## Abstract

The way in which a frictional interface fails is critical to our fundamental understanding of failure processes in fields ranging from engineering to the study of earthquakes. Frictional motion is initiated by rupture fronts that propagate within the thin interface that separates two sheared bodies. By measuring the shear and normal stresses along the interface, together with the subsequent rapid real-contact-area dynamics, we find that the ratio of shear stress to normal stress can locally far exceed the static-friction coefficient without precipitating slip. Moreover, different modes of rupture selected by the system correspond to distinct regimes of the local stress ratio. These results indicate the key role of nonuniformity to frictional stability and dynamics with implications for the prediction, selection, and arrest of different modes of earthquakes.

The relative motion of two contacting bodies under an imposed shear is governed by the ensemble of discrete contacts composing their interface (*1*). Although frictional slip (*2*, *3*) is initiated by the rapid rupture of these contacts, understanding of the mechanisms of how interface rupture takes place is limited by our knowledge of the properties, strength, and stability of this rough interface.

Frontlike rupture modes bridge the gap between the microscopic interactions that define local frictional resistance and the resulting macroscopic motion imbued in the slip of large bodies (*2*–*4*). Laboratory experiments reveal three distinct modes of rupture: (i) slow ruptures, which propagate far below material wave speeds (*5*–*8*), (ii) “sub-Rayleigh” ruptures (*5*, *6*, *9*–*12*) that propagate up to the Rayleigh wave speed, and (iii) “supershear” rupture modes that surpass the shear wave speed *C*_{S} (*5*, *10*, *12*). Sub-Rayleigh ruptures are related to shear fracture, which is well understood theoretically (*2*); however, our understanding of the other rupture modes is much less clear. Supershear modes, which have long been considered theoretically possible in shear fracture (*13*–*16*), have only recently been observed (*12*), whereas our understanding of slow rupture modes is still very much in its infancy (*17*, *18*). Each of these rupture modes may occur in earthquakes, but if, where, and how they are selected are still open questions. Although sub-Rayleigh modes are considered the most general mode of earthquake propagation (*2*, *4*, *15*), there is mounting evidence for the importance of both slow (*19*–*21*) and supershear (*22*, *23*) rupture modes along natural faults.

To understand how properties of frictional rupture couple to the profiles of local stresses along a frictional interface, we conducted experiments with acrylic poly(methyl-methacrylate) blocks in two qualitatively different loading systems (fig. S1) (*24*). At the start of each experiment, the blocks were pressed together by a normal force *F*_{N} (Fig. 1A). External shear forces, *F*_{S}, were applied to either the trailing edge (*x* = 0) of the top-block, uniformly along the bottom block, or in a mixture of the two. *F*_{S} was incremented quasi-statically, eventually triggering stick-slip sliding. We continuously performed simultaneous measurements of the real contact area, *A*(*x*,*t*), at every point *x* at rates of up to 250,000 samples/s (here, *t* is time). In parallel, shear and normal stress profiles adjacent to the interface, τ(*x*) and σ(*x*), respectively, were measured every second. All rupture events encompassing either partial sections of the interface (*6*) or the entire interface were measured.

In nearly all frictional systems, τ(*x*) and σ(*x*) are highly nonuniform, suggesting that nonuniformity is generic. Even in ideal laboratory systems, large stress nonuniformity can be produced by minute interface curvature, differences in the materials and/or geometries of the contacting bodies, or dynamically, by earlier slip events (*6*, *17*). In the example presented in Fig. 1, B and C, we show how geometrically dissimilar blocks generate large shear stress variations at an optically flat interface under uniform application of *F*_{N} (Fig. 1). Because the dimensions of the upper and lower blocks are not identical, differential Poisson expansion is countered by pinning of the blocks at the frictional interface (*25*). This generates shear stresses even for *F*_{S} = 0. Additional large stress variations result from any torque when *F*_{S} > 0 is applied.

Controlled variations of loading conditions (*24*) lead to spatial variations of τ(*x*) and σ(*x*) that produced the full range of rupture modes (Fig. 2). These include individual events (Fig. 2, A to C) where slow, sub-Rayleigh, and supershear ruptures or combinations of these modes (Fig. 2A) took place. Additionally, the dynamics of rupture can change in successive events within a single stick-slip sequence (Fig. 2, D to F). All events in this sequence nucleated at approximately the same location (*x* ~ 150 mm) and propagated in both directions. In the first stick-slip event (Fig. 2D), the left-traveling front initiated at near-shear velocity, then continually slowed until ultimately arresting at *x* ~ 50 mm. In the second event, the left-traveling front initiated beyond the shear-wave speed (~1600 m/s), slowed to sub-Rayleigh propagation (250 to 500 m/s) near the sample edge, but did not arrest. The left-propagating front in the third event initiated at a super-shear velocity (~2300 m/s) and traversed the entire interface without slowing down. The rupture direction need not correspond to the resulting slip, which is determined by the loading.

Comparison of the rupture velocities to the local stress ratio, τ(*x*)/σ(*x*), suggests that this quantity is strongly coupled to the local front dynamics (Fig. 2). The local propagation speeds consistently increase with τ(*x*)/σ(*x*), with front arrest occurring when τ(*x*)/σ(*x*) falls below ~0.5 (for example, Fig. 2A at *x* = 50 mm). This qualitative dependence is both local along the interface and independent of how the local stress ratios were imposed. Supershear fronts can also nucleate under quasi-static external loading (Fig. 2, C and F). These ruptures can abruptly initiate (Fig. 2C), even when preceded by a slow, gradual nucleation process.

The local propagation velocities *V*(*x*) of 287 different fronts as a function of τ(*x*)/σ(*x*), measured before slip initiation, demonstrate the generality of these observations (Fig. 3). Each front is part of a system-sized event, with each *V*(*x*) indicating the instantaneous rupture velocity as a front traverses a specific strain gage. If we consider only fronts traversing locations far from the system’s loading points or free edges, the data collapse onto a rough curve whose form indicates three distinct regimes of rupture dynamics: (i) slow fronts [τ(*x*)/σ(*x*) < 0.5], (ii) sub-Rayleigh fronts [0.5<τ(*x*)/σ(*x*) < 0.8], and (iii) supershear rupture [τ(*x*)/σ(*x*) > 0.8]. The data collapse occurred for widely different external loading conditions (*24*), including simple edge-loading, a combination of uniform shear and edge loading, and uniform shear loading (compare with Figs. 1 and 2). This suggests that rupture-mode selection is coupled to τ(*x*)/σ(*x*), though it is not explicitly dependent on how loads are applied. The data fail to collapse in regions (for example, near loading points) where stress gradients are so large that stresses at the interface do not mirror those relieved in the local vicinity of the rupture tip.

At loading levels that are even incrementally below that needed to precipitate each slip event in Fig. 3, no slip occurred. Thus, the system was stable for values of τ(*x*)/σ(*x*) that far exceeded the static-friction coefficient (*26*) μ_{S} = *F*_{S}/*F*_{N} ~ 0.5. This is surprising, as it is generally believed (*3*) that the value of μ_{S} is the criterion for stability to frictional motion at any point along a frictional interface. Our experiments demonstrate that this assumption is not valid; interfaces are locally stable even for local values of τ(*x*)/σ(*x*) that exceed 4μ_{S} (Fig. 3). Although τ(*x*)/σ(*x*) can locally exceed μ_{S}, the integrated values of τ(*x*) and σ(*x*) are consistent with the value of μ_{S} in each experiment.

When considering a frictional interface, it is tempting to simplify the problem by considering uniform stress profiles along the interface (*1*). Any geometrical or material mismatch, however, as well as the existence of edges, will lead to substantial nonuniformity (see, for example, Fig. 1). One can approximate uniform stress profiles only under a limited set of carefully controlled conditions (*12*, *26*). Thus, nonuniform stress profiles are the rule and are present on nearly any naturally occurring (*4*) or engineered frictional system. Along natural faults, heterogeneous profiles of σ(*x*) or τ(*x*) can have a variety of additional origins. These include material heterogeneity of either shear strength or elastic moduli of the host material, inelastic deformations near fault tips, or spatial gradients of applied stress fields (*3*). Stress inhomogeneity can further evolve dynamically, through the partial release and transfer of stresses between slip patches (*2*, *4*, *15*) or by finite slip events driven by inhomogeneous stress application (*6*, *17*).

We can intuitively understand why the local stress ratio is tightly coupled to the rupture mode (*15*). We know that a crack propagates when the strain energy released in the bulk medium exceeds its fracture energy (*27*): the energy needed to create a new unit surface. At a frictional interface, this effective energy cost is proportional to the real area of contact *A*(*x*), which is, in turn, locally proportional (*1*) to σ(*x*) at each point *x*. Here, the effective fracture energy Γ is not a material-dependent quantity as in the fracture of bulk materials, but instead reflects the local strength of the interface, as determined by σ(*x*). On the other hand, τ(*x*) is a locally measured quantity that mirrors the density of strain energy stored, before the arrival of a rupture front, within a region of finite size surrounding the point, *x*. Thus, at locations sufficiently removed from regions with high stress gradients, τ(*x*)/σ(*x*) reflects the balance between the potential energy available, before rupture, in the vicinity of each point and the energy needed to rupture the interface.

The idea of a material-dependent “peak” stress is frequently used to model material response at the tip of rapid ruptures (*2*, *16*, *28*). The peak stress, commonly denoted as τ_{p}, is often considered to be determined by the static-friction coefficient τ_{p} = σ(*x*) · μ_{S} (*3*). τ(*x*) can be considerably greater than this value (Fig. 3). Thus, if a characteristic τ_{p} exists, our peak results pose a lower limit for the peak strength and question the relevance of relating μ_{S} to local material strength.

Our results suggest that, once a rupture is nucleated, it will transition to supershear when a highly pre-stressed region is encountered. This is in agreement with numerically observed transitions to supershear rupture (*15*, *28*, *29*) in spatially nonuniform systems when highly stressed regions are encountered; however, we do not yet fully understand what causes rupture nucleation in our system. We find that nucleation locations are often regions where τ(*x*)/σ(*x*) is maximal (for instance, see Fig. 2C). Hence, either low σ(*x*) or high τ(*x*) can influence the location and initiation of rupture fronts. Increased σ(*x*), as at the leading edge in Fig. 2F, will serve to suppress nucleation. Likewise, values of τ(*x*) that oppose the applied shear, such as at the trailing edges of Fig. 2, will have the same effect. The high values of both σ(*x*) and τ(*x*) associated with corners can either make edges susceptible to rupture nucleation (Fig. 2A) or suppress rupture, depending on the competition between them.

Once a rupture front is nucleated, knowledge of the local stress profiles along the interface allows us to predict the rupture mode and could indicate when a rapid mode will either arrest to a complete stop (Fig. 2A) or evolve into a slow front (Fig. 2B). Thus, the initiation/transition locations of the slow fronts observed in previous studies (*5*, *6*) become clearer. Such questions of predictability are important when applied to understanding earthquake dynamics (*3*). Although τ(*x*)/σ(*x*) is an elusive quantity to measure along natural faults, indirect measurements may be possible by coupling precise measurements of spatial variations of *V*(x) to laboratory measurements of the τ(*x*)/σ(*x*) dependence on *V*, as in Fig. 3. This estimate of this otherwise inaccessible quantity could provide some measure of predictability of the eventual size and dynamics of fast earthquakes along natural faults.

## Supporting Online Material

www.sciencemag.org/cgi/content/full/330/6001/211/DC1

Materials and Methods

Fig. S1

References

## References and Notes

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- We thank E. Bouchbinder and A. Sagy for insightful comments and S. Kimhi for help with the system design. We acknowledge the support of grant 2006288 awarded by the U.S.-Israel Binational Science Foundation and the James S. McDonnell Foundation. We also acknowledge the European Science Foundation EUROCORES program FANAS for support via the Israel Science Foundation (grant 57/07). J.F. thanks the Max Born Chair for Natural Philosophy for support.