Non-hypersingular boundary integral equations for 3-D ... - Springer Link

Computational Mechanics 25 (2000) 613±626 Ó Springer-Verlag 2000

Non-hypersingular boundary integral equations for 3-D non-planar crack dynamics T. Tada, E. Fukuyama, R. Madariaga

The BIEM has been extensively applied to various classes of 2-D and 3-D crack-analysis problems (e.g. Beskos 1997). According to the displacement discontinuity method, which has produced the most successful results to date, the stress ®eld over the model space is expressed as a convolution, in time and space, of the slip along the crack and a set of integration kernels. Then a limiting process is so applied that the receiver point approaches the crack face, producing a set of boundary integral equations (BIEs) that relate the traction on the crack surface to the slip on it. The traction BIEs, thus derived, are hypersingular, and are not immediately amenable to numerical implementation. One of the most popular and successful methods to circumvent the hypersingularities is the approach of regularization. This consists in rewriting, most often through integration by parts, the hypersingular integrals in an 1 equivalent form which involves only weakly singular inIntroduction tegrals, at most integrable in the sense of Cauchy principal In seismology, numerical modeling of the dynamics of values. rupture propagation on faults with a realistic geometry is The integration by parts technique allowed SlaÂdek and essential in the efforts to better understand the complex SlaÂdek (1984) and Nishimura and Kobayashi (1989) to nature of earthquake phenomena. However, numerical derive a non-hypersigular BIE for 3-D non-planar cracks, analysis of rupture on non-planar faults has faced many in the Laplace domain and the frequency domain, retechnical dif®culties. In the present paper, we present a spectively. These formulations, however, could deal only new theoretical framework, based on the boundary intewith the transient response of stationary cracks. Later, an gral equation method (BIEM), to describe the dynamics of alternative technique, based on path-independent conser3-D cracks with arbitrary geometry, which is expected to vation integrals, was used by Zhang and Achenbach (1989) provide an important basis for future advances in the and Zhang (1991) to derive regularized elastodynamic practical numerical modeling of non-planar cracks. BIEs for 3-D non-planar cracks, in the frequency domain and the time domain respectively. However, it was not evident how their BIEs could be implemented to nonplanar crack problems, because they were written in a 1 global Cartesian coordinate system, which does not necT. Tada (&) , R. Madariaga essarily agree with the curved surface of the crack, on Laboratoire de GeÂologie, EÂcole Normale SupeÂrieure 24 rue Lhomond, 75231 Paris Cedex 05, France which slip is de®ned. Because of this disadvantage, these earlier BIEs for E. Fukuyama the 3-D non-planar crack have been con®ned to conNational Research Institute for Earth Science ceptual expressions, and they have been reduced to a and Disaster Prevention, 3-1 Tennodai, Tsukuba, numerically implementable form only for the special Ibaraki 305-0006, Japan case of the planar crack. Numerical study of the dynamics of the 3-D planar crack has been conducted by Present address: 1 Earthquake Research Institute, The University of Tokyo, 1-1-1 Yayoi, Bunkyo-ku, Zhang and Gross (1993), who used Zhang's (1991) Tokyo 113-0032, Japan equations, as well as in an independent work by Fukuyama and Madariaga (1995, 1998), who used the integration by parts technique. Meanwhile, there have We thank Hideo Aochi and Teruo Yamashita for discussions and communications. One of the authors (TT) was partly supported been alternative formulations of the elastodynamic BIEs by a Postdoctoral Fellowship for Research Abroad of the Japan for the time-domain analysis of the 3-D planar crack. Das' (1980) pioneering formulation was based on the Society for the Promotion of Science (JSPS). Abstract We derive a non-hypersingular boundary integral equation, in a fully explicit form, for the time-domain analysis of the dynamics of a 3-D non-planar crack, located in an in®nite homogeneous isotropic medium. The hypersingularities, existent in the more straightforward expression, are removed by way of a technique of regularization based on integration by parts. The variables are denoted in terms of a local Cartesian coordinate system, one of the axes of which is always held locally perpendicular to the potentially curved surface of the crack. Also given, in a fully explicit form, are the expressions for the off-fault stress and displacement ®eld, as well as the special form of the equations for the case in which the fault is planar.

613

614

traction method, in which the displacement ®eld was expressed as a convolution, in space and time, of the traction on the crack and a set of integration kernels. Geubelle and Rice (1995) followed the displacement discontinuity method, but they formulated their BIEs based on a Fourier transform in space. A recent study by Krysl and Belytschko (1999) illustrates an original method for a time-domain elastodynamic analysis of a 3-D crack of arbitrary geometry, based on the Element-Free Galerkin method which obviates the need for re-meshing. In the present paper, we enlarge Fukuyama and Madariaga's (1998) integration by parts technique to derive, for the ®rst time, a non-hypersingular time-domain BIE for the 3-D non-planar crack in a fully explicit form. In order to represent the BIE in terms of the one opening and two shear components of the slip on the crack surface, we introduce a local Cartesian coordinate system ®rst used by Tada and Yamashita (1997). The present article comprises: (1) the BIE describing the traction-slip relation for the 3-D non-planar crack; (2) a similar expression for the off-fault stress ®eld, and its special form for the planar crack; and (3) a similar expression for the off-fault displacement ®eld, and its special form for the planar crack.

displacement

u(x)

normal vector

n (ξ)

x receiver point

∆ u (ξ) Γ

ξ

slip

source point

crack surface

Fig. 1. Nomenclature for symbols

uk …x; t† Zt Z ˆ ÿ dS…n† dsDui …n; s†nj …n†Rij=k …x; t ÿ s; n; 0† ; C

0

…2† where, if the medium is isotropic,

Rij=k …x; t ÿ s; n; 0†  cijpq

o Gpk …x; t ÿ s; n; 0† oxq

ˆ kdij op Gkp ‡ l…oi Gjk ‡ oj Gik † ; …3†

with k and l being the Lame constants, is the stress Green's function, representing the ij-component of the 2 stress at receiver point x and time t ÿ s due to a unit force Representation theorem in the k-th direction applied at source point n and time 0. We start from the representation theorem of elasticity that The Green's functions satisfy the equations of motion expresses the elastic displacement ®eld over the entire o2 medium in terms of the slip distribution along the crack. o R ˆ q Gik ; …4† ij=k Assuming that the medium is at rest with no slip for time oxj ot 2 t  0, and that the traction is continuous across the crack, we have, for a crack located in an in®nite homogeneous with q denoting the density. In the following, an abbreviated notation isotropic elastic medium,

Z uk …x; t† ˆ ÿ

Zt dS…n†

C

oi  dsDui …n; s†cijpq nj …n†

Dui;j 

o Dui …n; s† onj

…5†

will be used for partial derivatives with respect to space, and time derivatives will be denoted by dots, especially by:

0

o Gkp …x; t ÿ s; n; 0† ; oxq

o ; oxi

o2 o2 ui  2 Dui …n; s† : …6† Gij  2 Gij …x; t ÿ s; n; 0†; D ot os Also, c and c are the Pand S-wave velocities respecL T where uk …x; t† is the displacement in the k-th direction tively. Roman subscripts are supposed to run over 1, 2 and at receiver point x and time t, C the crack surface, n 3, while Greek subscripts run over 1 and 2, and summation the source point on C, Dui …n; s† the slip across the crack over repeated indices will be implied wherever necessary. in the i-th direction at location n and time s as de®ned Use is also made of the reciprocity relations by the relative displacement of one (positive) side of C with reference to the other (negative) side, cijpq the o o Gij …x; t ÿ s; n; 0† ˆ ÿ Gij …x; t ÿ s; n; 0† …7† elastic constants, and n…n† the unit vector normal to onk oxk the crack surface at location n pointing from the negao o tive to the positive side of C (Fig. 1). Gkp …x; t ÿ s; n; 0† …8† Gij …x; t ÿ s; n; 0† ˆ ÿ Gij …x; t ÿ s; n; 0† : os ot is the displacement Green's function, representing the displacement in the k-th direction at receiver point x and time t ÿ s due to a unit force in the p-th direction 3 applied at source point n and time 0. Summation over 3-D Green's functions repeated indices is implied and dab denotes the KroFor simplicity, we use notations necker's delta. r  jjx ÿ njj; ci  …xi ÿ ni †=r …9† Equation (1) can be rewritten as



…1†

for the length and orientation cosines of the source-receiver vector, and

p2 

c2T l : ˆ 2 cL k ‡ 2l

…10†

The displacement Green's function Gij is given by

Gij ˆ oi oj J ‡ dij GT

…11†

or

G11 ˆ o21 J ‡ GT G22 ˆ G33 ˆ

o22 J o23 J

G23 ˆ o2 o3 J G31 ˆ o3 o1 J ; G12 ˆ o1 o2 J

‡ GT ‡ GT

…12†

1 ÿ r GT 

tÿsÿr=c Z L

ds
sd…t ÿ s ÿ r=cL ÿ s† 0

2 2 2  cÿ2 T GT ˆ …o1 ‡ o2 ‡ o3 †GT

…19†

2 2 2  p2 cÿ2 T GL ˆ …o1 ‡ o2 ‡ o3 †GL ;

…20†

we obtain the identities: 2 2  o23 G33 ˆ p2 cÿ2 T GL ÿ …o1 ‡ o2 †…G33 ÿ GT ‡ GL †

…21†

2 2 2  o23 G11 ˆ cÿ2 T GT ‡ o1 …G33 ÿ GT † ÿ …o1 ‡ o2 †GT

…22†

o23 G22

ˆ

 cÿ2 T GT

‡

o22 …G33

ÿ GT † ÿ

…o21

‡

o22 †GT

;

…23†

which shall be utilized later in order to remove terms containing the derivative o23 . Although alternative forms of these equations of motion are possible, we choose to use the speci®c form given by Eq. (21) through Eq. (23), because these render the ®nal BIE in the simplest form.

with

" 1 1 J 4pq r

Combining the symmetry relation (12) with the equations of motion

#

tÿsÿr=c Z T

ds
sd…t ÿ s ÿ r=cT ÿ s†

…13†

0

1 d…t ÿ s ÿ r=cT † 4plr Z

…14†

4 Planar crack By way of introduction, we start with the simple case of the planar crack. The representation theorem states that:

Zt

u1 …x; t† ˆ ÿ

dS…n†

ds…Du1 R13=1 ‡ Du2 R23=1 ‡ Du3 R33=1 †

C

0

Z

Zt

ˆÿ

dS…n†

 ds Du1 …lo1 G13 ‡ lo3 G11 † ‡ Du2 …lo2 G13 ‡ lo3 G12 † ‡ Du3 ‰ko1 G11 ‡ ko2 G12 ‡ …k ‡ 2l†o3 G13 Š

0

C

Z ˆ ÿl

Zt dS…n†

C

0

   1 ÿ 2p2 ds Du1 o3 …2G11 ÿ GT † ‡ Du2
2o3 G12 ‡ Du3 o1 2G33 ÿ 2GT ‡ G L p2

Z

for the elastodynamic case and

p2 ÿ 1 r J 8pl 1 GT  4plr

…15† …16†

for the elastostatic case. Explicit forms of the Green's functions are given in Appendix A. Of particular importance in the present article is the linear combination

…17†

for the dynamic case and

in the static case.

u3 …x; t† ˆ ÿ

dS…n†

1 2 p 4plr

…18†

 ds Du1 …lo1 G33 ‡ lo3 G13 †

0

C

‡ Du2 …lo2 G33 ‡ lo3 G23 † ‡ Du3 ‰ko1 G13 ‡ ko2 G23 ‡ …k ‡ 2l†o3 G33 Š Z ˆ ÿl

GL  G11 ‡ G22 ‡ G33 ÿ 2GT 1 2 p d…t ÿ s ÿ r=cL † ˆ 4plr GL  G11 ‡ G22 ‡ G33 ÿ 2GT ˆ

Zt

…24†

Zt dS…n†

C



 ds …Du1 o1 ‡ Du2 o2 †…2G33 ÿ GT †

0

  1 ÿ 2p2 ‡ Du3 o3 2G33 ‡ ; G L p2

…25†

where we made use of the symmetry relation (12). The equation for u2 is obtained by considering the symmetry between the x1 - and x2 -coordinates. Accordingly,

615

616

  1 1 ÿ 2p2 r33 …x; t† ˆ l 2 o3 u3 ‡ …o u ‡ o u † 1 1 2 2 p p2   Zt  Z 1 ÿ 2p2 2 dS…n† ds …Du1 o1 ‡ Du2 o2 †
2o3 2G33 ÿ GT ‡ G ˆ ÿl L p2 0 C    1 ÿ p2 1 ÿ2 2 2  GL ‡ 2 cT Du3 GL ÿ 4Du3 …o1 ‡ o2 † G33 ÿ GT ‡ p2 p t      Z Z 1 ÿ 2p2 1 ÿ p2 G
4o G ÿ G ‡ G ÿ Du ˆ ÿl2 dS…n† ds Dua;a
2o3 2G33 ÿ GT ‡ L 3;a a 33 T L p2 p2 0

C

ÿ

Zt

Z

l2 cÿ2 T

dS…n†

dsD u3 0

C

1 GL ; p2

…26†

2

Z

Zt

 ds Dub;b oa …4G33 ÿ 3GT †

dS…n† where Eqs. (12), (20) and (21) have been used and in- Ta …s; t† ˆ ÿl 0 C tegration by parts has been carried out, so as to prevent hypersingular integrals from appearing in a later BIE. ÿ Dua;b ob GT ‡ Du3;a With the limiting process x ! s 2 C, r33 approaches the   1 ÿ 2p2 normal traction T3 across the crack at location s and
2o3 2G33 ÿ GT ‡ GL p2 time t: T3 …s; t† ˆ r33 …s; t† :

…27†

With the 3-D Green's functions behaving as 1=r in the limit of r ! 0, the integrals that involve the ®rst-order spatial derivatives of G33 , GT and GL are of the order of 1=r 2 as r ! 0 and thus are integrable in the sense of Cauchy principal values. Likewise, making use of Eqs. (12), (19) and (22) and integrating by parts,

ÿ

l2 cÿ2 T

Z

Zt dS…n†

C

dsD ua G T :

…29†

0

Equations Eqs. (27) and (29) constitute a set of non-hypersingular BIEs for the 3-D planar crack that express the traction in terms of the slip. This is the theory underlying Fukuyama and Madariaga's (1998) formulation. Substituting the explicit forms of

r31 …x; t† ˆ l…o1 u3 ‡ o3 u1 † ˆ ÿl

ˆ ÿl

2

2

Z

Zt dS…n†

C

0

Z

Zt dS…n† 0

C

ÿ

l2 cÿ2 T

  ds …Du1 o1 ‡ Du2 o2 †o1 …4G33 ÿ 3GT † ‡ Du1 ‰ÿ…o21 ‡ o22 †GT ‡ cÿ2 T GT Š   1 ÿ 2p2 GL ‡ Du3
2o1 o3 2G33 ÿ GT ‡ p2    1 ÿ 2p2 ds Dub;b o1 …4G33 ÿ 3GT † ÿ Du1;b
ob GT ‡ Du3;1
2o3 2G33 ÿ GT ‡ GL p2

Z

Zt dS…n†

C

dsD u1 GT :

…28†

0

The tangential traction T1 across the crack at location s is the Green's functions into Eqs. (27) and (29) and making given by T1 …s; t† ˆ r31 …s; t†. This, along with a similar use of the following identity (Fukuyama and Madariaga equation for T2 , may be generalized to 1995; Appendix B to the present article):

Z C

1 ui …n; t ÿ r=c† dS…n† D r Z c ˆ 2pcDu_ i …s; t† ÿ c dS…n† a Du_ i;a …n; t ÿ r=c† r C

…if s 2 C† …30† with

r  jjs ÿ njj ci  …si ÿ ni †=r ;

…31† …32†

we eventually arrive at:

3l T3 …s; t† ˆ ÿ p

Z C

c dS…n† 2a r

l ÿ …1 ÿ 2p2 † p

Z C

Zp 1

l dv
vDu3;a …n; t ÿ vr=cT † ÿ p

locally perpendicular to the crack surface (Tada and Yamashita 1997). This normal axis shall be denoted by xn , and the other two, locally tangential to the crack surface, shall be named xs and xt , in such a way that (xn ; xs ; xt ) forms a right-handed system (Fig. 2). The choice of xs and xt has one degree of freedom corresponding to rotation. We also denote the unit vectors in the three orthogonal directions by n, s and t respectively. The local and global coordinate systems are mutually interchangeable by transformation formulae including:

Dui …n; s† ˆ ni …n†Dun …n; s† ‡ si …n†Dus …n; s† ‡ ti …n†Dut …n; s† Z dS…n† C

617

…37†

ca Du3;a …n; t ÿ r=cT † r2

c l …1 ÿ 2p2 †2 dS…n† 2a Du3;a …n; t ÿ r=cL † ‡ r p 4pcT

Z dS…n† C

ca Du_ 3;a …n; t ÿ r=cL † r

l Du_ 3 …s; t† ÿ 2pcT Z Z Zp 3l ca 5l c dS…n† 2 dv
vDub;b …n; t ÿ vr=cT † ‡ dS…n† 2a Dub;b …n; t ÿ r=cT † Ta …s; t† ˆ r r p 4p 1 C C Z Z l 2 ca l c dS…n† 2 Dub;b …n; t ÿ r=cL † ‡ dS…n† a Du_ b;b …n; t ÿ r=cT † ÿ p r r p 4pcT C C Z cb l l dS…n† 2 Dua;b …n; t ÿ r=cT † ÿ Du_ a …s; t† : ÿ r 4p 2cT

…33†

…34†

C

Equations (33) and (34) coincide with Eq. (A1) and Eq. (3) on ˆ ni …n†oi ; os ˆ si …n†oi ; ot ˆ ti …n†oi : (or (C21)) of Fukuyama and Madariaga (1998) respecGst ˆ si …n†tj …n†Gij Gnn ˆ ni …n†nj …n†Gij tively. Static case The elastostatic counterpart of Eqs. (33) and (34) can be obtained by simply dropping the time dependence:

T3 …s† ˆ ÿ

l …1 ÿ p2 † 2p

l Ta …s† ˆ ÿ 4p

Z C

Z

dS…n† C

ca Du3;a …n† r2

…35†

 c dS…n† …1 ÿ 2p2 † 2a Dub;b …n† r

i cb …36† ‡ 2 Dua;b …n† : r Our Eq. (35) coincides with Eq. (5) of Fukuyama and Madariaga (1995), while our Eq. (36) is a simpli®cation over their Eq. (6).

Gss ˆ si …n†sj …n†Gij Gtt ˆ ti …n†tj …n†Gij

Gtn ˆ ti …n†nj …n†Gij : Gns ˆ ni …n†sj …n†Gij

…38† …39†

This system of local Cartesian coordinates allows us to formulate the BIEs in such a way that all the spatial

xn normal direction crack surface

x t second tangential direction

5 xs Local Cartesian coordinate system first tangential direction For the non-planar crack case, we de®ne a local Cartesian coordinate system, one of the axes of which is always held Fig. 2. Local Cartesian coordinate system

derivatives of the slip Du are those with respect to xs and with xt . This constitutes an advantage over the use of the global 2 2 coordinates …x1 ; x2 ; x3 † as in Zhang (1991), since the slip akl …n†  ‰…1 ÿ 2p †=p Šdkl ‡ 2nk …n†nl …n†; Du is de®ned only along the crack surface. bkl …n†  ‰…1 ÿ 2p2 †=p2 Šdkl ‡ 2sk …n†sl …n†; It should be noted that the symmetry relations (12), as 2 2 well as the equations of motion (19) through Eq. (23), hold ckl …n†  ‰…1 ÿ 2p †=p Šdkl ‡ 2tk …n†tl …n†; true after replacing the indices 1; 2; 3 with n; s; t respectively. dkl …n†  nk …n†sl …n† ‡ sk …n†nl …n† Hereafter we use the subscript z which runs over n, s or t. e …n†  n …n†t …n† ‡ t …n†n …n† kl

618

uk …x; t† ˆ ÿ C

0

Zt

l

Zt dS…n†

ds…Dun Rnn=k ‡ Dus Rsn=k ‡ Dut Rtn=k † 0

C

dS…n†

ˆÿ

Z dsDui …n; s†nj …n†Rij=k …x; t; n; s† ˆ ÿ

Z

k

Making use of the symmetry relation (12) and the equations of motion (19) through Eq. (23) and integrating by parts, this can be rewritten as

Zt dS…n†

l

fkl …n†  sk …n†tl …n† ‡ tk …n†sl …n† :

6 Non-planar crack In the non-planar 3-D crack case, the representation theorem states that: Z

k

…42†

ds‰Dun …nk Rnn=n ‡ sk Rnn=s ‡ tk Rnn=t † ‡ Dus …nk Rsn=n ‡ sk Rsn=s ‡ tk Rsn=t † 0

C

‡ Dut …nk Rtn=n ‡ sk Rtn=s ‡ tk Rtn=t †Š Zt Z ˆ ÿ dS…n† dsDun fnk ‰…k ‡ 2l†on Gnn ‡ kos Gsn ‡ kot Gtn Š ‡ sk ‰…k ‡ 2l†on Gns ‡ kos Gss ‡ kot Gts Š 0

C

Z ‡ tk ‰…k ‡ 2l†on Gnt ‡ kos Gst ‡ kot Gtt Šg ÿ

Zt dS…n†

C

dsfDus ‰nk …lon Gsn ‡ los Gnn † 0

‡ sk …lon Gss ‡ los Gns † ‡ tk …lon Gst ‡ los Gnt †Š ‡ Dut ‰nk …lon Gtn ‡ lot Gnn † ‡ sk …lon Gts ‡ lot Gns † ‡ tk …lon Gtt ‡ lot Gnt †Šg      Zt Z 1 ÿ 2p2 1 ÿ 2p2 GL ‡…sk os ‡ tk ot † 2Gnn ÿ 2GT ‡ GL ˆ ÿl dS…n† dsDun nk on 2Gnn ‡ p2 p2 C

0

Z

Zt dS…n†

ÿl

dsf…Dus os ‡ Dut ot †nk …2Gnn ÿ GT † ‡ ‰Dus sk on …2Gss ÿ GT † ‡ Dut tk on …2Gtt ÿ GT †Š 0

C

‡ 2…Dus tk ‡ Dut sk †on Gst g ;

…40†

where we made use of the symmetry relation (12). Accordingly, rkl …x; t† ˆ kdkl op up ‡ l…ok ul ‡ ol uk †    Zt Z 1 ÿ 2p2 2 dS…n† dsDun …akl on ‡ dkl os ‡ ekl ot †on 2Gnn ‡ G ˆ ÿl L p2 0 C     1 ÿ 2p2 1 ÿ 2p2 GL ‡ …ekl on ‡ fkl os ‡ ckl ot †ot 2Gnn ÿ 2GT ‡ GL ‡ …dkl on ‡ bkl os ‡ fkl ot †os 2Gnn ÿ 2GT ‡ p2 p2 Zt Z  2 dS…n† ds …Dus os ‡ Dut ot †…akl on ‡ dkl os ‡ ekl ot †…2Gnn ÿ GT † ‡ Dus …dkl on ‡ bkl os ‡ fkl ot †on …2Gss ÿ GT † ÿl C

0

‡ Dut …ekl on ‡ fkl os ‡ ckl ot †on …2Gtt ÿ GT † ‡ 2…Dus ekl ‡ Dut dkl †os ot …Gnn ÿ GT † ‡ 2…Dus fkl ‡ Dut bkl †on ot …Gss ÿ GT †  …41† ‡ 2…Dus ckl ‡ Dut fkl †on os …Gtt ÿ GT † ;

rkl …x; t† ˆ ÿl

   1 ds ÿ akl …Dun;s os ‡ Dun;t ot † 2Gnn ÿ 2GT ‡ 2 GL p

Zt

Z

2

dS…n† C

0

Z

Zt

  1 ÿ 2p2 GL ‡ ‰bkl Dun;s os ‡ ckl Dun;t ot ‡ fkl …Dun;s ot ‡ Dun;t os †Š 2Gnn ÿ 2GT ‡ p2   1 ÿ 2p2 G ‡ 2…dkl Dun;s ‡ ekl Dun;t †on 2Gnn ÿ GT ‡ L p2 ÿl

2

619

 ds …Dus;s ‡ Dut;t †‰akl on …2Gnn ÿ GT † ‡ …dkl os ‡ ekl ot †…4Gnn ÿ 3GT †Š

dS…n† 0

C

‡ Dus;s on ‰bkl …2Gss ÿ GT † ‡ 2ckl …Gtt ÿ GT †Š ‡ Dut;t on ‰ckl …2Gtt ÿ GT † ‡ 2bkl …Gss ÿ GT †Š ‡ Dus;t fkl on …4Gss ÿ 3GT † ‡ Dut;s fkl on …4Gtt ÿ 3GT †

ÿ …Dus dkl ‡ Dut ekl †…;s os ‡;t ot †GT ÿ

l2 cÿ2 T

Z

Zt dS…n†

C

De®ning the multiplying coef®cients

Az …s; n†  nk …s†zl …s†akl …n† ˆ 2nk …s†zl …s†nk …n†nl …n† ‡ ‰…1 ÿ 2p2 †=p2 Šdzn Bz …s; n†  nk …s†zl …s†bkl …n† ˆ 2nk …s†zl …s†sk …n†sl …n† ‡ ‰…1 ÿ 2p2 †=p2 Šdzn Cz …s; n†  nk …s†zl …s†ckl …n† ˆ 2nk …s†zl …s†tk …n†tl …n† ‡ ‰…1 ÿ 2p2 †=p2 Šdzn

ds‰D un akl GL ‡ …D us dkl ‡ D ut ekl †GT Š :

…43†

0

Dz …s; n†  nk …s†zl …s†dkl …n† ˆ nk …s†zl …s†…nk …n†sl …n† ‡ sk …n†nl …n†† Ez …s; n†  nk …s†zl …s†ekl …n† …44† ˆ nk …s†zl …s†…nk …n†tl …n† ‡ tk …n†nl …n†† Fz …s; n†  nk …s†zl …s†fkl …n† ˆ nk …s†zl …s†…sk …n†tl …n† ‡ tk …n†sl …n†† ; we have, after taking the limit x ! s 2 C, the following equation for the z-component of the traction across the crack at location s and time t:

Tz …s; t† ˆ nk …s†zl …s†rkl …s; t† ˆ ÿl

2

Zt

Z dS…n† C

0

Z

Zt

   1 ds ÿ Az …Dun;s os ‡ Dun;t ot † 2Gnn ÿ 2GT ‡ 2 GL p

  1 ÿ 2p2 GL ‡ ‰Bz Dun;s os ‡ Cz Dun;t ot ‡ Fz …Dun;s ot ‡ Dun;t os †Š 2Gnn ÿ 2GT ‡ p2   1 ÿ 2p2 GL ‡ 2…Dz Dun;s ‡ Ez Dun;t †on 2Gnn ÿ GT ‡ p2 2

dS…n†

ÿl

C

 ds …Dus;s ‡ Dut;t †‰Az on …2Gnn ÿ GT † ‡ …Dz os ‡ Ez ot †…4Gnn ÿ 3GT † ‡ 2Bz on …Gss ÿ GT †

0

‡ 2Cz on …Gtt ÿ GT †Š ‡ …Dus;s Bz ‡ Dut;t Cz †on GT ‡ Dus;t Fz on …4Gss ÿ 3GT † ‡ Dut;s Fz on …4Gtt ÿ 3GT † o

ÿ…Dus Dz ‡ Dut Ez †…;s os ‡;t ot †GT ÿ

l2 cÿ2 T

Zt

Z dS…n† C

ds‰D un Az GL ‡ …D us Dz ‡ D ut Ez †GT Š : 0

…45†

620

The right hand side of this equation is non-hypersingular, in that the integral terms involve at most ®rst-order spatial derivatives of Gij , GT and GL which behave as 1=r 2 as r ! 0, and are hence integrable in the sense of Cauchy principal values. Equation (45) is our non-hypersingular BIE for the 3-D non-planar crack problem, expressing the traction components in terms of the slip components. In the special case of the planar crack, it can be easily shown that (45) reduces to Eqs. (27) and (29). Substituting the explicit forms of the Green's functions into Eq. (45), we ®nally have, after some algebra,

l Tz …s; t† ˆ ÿ 4p ÿ ÿ

l 4p l 4p

Z C

1 dS…n† 2 r

Z

…48†

‡ 2Dz cn cs ‡ 2Ez cn ct †cn ‡ 2…Dus;t c2s ‡ Dut;s c2t †Fz cn

1

dS…n†

1 ‰ÿ12W4z ‡ 2U1z ÿ 2U2z ÿ 14U3z ‡ 12W10z ÿ 7U5z ÿ 2U6z ÿ 5U7z ÿ U8z ‡ U9z Š…n; t ÿ r=cT † r2

dS…n†

1 ‰12p2 W4z ‡ …1 ÿ 2p2 †U1z ÿ …1 ÿ 4p2 †U2z ÿ 2…1 ÿ 8p2 †U3z ÿ 12p2 W10z ‡ 6p2 U5z ‡ 2p2 U6z r2

C

Z C

l p 4pcT l 4pc2T

W10z …n; s†  …Dus;s ‡ Dut;t †…Az c2n ‡ Bz c2s ‡ Cz c2t

dv
v‰ÿ30W4z ‡ 6U1z ÿ 6U2z ÿ 36U3z ‡ 30W10z ÿ 18U5z ÿ 6U6z ÿ 12U7z Š…n; t ÿ vr=cT †

l ‡ 4p U7z Š…n; t ÿ r=cL † ÿ 4pcT

ÿ

U9z …n; s†  …Dus Dz ‡ Dut Ez †…;s cs ‡;t ct †

Zp

2

ÿ

U8z …n; s†  …Dus;s Bz ‡ Dut;t Cz †cn

Z C

Z C

Z C

1 dS…n† ‰ÿ2W_ 4z ÿ 2U_ 3z ‡ 2W_ 10z ÿ U_ 5z ÿ U_ 7z ÿ U_ 8z ‡ U_ 9z Š…n; t ÿ r=cT † r

1 dS…n† ‰2p2 W_ 4z ‡ U_ 1z ÿ …1 ÿ 2p2 †…U_ 2z ‡ 2U_ 3z † ÿ 2p2 W_ 10z Š…n; t ÿ r=cL † r

1 un …n; t ÿ r=cL †p2 Az ‡ …D dS…n† ‰D us Dz ‡ D ut Ez †…n; t ÿ r=cT †Š ; r

are combinations of the spatial derivatives of the shear components of the slip. In the special case of the planar crack on the x1 x2 -plane, Eq. (46) reduces to a set of equations equivalent to Eqs. (33) and (34).

where

U1z …n; s†  …Dun;s cs ‡ Dun;t ct †Az U2z …n; s†  Dun;s Bz cs ‡ Dun;t Cz ct ‡ …Dun;s ct ‡ Dun;t cs †Fz

…47†

U3z …n; s†  …Dun;s Dz ‡ Dun;t Ez †cn W4z …n; s†  …U1z ÿ U2z ÿ 2U3z †c2n are combinations of the spatial derivatives of the normal (or opening) component of the slip and

U5z …n; s†  …Dus;s ‡ Dut;t †Az cn U6z …n; s†  …Dus;s ‡ Dut;t †…Bz ‡ Cz †cn U7z …n; s†  …Dus;s ‡ Dut;t †…Dz cs ‡ Ez ct † ‡ …Dus;t ‡ Dut;s †Fz cn

…46†

Static case The elastostatic counterpart of Eq. (46) may be obtained by simply dropping the time dependence:

Tz …s† ˆ ÿ

l 4p

Z

dS…n† C

1 ‰3…1 ÿ p2 †W4z ‡ p2 U1z ‡ p2 U2z r2

‡ 2…1 ÿ p2 †U3z ÿ 3…1 ÿ p2 †W10z ‡ …2 ÿ 3p2 †U5z ‡ …1 ÿ p2 †U6z ‡ …1 ÿ 2p2 †U7z ÿ U8z ‡ U9z Š…n† : …49† In the special case of the planar crack on the x1 x2 -plane, Eq. (49) reduces to Eqs. (35) and (36).

7 Off-fault stress field The stress ®eld outside the crack is expressed in a form parallel to Eq. (46):

l rkl …s; t† ˆ ÿ 4p l ÿ 4p l ÿ 4p

Z

1 dS…n† 2 r

C Z

dS…n† C Z

Zp dv
v‰ÿ30W4kl ‡ 6U1kl ÿ 6U2kl ÿ 36U3kl ‡ 30W10kl ÿ 18U5kl ÿ 6U6kl ÿ 12U7kl Š…n; t ÿ vr=cT † 1

1 ‰ÿ12W4kl ‡ 2U1kl ÿ 2U2kl ÿ 14U3kl ‡ 12W10kl ÿ 7U5kl ÿ 2U6kl ÿ 5U7kl ÿ U8kl ‡ U9kl Š…n; t ÿ r=cT † r2

1 ‰12p2 W4kl ‡ …1 ÿ 2p2 †U1kl ÿ …1 ÿ 4p2 †U2kl ÿ 2…1 ÿ 8p2 †U3kl ÿ 12p2 W10kl r2 C Z l 1 2 2 2 ‡ 6p U5kl ‡ 2p U6kl ‡ 4p U7kl Š…n; t ÿ r=cL † ÿ dS…n† ‰ÿ2W_ 4kl ÿ 2U_ 3kl ‡ 2W_ 10kl 4pcT r C Z l 1 ÿ U_ 5kl ÿ U_ 7kl ÿ U_ 8kl ‡ U_ 9kl Š…n; t ÿ r=cT † ÿ p dS…n† ‰2p2 W_ 4kl ‡ U_ 1kl ÿ …1 ÿ 2p2 †…U_ 2kl ‡ 2U_ 3kl † 4pcT r C Z l 1 ÿ 2p2 W_ 10kl Š…n; t ÿ r=cL † ÿ dS…n† ‰D us dkl ‡ D ut ekl †…n; t ÿ r=cT †Š ; un …n; t ÿ r=cL †p2 akl ‡ …D 4pc2T r dS…n†

621

…50†

C

where W4kl , U1kl and other similar symbols are de®ned by uscule akl , bkl etc. Of interest here is the special form of equations parallel to Eqs. (47) and (48) where the subscript Eq. (50) for the planar 3-D crack lying on the x1 x2 -plane, which is given by: z should be replaced by kl and the coef®cients in majuscule Az , Bz etc. should be replaced by those in min-

l r11 …s; t† ˆ ÿ 4p l ÿ 4p l ÿ 4p

Z C Z

C Z

1 dS…n† 2 r

C

dv
v‰12Du3;1 c1 …5c23 ÿ 1† ‡ 12Dua;a c3 …5c21 ÿ 1†Š…n; t ÿ vr=cT †

1

1 dS…n† 2 ‰4Du3;1 c1 …6c23 ÿ 1† ‡ 4Dua;a c3 …6c21 ÿ 1† ÿ 2Du1;1 c3 Š…n; t ÿ r=cT † r dS…n†

l ÿ 4pcT

Zp

1 ‰ÿ4p2 Du3;1 c1 …6c23 ÿ 1† ‡ 2…1 ÿ 2p2 †Du3;2 c2 ÿ 2Dua;a c3 …12p2 c21 ‡ 1 ÿ 4p2 †Š…n; t ÿ r=cL † r2

Z

1 dS…n† ‰4Du_ 3;1 c1 c23 ‡ 4Du_ a;a c3 c21 ÿ 2Du_ 1;1 c3 Š…n; t ÿ r=cT † r C Z l 1 p dS…n† ‰ÿ4p2 Du_ 3;1 c1 c23 ‡ 2…1 ÿ 2p2 †Du_ 3;2 c2 ÿ 2Du_ a;a c3 …2p2 c21 ‡ 1 ÿ 2p2 †Š…n; t ÿ r=cL † ÿ 4pcT r C Z l 1 …1 ÿ 2p2 † dS…n† D …51† u3 …n; t ÿ r=cL † ÿ 2 4pcT r C

Zp l 1 r12 …s; t† ˆ ÿ dS…n† 2 dv
v‰6…Du3;1 c2 ‡ Du3;2 c1 †…5c23 ÿ 1† ‡ 12Du1;2 c3 …5c21 ÿ 1† ‡ 12Du2;1 c3 …5c22 ÿ 1†Š…n; t ÿ vr=cT † 4p r 1 C Z l 1 dS…n† 2 ‰2…Du3;1 c2 ‡ Du3;2 c1 †…6c23 ÿ 1† ‡ Du1;2 c3 …24c21 ÿ 5† ‡ Du2;1 c3 …24c22 ÿ 5†Š…n; t ÿ r=cT † ÿ 4p r C Z l 1 dS…n† 2 ‰ÿ…Du3;1 c2 ‡ Du3;2 c1 †…12p2 c23 ‡ 1 ÿ 4p2 † ÿ 4p2 Du1;2 c3 …6c21 ÿ 1† ÿ 4p2 Du2;1 c3 …6c22 ÿ 1†Š…n; t ÿ r=cL † ÿ 4p r C Z l 1 dS…n† ‰2…Du_ 3;1 c2 ‡ Du_ 3;2 c1 †c23 ‡ Du_ 1;2 c3 …4c21 ÿ 1† ‡ Du_ 2;1 c3 …4c22 ÿ 1†Š…n; t ÿ r=cT † ÿ 4pcT r C Z l 1 p dS…n† ‰ÿ…Du_ 3;1 c2 ‡ Du_ 3;2 c1 †…2p2 c23 ‡ 1 ÿ 2p2 † ÿ 4p2 …Du_ 1;2 c21 ‡ Du_ 2;1 c22 †c3 Š…n; t ÿ r=cL † …52† ÿ 4pcT r Z

C

l r31 …s; t† ˆ ÿ 4p l ÿ 4p l ÿ 4p

Z C Z

C Z

1 dS…n† 2 r

C

dv
v‰12Du3;1 c3 …5c23 ÿ 3† ‡ 12Dua;a c1 …5c23 ÿ 1†Š…n; t ÿ vr=cT †

1

1 dS…n† 2 ‰2Du3;1 c3 …12c23 ÿ 7† ‡ Dua;a c1 …24c23 ÿ 5† ‡ Du1;a ca Š…n; t ÿ r=cT † r dS…n†

l ÿ 4pcT

Zp

1 ‰ÿ2Du3;1 c3 …12p2 c23 ‡ 1 ÿ 8p2 † ÿ 4p2 Dua;a c1 …6c23 ÿ 1†Š…n; t ÿ r=cL † r2

Z

1 dS…n† ‰2Du_ 3;1 c3 …2c23 ÿ 1† ‡ Du_ a;a c1 …4c23 ÿ 1† ‡ Du_ 1;a ca Š…n; t ÿ r=cT † r C Z l 1 p dS…n† ‰ÿ2Du_ 3;1 c3 …2p2 c23 ‡ 1 ÿ 2p2 † ÿ 4p2 Du_ a;a c1 c23 Š…n; t ÿ r=cL † ÿ 4pcT r C Z l 1 dS…n† D ÿ u1 …n; t ÿ r=cT † 4pc2T r

622

…53†

C

l r33 …s; t† ˆ ÿ 4p l ÿ 4p l ÿ 4p

Z C Z

C Z

1 dS…n† 2 r

C

dv
v‰ÿ12Du3;a ca …5c23 ÿ 1† ‡ 12Dua;a c3 …5c23 ÿ 3†Š…n; t ÿ vr=cT †

1

1 dS…n† 2 ‰ÿ4Du3;a ca …6c23 ÿ 1† ‡ 2Dua;a c3 …12c23 ÿ 7†Š…n; t ÿ r=cT † r dS…n†

l ÿ 4pcT

Zp

1 ‰4Du3;a ca …6p2 c23 ‡ 1 ÿ 2p2 † ÿ 2Dua;a c3 …12p2 c23 ‡ 1 ÿ 8p2 †Š…n; t ÿ r=cL † r2

Z

1 dS…n† ‰ÿ4Du_ 3;a ca c23 ‡ 2Du_ a;a c3 …2c23 ÿ 1†Š…n; t ÿ r=cT † r C Z l 1 p dS…n† ‰4Du_ 3;a ca …p2 c23 ‡ 1 ÿ p2 † ÿ 2Du_ a;a c3 …2p2 c23 ‡ 1 ÿ 2p2 †Š…n; t ÿ r=cL † ÿ 4pcT r ZC l 1 u3 …n; t ÿ r=cL † : dS…n† D ÿ 2 4pcT r

…54†

C

These equations were also derived by Aochi, Fukuyama and The special form of Eq. (55) for the planar 3-D crack lying Matsu'ura (1999a, b). As c3 ! 0, Eqs. (53) and (54) reduce to on the x1 x2 -plane is: Z equations equivalent to Eqs. (34) and (33), respectively. l 1 Other components, not listed here, may be obtained by r11 …s† ˆ ÿ dS…n† 2 ‰…1 ÿ p2 †Du3;1 c1 …1 ÿ 3c23 † 2p r considering the symmetry between the coordinates x1 and C x2 . Static case The elastostatic counterpart of Eq. (50) may be obtained by simply dropping the time dependence:

rkl …s† ˆÿ

l 4p

Z dS…n† C

1 ‰3…1 ÿ p2 †W4kl ‡ p2 U1kl ‡ p2 U2kl r2

‡ 2…1 ÿ p2 †U3kl ÿ 3…1 ÿ p2 †W10kl ‡ …2 ÿ 3p2 †U5kl 2

2

‡ …1 ÿ p †U6kl ‡ …1 ÿ 2p †U7kl ÿ U8kl ‡ U9kl Š…n† : …55†

‡ …1 ÿ 2p2 †Du3;2 c2 ÿ …1 ÿ p2 †Dua;a c3 …1 ‡ 3c21 †

…56† ‡ Du2;2 c3 Š…n† Z    l 1 dS…n† 2 Du3;1 c2 p2 ÿ 3…1 ÿ p2 †c23 r12 …s† ˆ ÿ 4p r C

  ‡ Du3;2 c1 p2 ÿ 3…1 ÿ p2 †c23   ‡ Du1;2 c3 1 ÿ 2p2 ÿ 6…1 ÿ p2 †c21   ‡Du2;1 c3 1 ÿ 2p2 ÿ 6…1 ÿ p2 †c22 …n†

…57†

l r31 …s† ˆ ÿ 4p

Z dS…n† C

1 ‰2…1 ÿ p2 †Du3;1 c3 …1 ÿ 3c23 † r2

‡ 2…1 ÿ p2 †Dua;a c1 …1 ÿ 3c23 † …58† ÿ Du2;2 c1 ‡ Du1;2 c2 Š…n† Z l 1 r33 …s† ˆ ÿ …1 ÿ p2 † dS…n† 2 ‰Du3;a ca …1 ‡ 3c23 † 2p r C

‡ Dua;a c3 …1 ÿ 3c23 †Š :

…59†

where

U11k …n; s†  Dun nk cn U12k …n; s†  Dun …sk cs ‡ tk ct † W13k …n; s†  …U11k ‡ U12k †c2n ˆ Dun c2n ck U14k …n; s†  Dus …nk cs ‡ sk cn † ‡ Dut …nk ct ‡ tk cn †

As c3 ! 0, Eqs. (58) and (59) reduce to Eqs. (36) and (35), W15k …n; s†  …Dus cs ‡ Dut ct †cn ck respectively.

…61†

8 are combinations of the slip components. Of interest here Off-fault displacement field is the special form of Eq. (60) for the planar 3-D crack The displacement ®eld outside the crack is obtained by lying on the x1 x2 -plane, which is given by: substituting the explicit form of the Green's functions into Eq. (40):

1 uk …s; t† ˆ ÿ 4p 1 ÿ 4p 1 ÿ 4p

Z

1 dS…n† 2 r

C Z

Zp dv
v‰30W13k ÿ 18U11k ÿ 6U12k ‡ 30W15k ÿ 6U14k Š…n; t ÿ vr=cT † 1

1 dS…n† 2 ‰12W13k ÿ 8U11k ÿ 2U12k ‡ 12W15k ÿ 3U14k Š…n; t ÿ r=cT † r

C Z

dS…n† C

1 ‰ÿ12p2 W13k ÿ …1 ÿ 8p2 †U11k ÿ …1 ÿ 4p2 †U12k ÿ 12p2 W15k ‡ 2p2 U14k Š…n; t ÿ r=cL † r2

Z

1 ÿ 4pcT

1 dS…n† ‰2W_ 13k ÿ 2U_ 11k ‡ 2W_ 15k ÿ U_ 14k Š…n; t ÿ r=cT † r C Z 1 1 p dS…n† ‰ÿ2p2 W_ 13k ÿ …1 ÿ 2p2 †…U_ 11k ‡ U_ 12k † ÿ 2p2 W_ 15k Š…n; t ÿ r=cL † ; ÿ 4pcT r

…60†

C

1 u1 …s; t† ˆ ÿ 4p ÿ ÿ

1 4p 1 4p

Z C

1 dS…n† 2 r

Z

1

1 ‰2Du3 c1 …6c23 ÿ 1† ‡ 3Du1 c3 …4c21 ÿ 1† ‡ 12Du2 c1 c2 c3 Š…n; t ÿ r=cT † r2

dS…n†

1 ‰ÿDu3 c1 …12p2 c23 ‡ 1 ÿ 4p2 † ÿ 2p2 Du1 c3 …6c21 ÿ 1† ÿ 12p2 Du2 c1 c2 c3 Š…n; t ÿ r=cL † r2

C

1 ÿ 4pcT

dv
v‰6Du3 c1 …5c23 ÿ 1† ‡ 6Du1 c3 …5c21 ÿ 1† ‡ 30Du2 c1 c2 c3 Š…n; t ÿ vr=cT †

dS…n† Z C

Zp

Z C

1 p ÿ 4pcT

1 dS…n† ‰2Du_ 3 c1 c23 ‡ Du_ 1 c3 …2c21 ÿ 1† ‡ 2Du_ 2 c1 c2 c3 Š…n; t ÿ r=cT † r Z C

1 dS…n† ‰ÿDu_ 3 c1 …2p2 c23 ‡ 1 ÿ 2p2 † ÿ 2p2 Du_ 1 c3 c21 ÿ 2p2 Du_ 2 c1 c2 c3 Š…n; t ÿ r=cL † r

…62†

623

1 u3 …s; t† ˆ ÿ 4p 1 ÿ 4p 1 ÿ 4p 624

Z C Z

C Z

1 dS…n† 2 r

C

dv
v‰6Du3 c3 …5c23 ÿ 3† ‡ 6Dua ca …5c23 ÿ 1†Š…n; t ÿ vr=cT †

1

1 dS…n† 2 ‰4Du3 c3 …3c23 ÿ 2† ‡ 3Dua ca …4c23 ÿ 1†Š…n; t ÿ r=cT † r dS…n†

1 ÿ 4pcT

Zp

1 ‰ÿDu3 c3 …12p2 c23 ‡ 1 ÿ 8p2 † ÿ 2p2 Dua ca …6c23 ÿ 1†Š…n; t ÿ r=cL † r2

Z

1 dS…n† ‰2Du_ 3 c3 …c23 ÿ 1† ‡ Du_ a ca …2c23 ÿ 1†Š…n; t ÿ r=cT † r C Z 1 1 p dS…n† ‰ÿDu_ 3 c3 …2p2 c23 ‡ 1 ÿ 2p2 † ÿ 2p2 Du_ a ca c23 Š…n; t ÿ r=cL † : ÿ 4pcT r

…63†

C

The component u2 may be obtained by considering the symmetry between the coordinates x1 and x2 . Static case The elastostatic counterpart of Eq. (60) may be obtained by simply dropping the time dependence:

1 uk …s† ˆ 4p

Z

dS…n† C

1 ‰3…1 ÿ p2 †W13k ‡ p2 U11k r2

ÿ p2 U12k ‡ 3…1 ÿ p2 †W15k ‡ p2 U14k Š…n† : …64† The special form of Eq. (64) for the planar 3-D crack lying on the x1 x2 -plane is:

u1 …s† ˆ

1 4p

Z

dS…n† C 2

1 ‰3…1 ÿ p2 †…Du3 c3 ‡ Dua ca †c1 c3 r2

‡ p …Du1 c3 ÿ Du3 c1 †Š…n† Z 1 1 dS…n† 2 …Du3 c3 ‡ Dua ca †…n† u3 …s† ˆ 4p r

…65†

C

 ‰3…1 ÿ p2 †c23 ‡ p2 Š :

…66†

9 Discussion and conclusion In the present article, we have enlarged Fukuyama and Madariaga's (1998) integration by parts technique and combined it with Tada and Yamashita's (1997) local Cartesian coordinate system, to derive, for the ®rst time, a nonhypersingular time-domain BIE for the 3-D non-planar crack in a fully explicit form. We have given not only the BIE describing the traction-slip relation on the crack, but also similar expressions for the stress and displacement ®elds off the fault. Although we are not going to detail it in this article, we have con®rmed that, in the special case in which the crack con®guration is independent of one coordinate, all the equations that we have derived reduce to corresponding equations given in the Tada and Yamashita

(1997) paper on the 2-D non-planar crack theory. This con®rms the correctness of our algebra. The present study completes the set of non-hypersingular BIEM theory for crack dynamics, that is based on the displacement discontinuity method, the time-domain representation and the integration by parts technique. In practical implementation to non-planar crack analysis, however, direct use of the non-planar crack equation is expected to face many obstacles, not least the dif®culty of meshing the curved surface and the extremely complex expression of the equation. This fact favors the approach of Aochi, Fukuyama and Matsu'ura (1999a, b), who propose to numerically model 3-D non-planar crack dynamics problems by approximating the curved crack surface by a patchwork of small planar elements. In their approach, the in¯uence of slip on one planar element on the traction on another element is to be evaluated by way of simpler equations (51)±(54), which are special cases of the more complicated equation (46) or (50). In approximating a curved crack surface by a patchwork of small planar elements, however, there is a point that requires attention. The use of piecewise-constant interpolation is fairly common in the discretization of the slip distribution on the crack surface. Tada and Yamashita (1996) pointed out that, when a curved crack is modeled as a series of smaller planar elements and when piecewiseconstant interpolation is applied to slip, slip ``gets stuck'' at spurious joints of differently oriented elements and results in a smaller expected slip than if the crack is modeled as a series of curved elements that are smoothly joined. Seelig and Gross' (1997) modeling method, which they put to use in the 2-D case, hints at one way to prevent spurious suppression of slip in such a case. They numerically modeled shear slip on a non-planar crack, in the same way as they would have modeled a traction-free open crack that was allowed to slip both in the shear and opening modes. They then numerically penalized the negative opening component of the slip, which would have meant material penetration. If we follow their method and penalize both positive and negative opening components of slip, this

approach can be used to model pure shear cracks, free from the spurious suppression of slip at element joints. Our present theoretical study thus marks a major step toward a more realistic 3-D numerical analysis of earthquake fault dynamics. With a view to future numerical modeling, Aochi, Fukuyama and Matsu'ura (1999a, b) derived a set of discretization kernels for Eqs. (51)±(54), or the stress components that would be expected in response to a uniform slip of a unit slip-rate taking place during a unit length of time on a quandrangular fault patch of a unit area. These discrete kernels will be used when one numerically analyzes the behavior of a 3-D nonplanar crack, approximating the crack by a patchwork of small planar elements and based on the piecewise-constant approximation for the slip-rate. This or other sorts of practical application of the 3-D non-planar crack dynamics theory are much awaited, for a better understanding of the seismic rupture phenomena on faults with complex geometry.

 o 1 GT ˆ ÿ d…t ÿ s ÿ r=cT † oxi 4plr2  r _ ÿ d…t ÿ s ÿ r=cT † ci  cT o 1 GL ˆ p2 ÿ d…t ÿ s ÿ r=cL † oxi 4plr2  r _ ÿ d…t ÿ s ÿ r=cL † ci cL

…A4†

…A5†

for the elastodynamic case. In the elastostatic case, they are

Gij …x; n† ˆ

1 ‰…1 ÿ p2 †ci cj ‡ …1 ‡ p2 †dij Š 8plr

…A6†

and

o 1 h Gij ˆ ÿ3…1 ÿ p2 †ci cj ck ÿ …1 ‡ p2 †ck dij oxk 8plr2 i …A7† ‡…1 ÿ p2 †…ci djk ‡ cj dik † Appendix A: 3-D Green's functions o 1 The explicit forms of the displacement Green's functions G ˆ …ÿci † …A8† of 3-D elasticity theory Gij and its ®rst-order spatial deri- oxi T 4plr 2 vatives are: o 1 GL ˆ …ÿp2 ci † : …A9† 1  oxi 4plr2 Gij …x; t; n; s† ˆ ‰I ‡ d…t ÿ s ÿ r=cT †Šdij 4plr Appendix B: Proof to Eq. (30) ‡ ‰ÿ3I ÿ d…t ÿ s ÿ r=cT † Fukuyama and Madariaga (1995) derived their Eq. (D1), or (30) of the present article, in the following way. De®ne a 2 ‡ p d…t ÿ s ÿ r=cL †Šci cj …A1† 2-D polar coordinate system …r; u† on the surface of a plane crack, with the origin at x and the moving radius with n ÿ x. From the relation Zp     o o I  dv
vd…t ÿ s ÿ vr=cT † Du_ i …n; t ÿ r=c† ˆ ÿca Du_ i …n; t ÿ r=c† or u;t ona tÿr=c 1 1 Z1=cT ui …n; t ÿ r=c† …B1† ÿ D 2 c dk
kd…t ÿ s ÿ kr† ˆ ÿcT it follows that 1=cL Z 1 c2T …t ÿ s† ui …n; t ÿ r=c† dS…n† D ˆÿ ‰ ÿH…t ÿ s ÿ r=c † ‡ H…t ÿ s ÿ r=c † Š T L r r2 …A2†

and

C

du

ˆ

o 1  Gij ˆ ‰15I ‡ 6d…t ÿ s ÿ r=cT † oxk 4plr 2 r _ ÿ s ÿ r=cT † ÿ 6p2 d…t ÿ s ÿ r=cL † ‡ d…t cT r _ ÿ s ÿ r=cL †Šci cj ck ÿ p2 d…t cL ‡ ‰ÿ3I ÿ 2d…t ÿ s ÿ r=cT † ‡ p2 d…t ÿ s ÿ r=cL † r _ ÿ s ÿ r=cT †Šck dij ÿ d…t cT ‡ ‰ÿ3I ÿ d…t ÿ s ÿ r=cT † ‡ p2 d…t ÿ s ÿ r=cL †Š …A3†  …ci djk ‡ cj dik †

Z1

Z2p 0

drD ui …n; t ÿ r=c† 0

Z2p ˆ ÿc

du‰Du_ i …n; t ÿ r=c†Š1 rˆ0

0

Z2p ÿc

Z1 du

0

drca 0

Z

ˆ 2pcDu_ i …x; t† ÿ c

o Du_ i …n; t ÿ r=c† ona dS…n†

C

o c Du_ i …n; t ÿ r=c† a ; r ona …B2†

which was to be demonstrated.

625

References

626

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