Emergence of Cracklike Behavior of Frictional Rupture: The Origin of Stress Drops

The process of frictional rupture, i.e., the failure of frictional systems, abounds in the technological and natural world around us, ranging from squealing car brake pads to earthquakes along geological faults. A general framework for understanding and interpreting frictional rupture commonly involves an analogy to ordinary crack propagation, with far-reaching implications for various disciplines from engineering tribology to geophysics. An important feature of the analogy to cracks is the existence of a reduction in the stress-bearing capacity of the ruptured interface, i.e., of a drop from the applied stress, realized far ahead of a propagating rupture, to the residual stress left behind it. Yet, how and under what conditions such finite and well-defined stress drops emerge from basic physics are not well understood. Here, we show that for a rapid rupture a stress drop is directly related to wave radiation from the frictional interface to the bodies surrounding it and to long-range bulk elastodynamics and not exclusively to the physics of the contact interface. Furthermore, we show that the emergence of a stress drop is a transient effect, affected by the wave travel time in finite systems and by the decay of long-range elastic interactions. Finally, we supplement our results for rapid rupture with predictions for a slow rupture. All of the theoretical predictions are supported by available experimental data and by extensive computations. Our findings elucidate the origin of stress drops in frictional rupture; i.e., they offer a comprehensive and fundamental understanding of why, how, and to what extent frictional rupture might be viewed as an ordinary fracture process.


I. BACKGROUND AND MOTIVATION
Rapid slip along interfaces separating bodies in frictional contact is mediated by the spatiotemporal dynamics of frictional rupture [1,2]. Frictional rupture is a fundamental process of prime importance for a broad range of physical systems; e.g., it is responsible for squealing in car brake pads [3], for bowing on a violin string [4], and for earthquakes along geological faults [5][6][7], to name just a few well-known examples. The essence of frictional rupture propagation is that a state of relatively high slip rate (the rate of interfacial shear displacement discontinuity) behind the rupture edge propagates into a low or vanishing slip-rate state ahead of it; cf. Fig. 1. As such, frictional rupture appears to be essentially similar to ordinary tensile (opening) cracks, where a finite tensile displacement discontinuity (broken material) state behind the crack edge propagates into a zero tensile displacement discontinuity (intact material) state ahead of it [8].
There is, however, an important and fundamental difference between frictional rupture and ordinary tensile cracks that manifests itself in the stress states associated with these two processes. A tensile crack, i.e., a crack subjected to opening forces, is composed of surfaces that cannot support stress, so the stress behind its edge vanishes. Consequently, tensile (opening) crack propagation is a process in which the applied stress ahead of the crack edge drops to zero behind it. This stress drop, which accompanies tensile crack propagation, has dramatic implications. Most notably, the loss of stress-bearing capacity along the crack surfaces is compensated by a large concentration of deformation and stress near the crack edge, oftentimes in a way that mimics a mathematical singularity, whose intensity increases with increasing stress drops [8]. Frictional rupture is different from tensile cracks, because the finite frictional interaction between the two bodies in contact behind the rupture edge generically implies that the stress there cannot drop to zero but rather remains finite.
The close relations between frictional rupture and tensile cracks can be maintained if, as is widely assumed, the stress behind the frictional rupture edge-the residual stress τ res -is well defined and is generically smaller than the far-field stress τ d that is required to drive a rupture. Moreover, the residual stress τ res is generally assumed to be an intrinsic interfacial property of the slipping contact interface, typically related to the kinetic friction coefficient. Under these assumptions, a finite stress drop Δτ ≡ τ d − τ res > 0 exists, and effective cracklike properties of frictional rupture, e.g., edge singularity, are expected to emerge. These assumptions have been adopted in an extremely broad range of theoretical and numerical studies [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27], and their implications have been consistent with geophysical observations [26,28] and have been confirmed in some recent laboratory experiments [29][30][31][32][33][34][35][36]. In fact, stress drops are among the few remotely observable parameters in earthquake science, providing a key link to the frictional properties of faults, which can be only indirectly inferred.
Yet, to the best of our knowledge, currently there is no basic understanding of how and under what conditions the effective cracklike behavior of frictional rupture emerges from fundamental physics. More specifically, there is a need to understand what the physical origin of a finite stress drop Δτ > 0 is and under what conditions it emerges. Here, we address these basic questions; first, we show that for a rapid rupture a finite and well-defined stress drop is not an interfacial property, as is widely assumed, but rather it is directly related to wave radiation from the frictional interface to the bodies surrounding it (the so-called radiation damping effect [37][38][39][40]) and to long-range bulk elastodynamic interaction effects. Second, we show that the emergence of a stress drop is a finite-time effect, limited by the wave travel time in finite systems. Third, we show that for a slow rupture, i.e., a rupture that is significantly slower than the elastic wave speeds [41][42][43][44], stress drops are transiently controlled by the long-range quasistatic elasticity of the bodies surrounding the frictional interface. A reanalysis of very recent experimental results, reported by two different experimental groups, provides strong support to our theoretical predictions, for both a rapid and a slow rupture. All in all, our findings elucidate the origin of stress drops in frictional rupture; i.e., they offer a comprehensive and fundamental understanding of why, how, and to what extent frictional rupture might be viewed as an ordinary fracture process.

II. THE PHYSICAL ORIGIN AND MAGNITUDE OF THE STRESS DROP ASSOCIATED WITH FRICTIONAL RUPTURE
The starting point for our discussion is a physically motivated interfacial constitutive law, i.e., a relation between the dynamical and structural variables that characterize a frictional interface and the frictional resistance stress τ [45]. A frictional interface is formed when two bodies come into contact. Each of them satisfies its own continuum momentum balance equation ρüðr; tÞ ¼ ∇ · σðr; tÞ, where ρ is the mass density, u and r ¼ ðx; yÞ (in two dimensions) are the displacement and position vector fields, respectively, and σ is the stress tensor field (a superposed dot represents a time derivative). σ in each body is related to u through a bulk constitutive law (i.e., a constitutive law that characterizes the bodies forming the interface), oftentimes Hooke's law of linear elasticity, adopted below as well. Note that body forces are neglected in the momentum balance equation.
The interfacial constitutive law, at any position x along the interface and at any time t, is described by the following local relation: which must be supplemented with a dynamical equation for the evolution of ϕ. Extensive evidence indicates that ϕ physically represents the age or maturity of the contact [45,[47][48][49][50][51][52] and that its evolution takes the form FIG. 1. A schematic representation of the spatial slip-rate v profile of a frictional rupture propagating at a velocity c r from left to right. A slipping or sliding state with a relatively high slip velocity v > 0 characterizes the interface behind the propagating rupture edge, and a low or vanishing slip-rate state, v ≈ 0, characterizes the interface ahead of it.
with gð1Þ ¼ 0 and where ϕ is of time dimension. The characteristic slip displacement D controls the transition from a stick state v ≈ 0, with a characteristic structural state ϕ ¼ ϕ 0 , to a steadily slipping or sliding state v > 0, with ϕ ss ¼ D=v. The precise functional form of gð·Þ [with gð1Þ ¼ 0] plays no role in what follows. The function fðjvj; ϕ ss ¼ D=vÞ ¼ τ ss ðvÞ=σ, under steady-state sliding conditions and a controlled normal stress σ, has been measured over a broad range of slip rates v for many materials [45]. Together with general theoretical considerations [59], it is now established that the steadystate frictional stress τ ss ðvÞ is generically N shaped, as shown in Fig. 2(a). Consider then a frictional system driven by a shear stress τ d , which is larger than the minimum of the τ ss ðvÞ curve; cf. Fig. 2(a). What are the generic properties of frictional rupture that might emerge under these conditions?
As explained in Sec. I, frictional rupture is a propagating spatiotemporal object that features a relatively high slip state v > 0 behind its edge and a stick (no or slow slip) v ≈ 0 state ahead of it, as shown in Fig. 1. This spatiotemporal dynamical process can be directly related to Fig. 2(a), where the driving stress τ d intersects the τ ss ðvÞ curve at three points. The leftmost intersection point features an extremely small slip velocity v ≈ 0, which corresponds to the state ahead of the rupture edge. We stress that, in general, the state of the interface ahead of the rupture edge may be far from steady state, and the results to follow are largely independent of its detailed properties. The rightmost intersection point features a relatively large slip velocity v > 0, which corresponds to the state behind the rupture edge. The transition between these two states takes place in the edge region and is controlled by _ ϕ in Eq. (2) and by spatiotemporal bulk dynamics. In this transition region, the slip velocity v also goes through the intermediate intersection point, which is not a stable fixed point as the other two. The crucial observation is that the stress behind a steadily propagating frictional rupture is τ d ; i.e., the residual stress equals the driving stress, τ res ¼ τ d . This observation implies that we expect no stress drop to emerge at all, Δτ ¼ τ d − τ res ¼ 0, and consequently no cracklike behavior.
In many studies available in the literature, a steady-state friction curve τ ss ðvÞ that does not feature a minimum is adopted [38,39,[62][63][64]. We consequently discuss here such a no-minimum friction curve and plot an example of it in Fig. 2(a). In this case, the driving stress τ d behind the rupture edge cannot be balanced by the friction stress, and the slip velocity in this region is expected to continuously accelerate. As such, we cannot expect a well-defined steady-state stress drop to emerge, though the stress will definitely drop below τ d behind the rupture edge.
The discussion above, for both the N-shaped and the nominimum steady-state friction curves, seems to lead to the quite remarkable conclusion that based on basic physics considerations we expect no finite and well-defined steadystate stress drops to emerge at all in the context of friction rupture and, hence, no cracklike behavior as well. This conclusion appears to be in sharp contrast to ample evidence indicating the existence of finite and well-defined steady-state stress drops in various frictional systems [1,29,30,32,33,35,65,66]. How can one reconcile the two apparently conflicting conclusions? To address this question, let us write down in more detail the general expression for interfacial shear stress σ xy ðx; y ¼ 0; tÞ, valid also out of the steady state, when a constant driving stress τ d is applied at the far boundaries of the systems, say, at y ¼ AEH (H is the height of each of the two bodies in contact). For bulk linear elastodynamics, we have σ xy ðx; y ¼ 0; tÞ ¼ τ d þsðx; tÞ, wheresðx; tÞ is a spatiotemporal integral that quantifies the long-range (in both space and time) elastodynamic interaction between different parts of the interface [67][68][69]. Under strict homogeneous steady-state conditions, we havesðx;tÞ → 0 and, consequently, σ xy ðx; y ¼ 0; tÞ → τ d , which corresponds to the rightmost intersection point in Fig. 2(a) attained far behind the rupture edge (see the discussion above). At finite times, before strict steady-state conditions are attained, the spatiotemporal integral termsðx; tÞ makes a finite contribution to σ xy ðx; y ¼ 0; tÞ, which quantifies the deviation from the steady state.
Under these conditions, and, in particular, for times in which information regarding the evolution of the slip velocity vðx; tÞ relative to some initial or reference slip velocity v 0 does not have enough time to propagate to the boundaries at y ¼ AEH and back to the interface, the spatiotemporal integral termsðx; tÞ can be decomposed into two contributions; one is a local contribution of the form ðμ=2c s Þ½vðx; tÞ − v 0 , where μ is the linear elastic shear modulus and c s is the shear wave speed, and the other is a nonlocal (in space and time) contribution sðx; tÞ [67][68][69]. This decomposition is valid for times shorter than OðH=c s Þ, before wave interaction with the boundaries is possible, and for these times the interfacial shear stress takes the form [55,57,[67][68][69][70] In many studies available in the literature, the idealized infinite system limit H → ∞ is considered, for which Eq. (3) is valid at all times. The term ðμ=2c s Þ½vðx; tÞ − v 0 physically represents wave radiation from the interface to the bodies that form it and is, therefore, known as the radiationdamping term [37][38][39][40]. It is associated with "damping" because, from the perspective of the interface, it acts as a viscous stress with μ=2c s being the effective viscosity. This term makes an important contribution to stress drops in frictional rupture, as is shown next. Consider a point along a frictional interface that is initially located ahead of a propagating frictional rupture and whose slip velocity is v ≈ 0, which represents v 0 in Eq. (3). When the frictional rupture goes through this point, the stress and slip velocities vary significantly. Suppose then that the system height H is sufficiently large such that the spatiotemporal integral decays once the rupture goes sufficiently far ahead, sðx; tÞ → 0, but the radiation damping contribution is still valid (i.e., shear waves do not have enough time to propagate to the far boundaries and back). Under these conditions, the slip velocity v res at the spatial point under consideration, now far behind the frictional rupture, is determined by the shear stress balance σ xy ¼ τ and satisfies where v 0 res ≫ v 0 and sðx; tÞ → 0 are used. Note that the superscript "0" in v 0 res represents the fact that it is the theoretically predicted residual slip velocity under the assumption that sðx; tÞ ¼ 0 far behind the propagating rupture front. The residual stress at this point takes the form τ res ¼ τ d − ðμ=2c s Þv 0 res , and, consequently, a finite stress drop of magnitude is expected to emerge on times shorter than OðH=c s Þ.
A geometric representation of Eqs. (4) and (5) is shown in Fig. 2(b) for the N-shaped τ ss ðvÞ and in Fig. 2(c) for the no-minimum τ ss ðvÞ, both shown in Fig. 2(a). In Figs. 2(b) and 2(c), the left-hand-side of Eq. (4) τ ss ðvÞ þ ðμ=2c s Þv is regarded as an effective steady-state curve and is plotted by a dashed line. The radiation-damping contribution ðμ=2c s Þv in Fig. 2(b) shifts the location of the effective (finite-time) steady-state slip rate to lower rates [compared to the strict steady state represented by the rightmost intersection point in Fig. 2(a)]. In Fig. 2(c), it gives rise to an effective (finite-time) steady-state slip rate, which simply does not exist for the no-minimum curve in Fig. 2(a). This result shows that Eq. (5) is valid independently of the properties of τ ss ðvÞ. Note that relevant solutions v 0 res to Eq. (4) exist only if τ d is larger than the effective (finite-time) minimum of the steady-state friction curve, as is highlighted in Figs. 2(b) and 2(c). The very same condition plays a central role in the analysis of Ref. [39], where the conditions for rupture mode selection (selfhealing pulses versus cracklike) are extensively discussed. Somewhat related issues are also discussed in Ref. [71].
We note that, once Δτ is assumed to exist, the relation in Eq. (5) between Δτ and v res as given quantities is a known elastodynamic relation [72], which has previously received some experimental support (e.g., see Fig. 8 in Ref. [73]) and is strongly supported by recent experimental data we have extracted from recent experimental works [35,36]; see Sec. IV. To the best of our knowledge, however, none of these works (except for Ref. [39], as mentioned above) predict the residual slip velocity v res to emerge from the combined effect of the steady-state friction curve τ ss ðvÞ and of the radiation-damping term ðμ=2c s Þv as predicted in Eq. (4), and none of them interpret Eq. (5) as a major contribution to stress drops Δτ in frictional rupture. In the context of a quasistatic fault model of Ref. [74], the radiation-damping term ðμ=2c s Þv is added in an ad hoc manner in order to avoid unbounded slip velocities from emerging during frictional instabilities in the quasistatic formulation. It is noted in this context [74] that the effective viscosity μ=2c s affects the magnitude of the stress drop Δτ when τ ss ðvÞ has no minimum [cf. Fig. 2(c)]; nevertheless, the general physical picture in which the radiation-damping term significantly contributes to stress drops in frictional rupture, independently of the properties of τ ss ðvÞ, has not been discussed.
When the radiation-damping term ðμ=2c s Þv in Eq. (4) does not faithfully represent the physics of a given system, no well-defined finite stress drop is expected to accompany a rapid frictional rupture. This situation can happen in two generic cases; first, in the limit of thin bodies H → 0, where essentially there is no bulk to radiate energy into, the radiation-damping term ðμ=2c s Þv simply does not exist to begin with. In this case, frictional rupture exists, but it is not accompanied by any stress drop, as shown previously in Ref. [75] and here in Fig. 6. Second, the radiation-damping term ðμ=2c s Þv, which exists at relatively short times [shorter than OðH=c s Þ], is expected to vanish in the long time limit t ≫ H=c s . This limit can be probed by performing experiments or simulations with long enough systems for long enough times. Indeed, simulations of effectively long systems yield rupture fronts with no stress drop; see Fig. 6(a) in Ref. [76] and details therein. Finally, note that the radiation-damping term ðμ=2c s Þv is expected to decrease to zero in discrete steps corresponding to each wave reflection from the system's boundaries. Experimental evidence for the stepwise nature of the decrease in the radiationdamping term (associated with discrete wave reflections) is discussed below.
A. A perturbative approach for rapid rupture and the slow rupture limit The main theoretical prediction in Eqs. (4) and (5) is obtained under the assumption that the spatiotemporal integral term sðx; tÞ in Eq. (3) vanishes well behind the propagating rupture. While this idealized assumption is physically sensible and is reasonably supported by simulational and experimental results for a rapid rupture discussed below, the long-range spatiotemporal nature of sðx; tÞ may suggest that it does not strictly vanish in many realistic situations. Consequently, our goal here is to understand how v 0 res and Δτ of Eqs. (4) and (5) change in the presence of a finite, yet small, sðx; tÞ.
To address this question, we denote the typical value of sðx; tÞ at the tail of a rapidly propagating rupture front by s and consider a perturbed solution of the form v ¼ v 0 res þ δv res . We then expand Eq. (3), which in the present context takes the form ΔτðvÞ ¼ ðμ=2c s Þv − s, to linear order in δv res (it is already linear in s) to obtain Note that the internal state field ϕ is assumed above to be slaved to v and that f ss ðvÞ ¼ τ ss ðvÞ=σ is the steady-state friction coefficient. Equation (6) reveals an interesting result; while both v 0 res and Δτ attain corrections that are linear in s, as expected from a linear perturbation approach, the actual smallness of the corresponding corrections may be quite different due to the appearance of ϵ. ϵ is a product of c s =v 0 res ≫ 1, σ=μ ≪ 1, and jf∂f ss =½∂ logðv 0 res Þgj ≪ 1 [5,45], where the latter two contributions are expected to dominate the first one, leading to jϵj ≪ 1. Consequently, while ϵ is expected to have a negligible effect on δv res , due to the appearance of the factor ð1 þ ϵÞ −1 , it implies that δðΔτÞ=Δτ is a factor of ϵ smaller than δv res =v 0 res . We thus expect the stress drop Δτ to be far less sensitive to finite values of sðx; tÞ compared to the residual slip velocity v res . This difference is demonstrated below.
It is important to note that sðtÞ of Eq. (8), and, consequently, also the stress drop Δτ, may feature a nontrivial dependence on the rupture size and may attain finite values for long times in practical applications. In fact, a simple self-consistent dimensional analysis in the limit of a large rupture in an infinite system indicates that s, and, hence, also Δτ, inversely scales with the square root of the rupture size. Furthermore, note that, while the stress drop Δτ predicted for a slow rupture in Eq. (7) is not related to the radiation-damping term, it is still not a purely interfacial property, but rather it involves the long-range elasticity of the bodies surrounding the interface.

B. Estimates of the spatiotemporal integral contribution to the stress drop
In the previous subsection, we discuss the contribution of the long-range spatiotemporal term sðx; tÞ to the stress drop, denoted there by s. In most of the discussion, excluding the last part concerning a slow rupture, s is assumed to be small compared to the radiation-damping contribution. However, we do not provide quantitative estimates for the relative magnitude of the two contributions, which is shown to be quantified by the ratio δv res =v 0 res [cf. Eq. (6)]. In this subsection, we provide such estimates in some strongly dynamic limiting cases. The basic idea is that, since s emerges from long-range linear elastodynamics, one can use some benchmark linear elastic fracture mechanics (LEFM) crack solutions [78], in which a finite stress drop Δτ is prescribed, to estimate its relative magnitude.
While the whole purpose of the present paper is to show that τ res is not an a priori known interfacial quantity, and, hence, also Δτ ¼ τ d − τ res is not a priori known, LEFM solutions are still useful in estimating the relative magnitude of s through δv res =v 0 res . The point is that one can use the prescribed Δτ to obtain v 0 res according to Eq. (5) for s ¼ 0, then extract from the benchmark LEFM crack solution v res for s ≠ 0, and finally obtain δv res =v 0 res ¼ ðv res − v 0 res Þ=v 0 res as an estimate for the relative magnitude of s due to long-range linear elastodynamic interactions, which are properly captured by the crack solution.
To apply this procedure, we consider Broberg's selfsimilar crack solutions in infinite media [78]. In these solutions, two crack tips are assumed to symmetrically expand at a constant velocity c r , starting from a zero crack length at t ¼ 0. Self-similarity, which significantly simplifies the problem, implies that all fields in the problem depend only on the dimensionless combination x=c r t. While in general, as well as in the present work, frictional rupture propagation is not a self-similar process and the propagation speed is typically not constant, when the propagation speed approaches the limiting or asymptotic speed, self-similar conditions are reasonably approximated. Self-similar solutions are also used in a related analysis in Ref. [79] (cf. Appendix C3 therein), with the aim of deriving precise conditions for rupture mode selection (self-healing pulses versus cracklike), originally discussed in Ref. [39].
Applying these ideas to antiplane shear (mode-III symmetry) self-similar crack solutions [see Eq. (6.9.145) in Ref. [78] ], we obtain in the limit c r → c s (c s is the limiting speed for mode III) δv res =v 0 res ¼ 2=π − 1 ≈ −0.36 (see Appendix G). This result, which is, in fact, quantitatively supported by numerical simulations in Sec. III, suggests that, while the relative magnitude of s is not extremely small under strongly dynamic mode-III conditions, it is still reasonably well within the range of validity of the linear perturbation theory of Eq. (6). Repeating the same procedure for in-plane shear (mode-II symmetry) self-similar crack solutions [see Eq. (6.9.85) in Ref. [78] ] in the limit c r → c R (c R is the Rayleigh wave speed, the limiting speed for mode II), we obtain that δv res =v 0 res varies between −0.613 and −0.417 (the exact value depends on Poisson's ratio); see Appendix G for details.
These latter estimates are quantitatively similar to the mode-III estimate in the corresponding limiting case, demonstrating that jsj is smaller than the radiation-damping contribution for an in-plane frictional rupture under strongly dynamic conditions. In both cases, the linear perturbation theory of Eq. (6) seems to be reasonably valid. We note that, while the estimates above are based on bilateral (self-similar) rupture propagation, similar results are expected to emerge for unilateral propagation. Finally, we stress again that the influence of s on Δτ through δv res =v 0 res in Eq. (6) is strongly reduced by the small dimensionless factor ϵ (defined there). This discussion concludes the presentation of our main theoretical predictions, encapsulated in Eqs. (4)- (8). In the next sections, we provide simulational and experimental support to these predictions.

III. SIMULATIONAL SUPPORT
At this point, we first set out to test the predictions in Eqs. (4)-(8) against extensive numerical simulations. To that aim, we consider two semi-infinite bodies in frictional contact. The advantage of considering infinite-height bodies, i.e., the H → ∞ limit, is that the interfacial relation in Eq. (3) becomes exact at all times, unlike for finite bodies. We also employ periodic boundary conditions, with periodicity W, in the sliding direction. We perform spectral boundary integral method [67][68][69] calculations under antiplane shear (mode-III symmetry) deformation conditions, which are similar to-yet somewhat simpler than-the inplane shear (mode-II symmetry) deformation conditions considered up to now [8]. The main simplification is that the displacement field in the mode-III problem uðx; y; tÞ ¼ u z ðx; y; tÞẑ (the unit vectors satisfyẑ ⊥x;ŷ) is essentially scalar. The basic field u z ðx; y; tÞ satisfies the bulk elastodynamic equation μ∇ 2 u z ¼ ρü z , together with vðx; tÞ ≡ _ u z ðx; y ¼ 0 þ ; tÞ − _ u z ðx; y ¼ 0 − ; tÞ and τðx; tÞ ≡ σ yz ðx; y ¼ 0; tÞ ¼ μ∂ y u z ðx; y ¼ 0; tÞ. Equation (3) remains valid, where σ xy ðx; y ¼ 0; tÞ is replaced by σ yz ðx; y ¼ 0; tÞ and the integral term sðx; tÞ corresponds to mode III; see Appendix A for more details.
The employed interfacial constitutive law features the generic properties discussed above, with fðjvj; ϕÞ of Eq. (1) that reduces under steady-state conditions to either the N-shaped or the no-minimum curves of Fig. 2(a) [the exact expressions for fðjvj; ϕÞ can be found in Appendix B] and with gð·Þ of Eq. (2) that is given by g ¼ 1 − jvjϕ=D [5,45,46,49,52]. The bodies are loaded by a constant driving stress τ d , as depicted schematically in Fig. 2(a), and the frictional rupture is nucleated by introducing Gaussian perturbations of the proper amplitude into a homogeneous state of very low slip velocity v 0 [that corresponds to the leftmost intersection point in Fig. 2(a)], following the theoretical framework of Ref. [80] (the details of the nucleation procedure are described in Appendices).
An example of the stress distribution of an emerging frictional rupture is shown in Fig. 3(a). The figure reveals two rapid rupture fronts propagating in opposite directions (at 84% of the shear wave speed c s ), where the stress ahead of the two fronts is the applied stress τ d . As the emerging rupture is rapid, i.e., propagating at a speed comparable to the elastic wave speed, the relevant prediction is given by Eq. (5). As predicted, the observed stress left behind the two rapid fronts, τ res , is constant and smaller than the driving stress τ d , giving rise to a finite stress drop Δτ. Following the discussion above, since in these calculations H → ∞, the finite stress drop persists indefinitely [while in finite size systems it persists for times ∼OðH=c s Þ; cf. the experiments of Ref. [35], discussed later]. The stress drop Δτ observed in Fig. 3(a) quantitatively agrees with the prediction in Eq. (5), as stated in the figure legend.
In Fig. 3(b), we present the slip-velocity distribution that corresponds to the snapshot shown in Fig. 3(a). As predicted, the slip velocity attains a plateau level v res behind the propagating rupture fronts, and the residual velocity v res is used to normalize the slip-velocity distribution. The simulationally measured residual velocity v res deviates by approximately 35% from the theoretical prediction in Eq. (5), in quantitative agreement with the estimate provided in Sec. II B for the spatiotemporal integral contribution to the stress drop in this case (see also Appendix G). Consequently, while the upper equation in Eq. (6) implies that s is, in fact, not very small, the prediction for Δτ works perfectly fine due to the ϵ ≪ 1 factor in the lower equation in Eq. (6).
In order to quantitatively test the latter prediction for a rapid rupture over a range of physical conditions, we first solve Eq. (4) to obtain v 0 res ðτ d Þ and then plug it into Eq. (5) to obtain Δτðτ d Þ, where the latter is plotted in solid lines in Fig. 3(c) [the two solid lines correspond to both the N-shaped or the no-minimum steady-state friction laws shown in Fig. 2(a)]. We then numerically calculate Δτ [as demonstrated in Fig. 3(a)] for various driving stresses τ d , for both the N-shaped or the no-minimum steady-state friction curves shown in Fig. 2(a). The numerical results (discrete symbols) are superimposed on the theoretical prediction in Fig. 3(c) [the lowest numerical data point on the lower curve corresponds to Figs. 3(a) and 3(b)].
The agreement between the theoretical prediction and the numerical results for Δτ is very good, where it is better for the N-shaped steady-state friction law (lower curve) than for the no-minimum steady-state friction law (upper curve). This difference is fully accounted for by the magnitude and  Fig. 2(a) and τ d ¼ 0.355σ (see the text for details). The snapshot reveals two rapid rupture fronts propagating at an instantaneous speed c r ≃ 0.84c s in opposite directions into regions characterized by the applied stress τ d and leaving behind them a welldefined residual stress τ res < τ d . Consequently, a well-defined and finite stress drop Δτ emerges. Note that the y axis is truncated at τ=σ ¼ 0.4 for visual clarity (the maximal value of τ=σ is 0.58) and that x is normalized by the system length W [the x axis is shared with (b)]. (b) The slip velocity vðxÞ that corresponds to the snapshot shown in (a), normalized by the simulationally measured residual velocity v res (see the text for a discussion). The y axis is also truncated for visual clarity, and x is normalized by the system length W. (c) The theoretical predictions of Eqs. (4) and (5) for Δτðτ d Þ of a rapid rupture (solid lines), both for the N-shaped steady-state friction law of Fig. 2(a)  The contribution of the spatiotemporal integral sðx; tÞ, which is smaller than the radiation-damping contribution for a rapid rupture, may be a dominant effect for a slow rupture, as discussed around Eqs. (7) and (8). To test this possibility, we generate a slow rupture by changing the frictional parameters, the loading level, and the nucleation procedure, as explained in Appendix E. An example of such a slow rupture is shown in Fig. 4 (solid red line), exhibiting a rupture front propagating at a velocity 2 orders of magnitude smaller than c s and leaving behind it a stress drop Δτ. We first verify that ð2c s Δτ=μv res Þ ≫ 1, as predicted by Eq. (7) (and as stated in the figure legend). To test whether indeed Δτ ≃ −sðtÞ [cf. Eq. (7)], we use the slip displacement gradient ∂ x δðx; tÞ obtained from the dynamical simulation to calculate the quasistatic integral in Eq. (8) for any point x along the interface at t corresponding to the snapshot in Fig. 4. The result for τ d þ sðx; tÞ is superimposed on τðx; tÞ in Fig. 4 (dashed green line), demonstrating excellent agreement with the fully dynamic result and, in particular, validating Δτ ≃ −sðtÞ for this t. Consequently, our simulations strongly support the theoretical predictions in Eqs. (4)-(8), for both a rapid and a slow frictional rupture. Next, we discuss direct experimental support for these predictions.

IV. EXPERIMENTAL SUPPORT
In the previous section, we provide strong simulational support to the theoretical predictions in Eqs. (4)- (8). Our goal here is to test these predictions against experimental data. Equation (4) predicts the slip velocity v res behind the frictional rupture once the steady-state friction curve τ ss ðvÞ is known. The latter is not always known a priori over a sufficiently wide range of steady-state slip velocities. In fact, measuring both the frictional stress τ res and the slip velocity v res behind rupture fronts allows one to extract τ ss ðvÞ. In this case, any triplet ðτ d ; τ res ; v res Þ is predicted to follow either Eq. (5) for a rapid rupture or Eq. (7) for a slow rupture, independently of the steady-state friction law τ ss ðvÞ.
Measurements of both τ res and v res behind rupture fronts for various τ d have been recently performed by two independent experimental groups using two different experimental systems and techniques [35,36]. The first focuses on the frictional dynamics along the interface between two blocks of Homalite probed through a novel ultrahigh full-field imaging technique [35]. The second focuses on the frictional dynamics along the interface between two blocks of poly(methylmethacrylate) probed through a combination of high-speed interfacial imaging (via a method of total internal reflection) and simultaneous measurements of the deformation fields slightly above the interface [36]. Here, we use the data reported in these works to quantitatively test our predictions.
We start with Ref. [35], where the observed rupture fronts are all in the rapid (fully inertial) regime, and, hence, the relevant prediction is given in Eq. (5). To test this prediction, we extract the relevant experimental data from Ref. [35] and plot in Fig. 5 the resulting Δτ versus v res against the linear Δτðv res Þ relation of Eq. (5), with a slope that corresponds to μ=2c s of Homalite (see the figure caption for additional details). The results reveal excellent agreement between the experimental data and the theoretical prediction, without any free parameter. The theoretical relation in Eq. (5) predicts v 0 res by assuming that the longrange spatiotemporal contribution sðx; tÞ is negligibly small, while in the experimental data we just use the measured v res . The fact that there are no significant horizontal deviations of the experimental data from the theoretical line of Eq. (5), i.e., that v res is close to v 0 res , indicates that sðx; tÞ is smaller in these experiments compared to the estimate provided in Sec. II B. The origin of the latter discrepancy is not yet clear.
The experimental data included in Fig. 5 are obtained in the short time regime, before any wave reflection from the system's boundaries, for which Eqs. (4) and (5) are valid. Yet, for a single case, measurements are reported after the first wave reflection (but before the second one); cf. Fig. 4(c) in Ref. [35]. Under these conditions, we expect the ordinary radiation-damping term ðμ=2c s Þv to be replaced by a smaller term. Consequently, Eq. (4) predicts that v res should increase, exactly as is experimentally observed in Fig. 4(c) in Ref. [35] (denoted as a "reflected rupture" in the figure), thus providing direct evidence for the stepwise reduction of the radiation-damping term with discrete wave reflections.
We now turn to the experiments of Ref. [36], which report on extensive measurements of τ res and v res ; cf. Fig. 3(a) there. In order to test our predictions, we need also measurements of the driving stress τ d , which allow us to extract the stress drop Δτ ¼ τ d − τ res . The corresponding extensive measurements of τ d are not presented in Ref. [36], though τ d is presented for two highly relevant examples in Fig. 2(b) there (where τ d is denoted by τ 0 ). One example corresponds to a very rapid rupture (in fact, it is a supershear rupture, propagating at nearly the dilatational wave speed) and the other to a slow rupture (propagating at about 10% of the Rayleigh wave speed). These data are exactly what is needed in order to test our predictions, in particular, regarding the change in behavior for a slow rupture. For a rapid rupture, we extract from Fig. 2(b) of Ref. [36] (blue data) Δτ ¼ 1.17 MPa and from Fig. 1(d) (bottom) v res ≃ 1.5 m=s (see additional details in Appendix F). Using the reported values ρ ¼ 1170 kg=m 3 and c s ¼ 1345 m=s [36], together with μ ¼ ρc 2 s , we obtain ð2c s Δτ=μv res Þ ¼ 0.99. This result is in great quantitative agreement with the prediction for a rapid rupture in Eq. (5) and is fully consistent with the independent experimental data presented in Fig. 5. This result indicates that the long-range spatiotemporal contribution sðx; tÞ is indeed small also in the experiments of Ref. [36].
We then extract the corresponding data for the slow rupture, obtaining from Fig. 2(b) of Ref. [36] (red data) Δτ ¼ 0.242 MPa and from Fig. 1(d) (top) v res ≃ 0.02 m=s. Using these data, we obtain ð2c s Δτ=μv res Þ ≃ 15 ≫ 1, in agreement with the theoretical prediction for a slow rupture in Eq. (7). The results presented in the past two sections provide strong simulational and experimental support for our theoretical predictions and, hence, for the proposed picture of the physical origin of stress drops in frictional rupture. As stress drops are important for the possible emergence of cracklike behavior in frictional rupture, future studies of the latter should be based on the results presented in this paper.

V. DISCUSSION AND CONCLUDING REMARKS
The possible deep relations between frictional rupture and ordinary fracture provide a powerful conceptual and quantitative framework to understand frictional dynamics in a wide variety of physical contexts. This framework is extensively used to interpret and quantify geophysical observations [26,28], as well as a broad spectrum of laboratory phenomena [29][30][31][32][33][34][35]. For example, a recent series of careful laboratory experiments [32][33][34] demonstrated that, when the analogy between frictional rupture and ordinary fracture holds, the dynamic propagation of laboratory earthquakes and their arrest can be quantitatively understood to an unprecedented degree [25]. Yet, the fundamental physical origin and range of validity of the analogy between frictional rupture and ordinary fracture are not yet fully understood. In this paper, we developed a comprehensive and fundamental understanding of why, how, and to what extent frictional rupture might be viewed as an ordinary fracture process.
An important ingredient in the analogy is the emergence of a finite and well-defined stress drop Δτ ¼ τ d − τ res , the difference between the applied driving stress τ d and the residual stress τ res , in frictional rupture. In the first part of the paper, we showed that, contrary to widely adopted assumptions, the residual stress τ res is not a characteristic property of frictional interfaces. Rather, for a rapid rupture, τ res is shown to crucially depend on elastodynamic bulk effects-mainly wave radiation from the frictional interface to the bodies surrounding it but also long-range elastodynamic bulk interactions [encapsulated in the integral term sðx; tÞ in Eq. (3)]-and the applied driving stress τ d itself, in addition to the contribution of the slip-rate dependence of the constitutive friction law. Notably, we showed that for a rapid rupture the deviation of τ res from τ d , i.e., the existence of a finite stress drop Δτ, is a finite-time effect, mainly limited by the wave travel time in finite systems. For a slow rupture, it is shown that, if a stress drop exists, it is intimately related to the long-range quasistatic elasticity of the bodies surrounding the interface, again not exclusively Homalite by the solid line, with the following high strain rate material parameters μ ¼ 1.96 GPa and c s ¼ 1263 m=s [35,81], which uniquely determine the slope of the stress drop Δτ versus the slip rate v res behind a rupture front. Experiments on the rupture of frictional interfaces composed of two blocks of Homalite are performed in Ref. [35], where a novel ultrahigh full-field imaging technique is employed to directly measure τ res and v res behind rupture fronts. Experiments for different values of the applied shear stress τ d are performed, which allow us to extract from Fig. 8(a) of Ref. [35] five triplets ðτ d ; τ res ; v res Þ (see details in Appendix F). For each triplet, we calculate Δτ ≡ τ d − τ res and then superimpose the resulting Δτ versus v res on the theoretical prediction (discrete symbols). The symbols (and colors) differentiate data obtained from high-resolution (full brown circles) and low-resolution (empty green diamonds) measurements, following the classification of Ref. [35]. The symbol size and/or error bar represent the full range of measurements reported on in Fig. 8(a) of Ref. [35] per imposed farfield conditions. to interfacial physics. Our theoretical predictions are supported by extensive computations and existing experimental data from two independent laboratory experiments. Our findings have important implications that go beyond their basic nature; first, the results show that the widely used slip-weakening models [9,10], in which the existence of a residual stress τ res < τ d is a priori assumed (as a fixed interfacial property), should be employed with care. In particular, as τ res has been shown to depend on the externally applied stress τ d , on the properties of the bodies surrounding the interface, and on the rate dependence of the frictional constitutive behavior, τ res cannot be assumed to be fixed. Rather, it should be self-consistently calculated from the coupled interface-bulk problem. Somewhat related conclusions in relation to slip-weakening models, based on measurements of evolving local friction during spontaneously developing laboratory earthquakes, have been drawn in Ref. [35]. We also note that the dependence of τ res on the bulk constitutive relation and on long-range bulk-mediated interactions may give rise to interesting effects for more complicated bulk constitutive relations (e.g., viscoelasticity) and in the presence of repeated rupture events [82]. Such effects should be explored in future work.
The existence of a finite stress drop Δτ generically leads to accelerating frictional rupture under stress-controlled farfield loading conditions τ d if Δτ is independent of the rupture size L. In these situations, an inertia-limited rapid rupture is expected to emerge on timescales for which wave interaction with the outer boundaries does not exist, and Δτ is controlled by elastodynamic bulk effects. These conditions are typically realized in many geological and laboratory earthquakes [26,[28][29][30][31][32][33][34][35]. On the other hand, slow rupture propagation-a widely observed, yet highly debated and elusive phenomenon [41][42][43][44]-is expected to feature smaller stress drops Δτ that may decrease with increasing rupture size L, such that rupture acceleration is limited.
The possible L dependence of Δτ and its possible relations to the emergence of a slow rupture should be further explored in the future. In addition, as the whole discussion in this paper is valid for stress-controlled farfield loading conditions (characterized by τ d ), future work should also consider velocity-controlled far-field loading conditions, where finite stress drops might emerge from different physical considerations [83]. Finally, as the existence of a finite stress drop Δτ does not in itself guarantee the cracklike behavior of frictional rupture, future work should clarify to what extent the analogy to ordinary cracks can, in fact, quantitatively account for the dynamics of frictional rupture. All in all, as stress drops are key quantities in frictional failure dynamics, we expect our results to provide a conceptual and quantitative framework to address various fundamental and applied problems in relation to the rupture dynamics of frictional interfaces, with implications for both laboratory and geophysical-scale phenomena. This research is made possible in part by the historic generosity of the Harold Perlman family. We are grateful to Ilya Svetlizky for pointing out the differences between rapid and slow rupture in the context of our work and the relevance of the experimental data of Ref. [36] to it. His comment has stimulated the discussion of slow rupture in this paper. In addition, we are grateful to him for pushing us to look at Broberg's self-similar solutions, eventually leading to the extraction of the estimates presented in Sec. II B.

APPENDIX A: THE MAIN NUMERICAL METHOD
The simulations discussed above rely on a spectral boundary integral formulation of the elastodynamic equations [67][68][69]. The latter relates the traction stresses acting along the interface between two linearly elastic half-spaces and the resulting displacements. For the mode-III (antiplane shear) elastodynamic problem studied in the manuscript, the interface is initially uniformly prestressed by τ d and is set to slide at an extremely small steady velocity v 0 , such that the shear tractions at the interface take the form The second right-hand-side term represents the instantaneous response to changes in the sliding velocity, the so-called radiation-damping term. As discussed in the manuscript, this term can be understood as the damping of interfacial energy due to elastic waves radiated into the infinite domain. The third term sðx; tÞ accounts for the history and spatial distribution of interfacial displacements uðx; tÞ. Both sðx; tÞ and uðx; tÞ are related in the spectral domain via a convolution integral, whose expression can be found in Ref. [69]. Because of the spectral nature of the formulation, the simulated domain is taken to be periodic in the lateral direction, with periodicity W. The latter is chosen to be large enough to prevent any effect of the periodicity on the results reported in the manuscript. A rupture is nucleated at the center of the domain by introducing a Gaussian perturbation of the slip velocity into an initial steady sliding state at v 0 . The sliding velocity is then computed by combining Eq. (A1) and the rate and state-dependent friction law τ ¼ σsgnðvÞfðjvj; ϕÞ (see Appendix B for more details). uðx; tÞ is then integrated in time using an explicit time-stepping scheme uðx; t þ ΔtÞ ¼ uðx; tÞ þ 0.5vðx; tÞΔt: Note that the factor 0.5 on the right-hand side of Eq. (A2) ensures that vðx; tÞ is indeed the slip velocity. In order to guarantee the stability and the convergence of the numerical scheme, Δt is defined as the time needed for a shear wave to travel a fraction 0.2 of one grid spacing, i.e., Δt ¼ 0.2Δx=c s . Additional information about the numerical scheme and the nucleation procedure can be found in Ref. [80], together with videos of similar rupture events.

APPENDIX B: THE FRICTION LAWS
The friction laws used in this work, and whose steadystate behaviors are plotted in Fig. 2, are related to the one used previously in Refs. [58,80]. A friction law is defined by the relation between the shear stress τ ≡ σ xy and the compressive normal stress σ ≡ −σ yy at the interface, τ ¼ σsgnðvÞfðjvj; ϕÞ, and by the evolution equation for state variable ϕ, _ ϕ ¼ gðjvj; ϕÞ. The constitutive functions fðjvj; ϕÞ and gðjvj; ϕÞ used in this work take the form where ϕ represents the typical age or maturity of contact asperities that compose the interface at a microscopic scale [45]. In Eq. (B1), f 0 sets the scale of the dimensionless frictional resistance (friction coefficient), b is the aging coefficient, and a is related to the thermally activated rheology of contact asperities [45]. The function ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 þ ðv Ã =vÞ 2 p that appears also in gðjvj; ϕÞ ensures that, for vanishingly small steady-state velocities, ϕ saturates after extremely long times to a finite value of D=v Ã rather than diverges. As discussed in Ref. [80], this regularization makes no significant difference in the results discussed above. For the sake of notational simplicity, the regularization is hence omitted in the main text (though it is included in the calculations). Equations (B1) and (B2) lead to the steady-state friction curve in Fig. 2 with a minimum at an intermediate v (brown solid line), while the nominimum steady-state friction curve (dash-dotted orange line) is obtained after neglecting the "þ1" in the b term.
The reader is referred to Refs. [58,80] for additional discussions about the formulations of Eqs. (B1) and (B2), which go beyond the conventional rate-and-state friction laws. Nevertheless, the results and conclusions discussed above are independent of the choice of the rate-and-state formulation.

APPENDIX C: 1D RUPTURE FRONTS
In this Appendix, we describe propagating steady-state rupture fronts in thin (quasi-1D) systems, where no stress drop emerges. We consider two long and thin linear elastic bodies of height H in frictional contact, such that the momentum balance equation ρü ¼ ∇ · σ reduces to [75,80,84] where u ≡ u x ,μ and c 1D are the effective shear modulus and wave speed [75,84], respectively, and τ d is a constant driving stress (see Fig. 2). Propagating 1D steady-state solutions then satisfy [80] where we define a comoving coordinate ξ ≡ x − βc 1D t, integrate out u, and eliminate partial time derivatives. Steady-state rupture propagation is a dynamical process in which a homogeneous V state invades a homogeneous v 0 ≪ V state [32,35,75,80], both shown in Fig. 2 as the intersections of the velocity-strengthening branches of the friction law with the driving stress τ d . We find these solutions for the friction law described in Sec. V, using a shooting method [85] (similar to that used in Refs. [75,84]). The solution is shown in Fig. 6. To normalize the stress fields, we use the definition τ m as the maximal value τ attains in the profile, and l ≡ ½ð1 − β 2 =ÞβfHVμ=½c 1D ðτ m − τ d Þg is the length scale over which the fields change, which can be calculated by a scaling analysis of Eq. (C2).
As seen in Fig. 6, the stress both ahead and behind the rupture front equals τ d ; i.e., there exists no stress drop. A corollary is that no singularity is observed in Fig. 6 [compare to Figs. 3(a) and 4]. r ¼ βc s , with β ¼ 0.144 (see the text for details on the employed normalization). Note also that τ d =σ ¼ 0.355, exactly as in Fig. 2, though in the latter 2D case a rupture front with β ¼ 0.84 emerges [cf. Fig. 3(a)].

APPENDIX D: PARAMETERS
The parameters used for all the calculations described in this work, except for the slow rupture discussed in Appendix E, are given in Table I. Note that the values of the listed parameters are characteristic of some laboratory experiments (see Ref. [59] for details). However, the generic properties of the derived results are independent of the exact numbers and are relevant to a broad range of materials and physical situations. For example, v Ã that controls the velocity scale below which the system is in the stick phase can be taken to be significantly smaller.

APPENDIX E: 2D SLOW RUPTURE FRONTS
The very same constitutive framework can give rise to slow rupture fronts (c r ≪ c s ), as demonstrated in Fig. 4. While the emergence of a slow rupture is of great interest, in general, in the present context we are interested in just generating a slow rupture and studying its properties in relation to the theoretical prediction in Eqs. (7) and (8). One way to generate a slow rupture within our constitutive framework is to use friction parameters that shift the steadystate friction curve to smaller slip velocities and to employ a different nucleation procedure.
In particular, using the different set of parameters listed in Table II, we obtain the dash-dotted orange steady-state curve in Fig. 7(a) [the solid brown line is identical to the one shown in Fig. 2(a)]. For these parameters and for the same value of the normalized driving stress τ d =σ, the effective steady-state friction curve shown in Fig. 7(b) (dashed orange line, obtained by adding the radiationdamping term ðμ=2c s Þv) is practically indistinguishable from the steady-state friction curve in the relevant slip velocities range. In addition, the rupture is nucleated by introducing a perturbation to the internal state field ϕ of the form with k ¼ 2π=W and ε ¼ 10 −4 , into an interface that slides homogeneously at a velocity v vw that corresponds to a fixed point on the velocity-weakening branch [it is marked by the black diamond in Fig. 7(b)]. This nucleation procedure is different from the one used elsewhere in the paper, where Gaussian perturbations are introduced into an essentially locked-in interface, as described in detail in Ref. [80]. These modifications are sufficient to generate the slow rupture shown in Fig. 4. The physics behind the emergence of a slow rupture, which is very interesting in itself, is not thoroughly discussed here. It deserves an investigation of its own, which we hope to pursue in the future.
In order to repeat this procedure for in-plane shear (mode-II symmetry) self-similar crack solutions, we start with the crack face displacement given in Eq. (6.9.85) in Ref. [78]: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc r tÞ 2 − x 2 p μc 2 d g 2 ðc r =c s ; c s =c d Þ for jxj < c r t, where the function g 2 ð·Þ is given in Eq. (6.9.87) in Ref. [78] and c d is the dilatational wave speed. Considering the limit c r → c R (c R is the Rayleigh wave speed, the limiting speed for mode II) and following the procedure described above, we obtain The latter is a function of c s =c d (note that c R =c s is also determined by c s =c d ). Evaluating g 2 ðc R =c s ; c s =c d Þ [88] for the whole range of admissible c s =c d values (which, in turn, depends on Poisson's ratio) and plugging in Eq. (G6) shows that δv res =v 0 res varies between −0.613 and −0.417, as reported on in Sec. II B.