A Cohesive Zone Model for Dynamic Shear Faulting Based on Experimentally Inferred Constitutive Relation and Strong Motion Source Parameters

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 94, NO. B4, PAGES , APRIL 10, 1989 A Cohesive Zone Model for Dynamic Shear Faulting Based on Experimentally Inferred Constitutive Relation and Strong Motion Source Parameters MITIYASU OHNAKA AND TERUO YAMASHITA Earthquake Research Institute, University of Tokyo, Bunkyo-ku, Tokyo To understand constitutive behavior near the rupture front during an earthquake source shear failure along a preexisting fault in terms of physics, local breakdown processes near the propagating tip of the slipping zone under mode II crack growth condition have been investigated experimentally and theoretically. A physically reasonable constitutive relation between cohesive stress r and slip displacement D, r = ( 'i - r,t)[1 + a log (1 + D)] exp (- rd) + rj, is put forward to describe dynamic breakdown processes during earthquake source tinlure in quantitative terms. in the above equation, ri is the initial shear stress on the verge of slip, rj is the dynamic friction stress, and a,, and rare constants. This relation is based on the constitutive features during slip failure instabilities revealed in the careful laboratory experiments. These experiments show that the shear stress first increases with ongoing slip during the dynamic breakdown process, the peak stress is attained at a very (usually negligibly) small but nonzero value of the slip displacement, and then the slip-weakening instability proceeds. The model leads to bounded slip acceleration and stresses at and near the dynamically propagating tip of the slipping zone along the fault in an elastic continuum. The dynamic behavior near the propagating tip of the slipping zone calculated from the theoretical model agrees with those observed during slip failure along the preexisting fault much larger than the cohesive zone. The model predicts that the maximum slip acceleration max be related to the maximum slip velocity Dmax and the critical displacement D, by)max = kd ' 2... D,., where k is a numerical parameter, taking a value ranging from 4.9 to 7.2 according to a value of rirp (rp being the pea. k shear stress) in the present model. The model further predicts that)max be expressed in terms of Dmax and the cutoffrequenc. y f nax of the power spectral density of the slip acceleration the fault plane as)max -- ( )Dmaxf a x and that ama x in terms of Dc andf nax as bma x = ( )D.f na x. These theoretical relations agree well with the experimental observations and can explain interrelations between strong motion source parameters for earthquakes. The pulse width of slip acceleration on the fault plane is directly proportional to the time Tc required for the crack tip to break down, andf ax is inversely proportional to T.. 1. INTRODUCTION The earthquake source is modeled as a dynamically propagating shear crack in the Earth with a rupture front near which the concentration of stresses is involved. The stresses near the crack tip exhibit the singularity of linear elastic fracture mechanics. This stress singularity is avoided if the cohesive zone (or the breakdown zone) over which the strength degrades with ongoing slip is assumed behind the propagating crack tip. The concept of the cohesive zone was first hypothetically introduced by Barenblatt [1959] to avoid the singular stresses near the tip of a tensile crack in linear elastic brittle materials, and later this hypothetical concept of the cohesive zone was applied to shear cracks by Ida [1972] and Palmer and Rice [1973] for geophysical purposes. The cohesive zone model for shear faulting assumes that the shear stress near a crack tip is a function of slip displacement. Recent laboratory experiments in which rock sample with a simulated fault that is large compared with the cohesive zone size is used have demonstrated that the shear strength actually degrades with ongoing slip near a tip of the propagating slip zone during shear failure along a preexisting fault [Okubo and Dieterich, 1981, 1984; Ohnaka and Yamamato, 1984; Ohnaka et al., 1986, 1987a, b]. This slip weakening behavior has been described approximately by a simple model that is often referred to as "slip-weakening Copyright 1989 by the American Geophysical Union. Paper number 88JB JB model" [e.g., Andrews, 1976; Rice, 1980, 1983; Rudnicki, 1980], in which the cohesive stress decreases linearly with ongoing slip from its peak value to the residual friction stress level (Figure 1). This simple slip-weakening model has successfully been used for seismological applications [Ida, 1972; Andrews, 1976; Burridge et al., 1979; Day, 1982]. Theoretical elasticity analyses for the simplified slipweakening instability model shown in Figure 1 predict, however, that slip acceleration is inevitably infinite at the tip of the cohesive zone [Ida, 1973]. This acceleration singularity at the crack tip is not physically reasonable. If the model is physically reasonable, it must involve the nonsingularity of acceleration at the crack tip as well; in other words, constitutive relation between shear stress and slip displacement must be chosen so as to give a finite acceleration as well as finite stresses at and near the crack tip. This is crucial when the strong motion source parameters such as the peak slip acceleration are discussed in terms of physics. Ida [1973] theoretically investigated the relation between the singularity of slip acceleration near the propagating tip of the cohesive zone and specific functional forms of the cohesive stress to the slip displacement and showed that the acceleration can be finite at the propagating tip if the cohesive stress is a suitable function of the slip displacement. Ida assumed mathematically simple relations between cohesive stress and slip displacement and was able to show one specific case for which the acceleration is finite at the crack tip [Ida, 1973]. However, Ida's specific model involving bounded acceleration at the crack tip does not satisfac- 4o89

2 4090 OHNAKA AND YAMASHITA: CONSTITUTIVE RELATION FOR DYNAMIC SHEAR FAULTING o Td,!! m!!!! Dc Displacement Fig. 1. The simplified slip-weakening model' -, is the initial shear stress, -, is the peak shear stress, 'a is the dynamic frictional stress level, D is the critical displacement, and G is the critical energy release rate or the fracture energy. torily explain the overall feature of slip-weakening instability behavior commonly observed during stick-slip shear failure in the laboratory, and in this sense his model may be called "mathematical" rather than "physical." Experimental observations show that such overall breakdown process is well-approximated by the above-mentioned slip-weakening instability model, and the slip-weakening instability concept itself seems to deserve special attention in the discussion about the earthquake source model. As pointed out above, however, such a simple slip-weakening model as shown in Figure 1 inevitably leads to the acceleration singularity of the type r- 2 at the distance r from the tip of the slipping zone along the fault plane. Recently, a rate- and state-dependent friction law has been proposed and developed by Dieterich [1978, 1979, 1981, 1986] and Ruina [1983]. This constitutive law is based on experimental studies on quasistatic frictional sliding in rock and the interpretation by Dieterich and Ruina. A specified simple model following a rate- and state-dependent friction law enables one to depict the slip motion during dynamic instabilities [Gu, ; Rice and Tse, 1986; Tse and Rice, 1986]. In particular, Rice and Tse [1986] analyzed dynamic motion for a single degree of freedom elastic system following the rate- and state-dependent friction law. Their analysis indicates that the model following the rate- and state-dependent friction law does not necessarily explain in quantitative terms observed relations between frictional stress and slip velocity during dynamic breakdown processes [Ohnaka et al., 1987a, b]. In fact, Ol<ubo and Dieterich [1986] point out that frictional strength is rateindependent at very high slip velocities. This practically means that the rate-dependent friction law is no longer applicable in the range of high slip velocities during dynamic breakdown processes. There might be an alternative interpretation that the rate- and state-dependent models can be modified to affect the rate independence at high slip velocities while maintaining their applicability at low velocities [Dieterich, 1986; Okubo and Dieterich, 1986]. This will be preferred if the same friction law can be used over the entire range of slip rates. However, virtually all the evidence for the rate- and state-dependent friction law comes only from experiments at very low slip rates, so that such a simple modification may not necessarily lead to the appropriate form of the friction law valid at high (or dynamic) speeds. In addition, it is unclear whether the model following the rateand state-dependent friction law does not involve any acceleration singularity near a dynamically propagating tip of the slipping zone along the fault in an elastic continuum. Although Gu [1985] discussed the breakdown (or slip- weakening) process on the basis of the rate- and statedependent friction law, he presumed that slip acceleration is constant during the entire breakdown process. Hence Gu's model yields no information concerning the peak slip acceleration, which is one of the key parameters characterizing earthquake source strong motion. An alternative way to discuss earthquake source strong motion in terms of physics may be to take an approach from the cohesive zone model into which details of the slip-weakening process suggested from careful experiments are incorporated. High-frequency strong motion in the earthquake source has attracted much attention of seismologists. Such highfrequency strong motion is related to inhomogeneity of the fault. The primary purpose of this paper is to find a physically reasonable relation between cohesive strength and slip displacement which does not give rise to any unrealistic singularities at and near the dynamically propagating tip of shear crack, on the basis of facts obtained from careful laboratory experiments, and to describe strong source motion characterized by high-frequency content comprehensively in terms of such a physically reasonable crack model. This is important when strong motion source parameters such as the peak acceleration are discussed in terms of physics. We further wish to show how strong motion source parameters such as the peak slip acceleration are related to characteristic parameters of the cohesive zone model, such as the breakdown stress drop, the breakdown time, and the critical displacement. To these ends, we have done a series of systematic laboratory experiments and theoretical studies, which will be described below. Once a specific, reasonable form of constitutive relation is found between cohesive stress and slip displacement, relations prescribing earthquake source strong motion can be derived theoretically from a model based on the constitutive relation. The theo- retical relations thus derived will be compared with experimental results to show how well the theoretical model explains experimental data and to check physical reasonableness of the basic assumptions used. In this paper the terms "cohesive zone" and "breakdown zone" are both synonymously used to describe the zone of shear stress degradation near a propagating rupture front. Methods 2. EXPERIMENTS To understand constitutive behavior near the rupture front during an earthquake source shear failure along a preexisting fault in terms of physics, local breakdown processes near the propagating tip of the slipping zone under mode II crack growth condition have been investigated in detail, using granite sample with a precut fault of which size is large compared with the breakdown zone size. The experimental methods and techniques have been described elsewhere

3 OHNAKA AND YAMASHITA: CONSTITUTIVE RELATION FOR DYNAMIC SHEAR FAULTING 4091 [Ohnaka et al., 1986, 1987a, b], so that only a brief description is given below. The same Tsukuba granite sample (28 cmx 28 cmx 5 cm), sawn in two along the diagonal, that had been used in previous experiments [Ohnaka et al., 1986, 1987a, b] was used for the present study to produce stick-slip failure instability along the precut fault. The fault length about 40 cm long is sufficiently large compared with the breakdown zone size, which has been found to be 5-6 cm in the zone of dynamic rupture propagation [Ohnaka et al., 1986, 1987a, b]. That the entire fault length is large compared with the breakdown zone size is essential for observing characteristic behaviors of local breakdown near the crack tip. The breakdown zone size depends on the roughness of fault surfaces [Okubo and Dieterich, 1984; Y. Kuwahara et al., unpublished manuscript, 1988]. The roughness and waviness of the fault surfaces in our sample were measured with a stylus profilometer, and the measured profile has been shown in the previous papers [Ohnaka et al., 1986, 1987a, b]. The physical parameters of the sample block used are as follows: the longitudinal wave velocity is 4.4 kms, the shear wave velocity is 2.9 kms, the rigidity is 2 x 10 4 MPa, and Poisson's ratio is Semiconductor strain gages with the effective length of 2 mm, mounted at 2.5 cm intervals along the fault at positions 5 mm from the fault (cf. Figure 3 in the work by Ohnaka et al. [1987a]) were used to monitor local dynamic shear strains and normal strains. Local dynamic stresses were determined from these dynamic strains; the shear stress -was given by the corresponding shear strain multiplied by the rigidity Ix of the sample, and the normal stress % was inferred from the relation: trn = [2Ix(1 + v)(1 - v)]en, where eis Poisson's ratio and en is the normal strain. Relative local displacements between the two sides of the fault were measured directly with metallic toil strain gages (effective length, 2 mm) by scaling measured strains to give displacements (strain sensitivity being higher than $ x 10-5). The transfer function f(zxx) of a strain gage sensor with the effective length L is given by [Ohnaka et al., 1986, 1987a] sin rzxx f( ¾) = (1) where zxx = (fl cos O)c. Here, 0 is the angle between the direction of the gage length and the direction of signal propagation, c the propagation velocity of the signal, f the signal frequency, and the signal has been assumed to be plane waves. Thus the frequency response of a strain gage sensor with L = 2 mm is flat from dc to $20 khz (-1.8 db), in case, for example, 0 = 0 and c = 3 kms. Strain and displacement signals were amplified and filtered, the resulting signals being sampled at the frequency of 1 MHz on 10-bit analog to digital converters and recorded finally on permanent disks. The overall frequency response of the entire measuring and recording system was flat from dc to 200 khz; hence the recorded signals provide enough resolution for studies of local dynamic breakdown processes near the propagating tip of the slipping zone. Signals from a slip failure event generated at a higher normal stress contain higher-frequency components [Ohnaka et al., 1986, 1987a]. A suitable transfer function of the filter operated for original signals from individual slip failure events has been determined, after preliminary trial- and-error smoothing operation, in such a way that the cutoff frequency of the transfer function of the smoothing operator is at least higher than the cutoff frequency of the power spectral density of the slip acceleration signals. The smoothing operator with a higher cutoff frequency was used for those data containing higher-frequency signal components to avoid removing significantly high-frequency signal components. Data An earthquake strong ground motion characterized by high-frequency content is related to the details of faulting. To construct an elaborate model that can explain such a strong motion, we must first of all understand detailed characteristic features near the propagating crack tip, and these features must be incorporated into the model. The resulting model will not give rise to singularities of slip acceleration as well as stresses at and near the propagating tip of a shear crack. For this purpose, we have done a great number of experiments on propagating stick-slip failure (mode II crack growth) and carefully observed local dynamic breakdown processes near the crack tip. Many of the findings in this series of experiments have already been published in the previous papers [Ohnaka et al., 1986, 1987a, b]. One of the important results relevant to the present concern is of the interrelations between local shear stress, slip displacement, slip velocity and slip acceleration near the dynamically propagating tip of the slipping zone during an unstable slip failure, and a typical example is reproduced in Figure 2 [Ohnaka et al., 1987b]. This slip failure event was generated at an average local normal stress of 4.0 MPa. All the signals shown in Figure 2 are smoothed by a digital filter with the following transfer function: sin (17,nf) sin (11,nf) 187 sin 2 (z-f) (2) where œ is the frequency. Figure 2a shows that the shear stress decreases from its peak value (or the breakdown shear strength) to a dynamic friction stress level with ongoing slip displacement. Such slip-weakening behavior has been commonly observed during not only stick-slip failure along a preexisting fault [Okubo and Dieterich, 1981, 1984; Ohnaka and Yarnarnoto, 1984; Ohnaka et al., 1986, 1987a, b], but shear failure of intact rock [Rice, 1980; Wong, 1982, 1986], and these overall breakdown processes are approximately described by the simple slip-weakening instability model shown in Figure 1. In Figure 2b the shear stress -is expressed as a function of the slip velocity V. Figure 2b shows that the shear stress first increases to its peak value with increasing slip velocity (d 'dt 0 and dvdt 0), and that the peak shear stress is attained at a small but nonzero value of the slip velocity. Once the peak value is attained, the shear stress decreases rapidly with increasing slip velocity to a level where the slip velocity has its maximum value, and slip failure instability is promoted during this accelerating phase (d 'dt 0 and dvdt y 0). The slip movement is stabilized by the subsequent decelerating phase where d 'dt 0 and dvdt O. Once the slip motion is arrested (V = 0), the mating surfaces are locked each other, and the shear strength on the fault increases along the - axis (V = 0) on the --V plane, if the time-dependent effect of restrengthening [e.g., Dieterich,

4 4092 OHNAKA AND YAMASHITA' CONSTITUTIVE RELATION FOR DYNAMIC SHEAR FAULTING EOBB40117 = 4.01MPe CHI (A) (B) DISPLRCEMENT (microns) O 2 O SLIP VELOCITY (cmsmc) (C) (D) u o w z o p_ w w u u ß DISPLRCEMENT (microns) DISPLRCEMENT (microns) Fig. 2. A typical example of the observed interrelations between shear stress, slip displacement, slip velocity, and acceleratio near the dynamically propagating tip of the slipping zone (or the cohesive zone) during stick-slip failure instability along the fault of which length is large compared with the cohesive zone size [Ohnaka et al., 1987b]. 1972, 1979; Ruina, 1983] plays a significant role during the continuous application of load. This is the restrengthening process during a stick period of stick-slip cycles. These basic features during the dynamic breakdown process have been commonly observed and confirmed in the present experiments as well, so that dynamic breakdown during an unstable slip failure may be modeled as shown in Figure 3 [Ohnaka et al., 1987a, b, with a slight modification near the crack tip]. In a later section, we will show theoretically that the peak shear stress is attained not at the crack tip, but at a short distance from the tip inside the crack. This feature has been incorporated into Figure 3. That the peak shear stress is attained at a nonzero value of the slip velocity shows that the shear stress reaches its peak value at a very small but nonzero value of the slip displacement. This can be demonstrated experimentally. Figure 4 shows an enlarged detail of the observed relation between shear stress r and tangential displacement Dii near the r axis (Dii-- 0) on the r - Dii plane for the same data shown in Figure 2. It is obvious from Figure 4 that the shear stress first increases to its peak value with ongoing slip near the dynamically propagating crack tip and that the peak stress is attained at a very small but nonzero value of the slip displacement (Oil = 0.026xm D,.). This behavior is not an artifact resulting from experimental errors, but a real feature. Although this feature has been discussed in an earlier paper [Ohnaka et al., 1987a], no clear experimental evidence has been presented in the previous paper. Figure 4 provides the first clear experimental evidence. To confirm this, another example is shown in Figure 5, for which the critical displacement is found to be 1.2 xm, and the peak stress is attained at Dii = 0.05 m Do. Careful experiments indicate that such a feature is commonly observed and that the peak stress is generally attained at Dll- ( )D,.. The present experiments further indicate that the dimensionless parameter S defined by S = (r, - r)(r,. - ra) [Das and Aki, 1977] approximately has a value of in the zone of dynamic rupture propagation (rupture velocity being roughly 3 kms, but the extending crack length was not possible to determine because of a

5 i oo OHNAKA AND YAMASHITA' CONSTITUTIVE RELATION FOR DYNAMIC SHEAR FAULTING 4093 A Model of Breakdown Zone Locked SI Dc ol of slipping zone x rp (A) rp T i Ti 03 T d,! Displacement Slip Velocity (C) (D) Db Displacement Displacement Fig. 3. A model of censtitutive behavior near the dynamically propagating tip of the slipping zone during the breakdown process of unstable slip failure. The peak shear stress rp is attained at a very (usually negligibly) small but nonzero value of displacement' in other words, the peak stress is attained very near the propagating tip of the slipping zone. The maximum slip acceleration)max is also attained very near the propagating front, while the maximum slip velocity bma x is attained closer to the middle of the cohesive zone. G, is the critical energy release rate, and x is the cohesive (or breakdown) zone size (Ohnaka et al. [1987a, b] with a slight modification). limited number of monitoring channels available), so that (r,. - rd)(rp - rd) = 1(1 + S) = This will be considered later when the numerical calculation is made on the basis of the theoretical model given in the next section. It seems physically important that the peak stress is attained at a nonzero value of slip displacement, and this may be the constitutive feature leading to bounded slip acceleration near a dynamically propagating crack tip. If this is the case, the simplified slip-weakening instability model (Figure 1) inevitably results in the acceleration singularity at the propagating crack tip, because such a detailed feature is overlooked in the model. It will be shown later that the observed feature must be incorporated into the model to enable one to discuss strong motion source parameters such as the peak slip acceleration in terms of physics. That the shear stress reaches its peak value at a nonzero value of the slip displacement means that the shear stress behind the propagating crack tip in the breakdown zone can increase with ongoing slip until its peak value is attained. This stress increase may be ascribed to an increase in frictional interaction between two mating rupture surfaces behind the crack tip with ongoing slip. Friction is basically

6 4094 OHNAKA AND YAMASHITA' CONSTITUTIVE RELATION FOR DYNAMIC SHEAR FAULTING i,, DISPLRCEMENT (microns) Fig. 4. An enlarged detail of the observed relation between shear stress and slip displacement in the neighborhood of the displacement Dii - 0 for the data shown in Figure 2. due to an interaction between two mating surfaces, and frictional strength is proportional to the real area of contact on the mating surfaces [e.g., Bowden and Tabor, 1954]. The observed increase in the cohesive stress with the increase in slip displacement can be explained if the real area of contact increases with increasing slip displacement regardless of the slip velocity, or alternatively, if the number of small asperities penetrated into the opposing surface increases with ongoing slip. In the next section, taking the observed feature into consideration, we attempt to construct a model suitable for explaining strong source motion which involves bounded slip acceleration and stresses at and near the propagating crack tip. Once such a model is constructed, relations between strong motion source parameters can be derived theoretically from the model, and theoretical relations derived will be compared in a later section with the corre- sponding experimental results to check physical reasonableness of our theoretical model. Experimental data to be used for such comparison are the maximum slip velocity5... the maximum slip acceleration bma x, the cutoff frequency œ ax of the power spectral density of a slip acceleration versus time record, the breakdown time To, and the pulse width T,, of slip acceleration. Figure 6 shows an example of recorded signals on local shear stress, slip acceleration, slip velocity, and slip displacement at a position along the fault during a propagating stick-slip õhear failure that was generated at an average (6.48 MPa) of local normal stresses. The slip velocity shown here is the time derivative of the slip displacement smoothed by a moving average of nine points, and the slip acceleration is the time derivative of the slip velocity. bma x and max are defined as the peak values of such slip velocity and acceleration records, respectively, and these peak values are obtained graphically from Figure 2 or 6. The breakdown time Tc is defined as the time required for the shear stress to decrease from its peak value to a dynamic friction stress level at one position (compare Figure 6); accordingly, the local breakdown time is evaluated from the shear stress-time curve recorded only at a position along the fault. There can be a number of different definitions for the pulse width. For simplicity, we here take the half value EI (a) (T. = 6.48MPa EI O'n = 5.80 mpa E o (c) ' (d) w loops Time DISPLRCœMENT (microns) Fig. 5. Another example of an enlarged detail of the observed relation between shear stress and slip displacement near the displacement Dll = 0. Fig. 6. An example of recorded signals at a position 5 mm from the fault during a dynamically propagating stick-slip shear failure (mode II crack growth) generated at an average normal stress of 6.48 MPa. (a) Dynamic shear stress. (b) Slip acceleration. (c) Slip velocity. (d) Slip displacement. The slip velocity and acceleration are the time derivatives of the displacement and the slip velocity, respectively. T c is the breakdown time and Tw is the half value width of slip acceleration pulse.

7 OHNAKA AND YAMASHITA: CONSTITUTIVE RELATION FOR DYNAMIC SHEAR FAULTING N J o z -G0-80 E CH1. 9SSM POWER SPECTRRL DENSITY fo.! 111 i I I! I11 I I I ' FREQUENCY! I!! I i I 100 khz MRX. POWER SPECTRRL DENSITY = E+O15 FREQUENCY RESOLUTION = E+02 Hz FOLDOVER FREQUENCY = 5.00OOE+05 Hz = POINT 1024 Fig. 7. An example of the power spectral density of a slip acceleration-time record. The upper limit of spectral band is denoted by f,ax- width T,,, as a characteristic pulse width of slip acceleration (Figure 6). Figure 7 shows an example of the power spectral density of a slip acceleration-time record, which has been shown in Figure 6. The cutoff frequency f' ax is defined as the upper limit of frequency of the spectral band of slip acceleration-time records (compare Figure 7), and it can also be estimated graphically. Experimental data on these parameters will be used later to be compared with theoretical relations that will be derived from the breakdown zone model following the constitutive relation (3) in the next section. 3. A CONSTITUTIVE RELATION BETWEEN COHESIVE STRESS AND DISPLACEMENT There are in general two processes involved in frictional sliding even at a constant sliding velocity: (1) slip-weakening process during which frictional shear strength degrades with ongoing slip and (2) slip-strengthening process during which frictional strength increases with increasing slip displacement. The shear strength in the breakdown zone behind the propagating crack tip may be ascribed to frictional interaction between two mating rupture surfaces. If this is the case, the shear stress in the breakdown zone can increase with slip displacement when the stress is less than the peak stress (or the breakdown strength). Once the peak stress is attained, however, the shear stress degrades with ongoing slip displacement from its peak value to a dynamic friction stress level. To describe these features in the breakdown zone we assume here a simple relation between the shear stress r and the tangential slip displaceme. nt Dii along the fault as r-- (ri- rd)f(dii) exp ( ) + d (3) wheref(dii) is a function of Dii, 97 is a constant, ri is the initial stress on the verge of slip, and rd is the dynamic friction stress. In the above equation, exp (-97011) represents the slip-weakening effect, and we assume that(dii) represents the slip-strengthening effect, where f(dll) must be chosen so as to satisfy the following conditions: f(0) = 1 f(dii) exp (- Dii) 0 (4) at sufficient amounts of slip displacement subsequently after the breakdown. The specific form of exponential for the slip-weakening relation has been assumed here because it can approximately explain our experimental data (compare Figure 2a). An exponential form similar to (3) has been abbuliiuo otuart [lyy, oj allu Stuart anu zulu " vko r, w, t l to model the breakdown process of earthquake instabilities. A specific form off(dii) is difficulto determine uniquely; however, preliminary frictional experiments indicate that the slip-strengthening effect in transitional phase during frictional sliding can approximately be described by a logarithmic law (M. Ohnaka, unpublished manuscript, 1988). This gives us a hint on a specific form of the function f(dii), and we thus assume that(dii) is expressed as (Dll) = 1 + a log (1 + Oll) (5) where a and are constants which are independent of Dii. The above equation satisfies the conditions (4), so that (5) is a possible solution. Frictional strength can increas even if Oil: 0. Dieterich [1972] found that static frictional strength increases with an increase in stationary contact time between two mating surfaces at a constant normal stress. This time dependency is considered to be due to time-dependent deformation at mating asperities [Dieterich, 1978], or alternatively, timedependent penetration of asperities into the opposing surface. Such asperities deform in the direction normal to the mating suff qes (Figure 8), and as a result, the real area of contact increases. Thus the above time dependency can be explained as a result of the time-dependent relative displacement D in the direction normal to the mating surfaces. This interpretation is supported by experiments by Scho& and Engelder [1976] on indentation creep which gives a timedependent increase in the real area of contact. Dieterich [1972, 1978] demonstrated that such apparent time dependency of static friction obeys the logarithmic law. Thus r in (3) can be expressed as ri: rs0[1 + y log (1 + 6t0)] (6) where r, 0 is the "instantaneous" static frictional stress at the normal stress %,, t o is the stationary contact time related to D, and y and 6 are constants independent of t o. In general, Fig. 8. (:Yn = constont Deformation Dl of asperities in the direction normal to the slipping surfaces and tangential displacement Dii.

8 4096 OHNAKA AND YAMASHITA: CONSTITUTIVE RELATION FOR DYNAMIC SHEAR FAULTING %o is lower than the maximum static frictional stress, but higher than r d. Substituting (5) and (6) into (3), we finally have an expression for r in terms of Dll and t o (or Dñ). Frictional stress is roughly proportional to the normal stress tr,,, and hence r may be written as x = [Xs0{1 + y log (1 + &0)}-Xd][1 + a log (1 + Dii)] ß exp (- rdii) +.L d (7) where x, %0, and x d are r, %o, and rd divided by %, respectively. Equation (7) incorporates the effects of both and Dñ (or to), and we have assumed that the effects of and Dñ are mutually independent. The constitutive relation (3) or (7) provides a description of a possible breakdown process for earthquake slip instabilities. We wish to show below that (3) or (7) can actually not only give finite slip acceleration and finite stress near a dynamically propagating crack tip during shear failure, but explain relations between strong motion source parameters in quantitative terms. 4. THEORETICAL ANALYSIS Slip Acceleration and Stress Near a Dynamically Propagating Crack Tip We assume a two-dimensional shear crack in an un- bounded elastic medium, and we use a Cartesian coordinate system with the (x, z) plane being the crack plane and the y axis normal to the crack plane. To make the problem more tractable we seek a solution for a semi-finite crack moving steadily in the positive x direction with a rupture speed v. The origin of the coordinate system is fixed at the moving crack tip, so that the crack is located on the negative x axis. The stationary contact time to can be regarded as constant during dynamic fault rupture, and hence ri in (6) will be regarded as constant during the dynamic breakdown process in the present analysis. This is a reasonable assumption, because the breakdown time is short enough, so that the time-dependent effect during the dynamic breakdown process is negligible, if compared with that during the restrengthening process. Tangential slip displacement will hereafter be expressed simply by D(=Di). Since the mathematical treatment has been presented by Ida [1972], only a brief description is given below. The relation between stress and relative displacement on the crack plane plays a central role in the present analysis. According to Aki and Richards [1980, p. 853], we have the relations kt l2p f0 AW'( :) c (8) pyz(x) = (1- v2vs 2) -o - x ds for an antiplane shear crack (mode III), and 2tXVs 2 [ -- v2vp 2) 12 (1 --VVs- v22vs2) 2 2 ) 122 ] Pxy(X) = rv2(1 ds c (9) 'P -o -x o for an in-plane shear crack (mode II), wherex is the rigidity, po(i, j = x, y, z) are the stress components, vp and vs are the P and S wave velocities, Au(x) and Aw(x) are the relative tangential displacement in the x and z directions, and Au'(x) and Aw'(x) denote the derivatives of Au and Aw with respect to x, respectively. The letter P before the integral sign denotes the Cauchy principal value. If we denote the preintegral factors associated with crack velocity v byzc(v), (8) and (9) are reduced to a simpler form as -fix) =zc(v)p fo -o D'(sc) se-'- ase (10) where 'r(x) is the shear stress, and D(½3 is the relative tangential displacement along the crack plane. The inverse expression of (10) is written as - xd'(x) - A 2 txc(v) 1 P ;o - x r( ) d (11) where the asymptotic relation D(x) - A - x with x -->-o has been considered (A being constant). Since a semi-infinite crack has been assumed in our analysis, the integral in (11) diverges if rd is nonzero at a distance from the crack tip. For simplicity, we assume for the present calculation that rd -- 0 all over the crack plane, so that (3) is reduced to r(d(x)) = ri[1 + a log (1 +3 D)] exp (-rd) x < 0 (12) The breakdown zone size xc is in general defined as the distance along rupture plane from the crack tip to the position where the shear stress degrades to a dynamic friction stress level rd, which has been assumed to be zero for the present analysis. For convenience's sake, we here define x. as the distance from the crack tip to the point where the relation r(-x0 = Ar v holds (r v being the peak stress and Arp a small fraction of r e, say, A = 0.01 or 0.15 being assumed), and we further introduce another parameter Xmax(>x0 at which the relation r(-xmax) = 0 is reasonably assumed to hold. The reason for introducing the parameter Xm x is to avoid the singularity at x = -x.. Integrating both sides of (11), we have D(x) = A - x 1 fo _ r( ) log " -' +N-.. _ s e-,v_ x ds c (13),rt2txC(v) Xrnax The condition of finite stress at the crack tip yields -- o Xmax V d T t C( )A (14) If we use nondimensional parameters X = -XXm,, and Y = -- :Xm x instead ofx and s c, then (13) is rewritten equivalently as D(X) = A (Xma xx) 12 Xmax C(v ) fo 1 r(y) log ' ' q- ' dy,) rr2p, Equation (14) becomes 2(Xmax ) T(Y) txc(v) - rt2a dr (14')

9 OHNAKA AND YAMASHITA: CONSTITUTIVE RELATION FOR DYNAMIC SHEAR FAULTING '( )[ I0' if(o), o 15: I.O 1=2 Fig. 9. Schematic illustration of the constitutive relation between the normalized cohesive stress a(tb) and displacement tb; a(tb) has its peak value at tb = e. Substituting (14') into (13'), and dividing both sides of the resulting equation by the critical displacement Dc, we finally have Parameters Prescribing Constitutive Relation Figure 9 shows a schematic illustration of the constitutive relation (17) employed for the present analysis. We assume that the dimensionless stress cr( b) has its peak value of unity at the dimensionless displacement b = e. Since Dc is the critical displacement, the relation A << 1 must hold. A set of the parameters (&,, ) uniquely prescribe the constitutive relation (17). Alternatively, it is also possible to describe the relation in terms of another set of the parameters (r e, ri, e), since &,, and can be regarded as functions of rp, ri, and e. Practically, rp, ri, and e are more convenient parameters, since these are evaluated directly from the observed cohesive stress-displacement curves during dynamic breakdown processes (compare Figures 2 and 4). Therefore we hereafter describe the relation (17) in terms of rp, ri, and e. Let A be a given parameter. We have (rirp)[1 + & log (1 + e)] exp (- e)= 1 (21) & (1 + e) - [1 + & log (1 + Be)] = 0 (22) (rirp)[1 + & log (1 + )] exp (- )= A (23) tb(x) = Dc where ß - - tr(d log dy (15) 01 F= dy (16) and 6(X) = D(-xmaxJOO c. Here, cr(d is the nondimensional stress near the crack tip, which is given by Since these are complex nonlinear simultaneous equations, strict analytical solutions are difficult to obtain. However, our experiments show that e has a value of the order of 10-2D,., and hence e is much smaller than unity, so that it will be reasonable to assume that e << 1. If this inequality is assumed, (21) is reduced to (rirp[1 + a log (1 + e)] = 1 + e From (21') and (22) we have where (21') & = so0 + ri0 (24) = [_Q _+(Q2 + 4PR)l2]2p - - [1 + a log (1 + 4)] exp (- 4) (17) where rp is the peak stress, and = a, = D,., and = D,., and,, and are constants independent of the dimensionless displacement 6. F is a dimensionless quantity related to the fracture energy or the critical energy release rate G by 1. = FroOt. ( 8) and G is written in terms of r and D as [Palmer and Rice, 973] Gc = r(d) dd 0 D(xmax) Equation (15) has the following constraint: (19) qb(1) = D( - Xmax)D c (20) The integral equation (15) is nonlinear, and hence it will be solved numerically in the next section. The slip velocity and acceleration are the first and second partial time derivatives of D(=Dc b) in the coordinate system fixed to the medium. so0 = [(rpri) - 1 ]log(1 + e) (25a) rio = e(ri rp)log (1 + e) (25b) e = + (25c) Q =lrio(1 + Be) so0 log (1 + Be) R = ri0 log (1 +3e) (25d) (25e) In the abov equations, & and are expressed in terms of. Substituting (24) into (23), we have the nonlinear equation for which can be solved numerically. Once is obtained, & and can be obtained from (24). If a set of solution thus obtained does not satisfy the presumed inequality e << 1, then the set of solution is discarded. Numerical Procedure To solve (15) numerically, we employ the method taken by Burridge et al. [1979]. The interval [0, 1] is divided into L subintervals. Very small subintervals are needed to study minute features of slip acceleration and stress near a crack tip. Let AX and A y be the sizes of subintervals near and at a distance from the crack tip, respectively, where AX < A y, NAY + MAY = 1, and N + M = L. If we assume that b takes a value at the center of individual subintervals, then

10 4098 OHNAKA AND YAMASHITA: CONSTITUTIVE RELATION FOR DYNAMIC SHEAR FAULTING _ Distonce from Crock-tip X Fig. 10. Plot of the dimensionless slip acceleration 0" against the dimensionless distance X from the crack tip. Numerical values in parentheses indicate sets of the parameters (rlrp, ) that give finite 0"(x). ()i = ()((i- 12)&Y) 1 < i < N &i=&(nzxx+(i-n- 12)AD N+ l <i <L Equation (20) is represented by an extrapolation from 0Land 0L to (3(1); that is, (26) 30r- 0L- = 2D(-Xmax)Dc (27) Since A(xmax)l2Dcin (15) is unknown, this is set to be The discretized form of (15) and (27) give simultaneous nonlinear equations for 0(i = 1, 2,..., L + 1). The slip velocity and acceleration are obtained by the numerical differentiation of Off = 1,..., L). D(-xmax)D c = 2 is assumed throughout the present calculation. An arbitrary set of values for (r irp, e) does not necessarily give finite acceleration at the crack tip, and hence we take the following approach to obtain the desired solution: a value for rirp is first assumed and then e is determined so that the second derivative of the slip function 6(23 is finite at the crack tip (X = 0). If the value for e thus determined satisfies the condition e << 1, the obtained set of values is a desired solution. L = 120, M = N = 60, = e10, and Ay = (1 - NzXX)M have been chosen for the present numerical computation. In practice, the parameter e is determined in the following way: for a given large e (which is denoted by one can examine whether the second derivative of 6(23 is finite at X = 0. If the desired solution is not obtained, then a further assumption e = e2(<e ) is made and the same computation is repeated. Such successive iteration is carried out until the desired solution is obtained. 0(X) = A(xmaxX)l2Dc has been assumed as an initial trial function for e = e, where (3(1) = A(xmax)l2Dr : D(-xmax)Dc = 2. For the ith step e = el, the slip function obtained at the preceding step e = ei_ is employed as an initial trial function. Figure 10 exemplifies that the solutions desired for finite accelerations exist for individual sets of (rirp, e); for instance, when r rp = 0.8 is given, the secon derivative of 0(X) is finite at e These sets of values which give finite 0"(23 are listed in Table 1. According to the present model, there is a correlation between r rp and e; that is, increases with decreasing r rp (Figure 11), though this has not been checked by experimental data. It may be too much to expect such a high resolution for the data to check a correlation between those two parameters experimentally. Our numerical calculation indicates that for a specific value for r,.7'p(=or(o)), 0" is finite at a suitable value of e, but goes infinite otherwise. In particular, 0" diverges if e - or if - O. Perhaps the obtained sets of solutions are unique under given conditions. It was not possible to check from our numerical calculation whether there is only a single value or a range of values of e that gives bounded accelerations for a given value of r 'p. Several sets of values for e, &,, and thus obtained as solutions for given r rp are listed in Table 1. THEORETICAL AND EXPERIMENTAL RESULTS: COMPARISON Figure 12 shows a typical example of results of numerical calculation, where interrelations between shear stress, slip displacement, slip velocity, and acceleration are presented. The overall features of these relations agree well with those obtained from the experiments (Figure 2). The shear stress increases to a peak value with increasing slip velocity and the peak shear stress is attained at a nonzero value of the slip velocity, and these features agree with the experimental observation [Ohnaka et al., 1987a]. The acceleration versus displacement curve shows a remarkably good agreement between theoretical and experimental results in the accelerating phase (compare Figures 2 and 12). There is some discrepancy between theoretical and experimental results in the decelerating phase, possibly because acceleration signals obtained from the experiments are masked by noises. There is also a slight quantitative difference between theoretical and experimental results: the shear stress has its peak value at a higher slip velocity in the theoretical model than in the experiments. This may be because the presumed constitutive relation (3) between r and D is subtly different from the relation observed experimentally in the neighborhood of D = 0. There is also a difference in residual stress level between Figures 2 and 12; however, this is because dynamic friction stress has been assumed to be zero in the theoretical model for simplicity. In Figure 13, the shear stress and slip acceleration are plotted against the distance from the crack tip along the crack plane. One can see that the peak shear stress as well as TABLE 1. Several Sets of Values for Parameters rlrp,, &, 1, and x x x X i i i Fig. 11. Relation between the parameters ' 'l ' 'p and e.

11 OHNAKA AND YAMASHITA: CONSTITUTIVE RELATION FOR DYNAMIC SHEAR FAULTING !! Displacement Slip velocity 5 2OO 150 o Displacement Displacement Fig. 12. A typical example of numerical calculation for relations between the normalized shear stress a(&), slip displacement &, slip velocity &', and acceleration &" near the dynamically propagating tip of the slipping zone along the fault in an infinite elastic medium. For the example shown, o 0) = 0.7 has been assumed. the peak acceleration is attained not at the crack tip, but at a short distance from the tip inside of the crack. This basic feature has been incorporated into the breakdown zone model schematically shown in Figure 3. If we employ the coordinate system fixed to the medium, the slip velocity and acceleration are written as 0 D(x vt) v (vt-x I -- - = Do&' (28) at Xma x,xmax -- x I Dc b", Xmax,] O(x - = vt (29) where the prime denotes the derivative with respect to the argument. Using (14'), (16) and the approximate relation A(xmax) 2D,. = 2.0 obtained from numerical calculation (compare Table 2), we have an expression for Xmax as Xma x = w21. C(v)Dc[' 'p (30) The nondimensional size of breakdown zone X,. is defined by X c ---- XcXmax (31) Substituting (30) and (31) into (28) and (29), we thus have O D(x vt) Fv rp&, XW (32) at rr2c(v) x. STRESS (33) Distonce Fig. 13. Plots of the normalized shear stress o![x) and slip acceleration &"(X) against the normalized distance X from the crack tip. Note that the peak shear stress as well as the peak acceleration is attained not at the crack tip, but inside of the crack. For the example shown, o![0) = 0.? has been assumed. Using the solution for &(, the quantity F can be numerically computed and tabulated in Table 2. F depends slightly on r rp and takes a value ranging approximately from 0.25 to 0.5 according to the value of r rp (Table 2); however, such a modest change in F may practically be neglected, and F may be regarded as virtually constant. The nondimensional breakdown zone size X., taking such a numerical value as listed in Table 2, is also considered to be virtually constant. We thus notice from (32) and (33) that the slip velocity depends only on rp and v and that the slip acceleration on rp, D. (or x0, and v; this has been pointed out by Ida [1973]. From (30) and (31) the relation between D. and x. is expressed in terms of rp and v as

12 4100 OHNAKA AND YAMASHITA: CONSTITUTIVE RELATION FOR DYNAMIC SHEAR FAULTING TABLE 2. Several Sets of Values for Parameters F, X, X,,,, (Dtmax, (D"max, and k (A = 0.15 Assumed) r, lrp AVr' maxoc r X c X,v (Dtmax (D"max k = (D"max((Dtmax) Dr. F rp -- = (34) (30) and (31) very simple relations between the half value width T,, of acceleration pulse and the breakdown time Tc A relation similar to (34) has been presented by Rice [1980] (remember that dynamic friction stress is neglected in the present analysis). From (28) and (29), we have the relation between the maximum slip velocity bmax and the maximum slip acceleration bma X as where k )max: cc (Dmax)2 (35) k -- qb"ma x(qb'max) 2 (36) The parameters &'max,( "rnax, and k have been numerically computed and also listed in Table 2. These numerical parameters depend slightly on r,.rp; for instance, k takes a value ranging from 4.9 to 7.2 according to a value of r,.rp, showing a modest increase in k with r,.rp. In practice, however, such a slight change may be negligible, and k may be regarded as virtually constant. The theoretical relation (35) is compared with the experimental result in Figure 14. Figure 14 shows the relation between the maximum slip velocity and acceleration, where individual)max and max are normalized to the corresponding characteristic quantities 15,, '. and5.,.2, respectively (5, and ',. being the mean critical displacement and the mean breakdown time). The white circles in the figure indicate experimental data obtained by using granite sample with a simulated fault that is large compared with the cohesive zone size. The straight line segments in the figure show the relation (35) with different values of k. The parameter k = 2-5 seems consistent with the experimental data (Figure 14), whereas k = 5-7 from the theoretical model if rrp = is assumed. The theoretical model gives slightly larger values than those estimated experimentally. However, such a modest difference between theoretical and experimental results may practically be insignificant, because errors accumulated during the process of numerical calculation could be involved in the theoretical results, particularly in acceleration data, and because experimental errors are also inevitably involved in the observed data. For instance, it is difficult at present to expect such a high resolution for experimental data as to detect a correlation between k and r,.rp suggested from the theoretical model (Table 2). We thus conclude that the theoretical result agrees satisfactorily with the experimental observations. The predominant period of acceleration waves near an earthquake fault will closely be related to the pulse width of slip acceleration on the fault plane. Hence the relation between the pulse width T,, of slip acceleration and the time T, required for the crack tip to break down (or the breakdown time) will be examined here. We find from Table 2 and if A = 0.01 is assumed, and T. Dc x T,,: - : 0.06z-2C(v) (37) v rp To. Dc tz = o. (38) v rp if A = 0.15 is assumed. These indicate that the pulse width of slip acceleration is directly related to the time required for the crack tip to break down. These theoretical relations are compared with experimental data in Figure 15, where a broken line represents the theoretical relation (38), and the solid circles show experimental data. These data are considerably scattered. Nevertheless, we can clearly see a correlation between T., and T.; that is, T,, increases linearly with increasing T,.. We find from Figure 15 that there is a good agreement between the theoretical and experimental results, and that the pulse width of slip acceleration on the fault plane is directly proportional to the local breakdown time To I0 % max Dc k ( ½ Dc max):' IO -I I0 ' I I0 Tcbmo, Fig. 14. Plot of the normalized maximum slip acceleration 'r2 ma xd r against the normalized maximum slip velocity P )ma X 5½: comparison between theoretical and empirical relations. Open circles are data from stick-slip experiments, and the solid star and the triangle are values of the order of estimate for actual earthquakes with moderate-to-large sizes. The theoretical model gives the straight lines with k = 5-7, when rlrp = is assumed.

13 ,,, OHNAKA AND YAMASHITA' CONSTITUTIVE RELATION FOR DYNAMIC SHEAR FAULTING O i i i. engineering. We have already revealed some aspects of strong motion in the source region by deriving relations between strong motion source parameters from the model (see equations (35), (37), or (38) and (39)). We here exclu- sively focus attention on specific relations between5... b... and f'max. From (32) and (33), Dma x and Dma x are written as o 2o IO - _ 0,I,,I,,, I,, 0 I0 20 holf-volue Width of Accelerofion Pulse (ps) Fig. 15. Relation between the breakdown time T and the slip acceleration pulse width Tw. Solid circles show experimental data on stick-slip, and the dashed line represents theoretical relation (38) in text. If the theoretical relation (37) is compared with the experimental data, a considerably larger systematic difference is found. This implies that D. determined experimentally from observed cohesive stress-displacement curves (compare Figure 2a) may be taken as the displacement at A in the theoretical calculation. This interpretation could be justified because noises inevitably superimposed on signals make it difficult to determine D. accurately. The cutoff frequency f nax Of the power spectral density of the slip acceleration versus time record at one position on the fault is also related to the local breakdown time T, at the same position. Figure 16 shows the relation empirically found between f' nax and 1T,. As described previously, the breakdown time T, is estimated from experimental data on shear stress, and f ax from data on slip acceleration, so that the data on both f nax and T, are mutually independent. Nevertheless, there is a good correlation between f nax and 1Tc. We thus conclude from Figure 16 that f nax = 1Tc (39) If the crack tip propagates at a constant velocity v along the fault, the relation (39) is equivalent to 5max = rr2c(v) ['17 'p tx (D,max = Xc (D' max Dc Tc (41) 5max- -- 7r4{C(v)}2Dc (D"max = X 2 b" ß Dr. cc (42) which show that the order of magnitudes for)max and can practically be estimated only from the two parameters D c and To, since the dimensionless parameters X, and 6"max are size- and time-scale independent and are roughly regarded as virtually constant (Table 2). Relations (41) and (42) may be useful for estimating the magnitudes of Dma x and bma x in case D and T are suitably evaluated. Our model predicts thatsma x be expressed in terms of bmax and f nax, and thatsma X in terms of D and f nax' that is, from (39), (41), and (42), and from (39) and (41), 5ma x = m c (D,'m x)omaxf ax = ( ' 4)Dmaxfmax s (43) )max = (Xc(D'max)Dcf nax = (0.56 '-' 0.91)Dcf' max (44) Theoretically, the dimensionless quantity mc( "max( 'max) depends slightly on the paramete r,.rp. Practically, however, the quantity is regarded as virtually constant (Table 2), and Xc((D"ma x(d'max) = when ri 'p takes a reasonable value ranging from 0.5 to 0.8. Similarly, XcCh'ma X takes a value ranging from 0.56 to 0.91 when r,.rp = Again, such a modest variation may practically be regarded as virtually constant. The theoretical relations (43) and (44) derived from the model are compared with the corresponding experimental 70 6O 5O -l- 40,, i i i i i - f nax = VXc (40) which has been assumed by Papageorgiou and Aki [1983a] in the analysis of seismological data to infer source parameters such as the cohesive zone size and the local stress drop on the basis of their own specific barrier model. Note that according to the relation (39), f nax is estimated from shear stress data only at one position near a fault. We turn our attention to interrelations between strong motion source parameters which are crucial for earthquake. : 30 2o % ß o ( i i i i 0 io ß IT½ (khz) Fig. 16. Observed relation between f ax and lto.

14 4102 OHNAKA AND YAMASHITA: CONSTITUTIVE RELATION FOR DYNAMIC SHEAR FAULTING E 10 e I I I I l0 s 10 4 :C: I0 Theoreticol Relotion intended to explain the rate-dependent effect experimentally observed at low slip rates by Dieterich [1979, 1981] and others, and in this sense the model is not a comprehensive one. Further elaborate and basic studies are needed to construct such a comprehensive model that is applicable over the entire range of slip velocities. This is beyond the scope of the present paper and will be left for a future study. The model following the constitutive relation (3) or (7) can explain the following empirical relation between )max and the breakdown stress drop Dma xv AT bhe (45) which has been obtained under mode II crack growth condition using granite rock sample much larger than the cohesive zone size [Ohnaka et al., 1987a]. If we assume v = kms, then we have ['( tma x7r2c(7j) = , so that from (41) l )ma xv = (0.74 (46) i0 -I I0 ' I I0 I0 ' I0 I0 ' I0 e max f ax (ms a) Fig. 17. Relation between Omax and 10maxf ax: comparison between experimental and theoretical results. Solid circles show experimental data, the thick line indicates the theoretical relation, and the star and the triangle represent values of the order of estimate for earthquakes with moderate-to-large sizes. data in Figures 17 and 18, respectively. Figure 17 shows the relation between l ma x and l )maxf nax. In the figure, a thick solid line represents the theoretical relation (43), and the solid circles indicate experimental data. It is found that the theoretical relation agrees well with the experimental results. Figure 18 shows the relation between bma x and Dc f ax' The theoretical relation (44) is represented by solid lines in Figure 18, and the solid circles in the figure indicate experimental data. In this case, the theoretical relation gives slightly larger values than the experimental data; however, both trends are nearly the same. We thus conclude that the basic relation (3) or (7) assumed in this paper is a reasonable constitutive relation which does not involve unrealistic sin- gularities of slip acceleration and stresses, and which can explain experimental data on dynamically propagating slip failure quantitatively. The constitutive relation can explain earthquake source strong motion as well, as will be discussed in the next section. 6. DISCUSSION As pointed out in foregoing sections, discussion of an earthquake source strong motion characterized by highfrequency content in terms of a physically reasonable crack model requires understanding characteristic features of the breakdown zone near the propagating crack tip, and a model with bounded slip acceleration and stresses at and near the crack tip. We have put forward a specific form of constiturive relation (3) or (7). Although the rate-dependent effect is not incorporated in the present model, the relation (3) or (7) can describe sequences of repeated slip instabilities on the same fault surfaces, because the stress ri is time dependent (compare equation (6)). However, the present model is not for an in-plane shear crack (mode II). This agrees with Ida's [1973] theoretical result. In the theoretical model, the dynamic friction stress has been assumed to be zero for simplicity. If, however, the breakdown (or cohesive) stress r b is replaced by the breakdown stress drop Arb, then the relation (46) agrees well with the empirical formula (45) obtained from the laboratory experiments. The rupture velocity v has been assumed to be sub-rayleigh to derive the theoretical relation (41). It has been observed in the laboratory experiments that v can be as high as shear wave velocity or super-shear [Ohnaka and Kuwahara, 1989]. In particular, the observed rupture velocities ranged from 0.7 to 4 kms for a set of experimental data used to derive the relation (45) [Ohnaka et al., 1987a]. Numerical experiments by Andrews [1976], Das and Aki [1977], and Okubo [1986] also show that supershear rupture velocities are possible for mode II crack tip propagation. It is not known at present, however, by what theoretical relation the equation (46) be replaced for supershear crack propagation under mode II condition. This will also be left for a future study. An earthquake source strong motion is prescribed by such characteristic physical parameters as)... D... and f nax, and these in turn are governed by the physical state in the source region in the Earth. What physical factors in the Earth prescribe these parameters? We find from (38), (39), (41), and (42) that D and f nax are related to v, -p, and D. as Dmax oc (47) C(v) V Dmax Z{S(v)}2Dc (48) v rp (49) fr axøcc(v)oc which show that)max is independent of the critical displacement D., and that both Sma X and f nax depend on Do. The critical displacement D is sensitive to the roughness of fault surfaces; D. becomes larger on rougher surfaces [Okubo and Dieterich, 1984; Y. Kuwahara et al., unpublished manuscript, 1988]. It follows from (48) and (49) that bothsma X and f ax have lower values on the fault with rougher surfaces.

15 OHNAKA AND YAMASHITA: CONSTITUTIVE RELATION FOR DYNAMIC SHEAR FAULTING 4103 I0_ 0,01 I I I llllj I I I I IIIII I ' I III I I0 bmo (ms) Fig. 18. Relation between )max and Dcjr, ax: comparison between experimental and theoretical results. Solid circles show experimental data, straight lines indicate the theoretical relations, and the star represents a value of the order of estimate for earthquakes with moderate-to- large size. Since vc(v) increases with increasing v, the parameters and f nax all increase with increasing v and rp or A r, (note that the breakdown stress drop A r, is roughly proportional to the breakdown stress rp). Both the breakdown stress and the breakdown stress drop become higher at higher normal stress. The rupture velocity tends to increase with increasing normal stress [Okubo and Dieterich, 1984], and the critical displacement is insensitive to the normal stress [Dieterich, 1981; Okubo and Dieterich, 1984; Ohnaka et al., 1987b]. We thus conclude from (47), (48), and (49) that b... D... and fi ax become higher at higher normal stresses in the brittle or semibrittle regime when the other conditions are equal. These theoretical predictions quite agree with the experimental observations [Ohnaka et al., 1986, 1987a, b]. As inferred from the discussion done so far, the generation of short-period seismic waves is directly related to the time T, required for local shear stress near the propagating tip of the breakdown zone to drop from its peak value (or the breakdown strength) to the dynamic friction stress level. From (37) or (38), r.,c --De. (50) from which we find that T, becomeshorter as both v and Vp get higher and as D, becomes smaller. In other words, short-period seismic waves are predominantly generated at a fault zone where the local breakdown strength (or the breakdown stress drop) and the rupture velocity are high and where the critical displacement is small. In view of the effect of the normal stress mentioned above, we can conclude that in the brittle or semibrittle regime, short-period seismic waves are predominantly generated at a local fault zone where the normal stress is high. This has been demonstrated experimentally [Ohnaka et al., 1986, 1987a]. The parameters T., x,, D., Ar,, 5... and )max characterize the cohesive zone model, and of these characteristic parameters, some are size- (or time-) scale independent and the others are scale dependent. According to Ohnaka et al. [1987a, b], for instance, T,, x,, D,, and Dma x are scale dependent. Size scale dependent parametersuch as Sma X can be scaled to the characteristic quantities of the cohesive zone model to extend the results in the laboratory to an earthquake failure in the Earth [Ohnaka et al., 1987a]. Papageorgiou and Aki [1983b] have inferred D and x, for several moderate-to-large earthquakes on the basis of their own specific barrier model. According to them, the critical displacement for actual earthquakes fell in the range 40 cm to 4 m and the breakdown zone size was in the range 500 m to 2 km. Thll we may reasonably he able to a llme that D. is of the order of 1 m and x, of the order of 1 km for actual earthquakes with moderate-to-large sizes. If we further assume for such earthquakes that v = 3 kms,)max = 2 ms, andsma x = 10 ms 2 (--- 1 g), then we have T.l)ma xd = 0.66 and T.2 JmaxD. = 1.09, and this has been plotted as a star mark in Figure 14. The triangle in the figure is the same plot in case max = 20 ms 2 has been assumed instead ofsma x = 10 ms 2. The corresponding plots are also given in Figures 17 and 18. From Figures 14, 17, and 18 it seems that (35), (43), and (44) represent relations between strong motion source parameters for both earthquakes in the Earth and stick-slip in the laboratory, and these theoretical relations may be useful for estimating strong motion source parameters such as the maximum slip acceleration and f,ax for earthquakes. The cutoff frequency f'max defined and discussed here may not necessarily be the same as the cutoff frequency fmax defined by Hanks [1982] as the upper limit of frequency of the spectral band of strong motion accelerograms recorded at distances from an earthquake source. There are three possible causes for the cutoff frequency fmax: (1) the source effect, (2) the recording site effect, and (3) the effect of path attenuation [e.g., Hanks, 1982]. The cohesive zone model inevitably leads to the conclusion that acceleration spectra cannot be flat at frequencies higher than the spectral corner frequency prescribed by fault duration or source dimension [e.g., Brune, 1970], but that the cutoff frequency f'max of acceleration spectra of source origin must exist, and this cutoff frequency is prescribed by the breakdown zone size. This has been discussed by earlier authors [e.g., Papageorgiou and Aki, 1983a; Aki, 1985]. If observed fmax is determined by the recording site effect andor the effect of path attenuation, then fmax < f 'max. However, iffmax is prescribed by the source effect, then fmax = f,ax, and in this case the relations (43) and (44) may be useful in determining strong motion source parameters. Acknowledgments. We are grateful to T. Hirasawa for his courtesy in providing experimental facilities, and to K. Yamamoto and Y. Kuwahara for their help in performing experiments presented in this paper. Pertinent suggestions and critical comments by an associate editor and two reviewers were helpful in revising the manuscript, for which we are grateful to them. This research was supported in part by a grant from the Ministry of Education, Science, and Culture of Japan (project number ). REFERENCES Aki, K., Origin of fmax, paper presented at the 5th Maurice Ewing Symposium on Earthquake Source Mechanics, AGU, Harriman, N.Y., May 20-23, 1985.

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