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A Supershear Transition Mechanism for Cracks

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Science  07 Mar 2003:
Vol. 299, Issue 5612, pp. 1557-1559
DOI: 10.1126/science.1080650

Abstract

Seismic data indicate that fault ruptures follow complicated paths with variable velocity because of inhomogeneities in initial stress or fracture energy. We report a phenomenon unique to three-dimensional cracks: Locally stronger fault sections, rather than slowing ruptures, drive them forward at velocities exceeding the shear wave speed. This supershear mechanism differentiates barrier and asperity models of fault heterogeneity, which previously have been regarded as indistinguishable. High strength barriers concentrate energy, producing potentially destructive pulses of strong ground motion.

Earthquakes are modeled as shear cracks propagating along fault planes. Heterogeneous stress and strength distributions lead to complex rupture histories. Because elastic waves radiate from the concentrated region of maximum sliding velocity at the rupture front, perturbations to its motion, particularly those inducing propagation faster than the shear wave speed, are the source of high-frequency ground motions (1). Material heterogeneity is characterized by regions of increased fracture energy, i.e., cohesive resistance to sliding, known as barriers (2). Alternatively, ruptures in the asperity model follow regions of high initial stress close to failure (3). This prestress variability is set by the residual stress field remaining from past events, combined with tectonic loading.

Spontaneous rupture occurs when the energy release rate (the flux of kinetic and strain energy from the prestress field into the rupture front) overcomes the local fracture energy. The trade-off between the energy that is supplied from the prestress field and the fracture energy results in an underdetermined problem, making it difficult to establish the source of fault heterogeneities (4,5). In fact, radiation from dynamic two-dimensional (2D) cracks is identical in barrier and asperity models (1). Recent advances in parallel computing enable us to investigate complex 3D dynamic rupture propagation.

Large strike-slip earthquakes occur predominantly in mode II (in which the slip is parallel to the direction of rupture propagation), for which the limiting velocity is usually the Rayleigh surface wave speedv R. However, for a homogeneous medium with high prestress or low fracture energy, speeds between the shear wave speedv S and longitudinal wave speedv P are allowed (6–10). Intersonic propagation, although theoretically predicted (7), was only recently observed (11). Stable velocities are centered around2 v S, a peculiar speed at which crack growth is similar to subsonic propagation (for example, the Mach cones extending from the rupture front vanish) (9, 10,12, 13).

Seismic data indicate that most earthquakes occur slightly belowv R. With the advent of strong motion seismograph networks in the near field, inversions and modeling of several major earthquakes suggest that supershear bursts occur on sections of the fault close to failure. These include the 1979 Imperial Valley (14), 1992 Landers (15), and 1999 Izmit, Turkey (16, 17) earthquakes.

We modeled bilaterally expanding ruptures (shear cracks) by solving the elastodynamic wave equation in a 3D medium surrounding a rectangular planar fault. Slip, the relative shear displacement between the two sides of the fault, was constrained to be horizontal. We nucleated ruptures by initially overstressing a vertical section of the fault. We placed periodic boundary conditions in the vertical direction to simulate an infinite-width mode II crack and we placed absorbing boundary conditions to prevent reflections from all other sides. Perturbations to the rupture front introduced a component of mode III failure (i.e., a slip parallel to the rupture front).

A uniform compressive stress acted on the fault, which was governed by a fracture criterion relating shear traction to slip. In particular, we used a slip-weakening cohesive zone model that constrains material failure processes to the fault plane (8,18). The fault started at a prestress value, σ0. Slip initiated when the shear stress reached a critical static friction stress level, σs, normalized to unity. The stress then decreased linearly with increasing slip over a characteristic distance (i.e., the slip-weakening distanced c), after which the surfaces were left at a constant sliding friction value, σf, that we scaled to zero. For the unit area of the new fracture, the work done against cohesion was the fracture energy, G = σs d c/2.

The supershear transition for 2D mode II cracks on homogeneous faults occurs only if the prestress is sufficiently high. A stress peak traveling at v S ahead of a sub-Rayleigh rupture exceeds the static stress, allowing the crack to jump to intersonic velocities (7, 8). Cracks become supershear after a distance (proportional to d c) that diverges as the prestress approaches a numerically obtained critical value: σ0 = 0.38 (8). For lower prestress values, the crack's terminal velocity is v R. This mechanism does not apply to 3D cracks. Instead, the dimensionless ratio κ of the energy release rate (∼σ0 2λ/μ, where μ is the shear modulus) to the fracture energy controls the supershear transition (19, 20). The length scale λ determines the stress concentration at the rupture front (i.e., the stress intensity factor) and is specific to the geometry of the problem (for example, the fault width for a rectangular planar fault).

We investigated cracks with prestress 0.3 ≤ σ0 ≤ 0.4, which remained sub-Rayleigh in the absence of any perturbations over the length of our fault. We introduced a circular obstacle to the fault, either a barrier, which requires more energy to break, or an anti-asperity, which is a region of lower prestress that decreases the energy release rate. Either obstacle locally reduced κ, suggesting that the energy balance would favor slower propagation or even arrest of the rupture. Model parameters, including the obstacle size, were consistent with observations (21).

The anti-asperity model only slightly delays the rupture front, which moves to regain its unperturbed shape after breaking the obstacle (Fig. 1 and movie S1). The barrier model is much more complex (Fig. 1 and movie S2). The stress concentration at the rupture front is initially insufficient to break the barrier. The rupture temporarily halts as the stress grows to the higher static level inside. Meanwhile, the faster-moving rupture front outside of the barrier encircles it, breaking it from all sides. This is similar to the rupture of a single asperity of high prestress, initiated from a point on the edge of the asperity (22). The rupture fronts within the barrier converge almost to a point, resulting in a concentrated region of high slip velocity at the far side of the barrier. When the barrier breaks, it emits an elliptical slip velocity pulse moving at v R in the direction parallel to slip and at v S in the perpendicular direction, as well as a weaker pulse moving at v P in the forward direction. Furthermore, the energy concentration induces the transition to a rupture velocity slightly less than2 v S (Fig. 2)

Figure 1

Consecutive illustrations showing slip velocity on the fault plane for the anti-asperity (left) and barrier (right) models, both having κ locally decreased by a factor of 5. Locked regions of the fault are uncolored, and the gray circle marks the obstacle's perimeter. The color scale changes at each time step.

Figure 2

Space-time plot of slip velocity along the symmetry axis through the center of the barrier, which lies atx = 15 km. The four black lines show wave speeds. Locked regions of the fault are uncolored.

To explore this phenomenon, we varied both initial stress and barrier strength. We found that maximum slip velocities within the barrier scale linearly with static stress, rather than with the prestress that has been found in homogeneous models and has been assumed for all ground motion predictions (23,24). We refer to this effect, resulting in slip velocities over an order of magnitude larger than at the unperturbed front, as barrier focusing.

Barriers slightly stronger than the surrounding fault (1 < σs < 1.3) only mildly perturb the rupture front, which returns toward the stable planar configuration at a sub-Rayleigh speed (Fig. 3). Above a critical strength (σs* ≈ 1.3), the restoring motion becomes supershear with a duration that, like the slip velocity, increases linearly with barrier strength until σs ≈ 3.5.

Figure 3

Dependence of supershear propagation on barrier strength and prestress. The background colors indicate the mechanism governing the supershear transition.

At the opposite extreme, when the barrier is much stronger than the surrounding fault (σs ≥ 7), the rupture front splits as it breaks the material above and below the obstacle and coalesces on the far side into a region of high slip velocity. This is caused by rupture front focusing, in which two colliding fronts cause a rapid stress drop (25). The split-front focusing effect causes a burst of supershear propagation, with a duration independent of the barrier strength.

When the barrier breaks, it releases a slip velocity pulse traveling atv P. If the rupture has not moved too far past the barrier, this pulse carries sufficient energy to enhance the duration of supershear propagation, which has already been activated by split-front focusing. Precise timing, occuring for 3.5 < σs < 7, results in a resonance during which the supershear front moves ahead of the unperturbed front.

Further numerical experiments showed that weak anti-asperities (σ0 ≤ 0) induce a supershear transition as a result of split-front focusing. Low prestress values are necessary because the energy release rate into the rupture front is a nonlocal function of the prestress history (26). The size of the anti-asperity relative to the crack length precludes large variations of the energy release rate, making anti-asperities dynamically much less of an obstacle than barriers. An anti-asperity that takes the same amount of time to break as a σs = 5 barrier will have σ0 = −1.2 and will produce only 0.65 times the spatial duration of supershear propagation. Furthermore, formation of a high-slip-velocity region within the anti-asperity does not occur as it does within the barrier (for which maximum slip velocity scales with the barrier strength). The anti-asperity with σ0 = −1.2 produces only 13% of the maximum slip velocity observed within the barrier. This limits the amplitude of seismic waves released from the anti-asperity, and thus prevents the rupture front from overtaking its unperturbed position.

Our results show that existing parameters governing the supershear transition on a homogeneous fault cannot characterize the complex dynamics of heterogeneous faults. Despite such idealizations as a perfectly circular barrier with uniform properties, our simulations reveal unexpected phenomena that may be common at many scales. Because ground motion is proportional to slip velocity (24), barrier focusing releases intense seismic waves. Near-field seismograms from the 1984 Morgan Hill earthquake show, in addition to the usual phases from the hypocenter, a second large pulse in ground motion arriving about 8 s later that can be traced to a region 14 km along the strike from the hypocenter (27, 28). Kinematic inversions suggest that the region failed after being encircled by the rupture front (28), making this a likely candidate for our mechanism. The extreme amplitudes of these pulses make them potentially destructive, suggesting that the identification of currently unbroken barriers (from inversions of past events) must become a part of seismic hazard analysis.

On a smaller scale, our results apply to dynamic shear fracture of brittle engineering materials. Laboratory experiments provide a convenient framework for analysis of our phenomena. Furthermore, barrier focusing, and the corresponding scaling of slip velocity with static stress rather than prestress, is unique to 3D cracks. This scaling, which implies a local independence from external loading, runs counter to fundamental fracture mechanics assumptions and likely extends to dynamic tensile failure of brittle materials that are common in engineering applications.

Supporting Online Material

www.sciencemag.org/cgi/content/full/299/5612/1557/DC1

Movies S1 and S2

  • * To whom correspondence should be addressed. E-mail: edunham{at}physics.ucsb.edu

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