A complete suite of closed analytical expressions is presented for the surface displacements, strains, and tilts due to inclined shear and tensile faults in a half-space for both point and finite rectangular sources. These expressions are particularly compact and free from field singular points which are inherent in the previously stated expressions of certain cases. The expressions derived here represent powerful tools not only for the analysis of static field changes associated with earthquake occurrence but also for the modeling of deformation fields arising from fluid-driven crack sources.

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Surface deformation due to shear and tensile faults in a half-space

This essential reference for graduate students and researchers provides a unified treatment of earthquakes and faulting as two aspects of brittle tectonics at different timescales. The intimate connection between the two is manifested in their scaling laws and populations, which evolve from fracture growth and interactions between fractures. The connection between faults and the seismicity generated is governed by the rate and state dependent friction laws - producing distinctive seismic styles of faulting and a gamut of earthquake phenomena including aftershocks, afterslip, earthquake triggering, and slow slip events. The third edition of this classic treatise presents a wealth of new topics and new observations. These include slow earthquake phenomena; friction of phyllosilicates, and at high sliding velocities; fault structures; relative roles of strong and seismogenic versus weak and creeping faults; dynamic triggering of earthquakes; oceanic earthquakes; megathrust earthquakes in subduction zones; deep earthquakes; and new observations of earthquake precursory phenomena.

The Mechanics of Earthquakes and Faulting

Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

This paper reviews rock friction and the frictional properties of earthquake faults. The basis for rate- and state-dependent friction laws is reviewed. The friction state variable is discussed, including its interpretation as a measure of average asperity contact time and porosity within granular fault gouge. Data are summarized showing that friction evolves even during truly stationary contact, and the connection between modern friction laws and the concept of “static” friction is discussed. Measurements of frictional healing, as evidenced by increasing static friction during quasistationary contact, are reviewed, as are their implications for fault healing. Shear localization in fault gouge is discussed, and the relationship between microstructures and friction is reviewed. These data indicate differences in the behavior of bare rock surfaces as compared to shear within granular fault gouge that can be attributed to dilation within fault gouge. Physical models for the characteristic friction distance are discussed and related to the problem of scaling this parameter to seismic faults. Earthquake afterslip, its relation to laboratory friction data, and the inverse correlation between afterslip and shallow coseismic slip are discussed in the context of a model for afterslip. Recent observations of the absence of afterslip are predicted by the model.

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Laboratory-derived friction laws and their application to seismic faulting

A review of recent developments concerning the structure, mechanics and fluid flow properties of fault zones

To understand constitutive behavior near the rupture front during an earthquake source shear failure along a preexisting fault in terms of physics, local breakdown processes near the propagating tip of the slipping zone under mode II crack growth condition have been investigated experimentally and theoretically. A physically reasonable constitutive relation between cohesive stress τ and slip displacement D, τ = (τi − τd)[1 + α log (1 + βD)] exp (−ηD) + τd, is put forward to describe dynamic breakdown processes during earthquake source failure in quantitative terms. In the above equation, τi is the initial shear stress on the verge of slip, τd is the dynamic friction stress, and α, β, and η are constants. This relation is based on the constitutive features during slip failure instabilities revealed in the careful laboratory experiments. These experiments show that the shear stress first increases with ongoing slip during the dynamic breakdown process, the peak stress is attained at a very (usually negligibly) small but nonzero value of the slip displacement, and then the slip-weakening instability proceeds. The model leads to bounded slip acceleration and stresses at and near the dynamically propagating tip of the slipping zone along the fault in an elastic continuum. The dynamic behavior near the propagating tip of the slipping zone calculated from the theoretical model agrees with those observed during slip failure along the preexisting fault much larger than the cohesive zone. The model predicts that the maximum slip acceleration be related to the maximum slip velocity and the critical displacement Dc by , where k is a numerical parameter, taking a value ranging from 4.9 to 7.2 according to a value of τi/τp (τp being the peak shear stress) in the present model. The model further predicts that be expressed in terms of and the cut off frequency fmaxs of the power spectral density of the slip acceleration on the fault plane as and that in terms of Dc and fmaxs as . These theoretical relations agree well with the experimental observations and can explain interrelations between strong motion source parameters for earthquakes. The pulse width of slip acceleration on the fault plane is directly proportional to the time Tc required for the crack tip to break down, and fmaxs is inversely proportional to Tc.

https://www.eri.u-tokyo.ac.jp/people/tyama/_userdata/Ohnaka&Yamashita(1989)JGR.pdf

A cohesive zone model for dynamic shear faulting based on experimentally inferred constitutive relation and strong motion source parameters

Summary. The Omori formula for aftershock activity can be shown to be a consequence of each of two models of a complex earthquake. In the first case. aftershocks are assumed to be caused by subsequent slip on asperities of a fault which are locked during the fracture in the main shock. In the second, aftershocks are caused by catastrophic coalescence of nearby small fractures with the fracture surface of the main shock. Hence, the first source model describes aftershock activity in the interior of a main-shock rupture zone, while the second describes aftershock activity near the outer boundary of a main-shock rupture zone. Stress corrosion cracking is assumed to be the physical cause of the quasistatic growth of the main fracture zone at the expense of the area of the locked patches in the first model and is also the cause of the growth of the small satellite fractures in the second.

Models of aftershock occurrence

—Spatio-temporal variation of rupture activity is modeled assuming fluid migration in a narrow porous fault zone formed along a vertical strike-slip fault in a semi-infinite elastic medium. Pores are assumed to be created in the fault zone by fault slip. The effective stress principle coupled to the Coulomb failure criterion introduces mechanical coupling between fault slip and pore fluid. The fluid is assumed to flow out of a localized high-pressure fluid compartment in the fault with the onset of earthquake rupture. The duration of the earthquake sequence is assumed to be considerably shorter than the recurrence period of characteristic events on the fault. The rupture process is shown to be significantly dependent on the rate of pore creation. If the rate is large enough, a foreshock–mainshock sequence is never observed. When an inhomogeneity is introduced in the spatial distribution of permeability, high complexity is observed in the spatio-temporal variation of rupture activity. For example, frequency-magnitude statistics of intermediate-size events are shown to obey the Gutenberg–Richter relation. Rupture sequences with features of earthquake swarms can be simulated when the rate of pore creation is relatively large. Such sequences generally start and end gradually with no single event dominating in the sequence. In addition, the b values are shown to be unusually large. These are consistent with seismological observations on earthquake swarms.

https://www.eri.u-tokyo.ac.jp/people/tyama/_userdata/Yamashita(1999)PAGEOPH.pdf

Pore Creation due to Fault Slip in a Fluid-permeated Fault Zone and its Effect on Seismicity: Generation Mechanism of Earthquake Swarm

SUMMARY The spontaneous growth of a dynamic in-plane shear crack is simulated using a newly developed method of analysis in which no a priori constraint is required for the crack tip path, unlike in other classical studies. We formulate the problem in terms of boundary integral equations; the hypersingularities of the integration kernels are removed by taking the finite parts. Our analysis shows that dynamic crack growth is spontaneously arrested soon after the bending of the crack tips, even in a uniformly stressed medium with homogeneously distributed fracture strengths. This shows that the dynamics of crack growth has a significant eVect on forming the non-planar crack shape, and consequently plays an essential role in the arrest of earthquake rupturing.

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Simulation of the spontaneous growth of a dynamic crack without constraints on the crack tip path

Laboratory and field observations suggest that dynamically propagating earthquake faults generate a large number of tensile microcracks in their vicinity, which will contribute to the formation of fault zones. Intense interactions are expected to occur between such microcracks and a dynamically propagating fault, which will complicate the fault growth. Near-field seismic waves will also be affected by the generation of microcracks. We numerically study how such tensile microcracks are generated and how dynamic growth of a macroscopic shear rupture and near-field elastic waves are affected by the distribution of generated microcracks. It is essential to consider a large number of microcracks in such studies, so that it is impractical to consider each microcrack individually from the viewpoint of computation time and memory. We overcome this difficulty by representing the microcrack distribution by anisotropic properties of the overall elastic coefficients on the basis of Hudson's (1980) study. Our simulations show that the decrease in the microcrack density is approximated well by a logarithmic function of the distance from the rupture plane. Microcracks on the dilational side of the rupture plane are shown to make larger angles to the rupture plane than on the compressive side. These are consistent with field and laboratory observations. It is also shown that dynamically generated microcracks tend to reduce the rupture-tip shear stress. This implies that the dynamic growth of a shear rupture is more decelerated when microcracks are generated than when the shear rupture is isolated in an isotropic and homogeneous medium. If the dynamic rupture growth is arrested suddenly, an abrupt expansion of the distribution zone of microcracks is shown to occur near the arrested rupture tip. Aftershocks are expected to cluster in this zone because of the shear stress enhancement there and the high density of distributed microcracks, which facilitates aftershock occurrence due to dynamic coalescence of microcracks. Our simulations also show that the component of radiated displacement waves perpendicular to the rupture plane is much more affected by the generation of microcracks than the parallel component.

/pdf/generation-of-microcracks-by-dynamic-shear-rupture-and-its-2el1z8beld.pdf

Teruo Yamashita

Papers

A cohesive zone model for dynamic shear faulting based on experimentally inferred constitutive relation and strong motion source parameters

Models of aftershock occurrence

Pore Creation due to Fault Slip in a Fluid-permeated Fault Zone and its Effect on Seismicity: Generation Mechanism of Earthquake Swarm

Simulation of the spontaneous growth of a dynamic crack without constraints on the crack tip path

Generation of microcracks by dynamic shear rupture and its effects on rupture growth and elastic wave radiation