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# Properties of the shear stress peak radiated ahead of rapidly accelerating rupture fronts that mediate frictional slip

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved November 30, 2015 (received for review September 2, 2015)

## Significance

The transition from stick to slip is governed by rupture fronts propagating along a frictional interface. It has long been suggested that rapid acceleration of these ruptures generates a shear stress peak that propagates ahead of the front. These peaked waves are strong; they can reach amplitudes that are large enough to trigger secondary supershear ruptures. We provide the first extensive quantitative experimental study to our knowledge of these highly directed waves and their relation to the rupture fronts driving them. Combining our experiments with finite-element simulations, we observe how these waves scale. This study provides insight into how rupture fronts accelerate beyond the shear-wave speed and may offer a possibility to obtain illusive information about propagating earthquakes.

## Abstract

We study rapidly accelerating rupture fronts at the onset of frictional motion by performing high-temporal-resolution measurements of both the real contact area and the strain fields surrounding the propagating rupture tip. We observe large-amplitude and localized shear stress peaks that precede rupture fronts and propagate at the shear-wave speed. These localized stress waves, which retain a well-defined form, are initiated during the rapid rupture acceleration phase. They transport considerable energy and are capable of nucleating a secondary supershear rupture. The amplitude of these localized waves roughly scales with the dynamic stress drop and does not decrease as long as the rupture front driving it continues to propagate. Only upon rupture arrest does decay initiate, although the stress wave both continues to propagate and retains its characteristic form. These experimental results are qualitatively described by a self-similar model: a simplified analytical solution of a suddenly expanding shear crack. Quantitative agreement with experiment is provided by realistic finite-element simulations that demonstrate that the radiated stress waves are strongly focused in the direction of the rupture front propagation and describe both their amplitude growth and spatial scaling. Our results demonstrate the extensive applicability of brittle fracture theory to fundamental understanding of friction. Implications for earthquake dynamics are discussed.

The onset of motion along a frictional interface entails rupture-front propagation. These rupture fronts have long been considered to have much in common with propagating cracks (1⇓–3). Recent friction experiments (4) have shown that the stresses and material motion surrounding the tip of a propagating rupture are indeed quantitatively described by singular linear elastic fracture mechanics (LEFM) solutions originally developed for brittle shear fracture. These singular fields are only regularized by dissipative and nonlinear processes in the vicinity of the rupture tip.

Nonsteady processes such as rapid rupture velocity variation during the nucleation or arrest phases result in the generation of stress-wave radiation (2, 5). In the study of earthquakes, understanding the source mechanism of those waves is of primary importance. Long-wavelength radiation is usually described by simple dislocation models (3, 6). High-frequency radiation, however, was proposed (2) to be controlled by the strong slip velocity concentrations at the rupture tip predicted by fracture mechanics. Descriptions that go beyond singular contributions to fracture involve significant analytical complications; full solutions of nonsteady dynamic crack problems are generally extremely difficult to obtain. Of the few full-field analytic solutions available, self-similar solutions of suddenly expanding shear cracks have provided much intuition (2, 5, 7, 8). These solutions, under shear loading (mode II), predict a localized shear stress peak that propagates ahead of the rupture tip at the shear-wave velocity. The results obtained in such solutions are generally considered as an upper bound for the realistic stress-wave radiation of smoothly accelerating ruptures. Radiated shear stress peaks have, for decades, drawn special attention, because they are thought to be an important vehicle for the nucleation of supershear ruptures, a class of ruptures that propagate beyond the shear-wave speed,

Despite their importance, experimental studies of radiated stress waves have been very limited. Here we present direct measurements of the stress fields surrounding the tips of rapidly propagating ruptures and show how a shear stress peak is formed. Supplementing experiments with finite-element calculations, we provide a detailed description of both the scaling and space–time structure of this phenomenon.

## Experimental Observations

Our experimental system is schematically presented in Fig. 1*A*. Two poly(methylmethacrylate) blocks with ∼1-μm rough surfaces are pressed together by an external normal force, *x* being the coordinate along the quasi-1D frictional interface), are obtained at 580,000 frames per second by means of direct optical imaging of *Supporting Information*. Once

The onset of frictional motion is marked by propagating crack-like rupture fronts that leave in their wake significantly reduced *B*, *Top*.

In what follows we consider the strain fields surrounding the propagating rupture. Owing to the linearity of the governing equations, constant values of the initial tensile strains

Fig. 1*B*, *Bottom* presents temporal measurements of shear strain variations, *B*, *Top*) reveals that they propagate at

The measurements of *B* reveal the sudden nucleation (

Overall relative motion of the two blocks occurs only after a rupture front has traversed the entire interface. Rupture fronts may arrest (Fig. 1*C*, *Top*), however, if they encounter either reduced shear stress regions or areas of increased interfacial strength (22⇓–24). Measurements of *C*, *Bottom*) indicate that the shear stress peak in Fig. 1*B*, in fact, persists, propagating far beyond the rupture arrest location while broadening and decaying in time. In addition, an inverted shear stress peak is generated that propagates at *C*, *Top*). This inverted shear stress peak is not to be confused with the violent oscillation at the rupture tip in Fig. 1*B*, *Bottom*. These experimental observations confirm predictions (2) that the nucleation and arrest stress-wave radiation are complementary phenomena, having the same form but with inverted signs.

The examples above demonstrate the general notion that generation of stress-wave radiation requires nonsteady rupture processes. Let us now consider the explicit form of this radiation. In some simplified cases, analytical solutions are available (5) that describe radiation patterns generated by accelerating shear cracks. One such solution (7, 8) describes bilaterally expanding ruptures that initiate with zero initial length and propagate at a constant velocity (*A*). In this problem, there is no characteristic time or length scale so self-similar propagating solutions can be found (*Supporting Information*). We will call this the “self-similar” solution, which was derived both for tension (25) and shear (1). The resulting normalized shear strain on the interface (*A*, *Bottom*. This solution describes a singular propagating crack tip that is preceded by a sharp and relatively localized shear stress peak. In crack tip vicinities all stress components (and strains) have the universal singular form *l* (26).

Experiments (4) have shown that under low shear stress loading, *B*.

Fig. 2*B* also shows that the self-similar solution of an expanding shear rupture can describe all of the measured strain components rather well. In particular, this solution can capture both the initial shear loading,

The fact that the form of the solution is so close to experiment suggests that the general form of the radiation pattern may be captured by this solution, but one should be careful not to take this comparison too far. The self-similar problem is, in many respects, unphysical; its core assumptions include a constant rupture front velocity propagating under constant background stress that yields a continually increasing *l*. None of these is generally satisfied in the experiments. To perform the comparison in Fig. 2*B* we needed to choose both *l* and *Supporting Information*). Although the self-similar problem might, consequently, be unrealistic (but see ref. 28), it nevertheless provides important physical intuition and has often been used to verify numerical methods (26).

## Finite-Element Simulations

To make quantitative comparisons with the experiments, we performed 2D finite-element calculations in which the plane-stress hypothesis, block dimensions, and their elastic moduli correspond to the experimental system. The experimental loading configuration is mimicked by applying homogeneous shear and normal stresses (Fig. 3*A*, *Inset*). To close the system, we choose the widely used (9, 13, 14, 23) linear slip weakening friction law. This is the simplest cohesive zone model that captures both measurements (4) of *d*_{c} (1.4 μm).

Spontaneous rupture nucleation in friction experiments and in natural faults has been the subject of extensive study (29⇓⇓–32). Previous numerical work has shown that different nucleation procedures can influence the propagating rupture fronts (12, 13, 33, 34). In particular, the more abrupt the initiation, the larger the amplitude of the radiated shear stress peak (13). Here we follow ref. 34 and induce a slowly propagating (∼0.1 *C*_{R}) initial “seed” crack; starting from *A*, *Bottom* and Fig. S2.

In Fig. 3*A* a snapshot of *B*) is obtained. In the numerical simulation,

We now use our numerical calculations to provide a detailed description of the shear stress peak structure. We concentrate, first, on the stress field on the interface (*A*, *Bottom*) and the numerical calculation (26) of a suddenly expanding shear crack. The shear stress peak can, therefore, not be easily defined at the early stages of rupture propagation, whereas the peak location becomes clear at later stages of its evolution. We define the shear stress peak initiation point by using backward linear extrapolation of the time-position curve of the peak (Fig. 3*A*, *Bottom*). As in the experiments, the shear stress peak consistently initiates at a location,

We are now in a position to consider the angular dependence of this propagating stress wave by using the full spatial form of the simulated fields. Two snapshots of the normalized shear strain *A*. Because *A*, *Top*). Its normalized amplitude, *B*, *Inset*). To account for this growth, to properly compare the angular form at successive time steps, we plot *B*. For small forward angles (up to *B*.

The observed increase of

## Stress-Wave Scaling

Let us now consider what determines the shear stress peak amplitude and scaling. We first consider the effect of preimposed shear stress. Fig. 5*A*, *Left*, *Inset* indicates that the dynamic strain drop, *A*, *Left* shows that the simulated stress peaks indeed collapse to a single function of the scaled propagation distance

Previous studies had suggested that once the shear stress peak, *A*, *Left*. Although, here, we concentrate on the dynamics of the propagating shear stress peak, we expect that the *A*, *Left* is, therefore, directly relevant to the supershear transition.

How large can *A*, *Left*). It is expected (9) that

Although the simulations provide us with access to the shear stress peak values at the interface, its sharp angular dependence (Fig. 4*B*) suggests that the experimental estimation of these values at *A*, *Right*, are indeed consistent with the numerics (Fig. 5*A*, *Left*) because they convincingly show that the shear peak roughly scales with the dynamic strain drop, *x*) from experimental uncertainties resulting from both spatial stress inhomogeneities and estimation of the measurements for *A*, *Right* fall below the slope

We further illustrate the scaling of the shear stress peak by considering (Fig. 5*B*) two different simulated rupture events. In each event, two typical snapshots of *x* are properly scaled, the space-time dependence of the on-fault (*w*, scales linearly with

## Discussion

We previously (4) demonstrated that the singular functions that were derived to describe shear fracture in the framework of fracture mechanics (7) provide an excellent description of near-tip stress field components—with the notable exception of the shear stress component at high rupture velocities. Here we have shown that this “discrepancy” with the singular solution is not a simple technical issue of accounting for nonsingular contributions to the singular description, but actually possesses a life (and extensive history) of its own.

In contrast to far-field acoustic data (6) that consider a rupture as a moving dipole source, we focused on the near-field radiation emitted by coherent accelerating ruptures. Transporting significant energy ahead of the rupture front, the stress radiation amplitudes can reach the strength of the frictional interface and trigger supershear (5, 9, 11⇓⇓–14). The radiated shear stress peaks have a characteristic near-field signature: high-amplitude radiation (comparable to the dynamic stress drop) that is both localized and strongly focused in the direction of rupture propagation. This signature is a general feature of nonsteady shear rupture (2, 5).

It is significant that both our experiments and simulations show that, during rupture propagation, the shear stress peak does not decrease in amplitude (cf. Fig. 1*B*). In contrast, upon rupture arrest we observe distinct shear stress peak decay (Fig. 1*C*). This decay is consistent with previous theoretical observations (2, 5, 35) of decaying radiation that was associated with abrupt changes of

How general is the shear stress peak scaling described by Fig. 5? Our simulations considered a particular, although important and fundamental, class of systems: (Griffith-like) quasistatic loading into a uniform initial stress distribution. Many experimental features are well-captured by the simulations (e.g., Figs. 3 and 5), despite the uncontrolled nucleation and

## Experiment

The experimental loading system, strain, and contact area measurements are described in detail in ref. 4. We specify here the main differences in the current study.

### System and Material Properties.

Our experiments were conducted using poly(methyl methacrylate) (PMMA) blocks of dimensions 200 × 100 × 5.5 mm (top block) and 250 × 100 × 5.5 mm (bottom block) in the *x*, *y*, and *z* direction, respectively (Fig. 1*A*). This is in contrast to ref. 4, where a 30-mm-wide bottom block was used. The contact faces of the blocks were diamond-machined to optical flatness. Material shear, *ρ* = 1,170 kg/m^{3}, yield dynamic values for the Poisson ratio of *E* is significantly different from the static value of

### Strain Measurements.

We use miniature Vishay 015RJ rosette strain gauges for local strain measurements. Each rosette strain gauge is composed of three independent active regions (each 0.34 × 0.38 mm size) which are separated in the *x* direction by ≈0.39 mm (for details see ref. 4). During rapid rupture propagation, this distance induces a small (≈0.31 μs) time delay between the components that are taken into account for proper calculation of

In these experiments we find that the measured strain fields are better described by the analytical solutions calculated for plane stress hypothesis

## Self-Similar Solution of an Expanding Shear Crack

The bilateral expansion of a crack from zero initial length under homogeneous shear loading at constant velocity is discussed in refs. 7 and 8. The solution to this problem is given by the self-similar stress fields *A* and Fig. S1, *Left*, respectively. Both fields show a pronounced peak arriving with the shear-wave front (*B*) and particle velocities and should be considered when experimental measurements are compared with LEFM solutions.

The comparison with experimentally measured strain components (Fig. 2*B*) was performed by specifying values of *l*, all within experimental error, to correspond to a snapshot of the self-similar solution. These parameters determine the local value of *l* are not, generally, satisfied in experiments, where a rupture will generally accelerate toward the Rayleigh wave speed. The solution is artificial in the sense that once *l*, forward propagation of the solution will no longer fit experiments, resulting in increasing deviations with time.

## Finite-Element Simulation

The elastic moduli, plane stress hypothesis, and plate dimensions that were used corresponded to the experimental system. The viscoelastic behavior of PMMA is not taken into account because we believe it has only a minor effect on the dynamics of propagating rupture fronts (the dynamic values of *E* were used). In addition, no attempt was made to exactly reproduce the experimental loading configuration; we applied a homogeneous stress configuration (where in the experiments, at times, significant stress gradients existed). We, instead, concentrated on the general features of mode II ruptures.

### Numerical Methods.

The finite-element simulations are based on an explicit Newmark-β integration scheme with a lumped mass matrix. The solids are discretized by regular quadrilateral elements with first-order interpolation and four Gaussian quadrature points. Each solid is discretized by

### Nucleation Procedure.

In the numerical work presented here we follow ref. 34 and induce a slowly propagating seed crack. Starting from *C*_{r}. In the nucleation zone the value of *Left*). When the system size is much larger and the dissipation zone is much smaller than the crack length, under homogeneous loading *Right*). This result is a general consequence of fracture mechanics (7).

What defines *A*)] is given by

### Shear Stress Peak Growth.

The supershear transition has been the subject of extensive studies. Let us relate the scaled growth of the shear stress peak in Fig. 5*A*, *Left* to estimates of the supershear transition. It was shown ref. 9 that if

In Fig. 5*A*, *Left* the normalized shear stress peak, *S*. Consequently, the supershear transition criterion (*x*, provides a prediction for the location of the supershear transition.

## Acknowledgments

We thank G. Cohen for fruitful discussions. This work was supported by James S. McDonnell Fund Grant 220020221, European Research Council Grant 267256, and Israel Science Foundation Grants 76/11 and 1523/15 (all to I.S. and J.F.); European Research Council Grant ERCstg UFO-240332 (to J.-F.M., D.S.K., and D.P.M.); and Swiss National Science Foundation Grant PMPDP2-145448 (to M.R.). This work was also supported by Cornell University (D.S.K.).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: jay{at}vms.huji.ac.il.

Author contributions: J.-F.M. and J.F. designed research; I.S. performed the experimental work; D.P.M., M.R., and D.S.K. performed the numerical work; and I.S., D.P.M., M.R., D.S.K., J.-F.M., and J.F. contributed to analysis of the data and to writing the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1517545113/-/DCSupplemental.

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