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Academic Commons Search Resultsen-usRelation between stress heterogeneity and aftershock rate in the rate-and-state model
http://academiccommons.columbia.edu/catalog/ac:162514
Helmstetter, Agnès; Shaw, Bruce E.http://hdl.handle.net/10022/AC:P:20798Wed, 19 Jun 2013 11:29:23 +0000We estimate the rate of aftershocks triggered by a heterogeneous stress change, using the rate-and-state model of Dieterich. We show that an exponential stress distribution Pτ(τ) ~ exp(−τ/τ0) gives an Omori law decay of aftershocks with time ~ 1/tp, with an exponent p = 1 − Aσn/τ0, where A is a parameter of the rate-and-state friction law and σn is the normal stress. Omori exponent p thus decreases if the stress “heterogeneity” τ0 decreases. We also invert the stress distribution Pτ(τ) from the seismicity rate R(t), assuming that the stress does not change with time. We apply this method to a synthetic stress map, using the (modified) scale invariant “k2” slip model (Herrero and Bernard). We generate synthetic aftershock catalogs from this stress change. The seismicity rate on the rupture area shows a huge increase at short times, even if the stress decreases on average. Aftershocks are clustered in the regions of low slip, but the spatial distribution is more diffuse than for a simple slip dislocation. Because the stress field is very heterogeneous, there are many patches of positive stress changes everywhere on the fault. This stochastic slip model gives a Gaussian stress distribution but nevertheless produces an aftershock rate which is very close to Omori's law, with an effective p ≤ 1, which increases slowly with time. We obtain a good estimation of the stress distribution for realistic catalogs when we constrain the shape of the distribution. However, there are probably other factors which also affect the temporal decay of aftershocks with time. In particular, heterogeneity of Aσn can also modify the parameters p and c of Omori's law. Finally, we show that stress shadows are very difficult to observe in a heterogeneous stress context.Mechanics, Plate tectonics, Geophysicsbes11Lamont-Doherty Earth ObservatoryArticlesMagnitude dependence of radiated energy spectra: Far-field expressions of slip pulses in earthquake models
http://academiccommons.columbia.edu/catalog/ac:162497
Shaw, Bruce E.http://hdl.handle.net/10022/AC:P:20793Tue, 18 Jun 2013 17:11:06 +0000We examine the radiated waves emitted by events on a model fault. The model deterministically produces a complex sequence of events, with a wide range of sizes, from a uniform frictional instability. The spontaneous rupture events emit a rich spectrum of radiated waves as they nucleate, propagate, and decelerate within the complex stress field left by previous events. Two model innovations, a new driving boundary condition on the fault and a new radiating boundary condition which allows a spatially varying prestress away from the fault, allow us to directly measure the radiation without problems from boundary reflections in our two-dimensional model. We quantify the radiation by first measuring the energy spectral density and then averaging over events of a similar size to examine the magnitude dependence. Assuming only a physics of the tractions on the fault, we obtain a full spectra of radiated waves for a complex population of events with a wide range of sizes. To quantify the resulting spectra, we consider two different spectral measures. One, the peak amplitude of the spectral energy density, occurs at a period which scales with the rupture length and corresponds with the classical corner frequency measurement. The other, the peak amplitude of the spectral average acceleration or the low-frequency corner in the case of a flat acceleration spectrum, occurs at a period that scales with the duration of slip of points on the fault. The period of the peak spectral acceleration saturates for large events. Looking at the rupture motions on the fault, we find that this spectral behavior corresponds with the behavior of slip pulses in the model. Intense narrow pulses of slip develop for very long rupture events. We quantify this by measuring the mean slip duration as a function of rupture length and show that it is has the same behavior as the peak period of spectral acceleration. Thus the duration of the slip pulses in these ruptures is directly expressed in their radiated spectra. Moreover, we find that these corner periods exhibit a nontrivial dependence on event magnitude for the different frictional instabilities that we have examined, suggesting that any observed dependence of these corner periods on earthquake magnitude might provide insight into the frictional instability of earthquakes.Mechanics, Geophysics, Plate tectonicsbes11Lamont-Doherty Earth ObservatoryArticlesExistence of continuum complexity in the elastodynamics of repeated fault ruptures
http://academiccommons.columbia.edu/catalog/ac:162484
Shaw, Bruce E.; Rice, James R.http://hdl.handle.net/10022/AC:P:20789Tue, 18 Jun 2013 14:38:25 +0000What are the origins of earthquake complexity? The possibility that some aspects of the complexity displayed by earthquakes might be explained by stress heterogeneities developed through the self-organization of repeated ruptures has been suggested by some simple self-organizing models. The question of whether or not even these simple self-organizing models require at least some degree of material heterogeneity to maintain complex sequences of events has been the subject of some controversy. In one class of elastodynamic models, previous work has described complexity as arising on a model fault with completely uniform material properties. Questions were raised, however, regarding the role of discreteness, the relevance of the nucleation mechanism, and special parameter choices, in generating the complexity that has been reported. In this paper, we examine the question of whether or not continuum complexity is achieved under the stringent conditions of continuous loading, and whether the results are similar to previously claimed findings of continuum complexity or its absence. The elastodynamic model that we use consists of a 1-D fault boundary with friction, a steady slowly moving 1-D boundary parallel to the fault, and a 2-D scalar elastic media connecting the two boundaries. The constitutive law used involves a pair of sequential weakening processes, one occurring over a small slip (or velocity) and accomplishing a small fraction of the total strength drop, and the other at larger slip (or velocity) and providing the remaining strength drop. The large-scale process is motivated by a heat weakening instability. Our main results are as follows. (1) We generally find complexity of type I, a broad distribution of large event sizes with nonperiodic recurrence, when the modeled region is very long, along strike, compared to the layer thickness. (2) We find that complexity of type II, with numerous small events showing a power law distribution, is not a generic result but does definitely exist in a restricted range of parameter space. For that, in the slip weakening version of our model, the strength drop and nucleation size in the small slip process must be much smaller than in the large slip process, and the nucleation length associated with the latter must be comparable to layer thickness. This suggests a basis for reconciling different previously reported results. (3) Bulk dispersion appears to be relatively unimportant to the results. In particular, motions on the fault plane are seen to be relatively insensitive to a wide range of changes in the dispersion in the bulk away from the fault, both at long wavelengths and at short wavelengths. In contrast, the fault properties are seen to be very important to the results. (4) Nucleation from slip weakening and time-dependent weakening showed similar large-scale behavior. However, not all constitutive laws are insensitive to all nucleation approximations; those making a model “inherently discrete” and hence grid-dependent, in particular, can affect large scales. (5) While inherent discreteness has been seen to be a source of power law small-event complexity in some fault models, it does not appear to be the cause of the complexity in the attractors examined here, and reported in earlier work, fortuitously in the pecial parameter range, with the same class of continuum fault models and same or very similar constitutive relations. Continuum homogeneous dynamic complexity does indeed exist, although that includes type II small-event complexity only under restricted circumstances.Plate tectonics, Mechanics, Physicsbes11Lamont-Doherty Earth ObservatoryArticlesExperimental Evidence for Different Strain Regimes of Crack Populations in a Clay Model
http://academiccommons.columbia.edu/catalog/ac:162481
Spyropoulos, Chrysanthe; Griffith, William J.; Scholz, Christopher H.; Shaw, Bruce E.http://hdl.handle.net/10022/AC:P:20788Tue, 18 Jun 2013 14:23:57 +0000We report results from clay extension experiments used as a model for the evolution of fault populations due to stress interactions. At yielding cracks begin to appear and the brittle strain due to them quickly reaches a rate matching the applied stretching rate. The crack density (number of cracks per unit area) initially increases apace, then reaches a maximum at a critical strain, decreasing thereafter. At low strains, where the crack population is dilute, a power law length distribution is observed, which at high strain, gradually transitions to an exponential. This agrees with fault populations data observed in low and high strain settings. These results indicate that fault populations ranging from power law to exponential size-frequency distributions reflect the population evolution with increased strain.Plate tectonics, Mechanicschs2, bes11Applied Physics and Applied Mathematics, Lamont-Doherty Earth ObservatoryArticlesPost seismic response of repeating aftershocks
http://academiccommons.columbia.edu/catalog/ac:162478
Schaff, David P.; Beroza, Gregory C.; Shaw, Bruce E.http://hdl.handle.net/10022/AC:P:20786Tue, 18 Jun 2013 13:59:02 +0000The recurrence intervals of repeating earthquakes on the San Andreas Fault in the Loma Prieta aftershock zone follow the characteristic 1/t decay of Omori's law. A model in which these earthquakes occur on isolated patches of the fault that fail in stick-slip with creep around them can explain this observation. In this model the recurrence interval is inversely proportional to the loading rate due to creep. Logarithmic velocity strengthening friction predicts 1/t decay in creep rate following the mainshock. The time dependence of recurrence is inconsistent with a simple viscous constitutive relationship, which predicts an exponential decay of loading rate. Thus, our observations imply postseismic slip at seismogenic depth under a power law rheology. The time dependence of postseismic deformation measured geodetically may be diagnostic of whether postseismic deformation is caused by creep or possible viscoelastic deformation at greater depths.Plate tectonics, Mechanicsbes11Lamont-Doherty Earth ObservatoryArticlesUniversality in selection with local perturbations in the Saffman-Taylor problem
http://academiccommons.columbia.edu/catalog/ac:162398
Shaw, Bruce E.http://hdl.handle.net/10022/AC:P:20764Mon, 17 Jun 2013 09:55:40 +0000An analytic theory using WKBJ methods for selection with local perturbations in the SaffmanTaylor [Proc. R. Soc. London, Ser. A 245, 312 (1958)] problem is presented. I obtain qualitative agreement with previously published phenomenology, including symmetric narrowed fingers for local reductions in the surface-tension parameter, narrowed asymmetric fingers for local increases,
and scaling of the tip curvature and asymmetry with the square root of the surface-tension parameter. The source of the universality in the perturbed problem is discussed, giving some explanation of why the experimental perturbations can be modeled by locally varying surface tension. Very good quantitative agreement between theory and a numerical simulation of the same perturbation is shown, with no adjustable parameters to fit. Finally, I outline experiments to test new behavior predicted by the theory; a quantitative prediction observable experimentally is given.Theoretical physics, Mechanicsbes11Lamont-Doherty Earth ObservatoryArticlesFinger narrowing under local perturbations in the Saffman-Taylor problem
http://academiccommons.columbia.edu/catalog/ac:162376
Zocchi, Giovanni; Shaw, Bruce E.; Libchaber, Albert; Kadanoff, Leo P.http://hdl.handle.net/10022/AC:P:20749Fri, 14 Jun 2013 11:45:56 +0000We present an experimental study and a numerical simulation of the effect of time-independent, localized perturbations applied to the interface in the Saffman-Taylor fingering problem. When the perturbation is applied at a specific spot near the tip of the finger, the selection of the steady-state shape is drastically changed. In particular, one can obtain fingers with a width well below λ=1/2. A perturbation applied far away from the tip has no effect. We observe the same behavior in the simulation and in the experiment.Theoretical physics, Mechanicsbes11Lamont-Doherty Earth ObservatoryArticlesA novel discrete damage zone model and enhancement of the extended finite element method for fracture mechanics problems
http://academiccommons.columbia.edu/catalog/ac:166527
Liu, Xiahttp://hdl.handle.net/10022/AC:P:14865Tue, 09 Oct 2012 16:38:53 +0000This research develops two novel numerical methods for applications in fracture mechanics: (I) A new crack tip enrichment function in the extended finite element method (XFEM), and (II) a discrete damage zone model for quasi-static and fatigue delamination in composites. The first method improves XFEM when applied to general nonlinear materials when crack tip analytical solutions are not available. For linear elastic materials, Branch functions are commonly used as crack tip enrichments. Typically, these are four functions derived from linear elasticity theory and added as additional degrees of freedom. However, for general inelastic material behavior, where the analytical solution and the order of singularity are unknown, Branch functions are typically not used, and only the Heaviside function is employed. This however may introduce numerical error, such as inconsistency in the position of the crack tip. Hence, a special construction of Ramp function is proposed as tip enrichment, which may alleviate some of the problems associated with the Heaviside function when applied to general nonlinear materials, especially ones with no analytical solutions available. The idea is to linearly ramp down the displacement jump on the opposite sides of the crack to the actual crack tip, which may stop the crack at any point within an element, employing only one enrichment function. Moreover, a material length scale that controls the slope of the ramping is introduced to allow for better flexibility in modeling general nonlinear materials. Numerical examples for ideal and hardening elasto-plastic and elasto-viscoplastic materials are given, and the convergence studies show that a better performance is obtained by the proposed Ramp function in comparison with the Heaviside function. Nevertheless, when analytical functions, such as the Hutchinson-Rice-Rosengren (HRR) fields, do exist (for very limited material models), they indeed perform better than the proposed Ramp function. However, they also employ more degrees of freedom per node and hence are more expensive. The second method developed in this thesis is a discrete damage zone model (DDZM) to simulate delamination in composite laminates. The method is aimed at simulating fracture initiation and propagation within the framework of the finite element method. In this approach, rather than employing specific cohesive laws, we employ damage laws to prescribe both interface spring softening and bulk material stiffness degradation to study crack propagation. For a homogeneous isotropic material the same damage law is assumed to hold in both the continuum and the interface elements. The irreversibility of damage naturally accounts for the reduction in material strength and stiffness if the material was previously loaded beyond the elastic limit. The model parameters for interface element are calculated from the principles of linear elastic fracture mechanics. The model is implemented in Abaqus and numerical results for single-mode as well as mixed-mode delamination are presented. The results are in good agreement with those obtained from the virtual crack closure technique (VCCT) and available analytical solutions, thus, illustrating the validity of this approach. The suitability of the method for studying fracture in fiber-matrix composites involving fiber debonding and matrix cracking is demonstrated. Finally, the DDZM method is extended to account for temperature dependent fatigue delamination in composites. The interface element softening is described by a combination of static and fatigue damage growth laws so as to model delamination under high-cycle fatigue. The dependence of fatigue delamination on the ambient temperature is incorporated by introducing an Arrhenius type relation into the damage evolution law. Numerical results for mode I, mode II and mixed mode delamination growth under cyclic loading are presented and the model parameters are calibrated using previously published experimental data. Then, predictions are made under varying mode mix conditions and are compared with numerical results in the literature.Engineering, Mechanicsxl2201Civil Engineering and Engineering MechanicsDissertationsNanofluidics: Fundamentals and Applications in Energy Conversion
http://academiccommons.columbia.edu/catalog/ac:152478
Liu, Linghttp://hdl.handle.net/10022/AC:P:14671Wed, 12 Sep 2012 11:58:14 +0000As a nonwetting liquid is forced to invade the cavities of nanoporous materials, the liquid-solid interfacial tension and the internal friction over the ultra-large specific surface area (usually billions of times larger than that of bulk materials) can lead to a nanoporous energy absorption system (or, composite) of unprecedented performance. Meanwhile, while functional liquids, e.g. electrolytes, are confined inside the nanopores, impressive mechanical-to-electrical and thermal-to-electrical effects have been demonstrated, thus making the nanoporous composite a promising candidate for harvesting/scavenging energy from various environmental energy sources, including low grade heat, vibrations, and human motion. Moreover, by taking advantage of the inverse process of the energy absorption/harvesting, thermally/electrically controllable actuators can be designed with simultaneous volume memory characteristics and large mechanical energy output. In light of all these attractive functionalities, the nanoporous composite becomes a very promising building block for developing the next-generation multifunctional (self-powered, protective and adaptive) structures and systems, with wide potential consumer, military, and national security applications. In essence, all the functionalities of the proposed nanofluidic energy conversion system are governed by nanofluidics , namely, the behavior of liquid molecules and ions when confined in ultra-small nanopores. Nanofluidics is an emerging research frontier where solid mechanics and fluid mechanics meet at the nanoscale. The complex interactions between liquid molecules/ions and solid atoms at the nanointerface, as well as the unique structural, thermal and electrical characteristics of fluids confined in nanocavities collectively represent an outstanding challenge in physical science. A thorough understanding of the science of nanofluids, in particular the detailed molecular mechanisms as well as the roles of various material and system parameters, does not only underpin the development and optimization of the aforementioned nanofluidic energy conversion system, but it also have broad impact on a number of other areas including environmental engineering, chemical engineering, bioengineering, and energy engineering, etc. This dissertation carries out a systematic computational study to explore the fundamental nanofluidic infiltration and transport mechanisms, as well as the thermal and electrical characteristics of the solid-liquid interface. New physical models describing the unique nanofluidic phenomena will be established, where critical parameters, such as the surface tension, contact angle, and viscosity, will be reinvestigated at the nanoscale. The effects of various material and system parameters, such as the solid phase, liquid phase, pore size and pore geometry, as well as the external thermal, electrical and mechanical loads, etc., will be systematically investigated and bridged with the nanofluidic energy conversion processes. The energy conversion efficiencies under various conditions will be evaluated via a synergy between simulation and experiment. Reverse analysis based on the revealed principles can guide the optimization of the various material and system parameters, which potentially may contribute to the design of highly efficient and sustainable nanofluidic energy conversion devices. Besides the direct impact on the nanofluidic energy conversion, the study is also directly relevant to biological conduction and environmental sustainability, in both of which infiltration and transport play important roles.Mechanics, Alternative energy, Energy, Materials sciencell2405Civil Engineering and Engineering Mechanics, Earth and Environmental EngineeringDissertationsScience of Nanofluidics and Energy Conversion
http://academiccommons.columbia.edu/catalog/ac:148244
Xu, Baoxinghttp://hdl.handle.net/10022/AC:P:13570Thu, 21 Jun 2012 12:12:49 +0000The emerging subject of nanofluidics, where solids and fluids interact closely at the nanoscale, has exhibited radically different from their macroscopic counterparts (and sometimes counterintuitive), and yet relatively less explored. On the other hand, the resulting unique properties may contribute to a number of innovative functions with fascinating applications. Among various exciting potential applications, an important and ever expanding one is to provide alternative solutions to energy conversion with high efficiency, including energy absorption, actuation and harvesting. In this dissertation, we first report a novel protection mechanism of energy capture through which an intensive impact or blast energy can be effectively mitigated based on a nonwetting liquid-nanoporous material system. The captured energy is stored in nanopores in the form of potential energy of intercalated water molecules for a while, and not necessarily converted to other forms of energy (e.g. heat). At unloading stage, the captured energy will be released gradually due to the hydrophobic inner surfaces of nanopores through the diffusion of water molecules out of nanopores, thus making this system reusable. Several key controlling factors including impacting velocity, nanopore size, nanopore structure, and liquid phase have been investigated on the capacity of energy capture. The molecular mechanism is elucidated through the study of water molecular distributions inside nanpores. These molecular dynamic (MD) findings are quantitatively verified by a parallel blast experiment on a zeolite/water system. During the transport of confined liquid molecules, the friction resistance exerted by solid atoms of nanopores to liquid molecules will dissipate part of energy, and is highly dependent of temperature of liquid molecules and wall morphology of nanopores. Using MD simulations, the effects of temperature and wall roughness on the transport resistance of water molecules inside nanopores are investigated in Chapter 3. The effective shear stress and nominal viscosity that dominate the nanofluidic transport resistance are extracted and coupled with the nanopore size, transport rate, and liquid property. The molecular-level mechanisms are revealed through the study of the density profile and hydrogen bonding of confined liquid molecules. A parallel experiment on a nanoporous carbon-liquid system is carried out and qualitatively verifies MD findings. Motived by the well-known thermo- and electro-capillary effect, Chapter 4 and Chapter 5 present a conceptual design of thermal and electric actuation system by adjusting the relative hydrophobicity of a liquid-nanoporous system through a thermal and electric field, respectively. The thermally and electrically dependent infiltration behaviors of liquids into nanopores are analyzed by using MD simulations. The fundamental molecular characteristics, including the density profile, contact angle, and surface tension of the confined liquid molecules, are examined to reveal underlying mechanisms. The energy density, power density, and efficiency of both thermal and electric actuation systems are explored and their variations with pore size, solid phase, and liquid phase are evaluated. Thermally and electrically controlled infiltration experiments on a zeolite-water /electrolyte solution system are performed accordingly to qualitatively validate these findings. These energy actuation systems can also become high density thermal or electric storage devices with proper designs. Energy harvesting by the flow of a hydrochloric acid-water solution through a nanopore is explored using atomistic simulations in the last chapter. Through ion configurations near the pore wall, an averaged ion drifting velocity is determined, and the induced voltage along the axial direction is obtained as a function of key material parameters, including the applied flow rate, environmental temperature, solution concentration and nanopore size. The molecular mechanism of ion hopping and motion is revealed. This study shed light on harvesting wasted thermal and mechanical energy from ambient environmental sources such as wasted heat in power plants. Nanofluidics is a novel and thriving research area, whose couplings with other disciplines such as material, mechanical, physical, chemical, electrical engineering are open.Engineering, Mechanics, Materials sciencebx2107Earth and Environmental EngineeringDissertations