Nucleation Zone Size Research Articles

Dynamic nucleation as a cascade-up of earthquakes depending on rupture propagation velocity

Earthquake dynamic rupture requires a nucleation process to provide sufficient energy to overcome the fracture energy. Large earthquakes may occur via a cascading rupture process, which includes many triggering processes that cascade from small to large sections of the fault system. During such a process, the nucleation of a large section of the fault plane may occur dynamically via the propagating rupture from a small section of the fault plane. A quasi-static view of seismic nucleation has been widely discussed in earthquake seismology; however, the dynamic nucleation process remains poorly known. Here, we investigate one aspect of the dynamic nucleation process by focusing on the rupture propagation velocity during the nucleation process. We simplify this process as self-similar crack propagation at a constant rupture velocity in a finite nucleation zone within a target region that possesses a uniform fracture energy. We numerically solve this elastodynamic problem in two dimensions for both the anti-plane and in-plane cases using the Boundary Integral Equation method. As the rupture velocity increases, the critical ratio of the fracture energy step to continue the rupture increases and the critical size of the dynamic nucleation zone decreases. The rapid increase in the ratio of the fracture energy step toward infinity could explain why earthquakes never propagate at slow rupture velocities. However, the effect on the size of the nucleation zone is rather limited, with the size of the dynamic nucleation zone decreasing to ~ 70% of the static nucleation zone size. However, such a small difference would result in a significant overall difference if such a dynamic nucleation process repeatedly occurred in the cascading rupture process of a large earthquake, which would be a difficult situation for earthquake early warning.Graphical

Numerical Simulation of Episodic Aseismic Slip Events as Incomplete Nucleation of Seismic Slip Due to Heterogeneous Stress Distribution

ABSTRACT Shear stress concentration at the deeper edge of a locked fault affects detachment of the fault, such as upward propagation of aseismic sliding, episodic aseismic slip events, and partial seismic rupture. Numerical simulations of earthquake cycles on a strike-slip fault were conducted using a rate- and state-dependent friction law to investigate the occurrence conditions of episodic aseismic slip events within a fault having uniform velocity-weakening friction. When the velocity-weakening zone is much wider than the critical nucleation zone size, a rupture that seismically or aseismically breaks a part of the velocity-weakening zone occurs during the interseismic period between large earthquakes. The partial seismic rupture results in a small earthquake, and the partial slow rupture results in an episodic aseismic slip event. The seismic or aseismic rupture is arrested in a low-shear-stress area, which is caused by the preceding large earthquake. The episodic aseismic slip events may be regarded as incomplete nucleation of an earthquake, because the rupture is arrested before acceleration to seismic slip, and this process may explain episodic aseismic slip events at seismogenic depths. The width of the area of preseismic sliding immediately before a simulated large earthquake is similar to that of episodic aseismic slip.

The relations between the corner frequency, seismic moment and source dynamic parameters derived from the spontaneous rupture of a circular fault

SUMMARYThe stress drop is an important dynamic source parameter for understanding the physics of source processes. The estimation of stress drops for moderate and small earthquakes is based on measurements of the corner frequency ${f_c}$, the seismic moment ${M_0}$ and a specific theoretical model of rupture behaviour. To date, several theoretical rupture models have been used. However, different models cause considerable differences in the estimated stress drop, even in an idealized scenario of circular earthquake rupture. Moreover, most of these models are either kinematic or quasi-dynamic models. Compared with previous models, we use the boundary integral equation method to simulate spontaneous dynamic rupture in a homogeneous elastic full space and then investigate the relations between the corner frequency, seismic moment and source dynamic parameters. Spontaneous ruptures include two states: runaway ruptures, in which the rupture does not stop without a barrier, and self-arresting ruptures, in which the rupture can stop itself after nucleation. The scaling relationships between ${f_c}$, ${M_0}$ and the dynamic parameters for runaway ruptures are different from those for self-arresting ruptures. There are obvious boundaries in those scaling relations that distinguish runaway ruptures from self-arresting ruptures. Because the stress drop varies during the rupture and the rupture shape is not circular, Eshelby's analytical solution may be inaccurate for spontaneous dynamic ruptures. For runaway ruptures, the relations between the corner frequency and dynamic parameters coincide with those in the previous kinematic or quasi-dynamic models. For self-arresting ruptures, the scaling relationships are opposite to those for runaway ruptures. Moreover, the relation between ${f_c}$ and ${M_0}$ for a spontaneous dynamic rupture depends on three factors: the dynamic rupture state, the background stress and the nucleation zone size. The scaling between ${f_c}$ and ${M_0}$ is ${f_c} \propto {M_0^{ - n}}$, where n is larger than 0. Earthquakes with the same dimensionless dynamic parameters but different nucleation zone sizes are self-similar and follow a ${f_c} \propto {M_0^{ - 1/3}}$ scaling law. However, if the nucleation zone size does not change, the relation between ${f_c}$ and ${M_0}$ shows a clear departure from self-similarity due to the rupture state or background stress.

Cascadia megathrust earthquake rupture model constrained by geodetic fault locking.

Paleo-earthquakes along the Cascadia subduction zone inferred from offshore sediments and Japan coastal tsunami deposits approximated to M9+ and ruptured the entire margin. However, due to the lack of modern megathrust earthquake records and general quiescence of subduction fault seismicity, the potential megathrust rupture scenario and influence of downdip limit of the seismogenic zone are still obscure. In this study, we present a numerical simulation of Cascadia subduction zone earthquake sequences in the laboratory-derived rate-and-state friction framework to investigate the potential influence of the geodetic fault locking on the megathrust sequences. We consider the rate-state friction stability parameter constrained by geodetic fault locking models derived from decadal GPS records, tidal gauge and levelling-derived uplift rate data along the Cascadia margin. We incorporate historical coseismic subsidence inferred from coastal marine sediments to validate our coseismic rupture scenarios. Earthquake rupture pattern is strongly controlled by the downdip width of the seismogenic, velocity-weakening zone and by the earthquake nucleation zone size. In our model, along-strike heterogeneous characteristic slip distance is required to generate margin-wide ruptures that result in reasonable agreement between the synthetic and observed coastal subsidence for the AD 1700 Cascadia Mw∼9.0 megathrust rupture. Our results suggest the geodetically inferred fault locking model can provide a useful constraint on earthquake rupture scenarios in subduction zones. This article is part of the theme issue 'Fracture dynamics of solid materials: from particles to the globe'.

A computational method for earthquake cycles within anisotropic media

SUMMARYWe present a numerical method for the simulation of earthquake cycles on a 1-D fault interface embedded in a 2-D homogeneous, anisotropic elastic solid. The fault is governed by an experimentally motivated friction law known as rate-and-state friction which furnishes a set of ordinary differential equations which couple the interface to the surrounding volume. Time enters the problem through the evolution of the ordinary differential equations along the fault and provides boundary conditions for the volume, which is governed by quasi-static elasticity. We develop a time-stepping method which accounts for the interface/volume coupling and requires solving an elliptic partial differential equation for the volume response at each time step. The 2-D volume is discretized with a second-order accurate finite difference method satisfying the summation-by-parts property, with boundary and fault interface conditions enforced weakly. This framework leads to a provably stable semi-discretization. To mimic slow tectonic loading, the remote side-boundaries are displaced at a slow rate, which eventually leads to earthquake nucleation at the fault. Time stepping is based on an adaptive, fourth-order Runge–Kutta method and captures the highly varying timescales present. The method is verified with convergence tests for both the orthotropic and fully anisotropic cases. An initial parameter study reveals regions of parameter space where the systems experience a bifurcation from period one to period two behaviour. Additionally, we find that anisotropy influences the recurrence interval between earthquakes, as well as the emergence of aseismic transients and the nucleation zone size and depth of earthquakes.

Dragon-Kings in rock fracturing: Insights gained from rock fracture tests in the laboratory

In order to shed some lights to the “dragon-kings” concept, this paper re-examines experimental results on rock fracture tests in the laboratory, obtained from acoustic emission monitoring. The fracture of intact rocks as well as rocks containing natural structures (joints, faults, foliations) under constant stress rate loading or creep conditions is generally characterized by typical stages with different underlying physics. The primary phase reflects the initial rupture of pre-existing microcrack population in the sample or in the fault zone. Sub-critical growth dominates the secondary phase. The third phases termed nucleation phase corresponds to the initiation and accelerated growth of the ultimate fracture. The secondary and nucleation phases in both intact rock and faulted rock show power-law (of time-to-failure) increasing event rate and moment release. Samples containing planar structures such as foliations and faults demonstrate very similar features to natural earthquakes including: 1) small number of immediate foreshocks by which fault nucleation zones could be mapped; 2) the critical nucleation zone size is normally a fraction of the sample dimension; 3) a lot of aftershocks concentrated on the fault ruptured during the main event; 4) stress drop due to the main rupture is of the order from a few tens to a few hundreds MPa; 5) b-value drops during foreshocks and recovers during the aftershocks. All these results agree with the suggestion that laboratory measurements require no scaling but can be applied directly to the Earth to represent local fault behavior. The ultimate failure of the sample, or fracture of major asperities on the fault surface, normally lead to extreme events, i.e., dragon-kings, which has a magnitude significantly greater than that expected by the Gutenberg-Richter power-law relation in the magnitude-frequency distribution for either foreshocks or aftershocks. There are at least two mechanisms that may lead to dragon-kings: 1) The power-law increasing event rate and moment release; and 2) Hierarchical fracturing behavior resulting from hierarchical inhomogeneities in the sample. In the 1st mechanism, the final failure corresponds to the end point of the progressive occurrence of events and thus the resulted dragon-king event can be interpreted as a superposition of many small events. While for the 2nd mechanism an event of extreme size is the result of fracture growth stepping from a lower hierarchy into a higher hierarchy on fault surface having asperities characterized by hierarchical distribution (of size or strength) rather than simple fractal distribution. In both mechanisms the underlying physics is that fracture in rocks is hard to stop beyond certain threshold corresponding to the critical nucleation zone size.

Postseismic variations in seismic moment and recurrence interval of repeating earthquakes

In laboratory experiments, longer stationary contact time leads to larger seismic moment during repeated ruptures. However, not all observations in natural fault systems agree with the prediction. We analyze a subset of 34 M−0.4–2.1 repeating earthquake sequences (RES) from 1987 to 2009 at Parkfield to examine the variation of their recurrence properties in space and time. Following a 2004 M6 earthquake, many of the repeating events have greatly reduced recurrence intervals ( Tr) that systematically increase with time. In addition to this change in timing, we also find systematic changes in seismic moment ( Mo), where most sequences experienced an immediate increase in Mo and subsequent decay as Tr approached pre-quake durations. The RES at shallower depth tend to have a larger range in both Tr and Mo, whereas deeper RES show smaller variation. The shallowest RES with the greatest magnitude (M1.8–2.1) among the events we studied reveal a large variation in Tr but small variation in Mo. These observations are qualitatively consistent with earthquake simulations in 3D continuum fault models with rate- and state-dependent friction. In the models, RES are produced on velocity-weakening patches surrounded by velocity-strengthening fault areas. The models show that the degree of postseismic variation in Mo and Tr is a function of radius ( r) and nucleation zone size ( h*) of the velocity-weakening patch. A ratio of r/ h* ~ 1 produces negative Mo– Tr slopes, whereas larger ratios of r/ h* yield weak positive slopes. Given the same nucleation size h* (i.e., the same frictional properties and effective normal stress), smaller radii and hence smaller seismic moments result in negative Mo– Tr slopes, whereas larger radii and hence larger moments lead to weak positive Mo– Tr slopes, which are consistent with observations. Conversely, with only a small percentage of its slip accumulated seismically, a small asperity appears to grow in Mo under high loading rate, which is contrary to the view that Mo should decrease due to a reduced strength recovery time.

Assessing the critical sizes for shear band formation in metallic glasses from molecular dynamics simulation

The size effect on the strength and fracture behavior of metallic glasses reported earlier by the authors is further examined in a mode I fracture from extensive molecular dynamics simulation using multimillion atoms and a method for atomic strain. In sharp contrast to the notch-free and notched samples, with small dimensions, in the large samples, shear bands easily initiate at the notch tips and grow into maturity. We identified two critical length scales for nucleating a shear band: the nucleation zone size d of about 10–20nm and the shear length ξ that is at least twice as large as d. The shape anisotropy of the critical nucleus described by the two length scales d and ξ may contribute to deformation anisotropy.

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 M o scales with the third power of the critical slip displacement D c and the critical size 2L c (L c , half-length) of the nucleation zone. This scaling relation quantitatively explains seismological data published, and it predicts that 2L c is of the order of 10 km for earthquakes with M o = 1021 Nm, 1 km for earthquakes with M o = 1018 Nm, and 100 m for earthquakes with M o = 1015 Nm, under the assumption that the breakdown stress drop Δτ b = 10 MPa. However, L c depends on not only D c but also Δτ b , so that the scaling relation between L c and D c may be violated by Δτ b , because Δτ b potentially takes any value in a wide range from 1 to 102 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.

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.

総合報告: 地震発生に至る過程と予測: その物理法則と地学的環境要因の重要性

This paper first reviews recent studies on constitutive formulations for shear failure. Noting that earthquake rupture is an general a mixed process between what is called frictional slip instability and fracture of initially intact rock in the fault zone, the paper then discusses how the constitutive law for earthquake rupture should be formulated to meet the physical principles and constraints to be imposed on the law. This leads to the conclusion that the constitutive law should be formulated as a slip-dependent law, which includes a scaling parameter Dc (critical slip displacement). Scale dependent physical quantities inherent in shear rupture, such as the breakdown zone size and the nucleation zone size, their duration, the slip acceleration, and the shear fracture energy, are consistently scaled by Dc, which in turn is prescribed by a characteristic scale representing the geometric irregularity of the fault zone. A physical model for earthquake generation is presented, which reconciles the observation that the mainshock seismic moment is scaled by the critical size of the nucleation zone. The constitutive law parameters depend on ambient conditions in the seismogenic layer, and therefore it is critical to evaluate the effects of the environmental factors on the constitutive law parameters in quantitative terms, in order that numerical simulation of a model earthquake could be useful for earthquake prediction in the real world.

Nonuniformity of the constitutive law parameters for shear rupture and quasistatic nucleation to dynamic rupture: a physical model of earthquake generation processes.

Based on the recent high-resolution laboratory experiments on propagating shear rupture, the constitutive law that governs shear rupture processes is discussed in view of the physical principles and constraints, and a specific constitutive law is proposed for shear rupture. It is demonstrated that nonuniform distributions of the constitutive law parameters on the fault are necessary for creating the nucleation process, which consists of two phases: (i) a stable, quasistatic phase, and (ii) the subsequent accelerating phase. Physical models of the breakdown zone and the nucleation zone are presented for shear rupture in the brittle regime. The constitutive law for shear rupture explicitly includes a scaling parameter Dc that enables one to give a common interpretation to both small scale rupture in the laboratory and large scale rupture as earthquake source in the Earth. Both the breakdown zone size Xc and the nucleation zone size L are prescribed and scaled by Dc, which in turn is prescribed by a characteristic length lambda c representing geometrical irregularities of the fault. The models presented here make it possible to understand the earthquake generation process from nucleation to unstable, dynamic rupture propagation in terms of physics. Since the nucleation process itself is an immediate earthquake precursor, deep understanding of the nucleation process in terms of physics is crucial for the short-term (or immediate) earthquake prediction.

Strain-rate effect on frictional strength and the slip nucleation process

Kato, N., Yamamoto, K., Yamamoto, H. and Hirasawa, T., 1992. Strain-rate effect on frictional strength and the slip nucleation process. In: T. Mikumo, K. Aki, M. Ohnaka, L.J. Ruff and P.K.P. Spudich (Editors), Earthquake Source Physics and Earthquake Precursors. Tectonophysics, 211: 269–282. The strain-rate dependence of frictional strength is investigated in relation to sliding behaviour by biaxial compression tests of large-scale granite samples. Frictional sliding is generated by increasing shear strain at a constant rate under a constant normal stress on the fault. The ram displacement is held constant for a prescribed time interval between two successive sliding events so as to have the same time effect of stationary contact between sliding surfaces. We obtained the following results. As the strain rate increases: (1) sliding becomes more unstable; (2) the nucleation zone size of stick-slip decreases; and (3) frictional strength logarithmically increases. Result (3) is consistent with the previous study by Dieterich (1979a), and (1) and (3) are similar to the strain-rate effects on failure characteristics of intact rocks reported in the literature. Result (3) can be explained by a rate- and state-dependent friction law. Our results suggest that the strain-rate dependence of the fault strength plays an important role in controlling the nucleation process of faulting.