i,m ~ (~o/~)c, ~. ~ (,7o/~Y (d/l)o)

Bulletin of the Seismological Society of America. Vol. 63, No. 3, pp. 959-968, June 1973 THE MAXIMUM ACCELERATION OF SEISMIC GROUND MOTION BY YOSHIAKI IDA* ABSTRACT A prediction of earthquake strong motion is attempted by studying the initial form of the seismic source-time function associated with a fracturing process described under idealized physical conditions. The effect of cohesive force on the source function is examined in detail, and the following relations are derived for the maximum particle velocity h M and the maximum acceleration//m i,m ~ (~o/~)c, ~. ~ (,7o/~Y (d/l)o) where ao is the strength of material, D o is the slip displacement required for the initial formation of crack surface, c is the rupture velocity, and p is the rigidity. Using these relations, it is concluded that the earthquake source motion is not directly governed by the interatomic cohesion, but rather by the gross strength of rocks. In fact, if 1 kbar for a o and 10 cm for D O are assumed, reasonable results, such as h M ~ 1 m/sec and//m "~ 1 g, are obtained, with the characteristic period and distance 0.1 sec and 100 meters, respectively. This result implies that the specific surface energy for the earthquake may be as large as 101 erg/cm 2, which is much larger than the typical values (10 s ) observed for rocks in the laboratory. INTRODUCTION In the vicinity of a seismic fault, it is expected that the ground motion is primarily governed by the fault-slip time function rather than the geometry and size of the fault plane. For the purpose of engineering seismology, it is thus very important to determine the accurate form of the fault-slip time function. Brune (1970) proposed a simple model of seismic source, assuming that the slip motion occurs instantaneously over the whole fault plane. In his model, the particle motion is related to the pre-existing stress field. It is physically more natural, however, to regard the earthquake occurrence as spontaneous rupture propagation associated with the stress concentration in the rupture front. If this is the case, the rupture cannot propagate supersonically, and Brune's time function is no longer applicable (Ida and Aki, 1972). In the case of a subsonic rupture propagation, the entire form of the source function involves several complicated effects, such as the initial state of the'displacement field and the dynamic processes of fracture and friction. It we restrict our Consideration to the initial rise of the source-time function, however, the analysis can be made simpler. For this purpose, we may focus our attention on the vicinity of the crack tip, since the effect of high stress concentration is so dominant there that the displacement field is almost determined by the material strength alone (Ida, 1972). The known solutions of dynamic crack propagation (Yoffe, 1951; Broberg, 1961; Kostrov, 1966) have a singularity of particle velocity or acceleration at the rupture front. This means that the initial form of the source function corresponding to the rupture front gives the most important contribution in the estimation of the maximum velocity and acceleration of the near-field ground motion. In this paper, we estimate these quantities, based on the analysis of cohesive force at the crack tip (Ida, 1972). * Now at the Institute for Solid State Physics, University of Tokyo, Roppongi, Minato-ku, Tokyo 106, Japan. 959

- (1 c2/fl 960 YOSHIAKI IDA COHESIVE FORCE AND PARTICLE MOTION The singularity at the crack tip is eliminated by considering an inelastic property of material in the fault plane near the tip. According to Barenblatt (1959), the inelasticity is formulated by the "cohesive force" that works across the crack tip against the fracture. Let us assume that the cohesive stress ac is given as a function of the displacement discontinuity D across the fault plane. The relation ac = a~(d) is usually called the "cohesive force diagram". The direction of D relative to the geometry of the fault plane depends on whether the crack is of plane shear, tensile, or longitudinal-shear type, and ac(d ) is generally a different function for the different types of crack. The displacement discontinuity D is an unknown function of time t and space coordinate x 1. We here choose as x 1 the coordinate axis that is placed along the fault plane, perpendicularly to the crack edge which is assumed to be an infinite line. The form of D in the vicinity of the crack tip is determined for a given ~rc(d), as has been studied in the case of the longitudinal-shear crack (Ida, 1972). The same discussion is also applicable to plane shear and tensile cracks, simply by replacing the factor involving the rupture velocity [see equations (7a), (7b) and (7c)]. To obtain practically useful expressions for particle motion, we introduce a "normalized" cohesive force diagram a(~b), as ac(d) = ao" a(d/do) (1) The constants ~o and Do are the representative values of stress and displacement. Let us choose as a o the stress of elastic limit (which is approximately equal to the yield strength) and as D o the displacement required to break down the cohesion. Then we have,,(o) ~ 1 cr(qs) --, 0 for q5 >~ 1. (2) If a(~b) is given, a normalized displacement discontinuity ~b is determined as a function of a normalized coordinate X (Ida, 1972). The nonnormalized displacement discontinuity D is obtained from the scaling relation [Ida, 1972, equation (17)] as D = Do$[-k(x 1 - ct)] (3) with the expression of X in terms of space and time X = - k(x I - ct). (4) Here c is the rupture velocity (c > 0), and k is defined by k = (327/np) (ao/doc) (5) where p is the rigidity, and ~2 is the normalized specific surface energy, i.e., 7 -= ½ S~ cr(~b) dqk (6) The dimensionless constant C in (4) is a universal function of c. In the case of the longitudinal-shear crack, C is simply (Ida, 1972) C = (1 - c2/fl2) 1/2 where fl is the shear-wave velocity. For the other types of cracks, C also involves the longitudinal-wave velocity ~. According to Weertman (1969), we have C = 4(fl/c) z (1 - cz/fl2) - 1/2 [(1 - cz/~ 2) (1-2) (7a) - c2/2fl2) 2] (7b)

THE MAXIMUM ACCELERATION OF SEISMIC GROUND MOTION 961 for the plane shear crack, and C = 4(fl/e) 2 (1 - e2/c~ 2)- 1/2 [(1 - ez/ot 2) (1 - c2/fl 2) -(1 -c2/2fl2) 2] (7c) for the tensile crack. In fact, the nature of cohesive force is not known very well either experimentally or theoretically. Here we try to estimate roughly the effect of a o and D o on the particle motion without detailed information on cohesion, remembering that the variation of o(qs) is greatly restricted by (2). This kind of crude approximation is practically useful, because the parameters a o and Do can be estimated more easily, even if it may be difficult to determine the complete form of cohesive force diagram. On the fault plane, the displacement u is related to the source function D, as u = ½D (8) and, thus, the particle velocity 5 and acceleration/~ are readily obtained from the function fi = ½ D okcgy i;t = ½Dok2c2d? " where ~b' and ~b" denote the derivatives of q~ with respect to X. If the function ~b' (X) has the maximum value ~b i' at a point X = X M, we have the following expression of the maximum particle velocity tim tim = (167~M'/~r) (aoc/#c). (9) Here let us choose the origin of X so that ~b(0) = 0. By calculating ~b(x) for several models of cohesive force diagram a(~b), it was shown that ~b'(x) actually has a maximum (Ida, 1972). The first five models in Table 1 cover the results in this calculation. In model 1, ~b'(x) is infinite at X = X M, because a(qs) is discontinuous at ~b = 1. In this model, the infinity of q5 M' is clearly caused by the artificial choice of o(qs). For the other models, q5 M' shows a finite value around unity in spite of the differences in details of the curves TABLE 1 THE MAXIMUM, (bm', OF THE NORMALIZED PARTICLE VELOCITY, ~'(X), ON THE FAULT PLANE FOR GIVEN MODELS OF COHESIVE FORCE DIAGRAM, O'(q~) Model o(4))* V cbm' X~cr 1 1 1/2 ov 4.0 2 1 - ~b 1/4 0.683 1.4 3 (1 - ~b) ~ 1/6 0.909 0.5 4 (1 - ~b)(1 + 2 ~b) 5/12 0.693 2.7 5 (1 - q)z)l/2 7r/8 0.712 2.9 6 2(~b+0.13)(~b- 1.37) 2 0.405 0.581 6.4 *cr(~b)=0for~b> lin the models 1 to 5, and forth > 1.37 in model 6. o'(q~). Roughly speaking, the dimensionless factor 16vqSM'/Tr in (9) is, thus, expected to be of the order of unity. From (4) and (5), we define XM and t M as XM = (~XM/32~) (PDo/aoC) (10) t M = XM/C (1 1)

962 YOSHIAKI IDA which give the dimension and duration associated with large particle velocity. In the same manner, we have the maximum acceleration/~m, if q$"(x) has a maximum qsm" i~m = (512?2 M"/rc z) (0.oc/#DoC)2Do (12) In fact, CM" is infinite for all of the models 1 through 5 in Table 1. The behavior of q$"(x) is thus examined in more detail in the next section. SINGULARITY IN ACCELERATION In the models 1 through 5 in Table 1, q$"(x) is proportional to X-1/2 in the vicinity of X = 0, and not finite at X -- 0. Furthermore, qs"(x) is also infinite at the point q5 = 1 in some models (1, 2, 4 and 5). Of these two singularities, the behavior around = 1 simply reflects a more or less unsmooth connection of 0.( ) at this point; we arbitrarily assumed that 0.( ) vanishes for > 1. Because the function '(X) is given by the Hilbert transform of 0.32(X) = 0.[q~(X)] (Ida, 1972), qy'(x)is finite at q$ = 1 if a'(q$) is continuous there. This condition is actually satisfied in model 3. To study the behavior of (X) in the vicinity of X = 0, we rewrite the relation between q$'(x) and aa2(x) given in Ida (1972), as @'(X) = - (X1/2il6 y) S~ ( Y- X)- 1 y-11210.32 ( y)_ 0.32(0)] dy. (13) From (13), it is readily found that the following relation is necessary so that ~b"(x) may be finite at X = 0 ~ Y-3/210.32(Y)--(732(O)] dy = 0. (14) Let us demonstrate through numerical calculation that (o"(x) can actually be finite at X = 0, if a suitable function is chosen as ~r(q$). We consider the following cohesive force diagram, for example a(~b) = ( a(d?+l 0 (~-l) 2 0 < (J < l 0 >l. (15) Putting a = 2 and l+ l 1 = 1.5 in (15), we keep l as an independent variable. The examples of the curve a(q$) are given for several values of l in Figure 1. Since a'( ) in (15) is continuous at = l, q$"(x) is finite except at X = 0 for any value of l. The distribution of q$(x) can be obtained in the same manner as described in Ida (1972). Figure 2 shows the result for q$'(x). When l is larger, the peak shifts to the right, and the steep rise of the curve around X = 0 is gradually removed. Finally, at 1 = 1.37, the condition (14) is approximately met, and 4b'(x) increases smoothly at X = 0 within the error of numerical calculation. The distribution of ~b"(x) is given in Figure 3. Corresponding to the behavior of (o'(x), c~"(o) approaches zero in the case of l = 1.37. The cohesive force diagram for l = 1.37 is denoted by model 6 in Table 1. Naturally, the cohesive force diagram that is physically realizable must give finite particle acceleration everywhere. Therefore, we understand that the condition (14) is a physical requirement for cr(q$). For such ~( ) that satisfies (14), equation (12) may be used in the estimation of the maximum acceleration. The condition (14) is regarded as the criterion to determine the critical stress o-(0), or at(0), above which the deformation is locally so concentrated that the displacement discontinuity appears along a crack surface in the continuous medium.

THE MAXIMUM ACCELERATION OF SEISMIC GROUND MOTION 963 I I e-1.0 0.5~ 0 0.5 1.5 D FIG. 1. The models of cohesive force diagram given by equation (15), in which a = 2 and l+l~ = 1.5 are assumed. The number attached to each curve denotes the value of/. I 1 [ [ 1.0 I 0.8-- 1.0 I.I i.2 g. 0.4 I 0 2.0 4.0 6.0 FIG. 2. The distribution of ~'(X) for the models of cohesive force in Figure I. The number attached to each curve denotes the value of I. THE EFFECT OF DISTANCE FROM THE FAULT When the observation point is not situated on the fault plane, the particle motion is generally affected by various factors of the sources, such as the dimension of the fault plane and the starting or stopping phases of rupture, other than the source-time function. For a distance that is sufficiently smaller than the source dimension, however, the stress field may still be approximated by our simple two-dimensional crack model, and we may estimate how the particle motion attenuates in the near-field. In this section, we consider this effect of distance in the case of the longitudinal-shear crack. For the other types of cracks, the mathematical treatment will be a little more complicated.

964 YOSHIAKI IDA 0.5 I.I 1.2 --e- o -0.5 I I I 2.0 4.0 6.0 FIG. 3. The distribution of ~b"(x) for the models of cohesive force in Figure 1. The number attached to each curve denotes the value of I. For the two-dimensional longitudinal-shear crack, the displacement u is given at an arbitrary point (x 1, x2) (Ida and Aki, 1972) by u = 1/(2zr) (1 - c2/fl2) 1[2 x 2 S zo~ [X 1 t2..}_ (1 - c2/f12)x221-1 D(x 1 - ct- x I ') dx( (16) where the coordinate x 2 is perpendicular to the fault plane and thus [Xz[ is the distance from the fault plane. Corresponding to (3) and (4), we introduce the normalized displacement U with the normalized coordinate X2, as U = u/do (17) Xz = (32 7Cro/Trl~Do)x 2. (18) Because of the symmetry of the problem, we have only to consider the positive side of )(2. The particle velocity fi and acceleration/~ are derived from U, as fi = Dokc(OU/~X) (19) gt = DokZ c2((32 U/ X(~ 2) (20) and, thus, the relative attenuations of ~ and /t are given by OU/OX and O2U/OX2 as a function of )(2. For the numerical calculation, the following relation that is derived with the use of equation (10') in the paper of Ida (1972) is more convenient than (16) ~W/~X = (¼) (2R)-1/2{(1 + X/R) 1/2 where - 1/(87) Sff [(1 - X/R)'/2X2 + (1 + X/R)'/2( Y- X)] yl/2[( y_ X)2-4- )(22]-1o"32(Y) dy} (21) R = (Xa2+ X22) 1/2

THE MAXIMUM ACCELERATION OF SEISMIC GROUND MOTION 965 The function 0"32 (Y) denotes the stress on the fault plane (at X2 = 0). The results of the calculation are shown in Figures 4 and 5 for model 6 of a(qs). These figures give the distribution of OU/~X and ~32U/OX2 as a function of X at various points of X2. The J 0.3 -- ~ X 2 = 0 J = o x (o ('o 0.2 0.1 X 2 = 100 o- ~ I I 0 5 I0 X FIG. 4. The normalized particle velocity ~U/~X at various distances X2 from the fault plane. X may be regarded as the coordinate along the fault, or the time, referring to (4). X2 = 0'1-'~ X2=O 0.1 x (1) fo 0 x X2 = I O0-0. 0 X J 5 k I0 FIG. 5. The normalized particle acceleration ~2U/OX2 at various distances )(2 from the fault plane. Xmay be regarded as the coordinate along the fault, or the time, referring to (4). maximum values of the normalized velocity and acceleration are listed for various values of X2 in Table 2. We may find that the acceleration//decreases rapidly with increasing )(2, and becomes much smaller than ~bm" when Xz exceeds X M. Therefore, the length

966 YOSHIAKI IDA X~ defined by (10) approximately gives the distance within which we expect as large an acceleration as on the fault plane. It is emphasized that the result of Table 2 involves only the effect of the time function and does not explain the actual ground motion for very large distances. TABLE 2 THE MAXIMUM VALUES OF THE NORMALIZED PARTICLE VELOCITY, ~U/~X, AND ACCELERATION, ~2U/~X2, AS A FUNCTION OF THE DISTANCE, X2, FROM THE FAULT PLANE FOR MODEL 6 X2 Max(OU]OX) Max(~2U/~X 2) Min(OZU[OX 2) 0 0.290 0.16-0.13 0.1 0.266 0.14-0.I0 0.5 0.212 0.085-0.048 1.0 0.173 0.053-0.024 5.0 0.0873 0.0087-0.0028 10.0 0.0624 0.0033-0.0010 50.0 0.0280 0.00031-0.000091 100.0 0.0198 0.00011-0.000032 ORIGIN OF STRONG GROUND MOTION It has been shown that the cohesive force diagram ere(d) determines the fault-slip time function at the initial stage, and yields the expressions (9) and (12) for the maximum velocity ~M and acceleration/~m on the fault plane. For a rough estimation, we may drop numerical factors involving y, q5 M' and qsm", and we have tim ~ (~o/la)c (22) i~m ~ (ao/#)z(ee/do). (23) In a similar manner, we obtain the following approximate relations from (10) and (11) xm ~ (~/Go)Do (24) t~t ~ (U/ao) (Do~c). (25) Let us try to substitute the values of a o and D O corresponding to the following two mechanisms of cohesion: (1) interatomic interactions (ao ~ P and Do ~ 1A), and (2) the gross strength of rocks broken by the displacement comparable with the thickness of the seismic fault (ao ~ 1 Kbar and D o ~ 10 cm). The results of the estimation are given in Table 3. In the first case, we have very large velocity and acceleration, but the duration or dimension associated with these high values is too small to account for seismic observations. In the second case, we have the velocity of 1 m/sec and the acceleration of 1 g, which are acceptable values (Brune, 1970;.Cloud and Perez, 1971). The associated time scale tm of 0.1 sec is also reasonable in view of the observed strong-motion spectra. The results for fi~t and tim in the second case agree with similar estimates made by Brune (1970). It is emphasized, however, that the physical bases for the estimation are distinctly different in the two studies. In our treatment, the maximum particle motion is determined by the material property concerning the strength against fracture, whereas the particle velocity or acceleration is governed by the ambient effective tectonic stress in Brune's model. In addition, the predominant frequency of 10 cycles is arbitrarily

THE MAXIMUM ACCELERATION OF SEISMIC GROUND MOTION 967 TABLE 3 THE MAXIMUM VELOCITY, /gm, AND ACCELERATION, /~M, FOR THE ASSUMED MECHANISMS OF FRACTURE Atomic Cohesion Strength of Rocks ao I Mbar 1 Kbar Assumed* Do 1 A 10 cm {ht~ loscm/sec l m/sec Obtainedt tim 1020 cm/sec 2 I g 1 A 0.1km 10-18 sec 0.1 sec *~ ~ 1 Mbar and c,,~ 1 km/sec are commonly assumed. ~'Equations (22), (23), (24) and (25) are used. assumed in Brune's estimate, but it was obtained in our study. Finally, our treatment is applicable to subsonic rupture propagation, whereas Brune's estimate is valid for a supersonic propagation which is not realistic for a spontaneous rupture (Ida and Aki, 1972). The result in Table 3 suggests that maximum seismic motion may not be directly related to atomic bonding, but rather governed by the bulk strength of rocks. The values of ~r o and D o in the second case yield the specific surface energy of 101 erg/cm 2, which is much larger than the laboratory measurements 103 erg/cm 2 for rocks (Brace and Walsh 1962). Such a large value of the specific surface energy for an earthquake was also obtained independently from the information on the time during which the rupture velocity approaches the upper limit (Kikuchi and Takeuchi, 1970). This problem should be examined more carefully because of the importance of specific surface energy in the fracture phenomena. The factor C specified by (6) is neglected in the approximate relations (22) and (23). This assumption is valid, unless the rupture velocity c is close to the critical sound velocity, i.e., the shear-wave velocity for the longitudinal-shear crack and the Rayleighwave velocity for the plane shear or tensile cracks. In many cases, the observed rupture velocity seems to be substantially smaller than the critical value (see Table 3 in Abe, 1972), even for deep-focus earthquakes (Fukao, 1972). In such cases, our rough estimation of ~M and/~m is not influenced very much, even if the factor C is taken into account. According to a theoretical calculation of dynamic crack propagation, the rupture velocity approaches the critical sound velocity (Kostrov, 1966). One possible explanation of the discrepancy between theory and observation is given by the dependence of ~ on c. According to (9), ~ should be infinitely large if c were the critical sound velocity. This suggests that the energy loss associated with ~, such as heat generation, would be enormous at the critical sound velocity, and that c is not easily increased unless such enormous energy is supplied. In other words, it is expected that the theory might give the correct observation of rupture velocity, if the energy dissipation is taken into account. For the practical purpose, it is highly desirable to predict the ground motion in more detail for individual event. At present, uncertain knowledge of the cohesive force diagram prevents us from determining more accurate values of the coefficients in (9) to (12). Therefore, better estimate cannot be made only by giving more suitable values for ao, Do, p and c. An expedient way to overcome this difficulty might be to regard these

968 YOSHIAKI IDA coefficients as empirical parameters. It is another problem whether such large faulting as an earthquake can be well described by the cohesive force diagram. Further theoretical and experimental studies are necessary to clarify this point. ACKNOWLEDGMENTS I would like to thank Dr. K. Aki for many useful suggestions. Drs. H. Kanamori and Y. Fukao read the manuscript and provided me with many helpful comments. This work was partially supported by the National Science Foundation under Grant GA 24268, and partially supported by the Advanced Research Projects Agency monitored by the Air Force Office of Scientific Research through contract F 44620-71-C-0049. REFERENCES Abe, K. (1972). Mechanisms and tectonic implications of the 1966 and 1970 Peru Earthquakes, Phys. Earth Planet. lnt. 5, 367-379. Barenblatt, G. I. (1959). The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses. Axially-symmetric cracks, Appl. Math. Mech. 23, 622-636. Brace, W. F. and J. B. Walsh (1962). Some direct measurements of the surface energy of quartz and orthoclase, Amer. Mineralogist 47, 1111-1122. Broberg, K. B. (1961). The propagation of a brittle crack, Arkiv Fysik 18, 159-192. Brune, J. N. (1970). Tectonic stress and the spectra of seismic shear waves from earthquakes, J. Geophys. Res. 75, 4997-5009. Cloud, W. K. and V. Perez (1971). Unusual accelerograms recorded at Lima, Peru, Bull. Seism. Soc. Am. 61,633-640. Fukao, Y. (1972). Source process of a large deep-focus earthquake and its tectonic implications--the Western Brazil Earthquake of 1963, Phys. Earth Planet. Int. 5, 61-76. Ida, Y. (1972). Cohesive force across the tip of a longitudinal-shear crack and Griffith's specific surface energy, J. Geophys. Res. 77, 3796-3805. Ida, Y. and K. Aki (1972). Seismic source-time function of propagating longitudinal-shear cracks, J, Geophys. Res. 77, 2034-2044. Kikuchi, M. and H. Takeuchi (1970). Unsteady propagation of longitudinal-shear cracks (in Japanese), Zisin 23, 304-312. Kostrov, B. V. (1966). Unsteady propagation of longitudinal-shear cracks, Appl. Math. Mech. 30, 1241-1248. Weertman, J. (1969). Dislocations in uniform motion on slip or climb planes having periodic force laws, in Mathematical Theory of Dislocations, T. Mura, Editor, American Society of Mechanical Engineers, New York, 178-202. Yoffe, E. H, (1951). The moving Griffith crack, Phil. Mag. 42, 739-750. DEPARTMENT OF EARTH AND PLANETARY SCIENCES MASSACHUSETTS INSTITUTE OF TECHNOLOGY CAMBRIDGE, MASSACHUSETTS 02139 Manuscript received October 18, 1972