Fracture Dynamics of Shallow Dip-Slip Earthquakes

Transcription

1 Fracture Dynamics of Shallow Dip-Slip Earthquakes K. Uenishi a *, T. Takahashi a and K. Fujimoto a a The University of Tokyo, Tokyo, Japan * uenishi@dyn.t.u-tokyo.ac.jp (corresponding author s ) Abstract While the fracture (rupture) dynamics of strike-slip earthquakes has been clarified to a practically acceptable level, the mechanical characteristics of shallow dip-slip seismic events remain unexplained owing to the shortage of the near-field seismological observations and the analytical difficulties at the rupture tip of an interface (fault plane) in the proximity of a free surface. n this contribution, utilizing the techniques of finite difference modeling and dynamic photoelasticity, the fracture dynamics of a dip-slip fault plane located near a free surface is studied numerically as well as experimentally. Each two-dimensional fracture model may contain a flat fault plane (initially welded interface) that dips either vertically or at an angle (e.g. 45 degrees) in a monolithic linear elastic medium (representing rocks). The time-dependent development of wave field associated with the crack-like rupture along every fault plane is recorded. Both numerical and experimental observations indicate when the fault rupture that is initiated at some depth approaches the free surface, four Rayleigh-type waves are produced. Two of them move along the free surface as Rayleigh surface waves into the opposite directions (in the hanging wall or footwall) to the far field, and the other two propagate back downwards along the fractured interface into depth. These downward interface waves may considerably govern the stopping phase of the dynamic fracture, and according to the seismological recordings, they seem to have existed during the rupture process of the 2011 off the Pacific coast of Tohoku, Japan, earthquake. n the case of an inclined fault plane, the interface and Rayleigh waves interact with each other and a shear wave possessing concentrated energy (corner wave) is generated and causes stronger disturbances in the hanging wall. The existence of the downward interface and corner waves, first numerically predicted in 2005 by Uenishi and Madariaga, seems to have been confirmed by this series of laboratory fracture experiments. Keywords: Rock dynamics, Earthquake dynamics, Fracture dynamics, Experimental mechanics, Computational seismology 1. ntroduction Understanding the mechanics related to the generation of earthquakes and seismic waves is one of the important research subjects not only in seismology and earthquake engineering but also in the field of rock dynamics. Since an idea that an earthquake may be produced by fracture (rupture) of a geological plane of weakness (interface; e.g. a fault plane in the Earth s crust) was accepted some fifty years ago, seismological investigations into the source mechanics of earthquakes and rockbursts have advanced rapidly together with the development of rock fracture mechanics (Uenishi, 2014). For example, the reasonable agreement between the theoretical and observational near-field seismological recordings of the 1966 strike-slip Parkfield earthquake in California, the United States (Aki, 1968), has brought remarkably sophisticated techniques to evaluate the near-field seismic disturbances induced by strike-slip faulting. By now it has become very common to inversely estimate the distribution of fault slip (displacement discontinuity), rupture history and their effects for large, shallow strike-slip earthquakes from seismograms (e.g. Olsen et al., 1997; Uenishi et al., 1999). However, the current state is totally different for shallow dip-slip faulting because of (1) the shortage of the near-field seismological recordings of this faulting type and (2) the theoretical difficulties in obtaining the mechanical characteristics, especially near the tip of a surface-breaking fault rupture (Madariaga, 2003). As pointed by Oglesby et al. (1998), seismic hazard risk owing to dip-slip earthquakes may be higher in the tectonically compressive areas such as Japan, Taiwan, Los Angeles in California, Central and South America, as well as in tectonically tensile, extensional regions like the Mediterranean and the Great Basin in the United States. Despite a large effort made to model

2 dip-slip earthquakes and their influence on induced strong motions (particle motions) (e.g. Burridge and Halliday, 1971; Boore and Zoback, 1974; Davis and Knopoff, 1991; Shi et al., 2003), previous study on shallow dip-slip earthquake mainly utilizes kinematical models, and the analysis based on the theoretical Cagniard-de Hoop (Niazi, 1975) or a numerical spectral method (Bouchon and Aki, 1977), for instance, becomes extremely complex and, due to analytical singularities, frequently it does not work rightly when the tip of the dip-slip fault rupture approaches the free surface of the Earth. Thus, the earlier works indicated the necessity for more careful treatment of the problem on the influence of the free surface near the tip of the shallow dipping fault rupture (Madariaga, 2003). One significant observation in shallow dip-slip faulting is the asymmetric ground motions in the proximity of the rupturing fault plane. n general, the strong motion in the hanging wall is much larger than that in the footwall, and for example, the 1971 San Fernando and the 1994 Northridge, California, earthquakes generated more serious structural damage or larger ground motion on the hanging wall in a systematic way (Oglesby et al., 1998). The more recent seismic events, the 1999 Chi-Chi, Taiwan (Kao and Chen, 2000; Oglesby and Day, 2001), the 2004 Niigata-ken Chuetsu (K-NET by the National Research nstitute for Earth Science and Disaster Prevention of Japan, and the 2008 wate-miyagi nland (Aoi et al., 2008), Japan, earthquakes seem to strengthen this point of view. The reason for the observed asymmetric ground motions is investigated by considering the trapped wave in the hanging wall (Oglesby et al., 1998), the large disturbance in the vicinity of the tip of moving rupture (rupture front wave) (Madariaga, 2003) or the nonequal mass distributions on the footwall and the hanging wall (Davis and Knopoff, 1991; Oglesby et al., 1998; 2000; Oglesby and Day, 2001), but the dynamic characteristics of shallow dip-slip faulting near a free surface have not been completely clarified yet. n this paper, in order to explain mechanically the causes for the asymmetry mentioned above, fracture dynamics of a dip-slip fault plane located in a two-dimensional, monolithic linear elastic medium (modeling rocks) is numerically and experimentally investigated. n the numerical simulations, a concise finite difference technique developed on a PC basis is utilized, and experimentally, dynamic interface fracture is initiated in a birefringent linear elastic material (polycarbonate plate) by an impinging spherical projectile. The time-dependent wave development is recorded for the crack-like rupture along the interface (fault plane) in the model. The dynamic fracture experiments are performed so as to check the existence of the downward interface and corner waves that has been first numerically predicted by Uenishi and Madariaga (2005). 2. Dip-Slip Faulting near a Free Surface: Prediction of the Existence of nterface and Corner Waves Based on Numerical Simulations 2.1 Problem statement n the numerical model, a fault plane with a dipping angle of either 90 (vertical) or 45 (inclined case) is prepared (Fig. 1). n both cases, the initial static shear stresses acting on the fault plane are assumed to be equal. For the vertical case (Fig. 1(a)), remote shear loading increases linearly with depth, and in the inclined case (Fig. 1(b)), the compressive normal stress makes the static shear stress on the fault plane increase linearly with depth. Utilizing a finite difference technique developed for a PC (Windows) (see e.g. Rossmanith et al., 1997; Uenishi and Rossmanith, 1998), the seismic wave field (isochromatic fringe patterns; contours of the maximum in-plane shear stress, max ) produced by fault rupture is observed and the free surface effect on dip-slip faulting is studied. As the problem is considered in the linear elastic framework, it may be assumed without losing generality that the longitudinal (P) wave speed V P in the elastic medium is 1. n the case Poisson s ratio is 0.25, the shear (S) wave speed V S is 1/ 3 (~ 0.58) and the Rayleigh (R) wave speed V R is some The orthogonal grid points are employed and displacements at each grid point is calculated at each time step with the second order accuracy. The constant grid spacing is 0.05, and the time step is Further, the energy absorbing boundary conditions are presumed on the outer boundaries except for the upper free surface where the vertical normal and tangential shear stresses are zero at any time. n all numerical simulations, crack-like rupture is assumed. That is, once a fault segment is ruptured, the accumulated static shear stress on that segment is released and that part of the fault plane remains fractured without healing. This crack-like rupture starts at time t = 0 and propagates along the fault plane with an ordinary constant subsonic speed V = 0.4 V P (~ 0.69 V S ) (V < V S < V P ) for a length L = 2 until it surfaces. n generating Fig. 1(b), thrust (reverse) fault slip is assumed, but the results shown are applicable also for normal dip-slip faulting.

3 Free surface Free surface Hanging wall 45 Footwall Vertical section V ~ 0.69 V S V ~ 0.69 V S Normalized time (a) (b) Fig. 1. (Top) Schematic sketch of the geometrical settings of the dip-slip fault models numerically investigated. For both (a) vertical and (b) inclined (45 dipping) cases, a monolithic, linear elastic medium is assumed. The fault rupture (interface crack) is initiated at the bottom at time t = 0 and propagated subsonically upwards to break the free surface. (Bottom) Sequences of snapshots of the dynamic isochromatic fringe patterns ( max stress field). Note that the geometrical symmetry is preserved in (a), but in (b), the symmetry between the free surface and the two sides of the fault plane (hanging wall and footwall) is broken.

4 (a) (b) Fig. 1 (continued). n the symmetric case (a), when subsonic fault rupture reaches the top free boundary, four surface-type waves may be produced: The two Rayleigh waves (labeled as R) travel along the free surface into the far-field and the interface waves () move downwards along the fractured fault surfaces. These interface waves merge to form a shear wave (S) at a later stage, normalized time 8.7. n the asymmetric case (b), a Rayleigh wave R h and the corner wave C may be recognized in the hanging wall. Strong dynamic disturbance and trapped wave energy may be found in the hanging wall behind the corner wave. On the contrary, the induced stresses are relatively small in the footwall. 2.2 Prediction of the existence of interface and corner waves: subsonic fault rupture n Fig. 1, snapshots of the time-dependent isochromatic fringe patterns associated with the subsonic fault rupture that is initiated at depth and approaching the free surface are shown. The fringe order is proportional to max ( 0), and in both vertical (a) and inclined (b) cases, at earlier stages strong rupture front waves develop near the moving tip of the fault rupture. Static stress singularities at the other lower tip of the ruptured fault plane is also recognizable, but with the subsonic (and sub-rayleigh, V < V R ) rupture assumption, Rayleigh-type waves propagating upwards along the ruptured interface cannot be produced (Compare Fig. 1 with Fig. 3 experimentally recorded for a higher rupture speed). When the rupture front reaches the free surface, four Rayleigh-type waves are generated. While two of them move along the free surface into the opposite directions to the far-field (marked as R or R h, R f in the figure), the other interface waves () propagate back downwards into depth along the ruptured interface. When the fault plane dips vertically and is geometrically symmetric (Fig. 1(a)), the downward interface waves may control the stopping phase of the dynamic rupture on the fault plane and have some influence on the magnitude of the earthquake. n the case the fault plane is asymmetrically inclined (Fig. 1(b)), it is immediately noticeable that the levels of the stresses induced in the hanging wall and in the footwall may become completely dissimilar. The downward-moving interface wave and the outward-travelling surface wave (R h ) interact with each other and a specific shear wave, corner wave (C), may be induced in the hanging wall. This corner wave can carry largely concentrated wave energy and may generate strong disturbances inside the hanging wall. n the footwall, however, the weaker surface wave (R f ) dominates the free surface

5 motions and the interaction between the surface wave and the interface wave propagating in the opposite direction () is very small. The P and S waves induced in the footwall upon fault surfacing (P f and S f ) seem relatively strong, but they are still much weaker than the corner wave in the hanging wall. Thus, the abovementioned asymmetric particle motions related to shallow dip-slip earthquakes may be caused. The downward surface-type waves are identifiable also in the numerical simulations of borehole blasting in solids in which the explosive charge is detonated at the bottom to make a detonation front move along an explosive column upwards to the free surface (bottom-to-top blasting). n these blasting simulations, Rayleigh waves may be produced upon detonation and propagated first upwards and later downwards along the explosive column (Rossmanith et al., 1997; Uenishi and Rossmanith, 1998). However, in earthquake source mechanics, the generation of the downward interface wave and the corner wave has not been well recognized so far, partly because the existence of these waves is not expected for an ordinary earthquake model with a fault rupturing only at depth. t is noteworthy that similar fracture pattern (downward rupture after initial upward one) has been reported (de et al., 2011) for the dynamic rupture development associated with the 2011 off the Pacific coast of Tohoku, Japan, earthquake. n the simulation of the inclined fault rupture, the shallowest part (length 0.15, i.e., three times the grid spacing; see Fig. 1(b)) is assumed to be vertical in order to numerically treat the corner effect appropriately in the finite difference framework. This geometry is chosen to avoid the problems of analytical singularities at the corner, but further computations show qualitatively same phenomena (generation of interface and corner waves) even without this short vertical section, i.e., even when the fault rupture arrests just below the free surface at a very shallow depth of Experimental Observation of nterface and Corner Waves 3.1 Setup n order to try to confirm the numerical results described above, dynamic laboratory photoelastic fracture experiments are performed using a birefringent linear elastic material and a projectile launched by a gun. A pre-cut interface is prepared in a polycarbonate plate (Makrolon, 880 mm 200 mm 10 mm). The plate is essentially subjected to no static stresses, and dynamic fracture is initiated by a spherical projectile (diameter 6 mm, mass 0.2 grams) impinging at a speed of 83 m/s upon the bottom intersection of the welded interface with the free surface (see Fig. 2). The fracture propagates along the interface that is (a) vertical or inclined at an angle of (b) 60, (c) 45 or (d) 30 degrees. The development of dynamic wave field for each inclination angle is recorded with a high speed digital video camera system (Photron FASTCAM SA5) at a frame rate of 75,000 fps. The observed area, shown in Fig. 2, is 90 mm 53 mm. According to the dynamic elasticity theory, the physical properties of the polycarbonate plate, the mass density = 1,200 kg/m 3, shear modulus = 786 MPa and Poisson s ratio = 0.38, roughly give the longitudinal wave speed V P = 1,840 m/s, shear wave speed V S = 810 m/s, and Rayleigh wave speed V R = 760 m/s, respectively. 3.2 Observation of supershear fault rupture and discussion Figure 3 shows typical snapshots of experimentally obtained isochromatic fringe patterns exhibiting the dynamic wave field that is induced by the interface rupture near a free surface. The photographs indicate, for all inclination angles, the rupture speed V is in the range of supershear (transonic), i.e. it is larger than the S wave speed V S in the medium but smaller than the P wave speed V P. The measured rupture speed is V ~ 1,600 m/s for dip angle 90 (a), 1,400 m/s for 60 (b) and 45 (c), 1,500 m/s for 30 (d), respectively. The corresponding Mach numbers are M P V/V P ~ (< 1) and M S V/V S ~ (> 1), and the rupture front waves at relative time 0 s form Mach cones, sometimes called shock waves or Mach waves. The rupture speed normally inferred from seismograms is in a subsonic range (as assumed in the above numerical simulations), but supershear fault rupture did exist, to name a few, during the dynamic process of the 1979 mperial Valley and the 1992 Landers earthquakes in California, the 2002 Denali event in Alaska (Uenishi, 2009). The wave patterns related to the initial stage of the interface rupture, represented by the rupture front waves, look different in the supershear fracture experiments (Fig. 3) than in the subsonic numerical simulations (Fig. 1). The reason is, as indicated by Rossmanith et al. (1997) as well as Uenishi and Rossmanith (1998), the initial wave patterns strongly depend on the speed of energy source (e.g. velocity of detonation in blasting, and in earthquake source mechanics, the rupture propagation speed). However, the basic features described in the previous chapter can be identifiable also in the photographs capturing supershear rupture. At an earlier stage, the rupture front wave, now moving

6 Photoelastic device High speed digital video camera nterface (welded) Observed area Specimen Projectile Photoelastic device Fig. 2. The setup and specimens prepared for the dynamic photoelastic fracture experiments. For both vertical and inclined models, the interfaces are welded. faster than shear and Rayleigh (interface) waves, is guided by the moving tip of the interface fracture. Then, the strong interface waves (), totally separated from the rupture front wave, move along the fractured interface. Upon reaching the free surface, the incident interface waves are mainly divided into two Rayleigh waves propagating along the free surface into the far field and the downward interface waves (see the snapshot taken at time s in the vertical interface case (a)). Or, in the inclined cases (photographs placed at the bottom of (b)-(d)), the Rayleigh and the downward interface wave are connected in the hanging wall to form relatively strong corner waves (C). The merging effect seems weaker in the case of larger dipping angle 60 degrees, but in each inclined case (b)-(d), the wave energy carried by the downward interface wave in the footwall looks smaller than that of the corner wave, and actually, in the case of dipping angle 30 (d), almost no dynamic disturbance can be found in the footwall. We believe these are the first photographs that clearly indicate the existence of corner and downward interface waves and the (a)symmetric wave patterns related to dip-slip faulting, which may be consistent with the seismologically observed phenomena. n the numerical simulations shown in Fig. 1, subsonic rupture speed is assumed because, as stated above, this speed level is ordinarily inferred from seismological recordings. For a reference purpose, supershear rupture is simulated for a vertical fault plane (Fig. 4), with the same geometrical and material settings as used for generating Fig. 1. Now, the crack-like rupture, starting at t = 0, propagates along the fault plane with a constant supershear speed V = (V P + V S )/2 ~ 0.79 (M P ~ 0.79, M S ~ 1.37) until it reaches the free surface. The figure shows, like the experimentally obtained snapshots, the Mach cone-type rupture front wave as well as the interface waves more slowly propagating upwards along the fractured interface. These interface waves may reach the free surface and divided into Rayleigh and downward interface waves, and the wave patterns obtained in the experiments may be well reproduced by numerical simulations.

7 Free surface front wave nterface nterface nterface nterface front wave front wave Time 0 s 0 s 0 s 0 s front wave s s s s s s s s s s s s s s s s s s s s R R s s s s C C C Weaker Weaker 10 mm s s s s (a) (b) (c) (d) Fig. 3. Typical experimentally obtained snapshots of isochromatic fringe patterns showing the dynamic wave field induced by supershear (transonic) rupture of an interface located near a free surface. The dip angle for each case is (a) 90, (b) 60, (c) 45 and (d) 30 degrees, respectively.

8 Free surface front wave direction; speed V ~ 1.37 VS Length 1 Fig. 4. A numerically generated snapshot of isochromatic fringe patterns associated with supershear rupture of a vertically dipping fault plane located near a free surface (normalized time at 1.8 after rupture initiation). 4. Conclusions We have investigated the dynamic seismic wave field generated by rupture of a shallow dip-slip fault plane situated near a free surface. Both numerical and experimental results show that fault surfacing may produce Rayleigh-type waves that propagate farther along the free surface or back downwards along the ruptured fault plane. The downward interface waves moving along the ruptured fault plane may have strong influence on the stopping phase of the dynamic fracture process, and they might have governed the seismic rupture related to the 2011 off the Pacific coast of Tohoku, Japan, earthquake. t has also been indicated that, when the inclined dip-slip fault rupture approaches the free surface, the dynamic stresses induced in the hanging wall may become larger than those in the footwall, because the interaction of the downward interface wave with the Rayleigh surface wave can produce a strong corner wave in the hanging wall. The existence of previously unrecognized downward interface and corner waves seems to have been confirmed by the dynamic laboratory photoelastic fracture experiments, and those waves may play an important role in comprehending the influence of the geometrical asymmetry on the dissimilar strong motions induced by shallow dip-slip faulting. Although the models employed here is very much simplified and further cautious numerical and experimental consideration is certainly needed, they may offer fundamental understanding of dynamic fracture process of shallow dip-slip faulting. Acknowledgements The authors would like to sincerely thank Dr. R. E. Knasmillner at TVFA of Vienna University of Technology in Austria for his kind advice regarding the selection of photoelastic materials suitable for dynamic fracture experiments. This study has been financially supported by the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT) through the KAKENH: Grant-in-Aid for Scientific Research (C) Program (No ). References Aki, K., 1968, Seismic displacements near a fault, J. Geophys. Res., 73, Aoi, S., Kunugi, T. and Fujiwara, H., 2008, Trampoline effect in extreme ground motion, Science, 322, Boore, D. M. and Zoback, M. D., 1974, Near-field motions from kinematic models of propagating faults, Bull. Seismol. Soc. Am., 64, Bouchon, M. and Aki, K., 1977, Discrete wave number representation of seismic-source wavefields, Bull. Seismol. Soc. Am., 67, Burridge, R. and Halliday, G., 1971, Dynamic shear cracks with friction as models for shallow focus earthquakes, Geophys. J. R. Astron. Soc., 25,

9 Davis, P. M. and Knopoff, L., 1991, The dipping antiplane crack, Geophys. J. nt., 106, de, S., Baltay, A. and Beroza, G. C., 2011, Shallow dynamic overshoot and energetic deep rupture in the 2011 M w 9.0 Tohoku-oki earthquake, Science, 332, Kao, H. and Chen, W.-P., 2000, The Chi-Chi earthquake sequence: Active, out-of-sequence thrust faulting in Taiwan, Science, 288, Madariaga, R., 2003, Radiation from a finite reverse fault in a half space, Pure Appl. Geophys., 160, Niazi, A., 1975, An exact solution for a finite, two-dimensional moving dislocation in an elastic half space with applications to the San Fernando earthquake of 1971, Bull. Seismol. Soc. Am., 65, Oglesby, D. D., Archuleta, R. J. and Nielsen, S. B., 1998, Earthquakes on dipping faults; the effects of broken symmetry, Science, 280, Oglesby, D. D., Archuleta, R. J. and Nielsen, S. B., 2000, Dynamics of dip-slip faulting; Explorations in two dimensions, J. Geophys. Res., 105, 13,643-13,653. Oglesby, D. D. and Day, S. M., 2001, Fault geometry and the dynamics of the 1999 Chi-Chi (Taiwan) earthquake, Bull. Seismol. Soc. Am., 91, Olsen, K., Madariaga, R. and Archuleta, R., 1997, Three-dimensional dynamic simulation of the 1992 Landers earthquake, Science, 278, Rossmanith, H. P., Uenishi, K. and Kouzniak, N., 1997, Blast wave propagation in rock mass - Part : Monolithic medium, Fragblast, 1, Shi, B., Brune, J. N., Zeng, Y. and Anooshehpoor, A., 2003, Dynamics of earthquake normal faulting: Two-dimensional lattice particle model, Bull. Seismol. Soc. Am., 93, Uenishi, K., 2009, Supershear rupture propagation in a monolithic medium, Proc. Twelfth ntl. Conf. on Fracture, T15.006, 10 pages. Uenishi, K., 2014, "Rock mechanics" and earthquakes - Recent fifty years, Rock Mechanics: Fifty Years of Progress and Views to the Future, (in Japanese). Uenishi, K. and Madariaga, R.., 2005, Surface breaking dip-slip fault: ts dynamics and generation of corner waves, Eos Trans. AGU, 86, Fall Meet. Suppl., Abstract S34A-03. Uenishi, K. and Rossmanith, H. P., 1998, Blast wave propagation in rock mass - Part : Layered media, Fragblast, 2, Uenishi, K., Rossmanith, H. P. and Scheidegger, A. E., 1999, Rayleigh pulse - Dynamic triggering of fault slip, Bull. Seismol. Soc. Am., 89,