Mitiyasu Ohnaka

Scaling of the shear rupture process from nucleation to dynamic propagation: Implications of geometric irregularity of the rupturing surfaces

A series of systematic, high-resolution laboratory experiments have been performed on the nucleation of propagating slip failure on preexisting faults having different surface roughnesses to demonstrate how the size scale and duration of shear rupture nucleation are affected by geometric irregularity of the rupturing surfaces. On the basis of the experimental results it has been discussed theoretically how consistently scale-dependent physical quantities inherent in shear rupture are scaled. The experiments led to conclusive results that the nucleation process consists of two phases (phase I, an initial, quasi-static phase, and phase II, a subsequent accelerating phase) and that the nucleation process is greatly affected by geometric irregularities on the rupturing surfaces. In phase I the rupture grows at a slow, steady speed which is independent of the rupture growth length L. In contrast, during phase II the rupture develops at accelerating speeds V, which increase with an increase in L, obeying a power law V/VS = α(L/λc)n, where VS is the shear wave velocity, λc is the characteristic length representing the geometric irregularity of the fault surfaces, and α and n are constants (α = 8.87 × 10−29 and n = 7.31). Scale dependency of scale-dependent physical quantities, including the nucleation zone size and its duration, is commonly ascribed to scale dependency of the slip-dependent constitutive law parameter Dc, which is in turn governed by λc. It has been discussed that a unified comprehension can be provided for shear rupture of any size scale if the constitutive law for shear rupture is formulated as a slip-dependent law.

Constitutive relations between dynamic physical parameters near a tip of the propagating slip zone during stick-slip shear failure

Abstract Constitutive relations between physical parameters in the cohesive zone during stick-slip shear failure are experimentally investigated. Stick-slip was generated along a 40 cm long precut fault in Tsukuba granite samples using a servocontrolled biaxial loading apparatus. Dynamic behavior during local breakdown processes near a tip of the slipping zone is revealed; the slip velocity and acceleration are given as a function of the slip displacement and the cohesive (or breakdown) shear stress as a function of the slip velocity. A cycle of the breakdown and restrengthening process of stick-slip is composed of five phases characterized in terms of the cohesive strength and the slip velocity. The cohesive strength can degrade regardless of the slip velocity during slip instabilities. The maximum slip acceleration u max and the maximum slip velocity u max are obtained experimentally as: u max = 2 u c u max 2 and u max = Δτ b G v where u c is the critical displacement, Δτ b the breakdown stress drop, G the rigidity and v the rupture velocity. These relations are consistent with Idas theoretical estimation based on the cohesive zone model. The above formula gives good estimates for the maximum slip acceleration of actual earthquakes. The cutoff frequency ƒ max of the power spectral density of the slip acceleration increases with increasing normal stress; in particular, ƒ max is found to be directly proportional to the normal stress σ n within the normal stress range less than 17 MPa as: ƒ max (kHz) = 4.0σ n (MPa) σ n ƒ max increase with an increase in u max or u max . All these results lead to the conclusion that u max , u max and ƒ max increase with increasing normal stress. This is consistent with a previous observation that τ b increases with increasing normal stress. The above empirical linear relation between ƒ max and σ n can be explained by a linear dependence of Δτ b on σ n . The size-scale dependence of physical parameters is discussed, and such size-scale dependent parameters as u max and apparent fracture energy are scaled to the breakdown zone size regarded as a characteristic length, to extend the results obtained in the laboratory to an earthquake failure in the earth.

Earthquake source nucleation: A physical model for short-term precursors

Abstract Ohnaka, M., 1992. Earthquake source nucleation: a physical model for short-term precursors. In: T. Mikumo, K. Aki, M. Ohnaka, L.J. Ruff and P.K.P. Spudich (Editors), Earthquake Source Physics and Earthquake Precursors. Tectonophysics, 211: 149–178. This paper deals with a quasistatic, leading to a quasidynamic at a later time, nucleation process that precedes the earthquake dynamic rupture, and models of the earthquake source nucleation are put forward based on physical principles. A quasistatic to quasidynamic rupture nucleation process is an intrinsic part of the ensuing earthquake dynamic rupture; in other words, the nucleation process itself is an short-term (or immediate) precursor that occurs in a localized zone. During the earthquake nucleation, premonitory slip proceeds in the localized nucleation zone, and shear stress also decreases gradually in the zone, since slip-weakening occurs during the nucleation. Immediate foreshock activity is a part of the nucleation process leading to the mainshock dynamic rupture; therefore, hypocentral locations of foreshocks are necessarily restricted to lie near the mainshock hypocentre (onset of the mainshock rupture). Whether or not foreshocks occur during the mainshock nucleation depends on how the rupture growth resistance varies on a local to small scale. Since patches of greater rupture growth resistance are considered to prevail on a local to small scale in the fault zone, carrying immediate foreshocks would be one of the major characteristics of earthquakes that nucleate within the brittle seismogenic layer. When a mainshock earthquake nucleates within the brittle seismogenic layer and its hypocentre is located near the base of the seismogenic layer, immediate foreshocks for this mainshock are necessarily restricted to lie within a localized region shallower than hypocentral depth of the mainshock. By contrast, the nucleation process below the base of the seismogenic layer is aseismic in nature, so that carrying no conspicuous foreshocks would be a common characteristic of interplate earthquakes that nucleate below the base of the brittle seismogenic layer. The breakdown strength τp , the breakdown stress drop Δτb and the critical slip displacement Dc are indicative of the rupture growth resistance. To estimate depth variations of these parameters, the effects of the normal stress σn and temperature T on those parameters are examined using available data so far published. The combined effects of σn and T on τp in the brittle to semibrittle regime are found to be represented empirically by: τ p (σ n , T) = τ pO (σ n )[1−(cosh 50 T ) + 40 sinh( 50 T ) exp(− 2000 T ] where τpo(σn) = 135.7 + 0.750σn, and T is measured in °K and σn in MPa. It is found that Dc increases sharply with T, but is insensitive to σn above 300°C, while Dc depends on σn but is insensitive to T below 300°C. On the basis of these results, variations in τp, Dc and Δτb at mid-crustal depths are estimated for quartzo-feldspathic rocks for a given geothermal gradient.

Characteristic features of local breakdown near a crack-tip in the transition zone from nucleation to unstable rupture during stick-slip shear failure

Abstract Characteristic features of local breakdown near a propagating crack-tip during slip failure nucleation and its transition process to unstable rupture under mode II conditions are experimentally studied using a rock sample with a simulated fault, the size of which is large compared with the size of the breakdown zone. In addition, distinctive features in the nucleation zone are compared with those in the zone of steady, dynamic rupture propagation. It is found that the local breakdown stress drop, the local dynamic stress drop, the local stress increase immediately before the slip-weakening process, the dimensionless parameter S defined by Das and Aki, and the shear fracture energy are functions of crack length in the transition zone from nucleation to unstable rupture; i.e., they all increase with crack growth in the nucleation zone. This is contrasted with the observations in the zone of steady dynamic rupture propagation, where these physical parameters do not depend on crack length. An increase in the magnitudes of these parameters with crack growth in the nucleation zone results in increasing resistance to crack extension with crack growth in the nucleation zone. The reason for the increase in crack growth resistance in this zone is that a slip failure nucleus is formed at a point (or zone) where the local strength and the shear fracture energy are at a minimum. This basic physical information can provide the key to earthquake prediction, since the nucleation itself is a precursory phenomenon. The ratio of the size of the breakdown zone to the crack length decreases abruptly beyond the critical crack length. Rupture velocity v accelerates with crack growth, and reaches the shear wave velocity or super-shear, and v is found to be expressed empirically as a function of crack length in the nucleation zone. A positive correlation is found between S and v in the nucleation zone (v increases with increasing S). However, no such correlation was observed in the zone of steady, dynamic rupture propagation, where v seems to have reached the shear wave velocity of super-shear for S = 0.5. This is consistent with Das and Akis numerical result. The interrelationships between shear stress, slip displacement, slip velocity and acceleration near the crack-tip during the breakdown in the nucleation zone are revealed, and they are compared with those in the zone of dynamic rupture propagation. The present experiments suggest that the relationship between shear stress and slip displacement is more fundamental during the breakdown process than the relationship between shear stress and slip velocity. A model is presented to describe the breakdown process in the nucleation zone

A Physical Scaling Relation Between the Size of an Earthquake and its Nucleation Zone Size

A specific model of the earthquake nucleation that proceeds on a non-uniform fault is put forward to explain seismological data on the nucleation in terms of the underlying physics. The model is compatible with Gutenberg-Richter’s similarity law for earthquake frequency-magnitude relation. A theoretical approach in the framework of fracture mechanics, based on a laboratory-based slip-dependent constitutive law, leads to the conclusion that the earthquake moment Mo scales with the third power of the critical slip displacement Dc and the critical size 2Lc (Lc, half-length) of the nucleation zone. This scaling relation quantitatively explains seismological data published, and it predicts that 2Lc is of the order of 10 km for earthquakes with Mo=10 21 Nm, 1 km for earthquakes with Mo=10 18 Nm, and 100 m for earthquakes with Mo=10 15 Nm, under the assumption that the breakdown stress drop Dtb=10 MPa. However, Lc depends on not only Dc but also Dtb, so that the scaling relation between Lc and Dc may be violated by Dtb, because Dtb potentially takes any value in a wide range from 1 to 10 MPa, depending on the seismogenic environment. The good agreement between the theoretical relation and observed results suggests that a large earthquake may result from the failure of a large patch of high rupture growth resistance, whereas a small earthquake may result from the breakdown of a small patch of high rupture growth resistance. The present result encourages one to pursue the prediction capability for large earthquakes.

A shear failure strength law of rock in the brittle‐plastic transition regime

As a first step to establish the law governing shear failure of typical crustal materials in the brittle-plastic transition regime under lithospheric conditions, and thereby to properly estimate a depth profile of lithospheric strength in quantitative terms, the effects of the normal stress σn across the fault surfaces and ambient temperature T on the shear failure strength of dry Westerly granite in the brittle to brittle-plastic transition regimes are evaluated quantitatively, using experimental data published by earlier authors. The empirical law proposed can predict the shear failure strength at strain rates of 10−4-10−5/S under any (σn, T) conditions in the brittle to brittle-plastic transition regimes.

A CONSTITUTIVE LAW FOR THE SHEAR FAILURE OF ROCK UNDER LITHOSPHERIC CONDITIONS

Abstract This paper first reviews recent studies on constitutive formulations for shear failure, which leads to the conclusion that revealing the constitutive property for not only frictional slip failure, but also shear failure of intact rock in the brittle to brittle-plastic transition regimes is critical to establish the constitutive law which governs earthquake shear failure that proceeds in the lithosphere. To this end, a unique, high stiffness testing apparatus capable of applying polyaxial stress fields to rectangular rock specimens, has been constructed, for which a brief description is given. The paper then presents new results of laboratory experiments on constitutive properties of shear fracture of intact granite at lithospheric conditions. In view of the physical principles and constraints to be imposed on the constitutive law for shear failure, the slip-dependency is a more fundamental property than the time- or rate-dependency, and hence the constitutive law for shear failure should primarily be slip-dependent. This slip-dependent constitutive law, which itself is self-consistent, can unifyingly and quantitatively treat the entire phases from stable, quasistatic ruture growth to unstable, dynamic fast-speed rupture propagation for shear failure of any type, whether it is frictional slip failure along a preexisting fault or shear fracture of intact materials. The slip-dependent constitutive law includes a scaling parameter Dc (critical slip displacement) explicitly, which in turn is scaled by a characteristic length λc representing geometrical irregularities of the fault surfaces. Incorporation of Dc (or λc) enables this constitutive law to provide a common interpretation for shear failure of any size scale as an earthquake source, from small scale in the laboratory to large scale in the Earth. The present experiments corroborate the slip-dependent constitutive relation and other constitutive properties over the entire process from slip-strengthening to slip-weakening of intact granite, under the conditions that confining pressure is in the range 440 to 500 MPa, pore pressure is in the range 30 to 300 MPa, temperature is in the range from room temperature to 456°C, and strain rate is in the range 10−4 to 10−7/s. It is found that some constitutive law parameters are not independent, but mutually related. The constitutive relation for shear failure and the parameters such as Δτp (breakdown stress drop) and Dc prescribing its relation, are affected by ambient conditions such as temperature and strain rate.

Evolution of microseismicity during frictional sliding

We have done frictional sliding experiments on Inada granite in double shear and monitored the acoustic emission (AE) produced and temporal changes in the microseismic b-value (where b is defined as the log-linear slope of the AE frequency-amplitude distribution), using both rough and smooth ground simulated fault surfaces. We have found, (i) the maximum amplitudes of AE events during stable sliding are strongly dependent on the surface roughness with smooth-ground surfaces giving smaller maximum AE amplitudes; (ii) b-values are related to the surface topographic fractal dimensions, so that in steady-state stable sliding smooth surfaces exhibit lower b-values than rough surfaces; (iii) the b-value falls before stick-slip instability. The change of b with slip we interpret in terms of evolving fractal crack damage during frictional sliding of the fault surfaces.

Nucleation process of unstable rupture in the brittle regime : A theoretical approach based on experimentally inferred relations

Ohnaka and Kuwahara investigated the nucleation process of unstable rupture in their elaborate laboratory experiment with a very high resolution, using a rock sample with a simulated fault. They found, among other things, that the crack growth resistance such as the stress σp — σi (σp being the peak stress at the crack tip and σi being the shear stress at the propagating tip on the verge of slip) and the breakdown stress drop increase with crack growth in the nucleation zone. They also found that the critical slip displacement is inhomogeneously distributed on the fault in the nucleation zone. We investigate whether inhomogeneous distribution of the crack growth resistance can necessarily cause the stable and quasi-static nucleation process in the brittle regime. The slip-dependent breakdown zone model has been assumed in the present theoretical analysis of crack. It has theoretically been shown that an increase in the critical slip displacement with the distance is required for the occurrence of stable crack growth. Its increase rate must be larger than a certain threshold value. The transition to unstable rupture growth occurs at a location where the increase rate falls short of the threshold value. The inhomogeneous distribution of the critical slip displacement may physically be attributed to that of fault surface roughness, as shown in laboratory experiments. The stress distributions on the crack plane prescribe the rate of stable crack growth. The stable rupture growth process observed by Ohnaka and Kuwahara can be explained quantitatively assuming inhomogeneously distributed stresses and critical slip displacement, which have also been observed in their experiments. The present theoretical study shows that both the distribution of the crack growth resistance, particularly that of the critical slip displacement, and the size of preexisting initial crack play important roles in creating stable and quasi-static nucleation in the brittle regime. We show that diverse rupture phenomena can be caused by the differences in the distribution of critical slip displacement and in the size of preexisting initial crack.

Earthquake cycles and physical modeling of the process leading up to a large earthquake

A thorough discussion is made on what the rational constitutive law for earthquake ruptures ought to be from the standpoint of the physics of rock friction and fracture on the basis of solid facts observed in the laboratory. From this standpoint, it is concluded that the constitutive law should be a slip-dependent law with parameters that may depend on slip rate or time. With the long-term goal of establishing a rational methodology of forecasting large earthquakes, the entire process of one cycle for a typical, large earthquake is modeled, and a comprehensive scenario that unifies individual models for intermediate-and short-term (immediate) forecasts is presented within the framework based on the slip-dependent constitutive law and the earthquake cycle model. The earthquake cycle includes the phase of accumulation of elastic strain energy with tectonic loading (phase II), and the phase of rupture nucleation at the critical stage where an adequate amount of the elastic strain energy has been stored (phase III). Phase II plays a critical role in physical modeling of intermediate-term forecasting, and phase III in physical modeling of short-term (immediate) forecasting. The seismogenic layer and individual faults therein are inhomogeneous, and some of the physical quantities inherent in earthquake ruptures exhibit scale-dependence. It is therefore critically important to incorporate the properties of inhomogeneity and physical scaling, in order to construct realistic, unified scenarios with predictive capability. The scenario presented may be significant and useful as a necessary first step for establishing the methodology for forecasting large earthquakes.