- Email: [email protected]
Optical methods in fault dynamics
Optics and Lasers in Engineering 40 (2003) 325–339
Optical methods in fault dynamics K. Uenishia,*,1, H.P. Rossmanithb a
Department of Earth and Planetary Sciences, Division of Engineering and Applied Sciences, Harvard University, 29 Oxford Street, Cambridge, MA 02138, USA b Institute of Mechanics, Vienna University of Technology, Wiedner Hauptstr. 8-10/325, A-1040 Vienna, Austria
Abstract The Rayleigh pulse interaction with a pre-stressed, partially contacting interface between similar and dissimilar materials is investigated experimentally as well as numerically. This study is intended to obtain an improved understanding of the interface (fault) dynamics during the earthquake rupture process. Using dynamic photoelasticity in conjunction with high-speed cinematography, snapshots of time-dependent isochromatic fringe patterns associated with Rayleigh pulse–interface interaction are experimentally recorded. It is shown that interface slip (instability) can be triggered dynamically by a pulse which propagates along the interface at the Rayleigh wave speed. For the numerical investigation, the finite difference wave simulator SWIFD is used for solving the problem under different combinations of contacting materials. The effect of acoustic impedance ratio of the two contacting materials on the wave patterns is discussed. The results indicate that upon interface rupture, Mach (head) waves, which carry a relatively large amount of energy in a concentrated form, can be generated and propagated from the interface contact region (asperity) into the acoustically softer material. Such Mach waves can cause severe damage onto a particular region inside an adjacent acoustically softer area. This type of damage concentration might be a possible reason for the generation of the ‘‘damage belt’’ in Kobe, Japan, on the occasion of the 1995 Hyogo-ken Nanbu (Kobe) Earthquake. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Acoustic impedance ratio; Contact mechanics; Dynamic photoelasticity; Earthquake dynamics; Hyogo-ken Nanbu (Kobe) Earthquake; Interface instability; Interface slip; Rayleigh wave
*Corresponding author. Fax: +1-617-495-9837. E-mail address: [email protected] (K. Uenishi). 1 On leave from Kobe University, Japan. 0143-8166/03/$ - see front matterr 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 0 2 ) 0 0 0 9 6 - 9
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1. Introduction Rayleigh and interface waves are considered to play a crucial role in slip, damage and failure of interfaces between two dissimilar or even similar materials, and it is reported that these waves can induce separation (delamination) of the interface surfaces [1–4]. The most familiar interface waves involving separation of interface surface are Schallamach waves that occur when two media of large differences in rigidity slide past one another [5]. Comninou and Dundurs [6–8] showed mathematically that a dynamic wave involving separation can stably propagate along an interface when two elastic media are compressed and simultaneously sheared. However, the validity of their mathematical solution was questioned by Freund [9] from an energy point of view, and the solution has mostly been ignored [10]. From the results obtained by frictional tests in the laboratory, Brune et al. [10–12] propose that dynamic rupture on a geological fault can be triggered by interface waves with separational sections propagating along the fault, or, in other words, by ripples which propagate along the fault, and slip occurs while the compressive stress is reduced [13]. It is suggested that the excitation of Rayleigh waves on a rupture surface can lead to pulses of separation [14]. Numerically, Day [15] indicated that, along a statically pre-stressed interface between similar materials, pulses of separation decay rapidly rather than propagate in a self-sustaining manner. However, Mora and Place [13] showed, again numerically, that such pulses can be sustained if interface surface roughness is present. Andrews and Ben-Zion [16] conducted 2D numerical simulations of dynamic rupture along a planar material interface governed by simple friction, showing self-sustaining propagation of slip pulse and spontaneous break-up of the propagating pulse into a number of smaller pulses. However, the mechanism of this separational pulse-induced dynamic interface instability has not been confirmed in a conclusive manner. In this study, the fundamental mechanisms of dynamic interface instability caused by a Rayleigh (R-) pulse will be investigated [17,18]. The main objectives will be: 1. To visualize and clarify qualitatively the complicated dynamic pulse-induced interface instability phenomena; 2. To evaluate quantitatively the pulse interaction process; and 3. To apply the model results to real earthquake problems. First, the basic characteristics of an R-pulse will be summarized in terms of stress field and particle motion. Second, some of the results of a series of 2D laboratory model experiments [17] utilizing dynamic photoelasticity will be presented. Experimentally obtained snapshots of isochromatic fringe patterns will show that an R-pulse can trigger instability of a partially contacting interface between similar materials. Third, the problem will be numerically analyzed using the finite difference wave simulator SWIFD [19,20] and the dynamic interaction process will be assessed quantitatively. The effect of acoustic impedance mismatch of the two contacting materials on the dynamic wave patterns will be described. Finally, the results obtained by the model investigation will be applied to explain the possible reason for the damage concentration caused by the 1995 Hyogo-ken Nanbu (Kobe), Japan, Earthquake.
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2. Basic characteristics of a Rayleigh pulse In a linear, isotropic homogeneous elastic material, an R-pulse propagates in a non-dispersive manner, with low geometrical damping at a smaller speed than the relevant shear (S-) wave and carries its energy mostly in a shallow layer adjacent to the surface [21,22]. The stress and displacement field associated with a plane R-pulse of arbitrary shape can be analytically expressed using one complex potential [17,23].
Fig. 1. A typical Rayleigh pulse that propagates from left to right along a free surface: (a) Isochromatic fringe pattern (contours of maximum in-plane shear stress) and (b) Particle movements at different depths.
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If the R-pulse is produced by a concentrated line load on the boundary of a halfspace, the complex potential can be determined explicitly from the known result that the surface normal stress distribution due to the R-pulse is proportional to the rate of loading [24]. Knowing the stress field from the complex potential, one can draw isochromatic fringe pattern pertaining to an R-pulse. A typical, theoretically predicted isochromatic fringe pattern is shown in Fig. 1(a) where the R-pulse, generated by a concentrated line load as a result of a detonating line charge, propagates from left to right along the free surface of the half-space. As indicated by the high fringe density in Fig. 1(a), the disturbance associated with an R-pulse is largely confined to a thin layer adjacent to the free surface. The particle movement induced by the same R-pulse is shown in Fig. 1(b) where the push and pull vertical (normal to the free surface) movement on the free surface, which becomes important in contact problems, is recognized. At depth, the horizontal particle movement is relatively small and the vertical movement is dominant.
3. Laboratory experiment 3.1. Setup Dynamic photoelasticity in conjunction with high-speed cinematography is utilized to investigate the interaction between an R-pulse and a statically prestressed, non-welded partially contacting interface. The experimental model is depicted schematically in Fig. 2. The model consists of two plates of Araldite B, which are initially in contact. The dimensions of the plates are chosen so as to prevent (unwanted) reflected waves from impinging upon the contact region and changing the results. The upper surface of the plate 2 (lower plate) is given a blunt double wedge cut so that only the central section of the surface would initially be in contact with the upper plate (plate 1). The wedge-type gap introduces contact singularities that are used for contact trace purposes. They have very little physical bearing on the phenomenon to be investigated. The two plates are statically preloaded in compression. The contact region is not glued (non-welded) and during the interaction process the contact can be weakened or strengthened depending on the relative position of the R-pulse with respect to the region of contact. A typical stress wave pattern is generated by detonating a small amount of explosive (240 mg of PbN6) on the lower surface of plate 1 at a distance 300 mm from the center of the contact region. The length of the generated R-pulse is 75 mm. For scale reasons, a grid with 25 mm spacing was drawn on the side of plate 1 facing the camera. A Cranz–Schardin type multiple spark gap camera is used to record the isochromatic fringe patterns that are produced by circularly polarized monochromatic light. 3.2. Results Fig. 3 shows a sequence of three experimentally recorded isochromatic fringe patterns of the dynamic pulse interaction process. The R-pulse propagates from left
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Fig. 2. The model setup for the Rayleigh pulse interaction experiment (all lengths in mm).
to right in the upper plate 1 and interacts with the region of contact. The times given in Fig. 3 correspond to the times elapsed from the moment of maximum stress amplification at the lhs edge of the contact region. Fig. 3(a) pertains to the instance where the incident R-pulse impinges upon the lhs edge of the contact region, showing that the strength of stress singularity indicated by the fringe order about the lhs edge is larger than that about the rhs edge where the dynamic effect is still negligible at this moment. This stress amplification about the lhs edge is due to the particle motion associated with the incident R-pulse [Fig. 1(b)]: its leading part induces a back- and downward movement of the particles [Fig. 3(a)]. The surface particles that are already statically in contact move together towards the lower plate 2 and thus amplify the stresses about the lhs edge. While the leading part of the retrograde particle motions in the incident R-pulse causes the stress increase about the lhs edge, the trailing part of the R-pulse induces back- and upwardly oriented movement [Fig. 1(b)], which may lead to stress reduction with possible cancellation if the surfaces separate, because the particles are now receding and opening the interface. In Fig. 3(b) the dynamic pulse interaction pattern changes considerably. The fringe pattern about the lhs edge shows little stress amplification. In a strict sense, an R-pulse does not exist in the contact region and there must be other kinds of generalized interface pulses that carry the energy across and along the region of contact. As the incident (generalized) R-pulse approaches the rhs edge of the contact region, as seen in Fig. 3(c), partial wave energy transmission can be found across the interface into the lower plate 2. Due to the separational movements of the surface particles in the trailing part of the R-pulse, there are no corresponding fringes in the lower plate 2 (interface separation). Fig. 3(c) thus indicates that
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Fig. 3. Sequence of snapshots of isochromatic fringe patterns obtained by the experiment (a) t=0 ms, (b) t=13 ms, (c) t=53 ms.
interface separation (instability) can be induced dynamically by an R-pulse even when the interface is statically pre-stressed. Fig. 4 summarizes the experimental results: the retrograde motion, especially the down- and upward movement of the
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Fig. 3 (continued).
Fig. 4. Summary of the dynamic Rayleigh (R-) pulse–interface interaction.
particles associated with the incident R-pulse, plays a crucial role in the pulseinduced interface separation.
4. Numerical simulations The same dynamic R-pulse interaction problem is numerically analyzed using the finite difference wave simulator SWIFD [19,20]. The contact problem is considered under plane stress conditions, and the R-pulse is assumed to interact with a contact region that is characterized by a Mohr–Coulomb friction criterion with the
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coefficient of friction equal to 0.3. The interface is pre-stressed and the tensile strength of the interface is set at a very low level. 4.1. Dynamic pulse interaction process Fig. 5(a) shows the time-dependent distribution of the normal stress acting perpendicular to the contact region during the dynamic interaction process. In the figure, the coordinate axes represent the position along the initially contacting region, the time and the normal vertical stress. This Lagrangian-type graphical representation has been chosen for improving the clarity and information about the dynamic interaction process. The 0 and 54.9 mm positions in Fig. 5(a) correspond to the lhs (incidence) and rhs (exit) edges of the contact region. The R-pulse, identified by the hump in the 3D surface, impinges on the contact region on the lower lhs of the diagram and travels to the upper rhs. The maximum stress amplification at the lhs of the contact region at time t ¼ 0 is clearly indicated. The decay of the amplitude of the R-pulse in Fig. 5(a) indicates that initially the tensile part of the incident R-pulse is strong enough to open the interface (tensile cut-off), but upon propagation along the interface, the pulse amplitude attenuates and the interface opening is no longer possible at a later stage. Fig. 5(b) shows the slip distribution along the initially contacting region. In the figure the axes correspond to the position along the initially contacting region, the
Fig. 5. The pulse interaction process from a quantitative point of view: (a) The normal stress acting perpendicularly to the contact region and (b) Development of slip along the region of contact.
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time elapsed with respect to the instance of maximum stress amplification at the lhs contact edge, and slip. Slip is defined by the difference of horizontal (tangential) displacements on the upper (1) and lower (2) surfaces of the interface, u1 u2 : It is shown that slip is initiated on the contact region on the lower lhs of the diagram and travels to the upper rhs at the speed of the R-pulse. This suggests that the initiation and propagation of slip is controlled by the incident R-pulse. 4.2. Partition of the pulse energy It is helpful to assess the relative amount of energy transmitted across and reflected at the contact region during the dynamic interaction process. In this study, R-pulse energy is calculated by Fourier decomposition and considering each decomposed harmonic wave individually [17]. Results are schematically depicted in Fig. 6. It is shown that over 50% of the energy initially carried in the incident R-pulse has been radiated into the far-field in the form of bulk waves and only 37% of the total energy is transmitted along the free surface in the form of a new R-pulse, Rt1. The energy contained in the reflected R-pulses is negligibly small compared with that carried by the transmitted R-pulses. 4.3. Effect of the acoustic impedance ratio The acoustic impedance ratio of the two contacting materials plays a crucial role in wave transmission and reflection at the interface (see e.g. [25]). Therefore, it is also important to investigate the influence of acoustic impedance ratio ðrcP Þ2 =ðrcP Þ1 on the pulse interaction process. Here, r is the mass density, cP is the longitudinal (P-) wave speed, and the subscripts 1 and 2 pertain to the upper and lower material, respectively. Fig. 7 shows numerically generated snapshots of isochromatic fringe
Fig. 6. Partition of the energy initially carried by the incident R-pulse.
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Fig. 7. Snapshots of isochromatic fringe patterns under two different contacting material combinations: (a) ðcS Þ2 oðcR Þ1 oðcP Þ2 and (b) ðcS Þ2 oðcP Þ2 oðcR Þ1 :
patterns taken at the same timing, 60 ms after the maximum stress amplification at the lhs edge of the contact region. Fig. 7(a) corresponds to the case where the incident R-pulse speed is between the shear (S-) and P-wave speeds of the acoustically softer, lower material 2: ðcS Þ2 oðcR Þ1 oðcP Þ2 : Since the slip pulse propagates at the Rayleigh wave speed of the acoustically harder material 1, ðcR Þ1 ; the pulse energy is transferred along the contact region at a transonic speed with respect to the acoustically more compliant material 2. A shear-type (S-) Mach (head) wave is then generated and propagated from the contact region into material 2. If the lower material 2 is sufficiently compliant, ðcS Þ2 oðcP Þ2 oðcR Þ1 [Fig. 7(b)], the energy is transmitted across the contact region supersonically into this layer, and both, longitudinal- (P-) and shear-type (S-) Mach waves are generated. The disturbance in the lower material 2 is largely confined to the region behind the P-Mach wave front. Both cases indicated in Fig. 7 suggest that an interface rupture pulse can propagate at the speed of Rayleigh wave of the acoustically stiffer material 1.
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5. Applications of the model 5.1. The model from a seismological point of view Seismologically, the model and the results obtained in the previous chapters can be interpreted as follows: In reality, the crustal rock nearby geological faults is fragmented by secondary faults over a wide range of scales. In modeling earthquakes, one consequence of this complexity is the introduction of the concept of asperities. Asperities, defined by ‘‘regions of very high slip’’, are considered to be of importance in earthquake hazard analysis because the fracture of asperities results in radiating most of the highfrequency seismic energy into the far-field. The geometrical explanation for asperities reflects the fact, that faults are not perfectly planar, but are rough on all scales and contain jogs and/or steps [26,27]. This study indicates that a relatively large amount of energy is radiated in the form of bulk waves from a contact region into the far-field. Interface slip can be found only inside the initially contacting region. These observations suggest that, when scaled up, the contact region in the model can be considered to represent an asperity on a geological fault. Therefore, the energy partition patterns obtained from the model study are of practical importance in the evaluation of the influence of asperities. 5.2. The ‘‘damage belt’’ associated with the 1995 Hyogo-ken Nanbu (Kobe) Earthquake in Japan On 17 January 1995, at 5:46 a.m. local time, an earthquake of moment magnitude 6.9 struck the region of Kobe and Osaka (Hanshin region) in the west-central part of the Japanese mainland. Seismic inversion [28,29] indicates that the rupture was initiated at a shallow depth on a fault system, which runs through the city of Kobe, and propagated bilaterally: along the Suma/Suwayama faults toward the city of Kobe and along the Nojima fault [Fig. 8(a)]. Strong ground shaking motions lasted for some 20 s and caused considerable damage within a 100 km radius from the epicenter, but Kobe and its neighboring region were most severely affected [30]. One of the puzzling phenomena observed in Kobe is the emergence of the strip of the most severely damaged zone, ‘‘damage belt’’ [the Japan Meteorological Agency Intensity 7 zone indicated in dark gray in Fig. 8(a)]. This strip, about 20 km long, is found close, but not parallel, to the Suma/Suwayama faults. Damage due to liquefaction was scarcely observed inside this strip, and it is suggested that the damage was caused directly due to the seismic waves [31,32]. A 2D model including a plane normal to the fault plane is regarded as appropriate for an approximate first-order analysis of the rupture mechanisms of the Hyogo-ken Nanbu Earthquake, because the Suma/Suwayama faults dip steeply, nearly 901, and a near-source, SH directivity pulse from strike-slip faulting was recorded in the city of Kobe [33,34]. Beneath the lhs edge [in Fig. 8(a)] of the strip, a region of relatively large slip (asperity) was found, and the arrival of a concentrated shear disturbance as well as
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Fig. 8. The 1995 Hyogo-ken Nanbu (Kobe), Japan, Earthquake and the numerical simulation for the case ðcS Þ2 oðcR Þ1 oðcP Þ2 : (a) The most severely damaged (the Japan Meteorological Agency Intensity 7) zone (dark gray region) associated with the Kobe Earthquake; Contours of (b) high PPV (peak particle velocity) and (c) high PPA (peak particle acceleration) obtained by the numerical study.
the resulting large (particle) velocity were recorded in central Kobe [28,29]. This indicates that the rupture-induced shear wave was of a Mach (head) wave type. As discussed above, an asperity on a geological fault corresponds to a contact region in the model. In Fig. 7(a), where an interface is located between dissimilar materials ½ðcS Þ2 oðcR Þ1 oðcP Þ2 ; an S-Mach wave is observed in the numerical simulation. In the model, material 1 fits the acoustically harder region in the foothills of the Rokko Mountains, where soils are very shallow or rock outcroppings are found and the damage tended to be relatively minor. Material 2 corresponds to the acoustically softer region (where soft alluvial soils primarily prevail) which includes the ‘‘damage belt’’. During dynamic interaction, each particle in the materials experiences a history of velocity and acceleration. The maximum values of these quantities, the peak particle velocity (PPV) and the peak particle acceleration (PPA), are very important and often used as practical design parameters in many applications, such as blasting in mines and quarries, and in engineering seismology. Fig. 8 shows the contours of high PPV [Fig. 8(b)] and PPA [Fig. 8(c)] obtained by the numerical simulation ½ðcS Þ2 oðcR Þ1 oðcP Þ2 : It is interesting to note, that the high PPV (or PPA) region is found in a narrow band, similar to the shape of the ‘‘damage belt’’ in Kobe [Fig. 8(a)]. The angle between the interface (fault) and the high PPV (PPA) region in
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this example is controlled by the propagation direction of the S-Mach wave generated during the interaction process [Fig. 7(a)]. This result indicates that a simple 2D model may be able to offer the information about the seismic rupture and the ensuing dynamic wave phenomena, although a more sophisticated study would have to be based on a 3D analysis, that includes local geological as well as topographical effect on the real fault rupture process.
6. Conclusions The purpose of this study was to offer an improved understanding of Rayleigh (R-) pulse interaction with a partially contacting interface between similar and dissimilar materials. The experiment has clearly shown that the interface separation (instability) can be dynamically induced by an R-pulse. The observed dynamic stress amplification and reduction at the interface are caused by the push and pull normal particle motion associated with an R-pulse. Numerically, the finite difference simulator SWIFD has provided quantitative information about the pulse–interface interaction process. It has been shown that a slip pulse can propagate at the Rayleigh wave speed of the acoustically harder material and that pulse interaction patterns are strongly controlled by the acoustic impedance mismatch of the contacting materials. If a contact region is located between dissimilar materials, Mach (head) waves can be generated and propagated. As a practical example, the mechanism of damage concentration associated with the January 17, 1995 Hyogo-ken Nanbu (Kobe), Japan, Earthquake has been studied. It has been suggested that the rupture-induced shear wave was a Mach (head) wave that propagated from a rupturing asperity (interface contact region) situated between dissimilar materials, and this wave could concentrate wave-induced high PPV (peak particle velocity) and PPA (peak particle acceleration) onto a narrow region such as observed in the city of Kobe. Due to a vast variety of application areas of Rayleigh pulses, the fundamental phenomena observed in this study may be of practical importance in various other fields of engineering and applied sciences.
Acknowledgements This work was financially sponsored by the Austrian National Science Foundation (FWF) through Research Project No. P10326-GEO.
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Optical methods in fault dynamics K. Uenishia,*,1, H.P. Rossmanithb a
Department of Earth and Planetary Sciences, Division of Engineering and Applied Sciences, Harvard University, 29 Oxford Street, Cambridge, MA 02138, USA b Institute of Mechanics, Vienna University of Technology, Wiedner Hauptstr. 8-10/325, A-1040 Vienna, Austria
Abstract The Rayleigh pulse interaction with a pre-stressed, partially contacting interface between similar and dissimilar materials is investigated experimentally as well as numerically. This study is intended to obtain an improved understanding of the interface (fault) dynamics during the earthquake rupture process. Using dynamic photoelasticity in conjunction with high-speed cinematography, snapshots of time-dependent isochromatic fringe patterns associated with Rayleigh pulse–interface interaction are experimentally recorded. It is shown that interface slip (instability) can be triggered dynamically by a pulse which propagates along the interface at the Rayleigh wave speed. For the numerical investigation, the finite difference wave simulator SWIFD is used for solving the problem under different combinations of contacting materials. The effect of acoustic impedance ratio of the two contacting materials on the wave patterns is discussed. The results indicate that upon interface rupture, Mach (head) waves, which carry a relatively large amount of energy in a concentrated form, can be generated and propagated from the interface contact region (asperity) into the acoustically softer material. Such Mach waves can cause severe damage onto a particular region inside an adjacent acoustically softer area. This type of damage concentration might be a possible reason for the generation of the ‘‘damage belt’’ in Kobe, Japan, on the occasion of the 1995 Hyogo-ken Nanbu (Kobe) Earthquake. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Acoustic impedance ratio; Contact mechanics; Dynamic photoelasticity; Earthquake dynamics; Hyogo-ken Nanbu (Kobe) Earthquake; Interface instability; Interface slip; Rayleigh wave
*Corresponding author. Fax: +1-617-495-9837. E-mail address: [email protected] (K. Uenishi). 1 On leave from Kobe University, Japan. 0143-8166/03/$ - see front matterr 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 0 2 ) 0 0 0 9 6 - 9
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1. Introduction Rayleigh and interface waves are considered to play a crucial role in slip, damage and failure of interfaces between two dissimilar or even similar materials, and it is reported that these waves can induce separation (delamination) of the interface surfaces [1–4]. The most familiar interface waves involving separation of interface surface are Schallamach waves that occur when two media of large differences in rigidity slide past one another [5]. Comninou and Dundurs [6–8] showed mathematically that a dynamic wave involving separation can stably propagate along an interface when two elastic media are compressed and simultaneously sheared. However, the validity of their mathematical solution was questioned by Freund [9] from an energy point of view, and the solution has mostly been ignored [10]. From the results obtained by frictional tests in the laboratory, Brune et al. [10–12] propose that dynamic rupture on a geological fault can be triggered by interface waves with separational sections propagating along the fault, or, in other words, by ripples which propagate along the fault, and slip occurs while the compressive stress is reduced [13]. It is suggested that the excitation of Rayleigh waves on a rupture surface can lead to pulses of separation [14]. Numerically, Day [15] indicated that, along a statically pre-stressed interface between similar materials, pulses of separation decay rapidly rather than propagate in a self-sustaining manner. However, Mora and Place [13] showed, again numerically, that such pulses can be sustained if interface surface roughness is present. Andrews and Ben-Zion [16] conducted 2D numerical simulations of dynamic rupture along a planar material interface governed by simple friction, showing self-sustaining propagation of slip pulse and spontaneous break-up of the propagating pulse into a number of smaller pulses. However, the mechanism of this separational pulse-induced dynamic interface instability has not been confirmed in a conclusive manner. In this study, the fundamental mechanisms of dynamic interface instability caused by a Rayleigh (R-) pulse will be investigated [17,18]. The main objectives will be: 1. To visualize and clarify qualitatively the complicated dynamic pulse-induced interface instability phenomena; 2. To evaluate quantitatively the pulse interaction process; and 3. To apply the model results to real earthquake problems. First, the basic characteristics of an R-pulse will be summarized in terms of stress field and particle motion. Second, some of the results of a series of 2D laboratory model experiments [17] utilizing dynamic photoelasticity will be presented. Experimentally obtained snapshots of isochromatic fringe patterns will show that an R-pulse can trigger instability of a partially contacting interface between similar materials. Third, the problem will be numerically analyzed using the finite difference wave simulator SWIFD [19,20] and the dynamic interaction process will be assessed quantitatively. The effect of acoustic impedance mismatch of the two contacting materials on the dynamic wave patterns will be described. Finally, the results obtained by the model investigation will be applied to explain the possible reason for the damage concentration caused by the 1995 Hyogo-ken Nanbu (Kobe), Japan, Earthquake.
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2. Basic characteristics of a Rayleigh pulse In a linear, isotropic homogeneous elastic material, an R-pulse propagates in a non-dispersive manner, with low geometrical damping at a smaller speed than the relevant shear (S-) wave and carries its energy mostly in a shallow layer adjacent to the surface [21,22]. The stress and displacement field associated with a plane R-pulse of arbitrary shape can be analytically expressed using one complex potential [17,23].
Fig. 1. A typical Rayleigh pulse that propagates from left to right along a free surface: (a) Isochromatic fringe pattern (contours of maximum in-plane shear stress) and (b) Particle movements at different depths.
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If the R-pulse is produced by a concentrated line load on the boundary of a halfspace, the complex potential can be determined explicitly from the known result that the surface normal stress distribution due to the R-pulse is proportional to the rate of loading [24]. Knowing the stress field from the complex potential, one can draw isochromatic fringe pattern pertaining to an R-pulse. A typical, theoretically predicted isochromatic fringe pattern is shown in Fig. 1(a) where the R-pulse, generated by a concentrated line load as a result of a detonating line charge, propagates from left to right along the free surface of the half-space. As indicated by the high fringe density in Fig. 1(a), the disturbance associated with an R-pulse is largely confined to a thin layer adjacent to the free surface. The particle movement induced by the same R-pulse is shown in Fig. 1(b) where the push and pull vertical (normal to the free surface) movement on the free surface, which becomes important in contact problems, is recognized. At depth, the horizontal particle movement is relatively small and the vertical movement is dominant.
3. Laboratory experiment 3.1. Setup Dynamic photoelasticity in conjunction with high-speed cinematography is utilized to investigate the interaction between an R-pulse and a statically prestressed, non-welded partially contacting interface. The experimental model is depicted schematically in Fig. 2. The model consists of two plates of Araldite B, which are initially in contact. The dimensions of the plates are chosen so as to prevent (unwanted) reflected waves from impinging upon the contact region and changing the results. The upper surface of the plate 2 (lower plate) is given a blunt double wedge cut so that only the central section of the surface would initially be in contact with the upper plate (plate 1). The wedge-type gap introduces contact singularities that are used for contact trace purposes. They have very little physical bearing on the phenomenon to be investigated. The two plates are statically preloaded in compression. The contact region is not glued (non-welded) and during the interaction process the contact can be weakened or strengthened depending on the relative position of the R-pulse with respect to the region of contact. A typical stress wave pattern is generated by detonating a small amount of explosive (240 mg of PbN6) on the lower surface of plate 1 at a distance 300 mm from the center of the contact region. The length of the generated R-pulse is 75 mm. For scale reasons, a grid with 25 mm spacing was drawn on the side of plate 1 facing the camera. A Cranz–Schardin type multiple spark gap camera is used to record the isochromatic fringe patterns that are produced by circularly polarized monochromatic light. 3.2. Results Fig. 3 shows a sequence of three experimentally recorded isochromatic fringe patterns of the dynamic pulse interaction process. The R-pulse propagates from left
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Fig. 2. The model setup for the Rayleigh pulse interaction experiment (all lengths in mm).
to right in the upper plate 1 and interacts with the region of contact. The times given in Fig. 3 correspond to the times elapsed from the moment of maximum stress amplification at the lhs edge of the contact region. Fig. 3(a) pertains to the instance where the incident R-pulse impinges upon the lhs edge of the contact region, showing that the strength of stress singularity indicated by the fringe order about the lhs edge is larger than that about the rhs edge where the dynamic effect is still negligible at this moment. This stress amplification about the lhs edge is due to the particle motion associated with the incident R-pulse [Fig. 1(b)]: its leading part induces a back- and downward movement of the particles [Fig. 3(a)]. The surface particles that are already statically in contact move together towards the lower plate 2 and thus amplify the stresses about the lhs edge. While the leading part of the retrograde particle motions in the incident R-pulse causes the stress increase about the lhs edge, the trailing part of the R-pulse induces back- and upwardly oriented movement [Fig. 1(b)], which may lead to stress reduction with possible cancellation if the surfaces separate, because the particles are now receding and opening the interface. In Fig. 3(b) the dynamic pulse interaction pattern changes considerably. The fringe pattern about the lhs edge shows little stress amplification. In a strict sense, an R-pulse does not exist in the contact region and there must be other kinds of generalized interface pulses that carry the energy across and along the region of contact. As the incident (generalized) R-pulse approaches the rhs edge of the contact region, as seen in Fig. 3(c), partial wave energy transmission can be found across the interface into the lower plate 2. Due to the separational movements of the surface particles in the trailing part of the R-pulse, there are no corresponding fringes in the lower plate 2 (interface separation). Fig. 3(c) thus indicates that
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Fig. 3. Sequence of snapshots of isochromatic fringe patterns obtained by the experiment (a) t=0 ms, (b) t=13 ms, (c) t=53 ms.
interface separation (instability) can be induced dynamically by an R-pulse even when the interface is statically pre-stressed. Fig. 4 summarizes the experimental results: the retrograde motion, especially the down- and upward movement of the
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Fig. 3 (continued).
Fig. 4. Summary of the dynamic Rayleigh (R-) pulse–interface interaction.
particles associated with the incident R-pulse, plays a crucial role in the pulseinduced interface separation.
4. Numerical simulations The same dynamic R-pulse interaction problem is numerically analyzed using the finite difference wave simulator SWIFD [19,20]. The contact problem is considered under plane stress conditions, and the R-pulse is assumed to interact with a contact region that is characterized by a Mohr–Coulomb friction criterion with the
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coefficient of friction equal to 0.3. The interface is pre-stressed and the tensile strength of the interface is set at a very low level. 4.1. Dynamic pulse interaction process Fig. 5(a) shows the time-dependent distribution of the normal stress acting perpendicular to the contact region during the dynamic interaction process. In the figure, the coordinate axes represent the position along the initially contacting region, the time and the normal vertical stress. This Lagrangian-type graphical representation has been chosen for improving the clarity and information about the dynamic interaction process. The 0 and 54.9 mm positions in Fig. 5(a) correspond to the lhs (incidence) and rhs (exit) edges of the contact region. The R-pulse, identified by the hump in the 3D surface, impinges on the contact region on the lower lhs of the diagram and travels to the upper rhs. The maximum stress amplification at the lhs of the contact region at time t ¼ 0 is clearly indicated. The decay of the amplitude of the R-pulse in Fig. 5(a) indicates that initially the tensile part of the incident R-pulse is strong enough to open the interface (tensile cut-off), but upon propagation along the interface, the pulse amplitude attenuates and the interface opening is no longer possible at a later stage. Fig. 5(b) shows the slip distribution along the initially contacting region. In the figure the axes correspond to the position along the initially contacting region, the
Fig. 5. The pulse interaction process from a quantitative point of view: (a) The normal stress acting perpendicularly to the contact region and (b) Development of slip along the region of contact.
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time elapsed with respect to the instance of maximum stress amplification at the lhs contact edge, and slip. Slip is defined by the difference of horizontal (tangential) displacements on the upper (1) and lower (2) surfaces of the interface, u1 u2 : It is shown that slip is initiated on the contact region on the lower lhs of the diagram and travels to the upper rhs at the speed of the R-pulse. This suggests that the initiation and propagation of slip is controlled by the incident R-pulse. 4.2. Partition of the pulse energy It is helpful to assess the relative amount of energy transmitted across and reflected at the contact region during the dynamic interaction process. In this study, R-pulse energy is calculated by Fourier decomposition and considering each decomposed harmonic wave individually [17]. Results are schematically depicted in Fig. 6. It is shown that over 50% of the energy initially carried in the incident R-pulse has been radiated into the far-field in the form of bulk waves and only 37% of the total energy is transmitted along the free surface in the form of a new R-pulse, Rt1. The energy contained in the reflected R-pulses is negligibly small compared with that carried by the transmitted R-pulses. 4.3. Effect of the acoustic impedance ratio The acoustic impedance ratio of the two contacting materials plays a crucial role in wave transmission and reflection at the interface (see e.g. [25]). Therefore, it is also important to investigate the influence of acoustic impedance ratio ðrcP Þ2 =ðrcP Þ1 on the pulse interaction process. Here, r is the mass density, cP is the longitudinal (P-) wave speed, and the subscripts 1 and 2 pertain to the upper and lower material, respectively. Fig. 7 shows numerically generated snapshots of isochromatic fringe
Fig. 6. Partition of the energy initially carried by the incident R-pulse.
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Fig. 7. Snapshots of isochromatic fringe patterns under two different contacting material combinations: (a) ðcS Þ2 oðcR Þ1 oðcP Þ2 and (b) ðcS Þ2 oðcP Þ2 oðcR Þ1 :
patterns taken at the same timing, 60 ms after the maximum stress amplification at the lhs edge of the contact region. Fig. 7(a) corresponds to the case where the incident R-pulse speed is between the shear (S-) and P-wave speeds of the acoustically softer, lower material 2: ðcS Þ2 oðcR Þ1 oðcP Þ2 : Since the slip pulse propagates at the Rayleigh wave speed of the acoustically harder material 1, ðcR Þ1 ; the pulse energy is transferred along the contact region at a transonic speed with respect to the acoustically more compliant material 2. A shear-type (S-) Mach (head) wave is then generated and propagated from the contact region into material 2. If the lower material 2 is sufficiently compliant, ðcS Þ2 oðcP Þ2 oðcR Þ1 [Fig. 7(b)], the energy is transmitted across the contact region supersonically into this layer, and both, longitudinal- (P-) and shear-type (S-) Mach waves are generated. The disturbance in the lower material 2 is largely confined to the region behind the P-Mach wave front. Both cases indicated in Fig. 7 suggest that an interface rupture pulse can propagate at the speed of Rayleigh wave of the acoustically stiffer material 1.
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5. Applications of the model 5.1. The model from a seismological point of view Seismologically, the model and the results obtained in the previous chapters can be interpreted as follows: In reality, the crustal rock nearby geological faults is fragmented by secondary faults over a wide range of scales. In modeling earthquakes, one consequence of this complexity is the introduction of the concept of asperities. Asperities, defined by ‘‘regions of very high slip’’, are considered to be of importance in earthquake hazard analysis because the fracture of asperities results in radiating most of the highfrequency seismic energy into the far-field. The geometrical explanation for asperities reflects the fact, that faults are not perfectly planar, but are rough on all scales and contain jogs and/or steps [26,27]. This study indicates that a relatively large amount of energy is radiated in the form of bulk waves from a contact region into the far-field. Interface slip can be found only inside the initially contacting region. These observations suggest that, when scaled up, the contact region in the model can be considered to represent an asperity on a geological fault. Therefore, the energy partition patterns obtained from the model study are of practical importance in the evaluation of the influence of asperities. 5.2. The ‘‘damage belt’’ associated with the 1995 Hyogo-ken Nanbu (Kobe) Earthquake in Japan On 17 January 1995, at 5:46 a.m. local time, an earthquake of moment magnitude 6.9 struck the region of Kobe and Osaka (Hanshin region) in the west-central part of the Japanese mainland. Seismic inversion [28,29] indicates that the rupture was initiated at a shallow depth on a fault system, which runs through the city of Kobe, and propagated bilaterally: along the Suma/Suwayama faults toward the city of Kobe and along the Nojima fault [Fig. 8(a)]. Strong ground shaking motions lasted for some 20 s and caused considerable damage within a 100 km radius from the epicenter, but Kobe and its neighboring region were most severely affected [30]. One of the puzzling phenomena observed in Kobe is the emergence of the strip of the most severely damaged zone, ‘‘damage belt’’ [the Japan Meteorological Agency Intensity 7 zone indicated in dark gray in Fig. 8(a)]. This strip, about 20 km long, is found close, but not parallel, to the Suma/Suwayama faults. Damage due to liquefaction was scarcely observed inside this strip, and it is suggested that the damage was caused directly due to the seismic waves [31,32]. A 2D model including a plane normal to the fault plane is regarded as appropriate for an approximate first-order analysis of the rupture mechanisms of the Hyogo-ken Nanbu Earthquake, because the Suma/Suwayama faults dip steeply, nearly 901, and a near-source, SH directivity pulse from strike-slip faulting was recorded in the city of Kobe [33,34]. Beneath the lhs edge [in Fig. 8(a)] of the strip, a region of relatively large slip (asperity) was found, and the arrival of a concentrated shear disturbance as well as
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Fig. 8. The 1995 Hyogo-ken Nanbu (Kobe), Japan, Earthquake and the numerical simulation for the case ðcS Þ2 oðcR Þ1 oðcP Þ2 : (a) The most severely damaged (the Japan Meteorological Agency Intensity 7) zone (dark gray region) associated with the Kobe Earthquake; Contours of (b) high PPV (peak particle velocity) and (c) high PPA (peak particle acceleration) obtained by the numerical study.
the resulting large (particle) velocity were recorded in central Kobe [28,29]. This indicates that the rupture-induced shear wave was of a Mach (head) wave type. As discussed above, an asperity on a geological fault corresponds to a contact region in the model. In Fig. 7(a), where an interface is located between dissimilar materials ½ðcS Þ2 oðcR Þ1 oðcP Þ2 ; an S-Mach wave is observed in the numerical simulation. In the model, material 1 fits the acoustically harder region in the foothills of the Rokko Mountains, where soils are very shallow or rock outcroppings are found and the damage tended to be relatively minor. Material 2 corresponds to the acoustically softer region (where soft alluvial soils primarily prevail) which includes the ‘‘damage belt’’. During dynamic interaction, each particle in the materials experiences a history of velocity and acceleration. The maximum values of these quantities, the peak particle velocity (PPV) and the peak particle acceleration (PPA), are very important and often used as practical design parameters in many applications, such as blasting in mines and quarries, and in engineering seismology. Fig. 8 shows the contours of high PPV [Fig. 8(b)] and PPA [Fig. 8(c)] obtained by the numerical simulation ½ðcS Þ2 oðcR Þ1 oðcP Þ2 : It is interesting to note, that the high PPV (or PPA) region is found in a narrow band, similar to the shape of the ‘‘damage belt’’ in Kobe [Fig. 8(a)]. The angle between the interface (fault) and the high PPV (PPA) region in
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this example is controlled by the propagation direction of the S-Mach wave generated during the interaction process [Fig. 7(a)]. This result indicates that a simple 2D model may be able to offer the information about the seismic rupture and the ensuing dynamic wave phenomena, although a more sophisticated study would have to be based on a 3D analysis, that includes local geological as well as topographical effect on the real fault rupture process.
6. Conclusions The purpose of this study was to offer an improved understanding of Rayleigh (R-) pulse interaction with a partially contacting interface between similar and dissimilar materials. The experiment has clearly shown that the interface separation (instability) can be dynamically induced by an R-pulse. The observed dynamic stress amplification and reduction at the interface are caused by the push and pull normal particle motion associated with an R-pulse. Numerically, the finite difference simulator SWIFD has provided quantitative information about the pulse–interface interaction process. It has been shown that a slip pulse can propagate at the Rayleigh wave speed of the acoustically harder material and that pulse interaction patterns are strongly controlled by the acoustic impedance mismatch of the contacting materials. If a contact region is located between dissimilar materials, Mach (head) waves can be generated and propagated. As a practical example, the mechanism of damage concentration associated with the January 17, 1995 Hyogo-ken Nanbu (Kobe), Japan, Earthquake has been studied. It has been suggested that the rupture-induced shear wave was a Mach (head) wave that propagated from a rupturing asperity (interface contact region) situated between dissimilar materials, and this wave could concentrate wave-induced high PPV (peak particle velocity) and PPA (peak particle acceleration) onto a narrow region such as observed in the city of Kobe. Due to a vast variety of application areas of Rayleigh pulses, the fundamental phenomena observed in this study may be of practical importance in various other fields of engineering and applied sciences.
Acknowledgements This work was financially sponsored by the Austrian National Science Foundation (FWF) through Research Project No. P10326-GEO.
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