Elastic Wave Propagation and Generation in Seismology

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Generally, the fault is assumed to be planar, and it is presumed that rupture on the fault plane plane of weakness or interface is controlled by friction. A typical constitutive relation that holds inside the rupture region is the slip-weakening friction law where the stress inside the rupture region drops as slip displacement discontinuity develops on the fault plane. Another often employed friction law assumes an established fault surface and is described by a laboratory-derived rate- and state-dependent friction law typically with slowness or ageing state evolution, 12 and analytically as well as numerically, models based on slip-weakening or rate- and state-dependent constitutive relations have been proposed to study the earthquake initiation process, and in particular, the transition from stable quasi-static enlargement of the rupture region to unstable fast propagation of rupture.

Hence, instead of handling the quasi-static state, the initiation of dynamic anti-plane slip instabilities of a slip-weakening fault has been investigated by means of a spectral analysis for a homogeneous linear elastic medium that is preloaded uniformly up to the frictional threshold, and the effect of slip-weakening on the duration of the nucleation phase and the critical fault length has been evaluated. For a more quantitative description of the difficulty in incorporating mechanical destabilization and initiation of dynamic stage of fault rupture in dynamics-based seismological investigations, therefore, the factors that govern the nucleation of slip-weakening instabilities have been identified for a two-dimensional planar fault Fig.

The size of the rupture region on the fault in this linear elastic model stably grows quasi-statically under increasing load until it reaches a critical nucleation length at which no further solution for quasi-static elastic equilibrium exists.

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This corresponds to the start of a dynamically controlled instability. For the case of a widely used linear slip-weakening law Fig. It is dependent on an elastic parameter of the medium and the slip-weakening rate W only, and it is not affected by the strength of the fault plane, rate of increase of loading stress, or by the exact shape of the loading stress distribution along the fault.

The nucleation length is given by Similar expressions can be found also for the three-dimensional case. Contrary to our intuition, earthquakes owing to linear slip-weakening see b instabilities may spontaneously occur regardless of the precise distribution of the loading stress acting on the fault. Here, a displacement field is depicted for anti-plane shear mode III rupture in an infinite, homogeneous linear elastic medium. Similarly, the problem for in-plane shear mode II or tensile mode I rupture can be defined.

In the formula for the universal nucleation length, Eq. This size seems irreconcilable with presence of small-magnitude earthquakes.

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The result implies that the slip-weakening process may occur at slips much larger than those at initiation and the model with a single slip-weakening process may be insufficient. For example, it can be assumed that a laboratory-like weakening at small slips with a larger value of W may govern nucleation, and then further weakening with smaller W e. However, even when the linear slip-weakening law may be assumed, there are reasons why such often inferred large values of D c must be cautiously handled. First, from the observational viewpoint, it has been shown that assessments of D c inferred from kinematic seismic inversions tend to be biased towards large due to the influence of spatiotemporal smoothing constraints.

It has been also indicated that D c is not uniquely determined by seismic inversions, 34 and so as to constrain D c , ground motion spectra at frequencies higher than those currently normally possible must be modeled. Second, from the forward modeling viewpoint, D c is likely to be large owing to the technical problem in simulating earthquakes.

Elastic Wave Propagation and Generation in Seismology eBook: Jose Pujol: esanspelinbe.gq: Kindle Store

A fault region, at least tens to hundreds of kilometers, must be simulated while slip as well as rapid stress change at the tips of propagating ruptures is being rightly resolved. Thus larger values of D c are likely to be used in regular numerical simulations. Once a rupture region loses its mechanical stability, its size extends dynamically. Laboratory experimental observations indicate that when rupture extends in brittle solid materials under suitable stress conditions and its propagation speed exceeds a certain value, it oscillates rupture surface roughening and afterwards divides into two or more branches.

For tensile rupture in brittle amorphous solids e. The rupture surface roughens severely at higher speeds and the rupture bifurcates at the highest speeds. Classical fracture mechanics assumes that rupture extends in a monolithic, homogeneous linear elastic medium without any preexistence of other defects. If a precut interface like a geological fault plane exists in the medium and rupture runs along that interface, the upper limit of the rupture propagation speed may change.

However, due to the fundamental difficulties in dynamic interface fracture mechanics, only a few theoretical studies have been conducted, and there is a deep disagreement even concerning the theoretically predicted propagation speed of interface rupture, e. According to numerical simulations related to slip pulses, along a statically preloaded interface between similar materials waves involving separation of interface surface decay quickly rather than move in a self-sustaining manner, 50 but slip pulses can be sustained if interface surface roughness is present 51 and slip pulses propagating in a self-sustaining manner may break up into a number of smaller pulses.

In its leading right part, partial wave energy transmission occurs from the upper plate 1 across the interface into the lower plate 2, but in the trailing left part, owing to the separational vertical particle movements from the interface in the plate 1, the corresponding fringes in the plate 2 are missing, indicating the existence of dynamic interface slip fault rupture. The isochromatic fringe pattern in Fig.

In the beginning, the slip pulse is running at the Rayleigh wave speed of the acoustically stiffer material 1, V R 1 , which lies between the S and longitudinal P wave speeds of the lower material 2, V S 2 and V P 2 , i. This relatively high rupture propagation speed seems consistent with seismological observations, suggesting that in the linear elastic framework, there is an essential difference between rupture along an interface of a bimaterial system and that in a homogeneous monolithic medium.

The most severely affected areas are marked in red. The rupture front moving from left to right in this nonlinear elastic material captures the shear wave front indicated by red broken lines, clearly indicating that dynamic rupture can accelerate to a supershear speed even without existence of precut interfaces.

Elapsed time after rupture initiation by pricking the sheet is 0, If a slip pulse can propagate at depth in the Earth where very high compressive stress is expected to be applied, the existence of such a pulse, together with the universal nucleation length mentioned above, may be a solution to the earthquake-related paradoxes, e.

Damage due to liquefaction was hardly observed inside this severely affected area, and therefore it is suggested that the damage was caused directly by the seismic waves. Hence the peak particle velocity PPV has been selected as the design parameters in engineering applications. For the dynamic rupture model in a bimaterial system in Fig.

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The model material 1 fits the acoustically stiffer area in the foothills of the Rokko Mountains where soils are very shallow or rock outcroppings prevail and the damage by subsonic slip pulse propagation tended to be relatively minor. The material 2 corresponds to the acoustically more compliant region in which primarily soft alluvial soils are found and the slip pulse, moved at a supershear speed locally, induced a Mach wave and the damage belt.

Exceptionally, as stated above, several different types of laboratory experiments of dynamic rupture on precut interfaces have indicated the existence of dynamic rupture moving at the Rayleigh wave speed or higher, but the inconsistencies of rupture speeds between theories, seismological observations and experiments and numerical simulations cannot always be attributed to the microscopic observations of materials that real solids have all kinds of defects such as preexisting discontinuities or microcracks generated during rupture propagation, because similar discrepancies may also emerge in molecular dynamics simulations of rupture running in perfect atomic lattices.

Large-scale atomistic simulations 56 have suggested that nonlinear elasticity of large strains, nonlinear hyperelastic stress-strain relation in solids, may govern the dynamics of brittle fracture and it may, instead of widely assumed linear elasticity, resolve the contradictions.

Real solids like the crust, indeed, have elastic characteristics that are notably dissimilar at various scales, for both small and large deformations. While quasi-static loading may give crustal rocks nonlinear strain-softening effect stress reduction after a peak value with increasing strain , hyperelastic rubber-like strain-hardening behavior of rocks nonlinear increase of stress with strain may prevail under dynamic loading, e. If the static normal uniform stress crack-parallel or T -stress acting parallel to the rupture propagation direction x -direction in Fig.

Mod-03 Lec-15 L15-3 Dimensional Wave Propagation, Waves in semi-infinite media, Rayleigh Wave

So far, fundamental mechanisms of earthquake nucleation, rupture propagation and wave radiation have been addressed. As described above, also the Hyogo-ken Nanbu earthquake may have generated high-frequency seismic waves. Unfortunately, however, the sensitivity of the SMAC-MDU strong-motion accelerometers accelerographs commonly deployed at that time dropped drastically above 20 Hz usually for technical reasons and waves over 10 Hz were filtered out, and therefore, it was difficult for them to detect higher frequencies.

Although failures of underground facilities owing to an earthquake hardly occur, in the seismic event, the Bantaki Tunnel in the Rokko Mountains has suffered unique structural failure Fig. The failure was found in the central section of the tunnel that had been completed about four years before the quake utilizing the New Austrian Tunneling Method NATM. On the sidewall, dynamic compression seemed to have caused exfoliation of lining concrete and buckling of reinforcing steel bars. In addition, the subgrade detached some 10 cm from the invert, but the invert itself was not deformed.

No damage existed at the crown on the ceiling , either, and no distinctive sign of permanent ground deformation was observed at the failure site. Failure of the columns supporting the roof at midspan central columns gave rise to the vertical collapse of the roof right. A simple dynamic analysis suggests that it is more straightforward to regard also this failure as a phenomenon caused by vertical vibrations in a higher frequency range, over 10 Hz.

For the essential comprehension of wave diffraction and scattering by structural inhomogeneities such as cavities tunnels and inclusions, the time-harmonic analytical elastodynamic approach based on the wave function expansion method has been repeatedly adopted. Typical problems are scattering of an incident plane harmonic P wave by a circular cavity, 62 an arbitrarily thick elastic cylinder 63 and a group of cylinders in parallel 64 in an infinitely extending elastic medium.

In the same way, dynamic response of a cylindrical cavity or rigid inclusion caused by incident harmonic in-plane shear SV wave 65 has been analyzed, and together with the image technique for the reflection of the waves at the ground surface, the incidence of an anti-plane shear SH wave on a single 66 or twin 67 circular tunnel s has been considered. These analyses do not seem to have been applied to real dynamic problems of failures of underground facilities, but, if a similar two-dimensional time-harmonic elastodynamic approach is utilized and diffraction of plane harmonic P vertical oscillations and SV horizontal vibrations waves by a simplified, uniformly lined circular tunnel situated in an infinitely extending linear elastic medium is analytically evaluated, it can be systematically shown that only P waves in a relatively high frequency range may be able to induce dynamic compression on the sidewall, impart large accelerations to the bottom of the circular tunnel, and as a result produce the failure pattern like that observed in the Bantaki Tunnel Fig.

Similarly, it can be demonstrated that in the Bantaki Tunnel, incident P waves consisting of higher-frequency components were diffracted at the bottom and there the particle acceleration was amplified about twice to result in the detachment of the more accelerated subgrade. This is not the case for low-frequency P waves that give more or less the same magnitude of particle accelerations at the bottom and on the ceiling of a lined tunnel. The failure in the Bantaki Tunnel, together with another underground failure pattern found in Kobe in , the collapse of the reinforced concrete columns at the Daikai underground station Fig.

Seaquakes are strong vertical shocks felt on board a floating body e. Unlike tsunamis, however, seaquakes are usually experienced only near the epicenter of an earthquake. In the Hyogo-ken Nanbu case, no less than four ferry boats, whose positions depicted in Fig. According to the ultrasonic wave height meters near the Port of Kobe and the Osaka-Kansai International Airport, tsunamis were only some 5 cm high and they were without any two clear peaks. A liquid layer seawater of finite thickness exists on top of the semi-infinitely extending solid seabed.

It is significantly dependent on the frequency of the incident P wave in the seabed. Book Description Cambridge University Press, Condition: New. New Book. Shipped from UK. Established seller since Seller Inventory FM More information about this seller Contact this seller. Language: English.

  1. Wave propagation in elastic solids.
  2. Elastic Wave Propagation and Generation in Seismology by Jose Pujol.
  3. Europes Languages on Englands Stages, 1590–1620 (Studies in Performance and Early Modern Drama);
  4. Brand new Book. Seismology has complementary observational and theoretical components, and a thorough understanding of the observations requires a sound theoretical background. This book bridges the gap between introductory textbooks and advanced monographs by providing the necessary mathematical tools and demonstrating how to apply them. Each seismological problem is carefully formulated and its solution is derived in a step-by-step approach.

    The text includes student exercises with hints , for which solutions are available on a dedicated website. This website also contains numerous downloadable programs for the computation of reflection and transmission coefficients, for the generation of P and S wave radiation patterns and synthetic seismograms, in infinite media. This book will therefore find a receptive audience among advanced undergraduate and graduate students interested in developing a solid mathematical background to tackle more advanced topics in seismology.

    Seller Inventory AAA Book Description Cambridge University Press , Book Description Cambridge University Press. New copy - Usually dispatched within 2 working days. Seller Inventory B Book Description Cambridge Univ Pr, By using our website you agree to our use of cookies. Dispatched from the UK in 1 business day When will my order arrive? Heiner Igel. Michael Dentith. Sean Carroll. Brian Clegg. Mark D. Carlo Rovelli.

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