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Journal Club Theme of January 2016: Extreme Mechanics on the Surface of our Planet

ahmedettaf's picture

Our experience with earthquakes is that they are violent events that take a heavy toll on our societies through life and property losses. However, earthquakes present us with some of the most challenging questions in mechanics. By better understanding the nucleation and propagation dynamics of earthquakes, we may make progress towards minimizing their negative impact. Insights from mechanics may help in the development of better seismic hazard models as well as in the construction of more efficient earthquake early warning systems. The purpose of this article is to give a brief overview of the multiscale nature of the earthquake mechanics problem, discuss contemporary research efforts, and introduce some of the outstanding questions in the field.

Background: Our current understanding of crustal earthquakes is that they nucleate as frictional instabilities on pre-existing fault surfaces (usually filled with crushed rocks commonly known as gouge) under slow tectonic loading. Once they outgrow their nucleation region, they propagate as shear fractures at speeds that are close to the Rayleigh wave speed (few Kilometers/second). As it propagates, an earthquake causes a permanent off-set between the two sides of the fault. This offset is known as earthquake slip.

 The San Andreas Fault in California accommodating the relative motion of the North American Plate and the Pacific plate.

 In what sense are earthquake extreme?

Speed: Earthquakes may propagate at a wide range of speeds. As predicted by fracture mechanics, a n earthquake as a mode II fracture will accelerate till a limiting speed that is set by the Rayleigh wave speed of the medium (~ 0.92 Shear wave speed). However, there are some notable exceptions. For example, some earthquakes may break the shear wave speed barrier through a Burridge-Andrews transition mechanism and propagate at speeds between the Eshelby speed and the pressure wave speed [Dunham and Archuleta, 2004 and references therein]. These are called inter-sonic ruptures [Rosakis et al., 1999]. On the other end of the spectrum, a recent discovery suggests that some earthquakes may propagate at just few kilometer/day. These are called slow earthquakes [Beroza and Ide, 2011, and references therein]. The mechanics of these slowly propagating fronts are not completely understood yet.

Scale For an earthquake to propagate there must be a change in the strain energy of the medium to provide the required energy flux. Part of this change is consumed as fracture energy at the crack tip and as frictional dissipation on the fault surface.  The rest of the energy is emitted as radiated energy through seismic waves. To give an idea of what order of magnitudes we are talking about, consider the 2011 Tohoku Earthquake. This earthquake ruptured an area that is almost 100,000 km2 (roughly 4 times the area of the state of Massachusetts), emitted radiated energy on the order of 10 MJ/m2 of the fault surface area (compare it to specific surface energy of most rocks that is in the range of 1-10 J/m2), accumulated slip up to 50 m in some areas, and accommodated most of the inelastic slip in a localized zone of fault gouge that may have not exceeded few millimeters.

Propagation: Earthquakes may propagate in either one of two modes: An expanding crack-like mode or a compact Pulse-like mode. In the crack-like case, each point on the fault continues to slip until the earthquake reaches a strong heterogeneity causing its arrest. Once this happens, arrest waves are reflected back from the termination points causing the slip in the interior of the crack to stop. Thus, each point on the fault surface slips for a period of time comparable to the overall earthquake duration. On the other hand, it may happen that points behind the rupture front cease to slip shortly after the passage of the rupture tip [Freund, 1979; Heaton, 1990; Zheng and Rice, 1998]. This is known as the slip pulse mode since the rupture in this case advances through a compacton of slip (similar to a soliton) rather than the continuously expanding crack. Understanding the conditions under which slip pulse form and their dynamics is still an area of active research. It is natural to contrast short rise time for slip in pulse like ruptures with dislocation dynamics. An interesting recent paper  from Prof. Acharya’s group  touches on some aspects of this analogy [Zhang et al., 2015: http://faculty.ce.cmu.edu/acharya/files/2013/07/layer_problem_final.pdf]

 Pulse-like vs Crack-like ruptures. [Left]: Slip rate on the vertical axis as a function of time at a specific point on the fault surface. [Right]: Slip rate as a function of position along the fault at a given instant of time. [Courtesy of Nadia Lapusta]

 How do we model earthquakes?

Earthquake processes span a wide range of length and time scales. Individually each scale is best described by a different model resolution. We adopt a multiscale approach (illustrated below) which begins on the left with the identification of key microscopic processes (such as granular rearrangements and grain breakage), transitions in the center to medium scale models (constitutive laws with shear banding), and ends on the right with large scale dynamic rupture simulations, retaining what is important from smaller scales and transferring it to larger scales. On the largest scale, Earthquakes are modeled as an initial-boundary value problem. Equations of motion are solved in the bulk given (i) boundary conditions on the fault surface emerging from subscale granular dynamics, and (ii) initial values of displacement, velocity and stress at the onset of rupture.

Multiscale earthquake problem. The system progressively increases in scale from top to bottom. (Top) Individual STZ rearrangements occur at the grain scale and produce plastic strain in the fault gouge [lieou et al., 2014]. (Center) Deformation within the fault gouge: a shear band that is much narrower than the gouge thickness accommodates plastic strain in the gouge. (Bottom) Fault scale, with a thin layer of fault gouge sheared between elastic rocks.

 Unlike in a typical engineering fracture mechanics problem, modeling an earthquake come with several challenges:

  1. Determination of the prestress: The state of initial stress before an earthquake is not known a priori.  Each earthquake event produces a residual stress field and the state of stress at any instant of time is the accumulation of the stress fields due to prior events in addition to the far field tectonic stress loading. While the latter may be estimated from geological and geodetic measurements, it is almost impossible to measure the former. The crack dynamics critically depends on the prestress field and thus one of the major challenges in computational earthquake physics is to come up with a reasonable estimate of the prestress distribution.

  2. Determination of the friction law governing fault dynamics: Current experimental techniques are capable of measuring rock/granular friction at either low strain rates and a wide range of pressures [Dieterich, 1979; Ruina, 1983] or at low pressures and a wide range of velocities [e.g. Beeler et al., 2008l Sone and Shimamoto, 2009]. However, a typical crustal earthquake nucleates at a depth where the  normal stress is of the order of hundreds of Migapascals and the slip during the event will accumulate at a rate of few meters per second (green circle in the opposite figure).  These conditions remain to be challenging to reproduce experimentally. Thus, theoretical models are required to fill this gap.

 A phase plot for the feasible conditions in rock and gouge friction experiments. The green circle represent conditions most relevant to crustal earthquakes but which are currently beyond the experimental capabilities.

 

  1. An exceptionally wide range of temporal and spatial scales: Earthquakes nucleate under slow tectonic loading that takes tens to hundreds of years to accumulate enough stress to trigger an event. Meanwhile, an earthquake event takes only seconds to minutes to complete. Thus, time scales from milliseconds to several years must be resolved. Spatially, the region near the earthquake tip through which sharp gradients of stress and deformation occur (i.e. the process zone) may be of the order of few millimeters [Noda et al., 2009] while a medium size earthquake will rupture tens or hundreds of kilometers. This requires resolving 8 to 9 decades of spatial scales. Innovations in computational methods are required to address this computational gap.

 

Examples of Progress at different Scales

Theoretical models for fault friction: Perhaps, the most successful model for rock friction thus far is the rate and state model [Dieterich, 1979; Ruina, 1983] in which the friction at a point is a function of not only the slip rate at that point but also the slip rate history through a set of empirically introduced state variables. This is an important paradigm shift from the widely used friction laws in engineering applications that are typically either dependent on the velocity only [Bowden and Tabor, 1956] or implement a binary static-dynamic model. The rate and state formulation has been successful in modeling rock friction at low slip rates (from  fractions of micrometers/sec to possibly mm/sec) and has been recently extended to include possible dynamic weakening effects introduced by localized shear heating at contact asperities (flash heating) [Rice, 2006]. It was also shown that the full features of the rate and state friction are essential for the well-posedness of the elastodynamic slip problem [Rice et al., 2001].

However, the empirical nature of the rate and state friction has its limitations when it comes to the inclusion of additional physics, particularly relevant to deformation and failure of gouge. For example, granular particles may break at high strain rates and pressure, distribute plastic shearing differently from rocks, and rearrange under the effect of external shear or vibration. These features are not modeled explicitly in the classical framework of Dieterich and Ruina laws.

An alternative way to approach the problem is to consider the fault zone for what it is: a layer of finite thickness filled with viscoplastic material capable of experiencing finite deformation in localized shear bands. The shear transformation zone (STZ) theory emerges as an attractive choice in that respect. The theory was first introduced by Falk and Langer [1998] to describe viscoplastic deformation in amorphous solids and has been extended recently to systems composed of hard spheres [Lieou and Langer, 2012]. The premise of the theory is that inelastic strain occurs only at rare localized spots known as shear transformation zones (STZs). These STZs are defect-like structures associated with extra free volume that facilitate irreversible non-affine rearrangements (e.g. sliding or rotation) of the gouge particles. Each STZ transition event generates a finite amount of local plastic strain. Overall, the macroscopic plastic strain is the cumulative result of many local events. The density of STZs is governed by an effective temperature that is in one-to-one correspondence with the system porosity. The evolution equation of the effective temperature is derived from the first and second laws of thermodynamics and thus is physically consistent. The paper by Falk and Langer, [2010] provides an excellent review for the fundamentals of the theory.

With just a few parameters, the STZ theory has successfully reproduced a large number of experiments and molecular dynamics simulations for glassy materials and granular systems including strain localization [e.g. Langer and Manning, 2007; Manning et al., 2009; Falk and Langer, 2010; Lieou and Langer, 2012] and strain rate dependence of shear strength in the dense flow regime [Elbanna and Carlson, 2014]. Due to its roots in statistical thermodynamics, the theory is flexible enough to be extended to include new physics. For example, in our recent work, in collaboration with Prof. Jean Carlson at UCSB and Dr. Charles Lieou (now at LANL), we have been successful in describing flow and strain localization in granular media with breakable particles [Lieou at al., 2014a], flash heating effects in granular flow [Elbanna and Carlson, 2014], and shear response in the presence and absence of external vibrations [Lieou et al., 2014b, 2015, 2016; Kothari and Elbanna, 2016].

Examples of successful applications of STZ theory that go beyond classical rate and state law. (Top)Predictions of STZ theory for the rheology of sheared granular system [Elbanna and Carlson, 2014]. Insert shows results from discrete element simulations [daCruz et al., 2005]. (Bottom) Predictions of STZ theory for volume changes in sheared granular system with angular grains in the [Lieou et al., 2014]. Discrete points represent experimental observations of van der Elst et al. [2012].

Dynamic Rupture models:  Recent progress in computational earthquake dynamics has made it feasible to explore the influence of friction and small scale material and rheological heterogeneities on the macroscopic rupture response.

  1. Simulations with strong velocity weakening friction: These have shown that ruptures may propagate as crack like or pulse like and in some cases as a train of pulses depending on the rate of friction weakening and the level of the prestress [Zheng and Rice, 1998; Lapusta et al., 2000; Lapusta 2001; Ampuero and Ben-Zion, 2008; Noda et al., 2009; Dunahm et al., 2011; Elbanna, 2011; Elbanna and Lapusta,2016]. Lower prestress favor pulse like ruptures while higher prestress values favor crack like ruptures. These conditions were made precise in Zheng and Rice [1998] for the case of antiplane shear in a linear elastic media. Elbanna [2011] developed a procedure for simulating steady pulses that propagate at a constant prestress level and investigated the stability properties of this steady solution. Results suggest that prestress heterogeneities  play an important role in regulating pulse dynamics in terms of their growth and arrest properties and that pulses, due to their compact nature, are more sensitive to these heterogeneities than cracks.  Gabriel et al. [2012] further investigated the effect of inelastic bulk response on rupture mode selection.

  2. Simulations with bulk elastic heterogeneities: Huang and Ampuero [2011] and Huang et al.  [2014] showed that pulse like ruptures may emerge on a fault interface embedded in a waveguide [e.g. a zone of low elastic modulus than the surrounding bulk] even in the absence of strong velocity weakening due to the wave reflection and diffraction at the soft zone boundaries. Ma and Elbanna [2015] showed that the existence of off-fault soft elastic heterogeneities may affect the crack propagation speed and facilitate supershear propagation under conditions not feasible in homogenous media. In particular, the transition length to supershear propagation gets shorter and may occur at lower prestress levels than in the homogeneous conditions.

  3. Simulations with pore fluids: The pioneering work from Prof. Rice’s group and coworkers [Rice, 2006; Dunham and Rice, 2008; Noda et al., 2009; Noda et al., 2010; Platt et al., 2014] provided a framework for modeling earthquake ruptures in fluid saturated fault zone. In particular they modeled thermal pressurization by coupling the elastodynamics with a 1D model for temperature evolution and pore fluid pressure diffusion in the direction normal to the rupture propagation. Frictional heating due to shearing causes the fluid, which is trapped by the densely packed solid particles, to expand in volume much more than would the solid cage. A pressure increase must then be induced in the pore fluid during slip unless the gouge dilates more than what the fluid thermal expansion requires. This mechanism leads to strength reduction as shear heating continues to raise temperature so that the pore pressure approaches the background normal stress. Using such models, it was possible to investigate the effect of poromechanical effects on fracture energy, slip mode selection (pulses vs cracks) and shear stress evolution. Furthermore, thermal pressurization has been hypothesized as a mechanism for explaining the lack of high frequencies in some Chi-Chi earthquake ground motion records [Noda and Lapusta, 2010] as well as a mechanism for controlling depth penetration of large earthquakes [Jiang and Lapusta, 2013].

  4. Beyond elasticity: Eric Dunham and co-workers [e.g. Dunham et al., 2011a, 2011b] pioneered simulations of earthquake ruptures on non-planar fault geometry. Work by Yehuda Ben-Zion (http://earth.usc.edu/~ybz/), Jean Paul Ampuero (http://web.gps.caltech.edu/~ampuero/publications.html) , and Harsha Bhat (http://www.ipgp.fr/~bhat/index/Publications.html), among others, has explored extensively rupture dynamics in a bulk with continuum damage rheology. Going beyond planar faults and linear elastic bulk is opening new opportunities for incorporating more realistic geometric and rheological constraints in earthquake simulations closing the gap between numerical predictions and seismological observations.

Results of dynamic rupture simulations may be partially validated by comparison to geodetic and seismological observations. A very exciting and relatively recent effort is to design dynamic rupture experiments mimicking real earthquakes. The pioneering work of Ares Rosakis and his collaborators (http://www.rosakis.caltech.edu/research/abstracts/le2.html) in this area has been of great value. It  confirmed several earlier numerical predictions and provided new insights into the dynamics of rupture propagation including: the possibility of supershear propagation, the existence of the slip pulse mode, the influence of off-fault damage on rupture dynamics, and directionality effects for propagating on bi-material interfaces. Extending these experiments to include gouge material in the fault zone will bring these experiments closer to natural fault zone topologies.

Challenges and Outlook

Multiphysics of fault gouge: Lots of work is still needed to incorporate multiple physical processes that may contribute to the evolution of strength in sheared gouge layers. This includes extending the current theories to incorporate silica gel formation in silica rich geologies [Rice, 2006], chemical decomposition in carbonate formations [Platt et al., 2015], athermal fluid pressurization due to gouge dilation and compaction, as well as fragmentation and pulverization [Yamashita et al., 2014]. While some of these processes have been incorporated in the context of 1D models, it is imperative to extend the formulation to higher dimensions and incorporate the viscoplastic response of gouge. Our ongoing work in this area suggests that 2D models of viscoplasticity in gouge do reveal interesting physics that are not apparent in 1D models such as complex strain localization patterns, brittle to ductile transition as a function of grain size and initial porosity, as well as spatially heterogeneous inelastic volume changes [Ma and Elbanna, 2016]. Coupling the Multiphysics 2D gouge viscoplastic models with the bulk medium in dynamic rupture models is expected to give a more consistent framework for the mechanics of deformations and failure in fault zones.

Some examples for Brittle to ductile transition in sheared gouge layers from 2D continuum viscoplasticity simulations using STZ theory [Ma and Elbanna, 2016] (Top) Effect of grain size: smaller grain size leads to more ductile response. (Bottom)Effect of disorder: Less initial disorder leads to brittle response.

Bridging scales for ground motion prediction: To reliably predict ground motion for seismic hazard calculations, earthquake models must be able to simulate several decades of spatial and temporal scales. The dynamic nature of the problem and the existence of wave mediated interactions make a naiive adaptive mesh refinement strategy unfruitful. High frequency waves will get trapped at the interface of coarse and fine meshes, reflected into the fine mesh domain, and spoil the crack dynamics. Recent advances in adaptive mesh refinement [e.g. Spring, 2015] or the increased interest in discontinuous Galerkin methods [e.g. work from SeisSol group http://www.seissol.org/publications] as well as more access to high performace computing [ https://www.scec.org/research]  offer new opportunities in that respect. Extending multiscale methods such as the quasi-continuum approaches [Shenoy et al., 1995] to dynamic application (i.e. wave/phonon transmission) will also open new paradigms in simulating the earthquake problem.

The question of prestress: As mentioned above, the determination of the the pre-existing state of stress before an earthquake rupture is a challenging task since prestress cannot be measured directly in the field. The only rational way to derive a consistent prestress field is to carry out a long term simulation for earthquake cycles so that the results become insensitive to the assumed conditions at the beginning of the simulation. This requires developing a computational paradigm for simulating fast dynamic rupture episodes interrupting slow long term tectonic loading. A breakthrough in that aspect was made by Lapusta et al. 2000 who developed a spectral boundary integral equation procedure with rigorous adaptive time stepping and mode dependent truncation of convolution kernels. Using this technique Lapusta et al. were able to explore for the first time in a fully consistent framework the different phases of the earthquake cycle including creep, nucleation, and fast dynamic propagation and after slip. Despite being a powerful approach, the method is limited to only planar faults in linear elastic media. Efforts are needed to extend this computational paradigm to more general situations including inelastic media and nonplanar fault geometries. Methods built on bulk discretization (such as finite element or finite difference) are currently too expensive – from the computational perspective- to carry out such long time simulation. An alternative approach, that is still at its infancy,  is to construct a reduced order model for the rupture dynamics that retains the main physical features of the probelm but offers compuattional efficiency [Elbanna and Heaton, 2012]. 

To summarize, earthquake mechanics presents an exciting opportunity for mechanicians. Its multiscale multiphysics nature demands combining approaches from physics, nonlinear dynamics, seismology and computational mechanics. Progress in understanding the different phases of earthquake nucleation, propagation and arrest has the potential of reducing risks associated with such  devastating natural hazard. Moreover, since the earth has a heterogeneous structure and rate dependent rheology, investigations on the mechanics of earthquakes may lead to progress in understanding mechanics of composite structures under dynamic excitations.

Happy New Year!

 

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Comments

Dear Ahmed,

Thank you very much for the wonderful & concise article. It will be a good reading for my train travel tomorrow morning!

But on a fast glance, I was wondering if there are detailed mechanistic models/understanding of how small earthquakes could impede larger ones. If I remember my reading from few years ago, there were also some theories that some of the these smaller earthquakes could lead to release of accummulated stresses? Do you think that these events could anyway impact bigger events? 

ahmedettaf's picture

Dear Ajay, Thank you for your comment. Your question is a very interesting and timely one. The answer, to the best of my knowledge, is inconclusive. Remember that fracture is a highly nonlinear phenomena. Add to this nonlinear friction that depends on the crack motion itself (friction decreases with increasing slip and slip rate) and you get a positive feedback loop that in most cases would lead to chaotic dynamics. In that respect a small earthquake may indeed trigger a large earthquake or may impede it ! Mechanistically, a small earthquake will release part of the fault stress within the region over which it has ruptured. However, since the small earthquake has stopeed, it would also leave a stress concentration at the boundary of the fracture zone and a stress field that goes like 1/sqrt(distance) in the unfractured region. Thus, small earthquakes redistribute the stresses and this redistribution may make it more probable to have a rupture at some other point which may not have occured otherwise. This will depend on the pre-existing stress field distribution and the frictional properties of the system.

Why is this a timely question? Because of this surge  in triggered seismicity due to energy extraction activities. It is interesting that Oklahoma in the past year has more small earthquakes than California becauase of gas shale extraction and the associated wastewater disposal activities. Injecting water under pressure at shallow and intermediated crustal depths lead to a reduction in effective normal stress and may lubricate faults that otherwise would have been stuck. Can these small induced earthquakes trigger a larger one? The answer is inconclusive and is a focus of a lot of current research. The key to addressing this question is to be able to do long term simulations of earthquakes coupled with poromechanical effects (to consider the effective stress changes due to pore pressure in the reservoir) and realistic models of friction that consiers the effect of vibrations. These are topics of contemporary active research.

This is a recent nice article in Science magazine on Injection-Induced Earthquakes: http://users.clas.ufl.edu/prwaylen/GEO2200%20Readings/Readings/Fracking/...

and you could find lots of references in it. You may find it relevant to our current discussion.

All the best,

ahmedettaf's picture

This is a related very recent paper in PNAS on intersonic cracks from Feinberg and J F Molinari groups: http://www.pnas.org/content/early/2016/01/02/1517545113.full.pdf

Ravindra Duddu's picture

Dear Ahmed,

Very nice article and thanks for the rigorous literature review. You mention a critical point (although it got buried into your article) that the mechanics behind slow earthquakes is not fully understood. While friction based theories (including those based velocity weaking friction) work well for simulating earthquake dynamics, alternative fracture based theories in semi-brittle media are required (perhaps) to investigate transient slip events such as slow earth quakes or aseismic creep. In this regard I would like to draw imechanica readers attention to the following publications:

1. Reber, J. E., L.L Lavier and N.W. Hayman (2015), Strain transients in semi-brittle systems. Nature Geoscience 8 (9), 712-715.

2. Reber, J. E., N. W. Hayman, and L. L. Lavier (2014), Stick-slip and creep behavior in lubricated granular material: insights into the brittle-ductile transition, Geophysical Research Letters, 2014GL059832, doi:10.1002/2014GL059832.

3. Lavier, L.L., R.A. Bennett, and R. Duddu (2013), Creep events at the brittle ductile transition, Geochemistry, Geophysics, Geosystems, 14(9), 3334-3351.

ahmedettaf's picture

Dear Ravindra, thank you for the excellent comment and great references. It is not unfair to say that the topic of slow earthquakes deserve a separate discussion topic on its own. It is one of the most fascianting observations in seismology in the last few years and the literature on it is diverse and rich. I admit that it is not covered as it should in the article. I hope the review article I cited above (e.g Beroza and Ide, 2011) would be a good starting point for those who would like to familarize themselves with the topci from both the observational and numercial perspectives (at least up to that date).

I agree with you that classical friction models may fail in explaining the rich behavior of the slip spectrum. As you correctly pointed out, understanding brittle to ductile transition in the crust (in rocks as well as in granular materials) is crucial. Indeed this has been one of the outstanding challenges in the field for few decades now. Integrated experimental and theoretical efforts, like in the references you included, do help us understanding these issues better and open new dimensions in investgating fault stability.

Now, if we go back to the issue of slow slip, there are few theoretical models in the literature in that respect. These include: rate and state friction faults behavior under nearly lithostatic conditions such that the effective stress on the fault is very low [Liu and Rice, 2007]; slip stabilization by fault gouge dilatancy that results in a reduction of pore fluid pressure and slip arrest [Rubin, 2008]; and stochastic Brownian fault models that model slip as random walk processes [Ide, 2008]. Indeed, fratcure mechanics or continuum viscoplasticity approaches may provide a fresh look into the problem since deformation in the crust occurs on a variety of scales and there is a transition between localized failures (e.g. cracks and shear bands) and distributed plasticity (e.g. microdamage, multiple cracks, yiedling, ...etc.). The feedback between these multisclae damage features and pore fluids are likely playing a crucial role in the process.

In a very recent paper [Lieou et al., 2015] we showed using the STZ theory of gouge vsicoplasticity that vibration may lead to non-monotonic rate dependency in gouge behavior.  This translates into changes in slip stability as a function of strain rate and may lead to the emergence or suppression of stick slip. Our newer results (to appear soon) suggest that increasing the strength of the applied vibration noise may suppress gouge rate weakening and lead to a gradual transition from fast stick slip motion (low vibration intensities) to slow slip (intermediate vibration intensities) and ultimately to creep (at high vibration intensities). We hypothesize that this could be an additional mechanism for slow slip and creep and may provide some insights into the correlations between tremors (high frequency vibration) and slow slip.

References:

Beroza, G. C., and S. Ide (2011), Slow earthquakes and nonvolcanic tremor, Annual Review of Earth and Planetary Sciences, 39 (1), 271–296, doi:10.1146/annurev-earth-040809-152531.

Liu, Y., and J. R. Rice (2007), Spontaneous and triggered aseismic deformation transients in a subduction fault model, Journal of Geophysical Research: Solid Earth, 112 (B9),B09,404, doi:10.1029/2007JB004930

Rubin, A. M. (2008), Episodic slow slip events and rate-and-state friction, Journal of Geophysical Research: Solid Earth, 113 (B11), B11,414, doi:10.1029/2008JB005642.

Ide, S. (2008), A brownian walk model for slow earthquakes, Geophysical Research Letters,35 (17), L17,301, doi:10.1029/2008GL034821.

Lieou, C. K. C., A. E. Elbanna, J. S. Langer, and J. M. Carlson (2015), Stick-slip insta-bilities in sheared granular flow: The role of friction and acoustic vibrations, Phys. Rev. E, 92, 022,209, doi:10.1103/PhysRevE.92.022209.

Some seminal papers on Field observations of slow slip :

Obara, K. (2002), Nonvolcanic deep tremor associated with subduction in southwest japan, Science, 296 (5573), 1679–1681, doi:10.1126/science.1070378

Obara, K., H. Hirose, F. Yamamizu, and K. Kasahara (2004), Episodic slow slip events accompanied by non-volcanic tremors in southwest japan subduction zone, Geophysical Research Letters, 31 (23), L23,602, doi:10.1029/2004GL020848.

Rogers, G., and H. Dragert (2003), Episodic tremor and slip on the cascadia sub-duction zone: The chatter of silent slip, Science, 300 (5627), 1942–1943, doi:10.1126/science.1084783.

 

 

 

Ravindra Duddu's picture

Dear Ahmed,

I agree with you completely that slow earthquakes deserve a separate discussion. As you know, numerical modeling of brittle ductile transition (BDT) in porous plastic solids has been done by Needleman and Tvergaard (1995, 2000) using the Gurson-Tvergaard-Needleman (GTN) model, which falls into the category of continuum damage mechanics approaches. However, BDT in metals occurs at high strain rates and high temperatures. This is perhaps fundamentally different from BDT in the crust. An extending question to explore is "Is a modified GTN model suitable for studying BDT in the crust?"

I agree that multiscale aspects of damage and presence of pore fluids constitute a formidable challenge. In this regard, I have a couple of questions. Is it physically accurate to define a critical strain energy release rate G_c for porous Earth (rock) media and use LEFM approaches. If we rely on phenomenological damage (GTN-type) models and how could we calibrate and validate them.

[1] Needleman, A. and Tvergaard, V. (1995). Analysis of a brittle-ductile transition under dynamic shear loading. International Journal of Solids and Structures, 32: 2571–2590.

[2] Needleman, A. and Tvergaard, V. (2000). Numerical modeling of the brittle-ductile transition, International Journal of Fracture, 101: 73 - 97. 

ahmedettaf's picture

Dear Ravi, I figured that I never posted the response to imechanica :) I believe you are raising some excellent points here. In partcular, I think Gc may still be used to characterize fracture toughness in porous materials if the process zone is large compared to the fundamental length scale associated with the microstructure. If this is not the case, Gc may still be used if the energetics are derived from a generalized continuum mechanics model that accounts for internal structure. Alternatively, fracture details at the microscales (e.g. fracture of ribs or walls in the proous media) could be accounted for explicitly in a multiscale framework. A couple of papers on fracture of foams that imechanics readers may find relevant are attached.

What complicates BDT further in crustal rocks and fault gouge is the coupling with pore fluids as well as dynamic effects that lead to transitent thermal, configurational and pore pressure build up which can significantly alter background conditions and brings around additional new physics (e.g. flash heating, pore fluid pressurization, and acoustic fluidization induced by vibrations).

 

 

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