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The article of record as published may be found at http://dx.doi.org/10.1785/0220170222 We describe a set of benchmark exercises that are designed to test if computer codes that simulate dynamic earthquake rupture are working as intended. These types of computer codes are often used to understand how earthquakes operate, and they produce simulation results that include earthquake size, amounts of fault slip, and the patterns of ground shaking and crustal deformation. The benchmark exercises examine a range of features that scientists incorporate in their dynamic earthquake rupture simulations. These include implementations of simple or complex fault geometry, off-fault rock response to an earthquake, stress conditions, and a variety of formulations for fault friction. Many of the benchmarks were designed to investigate scientific problems at the forefronts of earthquake physics and strong ground motions research. The exercises are freely available on our website for use by the scientific community. Southern California Earthquake Center (SCEC)

[1] We perform three-dimensional (3-D) numerical calculations of dynamic rupture along non-planar faults to study the effects of fault roughness on rupture propagation and resultant ground motion. The fault roughness model follows a self-similar fractal distribution over length scales spanning three orders of magnitude, from ~102 to ~105 m. The fault is governed by a strongly rate-weakening friction, and the bulk material is subject to Drucker-Prager viscoplasticity. Fault roughness promotes the development of self-healing rupture pulses and a heterogeneous distribution of fault slip at the free surface and at depth. The inelastic deformation, generated by the large dynamic stress near rupture fronts, occurs in a narrow volume around the fault with heterogeneous thickness correlated to local roughness slopes. Inelastic deformation near the free surface, however, is induced by the stress waves originated from dynamic rupture at depth and spreads to large distances (>10 km) away from the fault. The present simulations model seismic wave excitation up to ~10 Hz with rupture lengths of ~100 km, permitting comparisons with empirical studies of ground-motion intensity measures of engineering interest. Characteristics of site-averaged synthetic response spectra, including the distance and period dependence of the median values, absolute level, and intra-event standard deviation, are comparable to appropriate empirical estimates throughout the period range 0.1–3.0 s. This class of model may provide a viable representation of the ground-motion excitation process over a wide frequency range in a large spatial domain, with potential applications to the numerical prediction of source- and path-specific effects on earthquake ground motion.

SUMMARY We investigate the seismic velocity contrast across the San Andreas fault (SAF) in the Parkfield area using fault zone head waves (FZHW) that propagate along the bimaterial fault interface and direct P waves. We systematically analyse large data sets of near-fault waveforms recorded by several seismic networks over the period 1984–2005. Clear FZHW are observed at many stations on the NE side of the fault in the creeping section of the SAF north of Middle Mountain (MM). This indicates the presence of a sharp bimaterial interface and that the NE side of the fault has lower seismic velocities in that region. The obtained P-wave velocity contrast is about 5–10 per cent north of MM, and it systematically decreases to 0–2 per cent near Gold Hill (GH). The along-strike variations of the velocity contrast are consistent with geological observations of a sliver of high-velocity rock immediately to the NE of the SAF near GH, associated with the GH fault, and existing 3-D seismic tomography results. The obtained imaging results offer an explanation for the mixed rupture directions of the M6-type Parkfield earthquakes. The strong velocity contrast around MM is expected to produce a preferred propagation direction to the SE for earthquakes that nucleate near MM (e.g. the 1934 and 1966 Parkfield earthquakes). In contrast, the near-zero velocity contrast and multiple fault branches near GH imply that earthquakes that nucleate near GH (e.g. the 2004 Parkfield earthquake) are not expected to have a preferred propagation direction to the SE, and are likely to propagate in directions that are controlled by other factors such as structural and stress heterogeneities. The observed systematic reduction of the velocity contrast along the SAF from NW of MM to SE of GH provides a dynamic arrest mechanism for earthquakes that nucleate in the northern part of the Parkfield section and propagate to the SE, and a dynamic arrest mechanism for earthquakes that nucleate in the southern section and propagate to the NW.

SUMMARY We examine the characteristics of seismic radiation produced by localized fault-opening and shear motions in a homogeneous solid, along with the effects of having dissimilar solids across the fault on the seismic radiation. The study employs calculations of synthetic seismograms generated at various receiver locations by shear and tensile dislocation sources. The results indicate that, in contrast to the standard case of shear dislocation, the body wave amplitudes of the fault-parallel component generated at near-fault seismograms by tensile dislocation are considerably larger than those of the fault-normal component. The P and S arrivals from a shear dislocation at receivers on the opposite sides of a fault in a homogeneous solid have the same polarities on the fault-normal component and opposite polarities on the fault-parallel and vertical components. However, for a tensile dislocation source the situation is exactly reversed. The existence of a velocity contrast across the fault produces additional phases that mask somewhat the above signals. However, the generated amplitudes and other waveform characteristics may be used to distinguish between the physical fault plane and the auxiliary plane. The recording and analysis of the discussed signals for regular earthquakes that are dominated by shear motion will require high-resolution receivers located very close to the fault.

The effect of plasticity on dynamic frictional sliding along an interface between two identical elastic–viscoplastic solids is analyzed. The configuration considered is the same as that in Coker et al. (J Mech Phys Solids 53:884–992, 2005) except that here plane strain analyses are carried out and bulk material plasticity is accounted for. The specimens have an initial compressive stress and are subject to shear loading imposed by edge impact near the interface. The material on each side of the interface is modeled as an isotropically hardening elastic–viscoplastic solid. The interface is characterized as having an elastic response together with a rate- and state-dependent frictional law for its inelastic response. Depending on bulk material properties, interface properties and loading conditions, frictional slip along the interface can propagate in a crack-like mode, a pulse-like mode or a train-of-pulses mode. Results are presented for the effect of material plasticity on the mode and speed of frictional slip propagation as well as for the partitioning of energy components between stored elastic energy, kinetic energy, plastic dissipation in the bulk and frictional dissipation along the interface. Some parameter studies are carried out to explore the effects of varying the interface elastic stiffness and the impact velocity.

We study properties of dynamic ruptures and the partition of energy between radiation and dissipative mechanisms using two-dimensional in-plane calculations with the finite element method. The model consists of two identical isotropic elastic media separated by an interface governed by rate- and state-dependent friction. Rupture is initiated by gradually overstressing a localized nucleation zone. Different values of parameters controlling the velocity dependence of friction, the strength excess parameter and the length of the nucleation zone, lead to the following four rupture modes: supershear crack-like rupture, subshear crack-like rupture, subshear single pulse and supershear train of pulses. High initial shear stress and weak velocity dependence of friction favor crack-like ruptures, while the opposite conditions favor the pulse mode. The rupture mode can switch from a subshear single pulse to a supershear train of pulses when the width of the nucleation zone increases. The elastic strain energy released over the same propagation distance by the different rupture modes has the following order: supershear crack, subshear crack, supershear train of pulses and subshear single pulse. The same order applies also to the ratio of kinetic energy (radiation) to total change of elastic energy for the different rupture modes. Decreasing the dynamic coefficient of friction increases the fraction of stored energy that is converted to kinetic energy. General considerations and observations suggest that the subshear pulse and supershear crack are, respectively, the most and least common modes of earthquake ruptures.

SUMMARY We perform 2-D finite-difference calculations of mode II rupture along a bimaterial interface governed by slip-weakening friction, with the goal of clarifying rupture properties and the conditions leading to the development of unilateral wrinkle-like pulses. The simulations begin with an imposed bilateral rupture in a limited source region. Rupture properties outside the imposed source are examined for ranges of values of the degree of material contrast γ across the fault, the difference between static fs and dynamic fd coefficients of friction, and the difference between static friction and initial shear stress. The results show that mode II rupture evolves with propagation distance along a bimaterial interface, for broad ranges of realistic conditions, to a unilateral wrinkle-like pulse in the direction of slip on the compliant side of the fault. These conditions span in our calculations the ranges fs − fd 2‐ 5 per cent. When the difference between the static friction and initial shear stress is smaller, the evolution to unilateral wrinkle-like pulses occurs for smaller values of γ . The amount of slip increases with propagation distance, due to the incorporation of slip-weakening friction, in contrast to earlier results based on Coulomb and Prakash‐Clifton friction laws with slipindependent coefficient. In all cases leading to wrinkle-like pulses, the rupture velocity in the preferred (+) propagation direction is V + ≈ CGR, where CGR is the generalized Rayleigh wave speed. Simulations with imposed rupture speed in the source region close to the slower P wave speed P − can excite, in addition to the primary wrinkle-like pulse in the preferred direction with V + ≈ CGR, a weak pulse in the opposite (−) direction with V − ≈ P − . In some cases leading to bilateral crack-like propagation (e.g. fs − fd = 0.7), the rupture velocities in the opposite directions are V + ≈ P + (the faster P wave speed) and V − ≈ P − , with the initial supershear crack front in the + direction followed by a pulse with V + r ≈ CGR.

We discuss 2D simulations of in-plane ruptures on a fault between different solids in a model that includes dynamic generation of plastic strain off the fault. The fault is governed by a regularized friction with a gradual response to abrupt changes of normal stress, while plastic strain in the bulk is governed by a Coulomb yielding criterion. The occurrence of off-fault damage stabilizes the slip velocity on the fault. Stable self-sustaining ruptures propagate on the material interface as narrow unidirectional pulses in the direction of slip on the more compliant side of the fault. Plastic strain is generated only on the stiffer side of the fault (the tensional quadrant in the direction of rupture propagation) in a strip of approximately constant width that is correlated with the slip velocity on the fault. Simulations for various values of confining normal stress, material contrast and cohesion in the bulk suggest that significant generation of off-fault damage is limited to conditions that correspond to the top few kilometers of the crust. The results are compatible with field observations of shallow asymmetric damage patterns in the structure of several large faults.

A new model of an electric battery is established, based on the simplest battery model and Denno's battery model with an internal dissipation. The model is used to investigate the fundamental problem of how to maximize the work output of an electric battery in a given time. The important relation of the voltage across the battery terminals varying with the discharging time is derived by using variational calculus. The maximum work output of the battery is calculated. The optimal matching condition of the load resistance is determined. The reasonable range of the discharging time is given. The other characteristics of the battery at the maximum work output are discussed further. The results obtained here have some theoretical applications not only for raising the utilization of the electric energy stored in batteries, but also for improving the performance of some electric circuits.

A simple battery model established by Denno [Power system design and applications for alternative energy sources, Englewood Cliffs, NJ: Prentice-Hall: 1989] is used to investigate the basic question of how to maximize the work output of an electric battery. The discharge time is determined reasonably and the load resistance is matched optimally. The results obtained here can be used to explain quantitatively the example given by Gyftopoulos [Energy 24 (1999) 1035].

We study volumetric deformation structures in stepover regions using numerical simulations and field observations, with a focus on small-scale features near the ends of rupture segments that have opposite-polarity from the larger-scale structures that characterize the overall stepover region. The reversed-polarity small-scale structures are interpreted to be generated by arrest phases that start at the barriers and propagate some distance back into the rupture segment. Dynamic rupture propagating as a symmetric bilateral crack produces similar (anti-symmetric) structures at both rupture ends. In contrast, rupture in the form of a predominantly unidirectional pulse produces pronounced reversed-polarity structures only at the fault end in the dominant propagation direction. Several observational examples at different scales from strike-slip faults of the San Andreas system in southern California illustrate the existence of reversed-polarity secondary deformation structures. In the examples shown, relatively-small pressure-ridges are seen only on one side of relatively-large extensional stepovers. This suggests frequent predominantly unidirectional ruptures in at least some of those cases, although multisignal observations are needed to distinguish between different possible mechanisms. The results contribute to the ability of inferring from field observations on persistent behavior of earthquake ruptures associated with individual fault sections.