_{c}

characterize the size of the avalanches required to nucleate a macroscopic slip event.

We now propose a scaling relationship for

A_{c}

as a function of

σ - σ_{c}

. We posit that

σ_{c}

is the maximum stress that the frictional layer can locally carry in the presence of endogenous inertial mechanical noise. This noise is generated by the ballistic pulses of stress emitted by failing blocks when the interface is in the process of plastically rearranging locally. Now consider triggering an avalanche at

σ > σ_{c}

. Avalanches are compact objects (as

d_{f} > 1

), implying that each block yields many times, inducing a large inertial mechanical noise. On average, this will reduce the stress inside the avalanche to

σ_{c}

(sketch in Fig. 4A), while at large distances from the avalanche the stress remains

σ > σ_{c}

. This mismatch leads to stress concentrations at the avalanche’s edges proportional to a stress intensity factor

(σ - σ_{c}) \sqrt{A}

. As postulated by Griffith (54, 55), a fracture instability** will take place when the intensity factor reaches a threshold, implying

A_{c} \sim {(σ - σ_{c})}^{- 2}

[4]

(for any

d

). We confirm this result in Fig. 4B, supporting our hypothesis that the stress inside the avalanche on average drops to

σ_{c}

. The departure from scaling in Fig. 4B at small

Δ_{σ}

is due to the value of

A_{c}

being so large that our measurements suffer from finite-size effects (SI Appendix, section V). Note that we measure

A_{c}

using the ratio of successive moments to extract

A_{c} \equiv ⟨ A^{p + 1} ⟩ / ⟨ A^{p} ⟩

. In practice we use

p = 4

to be more sensitive to the biggest avalanches while still having good statistics, but our results are robust to different choices (SI Appendix, section V).

Fig. 4.

Macroscopic Slip

Macroscopic slip nucleates when, by chance, an avalanche exceeds the nucleation radius

A_{c}

. As stress increases, more and more avalanches are triggered, while concurrently the nucleation radius shrinks. Nucleation of macroscopic slip thus becomes more and more likely with increasing stress. Typically, macroscopic slip will have happened when the stress is sufficiently large such that

n_{a} P (A > A_{c}) \sim 1,

[5]

where

n_{a}

is the number of triggered avalanches that have occurred as a result of a stress increment

Δ_{σ} = σ - σ_{c}

, and

P (A > A_{c})

is the fraction of those avalanches that exceed the radius at which macroscopic slip is nucleated. Eq. 5 thus sets the typical value of stress,

σ_{n}

, at which macroscopic slip occurs.

The probability that an avalanche has a radius larger than

A_{c}

follows from Eqs. 3 and 4:

P (A > A_{c}) \sim A_{c}^{d_{f} (1 - τ)} \sim {(σ - σ_{c})}^{- 2 d_{f} (1 - τ)}

[6]

as verified in Fig. 4C.

The number of avalanches

n_{a}

follows from the distribution

P (x_{σ})

of the stress increment

x_{σ}

required for a given block to yield for the first time after a big slip event and trigger an avalanche. Let us assert for the moment (and confirm below) that this distribution follows a power law:

P (x_{σ}) \sim {(x_{σ})}^{θ} .

[7]

In that case, the fraction of blocks that triggers an avalanche upon increasing the stress by

Δ_{σ} = σ - σ_{c}

scales like

Φ_{a} \sim \int_{0}^{Δ_{σ}} {(x_{σ})}^{θ} d x_{σ} \sim {(σ - σ_{c})}^{θ + 1} .

[8]

This allows us to measure

θ

by counting the number of avalanches during the loading periods. We find a nontrivial exponent

θ ≃ 3.7

, as shown in Fig. 5. For the number of avalanches, we thus get

n_{a} = N Φ_{a} \sim N {(σ - σ_{c})}^{θ + 1} .

[9]

Inserting Eqs. 6 and 9 into Eq. 5 leads to

σ_{n} - σ_{c} \sim N^{\frac{- 1}{2 d_{f} (τ - 1) + θ + 1}} \sim N^{- 0.16} .

[10]

This argument results in the following: (i) The stress

σ_{n}

at which macroscopic slip nucleates is stochastic, as embodied by Eq. 6. (ii) The stick–slip amplitude

σ_{n} - σ_{c}

eventually vanishes as the number of asperities

N \to \infty

. Stick–slip is thus a finite-size effect, yet the decay is so slow that it is expected to persist in realistic systems. In a truly infinite system avalanches should be power-law distributed. (iii) The fracture nucleation radius diverges as

A_{c} (σ = σ_{n}) \sim N^{\frac{2}{2 d_{f} (τ - 1) + θ + 1}} \sim N^{0.32} .

[11]

Fig. 5.

Cumulative probability density of avalanches as a function of the stress increase $Δ_{σ} = σ - σ_{c}$ , measured by counting the number of avalanches during quasi-static loading. It provides a direct measurement of $P (x_{σ}) \sim {(x_{σ})}^{θ}$ characterizing the interface stability. The dashed line marks the power-law scaling with the indicated exponent.

Argument for Pseudogap $P (x_{σ}) \sim {(x_{σ})}^{θ}$

The stability distribution,

P (x_{σ})

, can in general obey one of three scenarios at small

x_{σ}

: (i) depinning, a finite number of blocks can yield after a small increase of stress, characterized by an exponent

θ = 0

(32, 36); (ii) a pseudogap, the number of blocks that can yield vanishes only at

x_{σ} = 0

, i.e.,

θ > 0

; and (iii) a gap, a small depleted region at small

x_{σ}

, such that

P (x_{σ} < D) = 0

for some small but finite

D

, thus requiring a finite increase of stress to destabilize any block. This scenario appears to be required to get true stick–slip as

N \to \infty

.

Our data in Fig. 5 and other measurements below support scenario (ii). We now exclude the depinning scenario based on a stability argument. In the presence of inertia, the temporary stress overshoot can destabilize blocks that would otherwise stop at small

x_{σ}

. Stability of the system requires that the number of blocks that are destabilized by one event does not diverge when the system size goes to infinity. This leads to the condition

θ > 0

as follows: When a block fails, it emits a temporary stress overshoot

σ_{I} \sim 1 / r

(in 2D). The probability that this will destabilize other blocks is

P (x_{σ} < σ_{I}) \sim r^{- (θ + 1)}

. Consequently, in a system of size

R

the number of destabilized blocks

n_{f} \sim \int_{h}^{R} r^{- (θ + 1)} d r

diverges as

R \to \infty

, unless

θ > 0

(see SI Appendix, section IV for a more general argument).

We currently do not have a theory for the value of exponent

θ

, but preliminary observations indicate that

θ

is nonuniversal. Building a theory to understand

θ

should explain the following observations (presented in detail in SI Appendix, section IV): (a) The blocks for which

x_{σ}

is very small following a macroscopic slip event typically lie in a shallow well followed by another shallow well in the block (56). (b) As a consequence, when triggered, they tend to lead to small slips and are less likely to trigger slip in other sites. As a result, there exists another exponent

θ' ≃ 2.5 \leq θ

characterizing the density of sites at a distance

x_{σ}

to yield, unconditioned to subsequently triggering an avalanche (our argument and measure in Fig. 5 is conditioned to sites triggering an avalanche). (c) The exponent

θ

is not universal and depends on the specific choice of disorder, in particular on the parameter

k

entering the Weibull distribution and characterizing the probability to find narrow wells. Using

k = 1.2

instead of

k = 2

, we find

θ' ≃ 1.4

.

The presence of the strong depletion of the number of the almost unstable blocks, induced by the macroscopic slips (

θ > 0

for Eq. 7), can be a possible explanation of the observed exponent

τ ≃ 1.5

that characterizes the distribution of the avalanche sizes in Eq. 1. For the depinning transition, in the overdamped limit, we know that

τ_{dep} = 1.28

(57, 58). However, we also know that each such avalanche is a collection of spatially disconnected slipping regions, called clusters. When treated as separate events, the distribution of avalanche sizes of individual clusters is also scale-free, but with a larger exponent,

τ_{clus} ≃ 1.56

(59), close to our measured

τ ≃ 1.5

. The existence of these disconnected clusters is a consequence of the long-range nature of the elastic interactions (60): A slipping block is a source of instability for the neighborhood, but also for blocks far away that are very close to their yield stress (have a small

x_{σ}

). The presently observed strong depletion of

P (x_{σ})

for small

x_{σ}

implies that there are very few blocks close to yielding, thus reducing the likeliness of triggering a “secondary,” disconnected, avalanche.

Discussion

We have introduced a model of a frictional interface that includes microscopic disorder at the asperity scale, long-range elastic coupling between local slip events, and the propagation of inertial waves. Our results support a description unifying collective avalanches of asperity detachments and fracture-like macroscopic slip events, in which the former nucleates the latter once a critical avalanche size is reached. These predictions are compatible with existing observations: The presence of avalanches is consistent with the measured distribution of acoustic emission (16, 17), while Griffith’s criterion is supported by a decreasing nucleation length with increasing stress (10). Two surprises emerge from our predictions. First, a key aspect of the interface is the distribution of asperities about to yield, which is very much depleted and characterized by a nontrivial exponent

θ

after a macroscopic slip event. Second, we find that the transition to sliding is a continuous transition in the thermodynamic limit, but that finite-size effects decay extremely slowly: The stress drop is a stochastic quantity whose typical scale decays as

N^{- 0.16}

and will thus persist in very large systems, leading to a slowly diverging nucleation radius

A_{c} \sim N^{0.32}

.

Our predictions could be quantitatively tested in nearly flat and homogeneously loaded samples, which may be achievable experimentally using the apparatus of refs. 61 or 62. In particular, microscopic slip events could be measured using (an array of) mechanical or acoustic sensors like those in refs. 7 or 16. Their cumulative number while quasi-statically loading the sample by a stress increment

Δ_{σ}

after a macroscopic slip event is proportional to

{(Δ_{σ})}^{θ + 1}

, thus allowing one to access empirically the pseudogap exponent

θ

. Moreover, well-separated avalanches could be acquired using our trick of triggering avalanches at different stress levels after macroscopic slip, for instance by supplying a focused acoustic signal to the system and measuring the magnitude of the mechanical or acoustic response. We expect the distribution of the magnitude to display a power law

P (S) \sim S^{- τ}

with a cutoff

S_{c}

decreasing as

S_{c} \sim A_{c}^{d_{f}} \sim {(Δ_{σ})}^{- 2 d_{f}}

. Beyond

τ

, such a measurement would thus also yield an estimate of the fractal dimension of the avalanches

d_{f}

, without the need to spatially resolve the avalanches. As a reference, we document the statistics needed to extract these exponents reliably using our model in SI Appendix, section VI.

There is an apparent opposition between the description presented here and rate-and-state models where velocity-weakening is assumed to hold in the continuous limit, and nucleation stems from a first-order transition. It would be very interesting to study how these two scenarios evolve when disorder is present at all scales (including the fact that the surface can have a roughness exponent and the loading can be very heterogeneous). It is possible that in our approach as well, the transition becomes first order for certain statistics of the disorder. We view it as an important extension of the present work. Another important extension is the inclusion of creep. It may be readily achievable by putting our model in contact with a thermal bath, since in that case individual asperities will age to find a deeper nearby well.

Finally, it is interesting to ask which class of dynamical transitions can become first order due to inertia and which cannot. The role of inertia has been studied recently in amorphous materials (63–67), where it leads to a large pseudogap exponent

θ

comparable to ours (63) (and much larger than the one present in the absence of inertia in these materials). It has been proposed that depending on the amount of damping, different universality classes could exist (64, 65), but that for strongly underdamped systems the transition appears to become first order (63). If confirmed, we speculate that the cause of the difference between amorphous solids and frictional interfaces is that avalanches are compact objects (having a fractal dimension

d_{f} > 1

) only in the latter case. If that were not true, our assumption that the inertial noise within an avalanche is comparable to that occurring in a macroscopic slip event may not hold, possibly leading to different physics.

Methods

We consider two ensembles, each consisting of independent realizations comprising an approximately square box characterized by

N

blocks along the weak layer, with

N = 3^{6}

(300 realizations) and

N = 3^{6} \times 2

(1,000 realizations). The mechanical response is approximately incompressible, which allows us to focus on the shear response. The box is assumed periodic in the horizontal direction. Quasi-static shear is applied by fixing the displacement of the bottom boundary to zero, while incrementing the displacement of the top boundary in very small steps (although we efficiently skip periods in which no yielding takes place, by homogeneously distributing the shear strain). Loading is stopped when the local strain exceeds a maximum.

After each step the energy is minimized according to the following equation of motion:

ρ \vec{a} (\vec{r}) = \vec{\nabla} \cdot σ (ε (\vec{r})) - α \vec{v} (\vec{r})

[12]

(see SI Appendix, section I for nomenclature). From left to right, Eq. 12 comprises (i) an inertial term, in which

ρ

is the mass density and

\vec{a} = \partial_{t}^{2} \vec{u}

is the acceleration (where

\vec{u}

is displacement and

t

is time); (ii) the divergence of the stress tensor

σ

; and (iii) a non-Rayleigh damping term, where

α

is the damping coefficient and

\vec{v} = \partial_{t} \vec{u}

is the velocity. The stress

σ

follows from strain

ε

(which is the symmetric gradient of the displacement

\vec{u}

) using the constitutive model outlined in the main text. We set

α

such that kinetic energy is effectively leaked at the (periodic) boundaries.

Eq. 12 is solved in the weak form by discretizing in space and time. In space, we discretize using finite elements. These elements coincide with the elasto-plastic blocks along the weak layer, while the elastic domain is discretized using elements that are conveniently chosen to increase in size with increasing distance to the weak layer to save computational costs. We discretize in time using the velocity-Verlet protocol.

Note that we formulate our model under the small strain assumption. To respect this assumption but still acquire a decently long steady-state response, we choose the yield strains to be very small. This fixes the absolute strain and stress values to be small, which we rescale for visualization to be of order one. See SI Appendix, section II for details. Furthermore, note that the numerical implementation is open source (68, 69) and that we have made all data underlying this paper freely available (70).

Data Availability

Data deposition: All data and codes underlying this paper are freely available at GitHub (https://github.com/tdegeus/GooseFEM and https://github.com/tdegeus/ElastoPlasticQPot) and Zenodo (DOI: https://doi.org/10.5281/zenodo.3477938).

Acknowledgments

T.G. was partly financially supported by The Netherlands Organization for Scientific Research (NWO) by a NWO Rubicon grant, number 680-50-1520. M.W. thanks the Swiss National Science Foundation for support under Grant No. 200021-165509, and the Simons Foundation Grant (

#

454953 to M.W.). We acknowledge an anonymous referee for useful comments on experimental validation.

Supporting Information

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References

1

C. H. Scholz, J. T. Engelder, The role of asperity indentation and ploughing in rock friction – I. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 13, 149–154 (1976).

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Argument for Pseudogap $P (x_{σ}) \sim {(x_{σ})}^{θ}$