Full text of DOI:10.1186/BF03353319

Earth Planets Space, 54, 1181–1194, 2002
Elastic property of damaged zone inferred from in-situ stresses and its role on
the shear strength of faults
Kiyohiko Yamamoto, Namiko Sato, and Yasuo Yabe
Graduate School of Science, Tohoku University, Sendai 980-8578, Japan
(Received January 4, 2002; Revised August 14, 2002; Accepted August 29, 2002)
The Nojima fault in Hyogo prefecture, Japan, ruptured during the 1995 Hyogo-ken Nanbu earthquake (MJMA
=
7.3). The stress measurements at sites close to this fault have revealed that the direction of the largest horizontal
stress is almost perpendicular to the strike of this sub-vertical fault and that, in the zone within about 100 m from the
fault core axis, the ratio of the largest shear stress to the normal stress is significantly small compared with that of the
outside. It is thus the logical consequence that the principal stress outside the zone tends to direct perpendicularly
to the fault plane. A model called the fracture process model is introduced for the relationship between fracture
strength and elastic property of rocks. Making use of this model on the assumption that the observed shear stress
equilibrates to the shear strength of damaged zone, it is found that the elastic wave velocities estimated from the
stress well explain the observed velocities of damaged zone. This model suggests further that the friction coefficient
of fault can be smaller than 0.15 due to the characteristic deformation of damaged zone and that the pressurized
fluid is not essential for the formation of weak faults.
1. Introduction
have found neither the evidence that the shear strength of
The strength of faults is essential for modeling the defor- the fault is normal in magnitude nor the evidence that the
mation process of the earth’s crust. The constitutive law for fault zone is saturated with pressurized fluid. Further, the
frictional sliding is substantial for modeling the earthquake deep drillings of 12.2 km at greatest that have been con-
generation process. For these studies, it is most basic to ducted worldwide so far have suggested that the upper crust
clarify the shear strength of faults that have been naturally is hydraulically permeable in general (Barton et al., 1995;
formed. The frictional coefficient of faults inferred from lab- Huenges et al., 1997; Zoback and Townend, 2001).
oratory experiments is equal to or more than about 0.6 (e.g.
The frictional strength of simulated faults is expressed to
Byerlee, 1978). On the other hand, the frictional strength be the product of the friction coefficient and the normal stress
for the San Andreas Fault inferred from the heat flow data to the fault plane. According to Byerlee (1978), the fric-
is very small (Brune et al., 1969). The recent data of well tion coefficient is about 0.85 for small normal stress on fault
borehole breakouts and of hydraulic fracturing measurement planes and about 0.60 for larger normal stress. The com-
appear to support that the San Andreas Fault is weak. The pressive strength of intact rock specimens is approximately
friction coefficient of the fault is estimated to be about 0.1 represented by the Coulomb’s criterion. The criterion is rep-
or so in appearance (Zoback et al., 1987). The San Andreas resented to be the sum of cohesion force and the frictional
Fault and some faults in the San Andreas Fault system have strength. The frictional strength is expressed to be the prod-
been shown to be weak from the seismological and geolog- uct of the internal friction coefficient and the normal stress.
ical data and some faults even for intra-plate earthquakes as The internal friction coefficient ranges from 0.5 to 1.5 (e.g.
well (Jones, 1988; Oppenheimer et al., 1988; Chester et al., Paterson, 1978). This coefficient is almost comparable in
1993; Iio, 1997).
magnitude to the coefficient of friction. This permits us to
Some ideas have been proposed to compensate the gap in suspect that the both friction coefficients are the appearances
the frictional strength between the San Andreas Fault and the of the same mechanical property of materials. This property
simulated faults in laboratories. It is well known that pres- has to be taken into consideration as one of the constraints
surized pore fluid reduces the compressive or shear strength for the discussion of the frictional strength.
of rocks (e.g. Handin et al., 1963). One of the ideas is that
The fracture strength is defined from the macroscopic
the pressurized pore fluid is sealed in fault zone (e.g. Sibson viewpoint, or in terms of average stresses, as a rule. In order
et al., 1988). Deep drilling was conducted at Cajon Pass at to compare the friction coefficient of a fault with that from
about 3 km from the San Andreas Fault to measure the stress laboratory experiments, the friction coefficient of the fault
and the pore fluid pressure directly at depths (e.g. Coyle and should be determined from the average stress over the fault
Zoback, 1988; Zoback and Healy, 1992). However, they plane at the time when the rupture starts to propagate at a
point. It is clear thus that the stresses only at a few sites are
c
Copy rightꢀ The Society of Geomagnetism and Earth, Planetary and Space Sciences
insufficient to determine the friction coefficient, even if they
are close to the fault.
(SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan;
The Geodetic Society of Japan; The Japanese Society for Planetary Sciences.
1181

1182
K. YAMAMOTO et al.: THE ROLE OF DAMAGED ZONE ON TNE STRENGTH OF FAULTS
The stress is one of the independent state variables. The in the vertical. If a fault parallel to the maximum shear plane
stress measurement is thus the only way to clarify the begins to rupture at r = rs, rs corresponds to the apparent
strength of faults in principle. The measurement can be friction coefficient of the fault. The parameter r is referred
generally performed only at shallow depths compared with to as r-value or potential stress hereafter.
the seismogenic depths and the spatial distribution of the 2.2 Stress orientation
sites for measurement may usually be sparse compared with
Figure 1 shows the directions θhmax of the largest horizon-
the spatial variation of stress. For these reasons, alternative tal stress determined by DRA (Sato et al., 1999). IKH is
methods are necessarily required to get the information about located near the southern end of the buried fault extended
the stress state at depths.
from the surface break. Sato (1999) has pointed out that the
This paper presents a simple model of faults in order rotation of θhmax at IKH occurs in the zone of small r-value
to discuss the frictional strength of faults from the macro- of about 0.2. According to Ito et al. (1997), there is a fracture
scopic viewpoint. The model shows that the elastic property zone around the depth of 670 m. Except for shallow depths
of damaged zone plays an important role on the strength. of IKH, θhmax approximately lie in the NW-SE direction.
Yamamoto (1995, 1998) has proposed a model called the They are almost perpendicular to the fault strike. The ap-
fracture process model that yields the shear strength of rocks parent friction coefficient at TSM, which is calculated from
microscopically fractured. Making use of the model together the stresses measured by DRA at depths about 310 and 415,
with the theories of effective elastic constants of composites, never exceeds the magnitude of 0.3 (Yamamoto and Yabe,
the elastic wave velocities of damaged zone are estimated 2001).
from the in-situ stresses measured at sites close to the No-
The directions determined by HF and the observation of
jima fault. The velocities are compared with those of the bore hole breakouts lie in the NW-SE direction, too (Ito et
damaged zone for some faults to examine the applicability of al., 1997; Tsukahara et al., 2001; Ikeda et al., 2001). Refer-
the model of faults. Further, the strength of fault is discussed ring to the horizontal strain from 1885 to 1985 observed by
on the elastic properties of damaged zone thus estimated.
Geographical Survey Institute, Japan (Geographical Survey
Institute, 1997), the direction θhmax is nearly equal to that of
the largest contraction of the Osaka bay area adjacent to the
sites. This implies that the NW-SE direction of θhmax has not
been formed after the faulting, but suggests that the stress re-
flects the tectonic situation of the fault even in the vicinity of
the fault in this case.
2. Brief Review of the Stress Field near the Nojima
Fault
Stresses have been measured in holes drilled at three sites
close to the Nojima fault immediately after the 1995 Hyogo-
ken Nanbu earthquake (MJMA = 7.3). Ikeda et al. (2001) and
Tsukahara et al. (2001) employed the hydraulic fracturing
technique (HF) for the in-situ measurement and Sato et al.
(1999) and Yamamoto and Yabe (2001) applied the deforma-
tion rate analysis (DRA) to rock core sample recovered from
the holes. DRA has been developed to measure the stresses
based on the rock property of stress memory (Yamamoto et
al., 1990). The stresses measured by DRA are expected to be
the stress before the earthquake, while the stresses by HF are
those after the earthquake. Some details of the DRA mea-
surement on the Nojima cores are described in our previous
paper (Yamamoto and Yabe, 2001).
The measurement sites are shown in Fig. 1. The HF and
the DRA measurements were performed for the same holes
at sites, TSM and IKH and for the different holes close
to each other at HRB. The holes are distinguished by the
subscripts GSJ and NIED, respectively. The results of DRA
are reviewed in this section, being compared with the results
by HF. The arrows show the directions θhmax of the largest
horizontal stress determined by DRA.
2.3 Shear stress magnitude
The r-values at TSM are determined to be about 0.5. This
means that in the vicinity of the fault there are areas where
r-value is close to the fracture strength of the intact rocks
that are free from major flaws, while the apparent friction
coefficient of the fault is not larger than 0.3 (Yamamoto
and Yabe, 2001). This is caused by the fault that is not
optimally oriented. Tsukahara et al. (2001) performed the
HF measurement at two depths near 1,500 m in the same
hole. The stresses at these depths are of almost pure strike-
slip regime and the r-values for these depths are about 0.16.
The small r-value and the strike slip regime mean that the
stress field at the depths reduces toward the lithostatic state
and that the apparent friction coefficient never exceeds 0.16.
Although these r-values at the large depths are very small
compared with those at small depths by DRA, the apparent
friction coefficients appear to be common to both the small
and the large depths.
Ikeda et al. (2001) carried out the measurement by HF in
Hole HRBNIED drilled by NIED. This hole locates close to
Hole HRBGSJ. Their measurement has revealed that the site
is in the reverse fault regime and the r-values are between
0.3 and 0.4 for the depth smaller than about 800 m. On the
other hand, the stresses are in the strike slip regime and the r-
values are nearly equal to or smaller than 0.2 at depths near
1,200 m. The characteristics of the small r-value and the
strike-slip regime for the depth near 1,200 m are quite similar
to those at the large depths of TSM.
2.1 Definition of r-value
In order to discuss the stresses in relation to the shear
strength of the earth’s crust, a parameter r defined by
r = (σL − σS)/(σL + σS)
(1)
is introduced, where σL and σS are the largest and the small-
est principal stress of compression. The shear strength of
rocks increases proportionally to the normal component of
the stress on the fault surface in general. Therefore, the pa-
rameter r may be thought as an index of the potential of the
stress field for shear fracture. The value of r is calculated
here on the assumption that one of the principal directions is
Sato (1999) obtained the stresses around 350 m in depth
for HRBGSJ by DRA. The stresses are characterized with the
strike slip regime and the relatively small r-values. These

K. YAMAMOTO et al.: THE ROLE OF DAMAGED ZONE ON TNE STRENGTH OF FAULTS
1183
Fig. 1. The site locations for the stress measurement by DRA along the Nojima fault in Awaji Is., Hyogo Pref., Japan. The arrows indicate the directions
of the largest horizontal stress of compression determined by DRA at the depths indicated near the respective arrows (data after Sato, 1999). The map
is modified from Awata et al. (1996).
0.6
r
a)
b)
0.4
0.2
0
HRBNIED (HF)
HRBGSJ (DRA)
TSM(DRA)
TSM(HF)
200
0
500
400
1000
depth, m
1500
0
distance from fault core axis, m
Fig. 2. Relationships of r-value to the distance from the fault core axis (a) and to depth (b). The difference in symbol means the difference in the method
of stress measurement. HRBNIED (HF) indicates the data of hydraulic fracturing at HRB by Ikeda et al. (2001), HRBGSJ (DRA) does the data of DRA
at HRBGSJ by Sato (1999). TSM (DRA) and TSM (HF) respectively mean the data of DRA at TSM by Sato (1999) and those of hydraulic fracturing by
Tsukahara et al. (2001). This figure is modified from Sato (1999).
characteristics are different from those at shallow depths of seen in this figure. It may be thus reasonable to conclude that
HRBNIED, but rather similar to those at greater depths of the small shear stress is caused not by the change in the stress
HRBNIED and TSM. The HRBGSJ locates approximately with depth but depends on the distance of measuring points
at a distance of about 50 m, while HRBNIED at a distance from the fault core axis. Further, the small r-value may be
of about 400 m from the fault on ground surface. Taking caused not by the stress drop due to the faulting associated
account of these distances relative to the fault, the small r- with the earthquake, because the small values are determined
value is considered to reflect the property of the fault zone. by the both methods of DRA and HF. The small magnitude
Sato (1999) showed the relationship of the r-value to the of shear stress may be one of the characteristics of the stress
distance of the measuring points from the fault core axis. The state in the zone close to the center of a fault. The width of
relationship is shown in Fig. 2 together with the relationship the zone of small r-value is about 100 m in the case of the
to depth. The small shear stress near the fault core is clearly Nojima fault.

1184
K. YAMAMOTO et al.: THE ROLE OF DAMAGED ZONE ON TNE STRENGTH OF FAULTS
3. Role of Damaged Zone on Shear Strength of or near the damaged zone. However, the small strength of
Fault
damaged zone is not a sufficient condition for weak faults
from a macroscopic viewpoint.
We make the three assumptions as follows to discuss the
3.1 Definition of damaged zone
Here, the terminology by Chester et al. (1993) or Tanaka
et al. (1999) is employed for fault structure as a rule. In the strength of faults as defined in Section 1. 1) Fault is com-
case of the Nojima fault, Ito et al. (1996) and Tanaka et al. pletely locked before faulting from the macroscopic view-
respectively have pointed out from the geophysical and the point. This permits us to take the frictional stress out of
geological data that there are damaged rocks of a few tens consideration and to take the deformation around the fault
meter width a side along the fault core axis. The r-value to be elastic. The frictional strength may be called the shear
is small in the zone within about 100 m from the fault core strength at this time. 2) Fault plane consists of asperity ar-
axis, as described above. The small r-value is considered eas and aperture areas as shown in Fig. 4. Asperities mean
thus to be originated from the small shear stress in the zone the parts of a fault that have the same strength as the host
containing the fault core and damaged rocks around the fault rock. Damaged zone means the apertures filled with damage
core. We call the zone as the damaged zone. Since the rocks. 3) The fracture of asperities means the fracture of a
measured stresses by DRA are considered to be those before fault.
the earthquake, the small r-values obtained by DRA imply
There may be many modes of rupture propagation in ac-
that the damaged zone has grown up with repeated faulting. tual, for example, abrupt slipping, stable sliding, those on
The stress state in the zone is schematically illustrated in a single fault plane, those on multiple fault planes and so
Fig. 3(a). The largest principal stress directs almost perpen- on. These modes of rupture propagation may be the ways to
dicularly to fault plane in order to contract the zone. This release the strain energy stored in rocks surrounding a fault
simplification of the stress state is considered to be justifiable and/or faults. The rupture propagation initiates at or just after
from the observational results of internal structure of faults the time when the applied stress reaches the fracture strength
by Chester et al. (1993) except for the simple shear zone of rocks. The rupture propagation is here presumed to be in-
in the fault core axis. According to them, the strain in the dependent from the loading process defined for the process
fault zone may be modeled as simple shear in the fault core just prior to the time when rupture propagation starts. The
and nearly fault normal contraction in the bounding damaged process is called the fracture process here. The rupture prop-
zone and host rock.
3.2 Bounds of shear strength of fault
Rocks around the fault core have heavily damaged. This
agation should be thus argued distinctively from the fracture
process.
Asperities are assumed to behave as intact rocks. Accord-
implies that the strain is concentrated around the fault core. ing to the Coulomb’s criterion for failure, shear strength in-
If the strain is elastic, large strain means large stress. Figure 2 creases with an increase in the normal stress on the shear
shows that the shear stress is decreased in spite of the large plane where shear fracture occurs. It is assumed for simplic-
strain concentrated near the fault core. It is reasonable thus ity that the effect of the geometry or the spatial distribution of
to consider that the small shear stress is caused by the small asperities on the stress concentration can be ignored. In the
shear strength of damaged rocks. Their small shear strength case that the apertures are void or hydraulically permeable,
can also explain the reason why one of the principal stress
axes is nearly perpendicular to the fault plane at least in
Fig. 3. a) Schematic illustration of the principal directions of compressive
stress acting to the damaged zone. The coordinates taken for the damaged
zone is shown together. One of the principal directions is taken to be Fig. 4. Image of the horizontal cross-section of a vertical fault for the
orthogonal to the fault plane. The principal stresses are denoted by σi .
For the Nojima fault, the x1 axis is taken to be orthogonal to the fault
plane and the x3 axis is taken to be in the depth direction and the stress
σ1 is the largest. b) A rock specimen under the loading test of tri-axial
compression and the coordinate defined for the specimen.
discussion of the macroscopic strength. Asperity means the part of a
fault zone where the rocks on the both sides of a fault plane completely
contact with each other. The fracture zone is defined as the aperture filled
with fractured rocks. σn and τ denote the average normal stress and the
average shear stress applied to the fault plane.

K. YAMAMOTO et al.: THE ROLE OF DAMAGED ZONE ON TNE STRENGTH OF FAULTS
1185
the average normal stress σas at asperities may be written as Here, the crack density c is defined by
σas = σn/S
(2)
c = φ/α
σn being the normal stress averaged over the fault plane and
S (ꢀ 1) being the fraction of the asperity area to the total
area of fault plane. Then, the strength of the fault τs may be
expressed by
φ and α, respectively, are porosity and aspect ratio of cracks.
u is the applied shear stress normalized by the shear strength
of the specimen. The expression (6) is derived based on the
experimental data for the confining pressure ranging from
0.040 to 0.355 GPa.
τs = Sτ0 + μSσas,
(3)
C f (σC ) is a function proportional to σ−p, p is approxi-
C
where τ0 and μ are the cohesion and the internal friction
coefficient of asperities. Therefore, when S is small, the
shear strength τs of the fault is expressed by
mately 1/2. When f (σC ) is taken to be unity at σC = 0.1
GPa, C is approximately 10. If the lithostatic pressure ap-
proximates the average stress at a depth, C f (σC ) may be
written by
(4)
τs ≈ μσn.
The strength is seen to be independent of S. This may be a
rough explanation for the frictional strength that is approx-
imately equal to the internal frictional coefficient of intact
rocks and is approximately constant independently of con-
tact area.
Consider the case that the apertures are filled with com-
pletely incompressible material with negligibly small rigid-
ity for simplicity, as illustrated in Fig. 4. In this case, the
normal stress in the asperities is not expected to change with
the change in S, while the shear stress τas at asperities builds
up with a decrease in S. The strength τs of fault is thus writ-
ten as
C f (σC ) ≈ 10 × {(ρrockgd − ppore)/0.1}−1/2
,
(7)
where ρrock, g and d, respectively, are the density of over-
burden rocks, the gravitational acceleration and the depth.
ppore denotes the pore pressure in the tensile cracks. When
the pore pressure is equal to the hydrostatic pressure, σC is
given by
C f (σC ) ≈ 10 × {gd(ρrock − ρpore)/0.1}−1/2
,
(8)
where ρpore is the density of pore fluid. When pore fluid is
pressurized or sealed, crack density may be larger than that
in the case of the hydrostatic pore pressure.
The function G(u) represents the fracture density. The
fracture density is defined to be the volume fraction occu-
pied in a specimen by the volume elements that have lost
their strengths by shear micro-fracture. Here, the following
are assumed; 1) all the elements have the same size, 2) their
shear strengths obey the power distribution, and 3) the frac-
tured elements do not support any load. Then, the expression
for G(u) is written as
τs ≈ Sμσn.
(5)
The strength τs decreases with a decrease in the fraction S
or an increase in the aperture area or the damaged zone.
This implies that the shear deformation of fracture zone,
which is larger than the contraction, makes the strength of
fault decrease. This simple model suggests that the elastic
property and structure of fracture zone play an important role
on the strength of faults.
G(u) = s0[u/(1 − G(u))]n,
(9)
4. Relationship between Stress and Elastic Prop-
erty
4.1 Fracture process model
s0 being the normalization factor. See Appendix A for the
derivation of G(u) in more detail.
The normalized shear stress u is written by u
∼
r/r f ,
=
Micro-cracks are increasingly produced in rock specimens
under tri-axial loading of compression when applied axial
stress is monotonously increased. This means that the ap-
plied stress is equal to the fracture strength of the specimen
that contains the micro-cracks produced so far. The relation-
ship of crack density to applied stress difference obtained
for the specimen may be thus interpreted as the relationship
between the strength of the specimen and the density of the
cracks included in the specimen.
where r is given by (1). r f denotes the shear strength of
specimens and the equality holds when the cohesion can
be neglected in the Coulomb’s criterion. The model for
Eq. (9) is called fracture process model. The experimental
data show that tensile cracks are kept open even under high
confining pressure. The expression (6) suggests that the
stress concentrations around tips of shear cracks produced
by shear fracture keep the tensile cracks open.
The behavior of G(u) for m = 2 is illustrated for example
in Fig. 5(a). The function G(u) yields the upper and the
lower bound of the fracture density for a specimen sustaining
the applied stress u. The hatched area enclosed by the curve
ABC and the line CA indicates thus the realizable domain of
(u, G) in the specimen. The value of u on the curve ABC
means the fracture strength of the specimen of which the
fracture density is G. The applied stress at B means thus
the ultimate strength of the specimen. In the case of G larger
than vc in Fig. 5(a), rupture starts to propagate throughout
the specimen just at the time when the increasing applied
stress reaches to the stress on the curve BC, provided that the
Figure 3(b) illustrates a specimen, where the Cartesian
coordinate is defined in order that x1 in the direction par-
allel to the loading axis. Matsushima (1960) measured the
P-wave velocity on rock specimens of a kind of granite un-
der confining pressure σC . From his experimental data, Ya-
mamoto (1995, 1998) has derived the relationship between
applied shear stress and density of tensile cracks, provided
that cracks orient their surfaces parallel to x1 and distribute
their normal lines symmetrically around x1. The relationship
is given as follows:
c = C f (σC )G(u).
(6)

1186
K. YAMAMOTO et al.: THE ROLE OF DAMAGED ZONE ON TNE STRENGTH OF FAULTS
Fig. 5. Fracture density G(u) as a function of applied shear stress u divided by the ultimate shear strength. a) Explanation of G(u). The stress at the point
B means the ultimate fracture strength of a specimen. The path BD means that all volume loses the strength at once or the path for the macroscopic
fracture. The path BC means the post-failure state. b) The behavior of the function G(u) for some values of m. r is the potential stress defined by (1) in
the text.
loading system is elastic. The state on the curve BC can be age zones should be confirmed by comparing the deductions
exactly realized in laboratories, only when a completely rigid from the model with the observed field data.
apparatus is used to apply the displacement to a specimen.
4.3 Calculation of elastic constants
In the case of G smaller than vc, G may increase tracing
The seismic wave velocities of damaged zones have been
the curve AB with an increase in u ≈ r/r f or r for rock determined for some faults by seismic reflection profil-
specimens that have suffered no damages. In the case of ing (e.g. Feng and McEvilly, 1983) and by analysis of
G larger than vc, a rock specimen has to deform until the trapped waves (e.g. Li et al., 1998, 2000; Nishigami, 2000;
applied stress decreases to the magnitude the specimen can Kuwahara and Ito, 1999). Although the elastic property is
sustain, that is the stress on the curve BC. The post-failure naturally expected to be anisotropic from the crack orienta-
state means here the state on the curve BC. The applied tion, the observations and the analyses have been performed,
displacement rather than the applied stress may control the provided that zones are isotropic. For the comparison, we
stress in the rocks under the post-failure state.
calculate the velocities of the zone for the crack density c
4.2 Estimation of tensile crack density in damaged zone estimated by the above procedure on the assumption that the
Rocks in the damaged zone have been heavily damaged isotropic property stands for the averaged one. The elastic
and have small compressive or shear fracture strength. The constants are calculated by making use of the method called
stress outside the zone should equilibrate to the strength of new self-consistent scheme (NSC) proposed by Yamamoto
the zone, because the strength outside the zone is larger than et al. (1981) and Norris (1985).
that inside. These mean that the damaged zone is under the
It has been suggested in the preceding section that elastic
post-failure state. For the present analysis, it is assumed that anisotropy of damaged zone plays an important role on the
the expression (9) is applicable throughout the range of frac- shear strength of fault. Especially small rigidity and large
ture density G. This expression is derived from the unique Young’s modulus to the stresses on the plane parallel to the
principle throughout the range of G that the strength distri- fault are expected to bring about the large concentration of
bution is expressed by a power function. For this reason, the the shear stress relative at asperities to the normal stress.
assumption may be justifiable, although the expression has Such concentration of shear stress cannot be expected for the
been experimentally confirmed to hold for G smaller than vc damaged zone where crack orientation is isotropic. For the
or on the AB branch.
reason, the elastic anisotropy of damaged zone is necessar-
If the r-value is determined from the stresses near the ily required to be known for the discussion of the strength of
damaged zone, the value of G is determined for u = r/r f , fault. The elastic constants of damaged zone are calculated
provided that the damaged zone is in the post-failure state. by approximation method for weakly interacted inclusions
Figure 5(b) shows the function G(u) calculated for some val- (WIA) by Yamamoto (1980, 1995), assuming that the sur-
ues of parameter m. The value of m has been estimated to face normals of cracks in damaged zones are parallel to the
range roughly from 5 to 10 for some kinds of rocks and the fault plane and orient symmetrically around the fault nor-
crust (Yamamoto, 1998). Assuming that the shear strength mal. The elastic property of damaged zone is thus symmet-
of the rocks outside the damaged zone is represented by ric around the x1-axis (see Fig. 3(a)). The method of WIA
r f = 0.6, the fracture density G(u) is estimated to be a value is introduced in Appendix B. The precision of the method
about 0.8 for the damaged zone where r = 0.2. The density and the effects of crack shape and pore material have been
c of tensile cracks at a depth can be estimated by making use introduced briefly in our previous paper (Yamamoto et al.,
of (7) or (8) on the assumption that the r-value is constant 2001).
for depth. The applicability of the present model to dam-
These methods, NSC and WIA are based on the theory

K. YAMAMOTO et al.: THE ROLE OF DAMAGED ZONE ON TNE STRENGTH OF FAULTS
1187
0
m = 5.0
α = 10-3
5
r = 0.20 (PF)
(G ≈ 0.8)
Damaged zone
r = 0.20 (PF)
(G ≈ 0.8)
10
15
Damaged zone
r
0.30
0.40
0.50
0.60 (Critical)
0.40 (Post Failure)
0.20
(Post Failure)
0
1.0
1.0
0.5
V /V
s s
0
0.5
Host
Host
V /V
p
p
Fig. 6. P- and S-wave velocities of the crust subjected to the potential stress r. The each curve shows the ratio of the velocity at the depth to that of the
host rock. The potential stress r equal to 0.2 (Post-Failure) corresponds to that in the damaged zone.
of Eshelby (1957). The cracks are theoretically isolated the elastic property of the wave-guide. According to them,
from one another. Therefore, the deformation of the zone the width is about 20 to 50 m and the velocity of S-waves is
pressurizes the pore fluid in cracks, when the cracks are 40 to 66% of that outside the wave-guide. Li et al. (2000)
saturated with fluid. Even if the zone is permeable, it needs also obtained the S-wave velocity structure of the Landers
a time for fluid to transfer from a crack to other cracks, fault zone from trapped waves. They determined the veloc-
although the time for the transference is unknown. The ity to be from 55% to 69% of the S-wave velocity of the host
fluid may be pressurized for the quick deformation of the rock for the depth from 1 to 10 km. Their observed velocity
zone such as caused by seismic waves, while the cracks appears to be larger by about 10% than that of the present
may behave as if they are void for the slow deformation of estimate at the Nojima fault.
the zone over years. In the present paper, the elastic wave
One easy way to diminish the disagreement in the case of
velocities of damaged zone are calculated for the isolated the Landers fault zone may be to reduce the fracture density.
cracks saturated with water, while its elastic response to Li et al. (1998) have pointed out that the ratio of the travel
tectonic stress change is done for the void cracks.
time of P-waves to that of S-waves recovered after the 1992
Landers, California, earthquake cannot be explained well by
the theory for isotropic orientation of cracks. This ratio may
imply the anisotropy in the elastic property of damaged zone.
Taking such ambiguities in fracture density and anisotropy
into consideration, it is not considered that the disagreement
is definitive. Taking account of the results by Li et al. (1998),
Kuwahara and Ito (1999) and by Nishigami (2000) together,
the S-wave velocity estimated by the present model appears
to approximate to the observed one of damaged zone. For the
further discussion in detail, the anisotropy should be taken
into consideration.
McGarr et al. (1982) reviewed the in-situ stresses near the
San Andreas Fault in the Mojave Desert. The site MOJ1 at
the nearest to the San Andreas Fault is about 2 km distant
from the fault. The measurement was performed to 218 m
deep at the greatest at this site. The values of r calculated
for the stress data are about 0.19 to 0.21 at depths larger than
100 m. These r-values are close to those for the damaged
zone of the Nojima fault. We assume that this small r-value
is caused by the small strength of the damaged zone of the
San Andreas Fault, although more stress measurements are
required to conclude this. On this assumption, the fracture
5. Seismic Wave Velocities of Damaged Zone
The fracture density G has been estimated to be about
0.8 for the Nojima fault, as stated in the preceding sec-
tion. Assuming that confining pressure and pore pressure
respectively are lithostatic and hydrostatic, the crack den-
sity c is determined as a function of depth by substituting
ρrock = 2.7 × 103 kg/m3 and ρpore = 1.0 × 103 kg/m3 into
(8). The seismic wave velocities are calculated for c thus de-
termined. The Poisson’s ratio, the incompressibility and the
density of the host rock respectively are taken to be 1/4, 60
GPa, and 2.7 × 103 kg/m3 in order to approximate the prop-
erty of granite. The calculated velocities are shown in Fig. 6
as the ratios to those of the host rock, provided that the aspect
ratio α of cracks is taken to be equal to 10−3
.
The velocity ratio is about 70% for the P-waves and about
33% for the S-waves at 3 km in depth. Although the veloc-
ities gradually increase with an increase in depth, the ratio
is about 80% for P-waves and about 60% for S-waves at 15
km in depth. Li et al. (1998), Kuwahara and Ito (1999), and
Nishigami (2000) have observed the waves that propagate
being trapped along a fault. They estimated the width and

1188
K. YAMAMOTO et al.: THE ROLE OF DAMAGED ZONE ON TNE STRENGTH OF FAULTS
Fig. 7. P-wave velocities of the damaged zone and the host rock shown on the P-wave velocity structure determined by Feng and McEvilly (1983). Solid
circles indicate the P-wave velocities at the depths taken as those of the damaged zone to be compared with the calculated ones, and the velocities at the
points where the lines HOST E and HOST W intersect the velocity contours are taken as the velocities of the host rock for the calculation. The figure
is modified from Feng and McEvilly (1983).
0
density is inferred to be about 0.8 for the San Andreas Fault
as well as the Nojima fault.
Feng and McEvilly (1983) interpreted seismic reflection
profiling data to get the P-wave velocity structure of the San
5
Andreas Fault zone. Figure 7 shows the structure they have
determined. It is seen that the low velocity zone reaches
the depth of about 25 km and the velocity in the fault zone
10
decreases to about 65% of the velocity outside the fault zone
at the smallest. Here, the smallest velocity is regarded as the
velocity of damaged zone. Assuming that the velocities on
15
the lines denoted by HOST W and HOST E in Fig. 7 are the
velocities of the host rocks at the depths, the velocities of
the damaged zone are calculated for the host rocks by using
ESTIMATED (HOST_W)
20
OBSERVED
ESTIMATED (HOST_E)
the velocity ratio calculated for G = 0.8, that is shown in
Fig. 6. The velocities of the damaged zone thus estimated are
25
shown in Fig. 8. Solid circle indicates the observed velocity.
Square and diamond respectively denote the calculated ones
for HOST W and for HOST E.
0
2
4
6
V , km/s
p
It is found that the calculated velocities are nearly equal to
the observed ones for the depth smaller than about 15 km.
This calculation was performed for hydrostatic pore pres-
sure as described earlier, provided that the fracture density is
constant for depth. The coincidence of the calculated veloc-
ity with the observed one suggests that the P-wave velocity
of the damaged zone of the San Andreas Fault is explained
by the fracture process model and no pressurized or sealed
fluid is required for the explanation. In order to keep the ten-
sile cracks open under pressure at depth, the stress but for
high pore pressure is needed. Shear fractures in the damaged
zone may have been produced under the compressive stress
nearly vertical to the fault plane. The stress concentration
caused by the shear fractures keeps tensile cracks open as
suggested from the tensile cracks in rock specimens under
tri-axial compression test. This may be one of the proofs to
Fig. 8. Comparison of the estimated P-wave velocity with observed one
for the damaged zone of the San Andreas Fault shown in Fig. 7. Circles,
rectangles and diamonds indicate the observed velocities, the estimated
velocities from the host rocks on the line HOST W, and those on the line
HOST E, respectively.
justify the application of the fracture process model to dam-
aged zones.
6. Discussion on Shear Strength of Fault
The Cartesian coordinate defined in Fig. 3(a) is employed
for the definition of the elastic constants of damaged zone.
Here, Eii denotes the stiffness for the relationship between
eii and τii , and μi j is the rigidity for the relationship between
ei j and τi j . We denote the shear stress, τ12 (= τ13), on the
fault plane by τ and the normal stress, σ11, to the fault plane

K. YAMAMOTO et al.: THE ROLE OF DAMAGED ZONE ON TNE STRENGTH OF FAULTS
1189
by σ. Then, the strength of fault may be expressed in terms
The Young’s modulus E11 and the rigidity μ12 or 13 of the
of τ/σ, as described in the previous section. We define γ as fracture zone are calculated making use of WIA for void
follows;
cracks of their aspect ratio α equal to 10−3. Figure 9 shows
γ ꢀ−1 as a function of depth for the fracture density G = 0.8
(r ≈ 0.2), which has been determined for the damaged zone
of the Nojima fault. Solid curve denotes the case of the hy-
drostatic pore pressure and dash curve does the case of the
void cracks. The value of γ ꢀ should be larger than that of
the solid curve, if the damaged zone is hydraulically imper-
meable. When the pore-pressure in the cracks is hydrostatic,
the value of γ ꢀ is larger than 10 at small depths and decreases
with an increase in depth to about 4 at the depth of 15 km.
Although the shear strength of fault becomes larger with an
increase in depth, the strength can be about 0.15 in terms of
μ even at the depth of 15 km, provided that μ is taken to be
0.6 for host rocks. It may be concluded that faults can be
weak in the upper crust even without pressurized or sealed
fluid in damaged zones.
γ ≡ ꢀ(τ/σ)ASP/ꢀ(τ/σ)TCT
,
(10)
where ꢀ(τ/σ)ASP indicates the variation of the potential
stress at asperities and ꢀ(τ/σ)TCT is that of the tectonic po-
tential stress. This index γ means the concentration rate of
the potential stress at asperities for an increase in the tectonic
potential stress.
If the shear strength of the asperities is constant, the shear
strength of faults may be proportional to γ −1, provided that
the fracture of asperities means the fracture of the fault. If the
fraction S of asperity areas to the whole fault plane area is
negligibly small, the average elastic shear strain on the fault
plane may be approximately equal to τTCT/μ12 or 13. Since
the strain at asperities is considered to be nearly equal to this
strain, the shear stress at asperities is expressed by
ꢀτTCTμHOST/μ12 or 13
.
7. Summary
By the similar way, the normal stress at asperities is ex-
The Nojima fault in Awaji Is., Hyogo prefecture, Japan,
ruptured during the 1995 Hyogo-ken Nanbu earthquake
(MJMA = 7.3). Stresses have been measured at sites close
to this fault. The measurements have revealed that the max-
imum horizontal stress is almost perpendicular to the strike
of this nearly vertical fault at all the sites except for shallow
pressed by ꢀσTCT HOST/E11, where μHOST and EHOST are
the rigidity and the Young’s modulus of the host rock, re-
spectively. Therefore, the ratio γ is expressed by
E
γ ≈ (μHOST/μ12 or 13)/(EHOST/E11) ≡ γ ꢀ.
(11)
The ratio γ cannot directly be known. The index γ ꢀ de- depths at a site. These orientations of stress suggest that the
fined by (11) is employed standing for γ to estimate the Nojima fault is weak, although this fault is not located at a
strength of fault on the assumption that the fraction S of as- plate boundary.
perity area is negligibly small. The ratio γ ꢀ is here called as
the concentration index of potential stress.
The other important result is that in the zone within about
100 m from the fault core axis, the shear stress is very small
compared with that at a distance from the fault. It can be
concluded that the principal direction of stress perpendicular
to the fault plane results from the small shear stress in the
zone. It is considered that the rocks near the fault core have
been heavily damaged to have small compressive or shear
strength. The small shear stress measured near fault core
can be interpreted thus as the reflection of the small shear
strength of damaged zone.
γ
'
Large
10
Small
2.5
2
5
0
5
m=5.0
G=0.8 (r ≈ 0.2)
Some problems seem however to be left in order to con-
clude that the fault is weak from the macroscopic viewpoint.
One of them is that the stress measurements have been lim-
ited at shallow depths. Another is that the measurements
have not been performed uniformly and densely along the
fault. It may be also the problem that the mechanism to
weaken faults is unclear. The main aims of the present pa-
per are to derive the answers to these problems based on the
above results of the stress measurements.
Damaged rocks have not only small fracture strength but
also small elastic constants. If a fault is locked from the
macroscopic viewpoint and the fault plane consists of asper-
ities and damage zone, the shear strength of the fault will
be controlled by the stress concentration at asperities, which
may be brought about by deformation of damaged zone. It is
thus important to determine the elastic constants of damaged
zone. The elastic property is also important for the investiga-
tion of the stress state in damaged zones at great depths that
cannot be known by the direct stress measurement.
ρ
σC = rockgd
10
15
20
25
Ppore
> watergd
ρ
ρ
ρ
water)gd
σC = (
-
rock
0
0.2
0.4
(γ- =(μ12/E11)/(μHost/EHost)]
' ) [
-1
Fig. 9. The variation of the concentration index γ ꢀ with depth. See the text
for the definition of γ ꢀ. The solid and dash lines respectively indicate the
hydrostatic pore pressure and the void cracks. The domain indicated by
small arrows are the range of γ ꢀ when fluid in cracks is pressurized or
sealed.
A model called the fracture process model is introduced
to estimate the density of tensile cracks in damaged zone

1190
K. YAMAMOTO et al.: THE ROLE OF DAMAGED ZONE ON TNE STRENGTH OF FAULTS
from the measured stresses near the zone. The theoretical the volume may lose their strengths by the faulting. We may
method called NSC has been established to calculate the reasonably to consider that the volume and the fault plane are
elastic wave velocities of cracked media. Making use of the approximately identical in linear dimension. The volume is
fracture process model and the NSC method, the elastic wave called a microfracture element. The strength of a microfrac-
velocities of damaged zone are estimated from the stresses ture element is thus identical to the strength of the weakest
measured near the Nojima fault.
physical elements in the microfracture element.
The seismic wave velocities are calculated on the assump-
It is assumed that a microfracture element consists of q
tions that the ratio r of the maximum shear stress to the nor- physical elements. When the strength distribution function is
mal stress is constant at 0.2 throughout depth and that the represented by Pe(us) for the physical elements, the strength
damaged zone is hydraulically permeable. Further, these as- distribution function P(us) for microfracture elements may
sumptions are presumed to hold for every fault for the com- be expressed by
parison of the estimated results with the field data. The cal-
P(us) = 1 − [1 − Pe(us)]q
(e.g. Freudenthal, 1968). Here, the strength us is defined in
the range where Pe(us) and P(us) have the values between 0
and unity. When the distribution function Pe(us) is expressed
by a power function,
(A.1)
culated velocity for P-waves is found to explain almost com-
pletely the smallest velocity at the center of the San Andreas
Fault zone to about 15 km in depth. Although the calculated
S-wave velocity explains the observed one not so completely
as that of P-waves, the observed seismic wave velocities that
are consistent with the calculated ones suggest that damaged
zone is hydraulically permeable in the upper crust.
Pe(us) = sums ,
(A.2)
The ratio γ of shear stress concentration to normal stress
concentration at asperities is theoretically estimated from the
elastic anisotropy of damaged zone on the assumption that
asperities occupy the negligibly small area of fault plane.
The ratio is estimated to be larger than 10 for about 3 km
deep and approximately 4 for 15 km deep. These results
imply that the shear strength of fault can be smaller than
0.15 in terms of friction coefficient. The pressurized fluid
may not be essential for weak faults in this context, although
the pressurized fluid might play an important role on the
propagation of rupture.
P(us) may be written by
P(us) ≈ qsums .
Here, we redefine the strength distribution function for
microfracture elements to be
P(us) = s0ums ,
(A.3)
where
s0 ≈ qs.
(A.4)
When the fractured volume has completely lost their strength
(kꢀ = 0 in Fig. A.2), the load to a specimen should be
sustained by the potential microfracture elements. For a
specimen where the fractured volume fraction is expressed
by G(u) at the applied stress u, the stress ue in the potential
elements may be expressed by
Acknowledgments. Prof. M. Ando, Kyoto University∗, provided
our stress measurement in the program for the research of earth-
quake fault by drilling. Dr. K. Takemura, Kyoto University, pre-
pared core samples for our measurement. Dr. H. Ito and Dr. Y.
Kuwahara, Geological Survey of Japan∗∗, readily permitted us to
use the core samples of Ikuha and Hirabayashi for our measure-
ment. Prof. Tsukahara, Shinsyu Univ. and Dr. Ikeda, NEID∗∗∗
,
u
prepared us the results of their stress measurements. Further, the
discussions with them and the suggestions from them are very use-
ful to progress this study. We have to thank all of them for their
helpful support to this study. We would like to express special
thanks to emeritus Prof. Hirasawa, Tohoku University, for his sug-
gestive comments and introductions on the problems included in the
process of faulting. The suggestions and the advices from Dr. Iio
and an anonymous reviewer are very helpful for us to improve this
manuscript. We also express very much thanks to them. (∗Nagoya
Univ. at present; ∗∗National Institute of Advanced Industrial Sci-
ence and Technology, ∗∗∗Hokkaido Univ. at present).
ue =
.
(A.5)
1 − G(u)
The volume fraction G(u) and the stress ue are called the
fracture density and the effective stress, respectively.
When applied stress u is increased from 0 to u, the effec-
tive stress is increased from 0 to ue and the elements of their
strength smaller than ue have been fractured. Since the vol-
umes are the same for all microfracture elements, G(u) may
be expressed by
ꢀ
ꢁm
u
G(u) = P(ue) = so
.
(A.6)
Appendix A. Fracture Process Model
1 − G(u)
We consider a rock specimen subjected to an axial stress
under confining pressure. It is assumed that the specimen
consists of small volume elements called physical elements
as shown in Fig. A.1. Further, the element volume is as-
sumed to be a characteristic constant proper to the rock. The
element volume can be taken thus to be a measure of speci-
men volume.
This function G(u) is a two valued function of u. The largest
value of u, that is the ultimate strength u f of the specimen,
is derived from du/dG = 0 as
u f = (1 − vc)vc1/m(so)−1/m
vc = 1/(1 + m).
(A.7)
When one of physical elements fractures by an increment
in applied axial stress, the rupture started at the element prop-
agates over some numbers of physical elements to form a
fault plane. The strain energy stored in the volume surround-
ing the plane may be released and the physical elements in
The ultimate strength u f of a specimen is attained at G = vc,
vc is called the critical volume fraction for fracture. Using
(A.4), the ultimate strength u f is written by
u f ≈ (1 − vc)vc1/m(qs)−1/m
.
(A.8)

K. YAMAMOTO et al.: THE ROLE OF DAMAGED ZONE ON TNE STRENGTH OF FAULTS
1191
SPECIMEN
MICROFRACTURE
ELEMENT
PHYSICAL ELEMENTS
M
q
MICROFRACTURE
ELEMENTS
Weakest Element
Fig. A.1. Schematic illustration of a rock specimen modeled for explanation of the fracturing process. A specimen consists of M microfracture elements
and a microfracture element consists of q physical elements. Rupture started at a physical element propagates over a number of physical elements to
form a microfracture element.
τ
MICROFRACTURE
ELEMENTS
POTENTIAL
ELEMENT
k'τ
FRACTURED
ELEMENT
(k' = 0)
τ
e
SPECIMEN
τ
Fig. A.2. Illustration for explanation of effective stress τe. Potential elements mean the microfracture elements that have not been fractured under the
stress τ applied to specimen. kꢀτ indicates the stress in fractured elements remaining after fracturing. kꢀ = 0 represents the complete loss of strength.
Since q represents the size of microfracture elements, this
We denote strain component by ep and stress component
expression means that the ultimate strength u f decreases by τq, elastic constants and elastic compliances, respectively,
with an increase in the size of microfracture. The expres- by Cqp and Spq, where p, q = 1–6. When we express strain
sion means that the ultimate strength decreases, when the and stress fields, respectively, by
microfracture size increases with an increase in specimen
e = {ep} and τ = {τq},
size. Therefore, Eq. (A.8) expresses the size effect of ulti-
mate strength.
the stress-strain relationship is written by
τ = Ce or e = Sτ,
where
The function G(u) represents the process to the macro-
scopic fracture and yields the definition of ultimate strength
in terms of the strength distribution for microfractures when
applied stress is increased up to u f . The fracture process
model means the model used for the derivation of the func-
tion G(u).
C = {Cpq
}
and S = {Sqp}.
Appendix B. Weak Interaction Approximation
We consider a composite in which k inclusions are embed-
ded in homogeneous isotropic matrix. When the strain and
the stress averaged over the composite are denoted by ea and
(WIA)
Basic Equations

1192
K. YAMAMOTO et al.: THE ROLE OF DAMAGED ZONE ON TNE STRENGTH OF FAULTS
τa, respectively, ea and τa may be written by
stresses in the composite (see Watt et al., 1976). The sec-
ond terms in the right hand sides of (B.4) correspond to the
interaction energy defined by Eshelby (1957).
k
ꢂ
ea = (1 − v)ekM
τa = (1 − v)τkM
+
vj eIj
Method for Approximation
j=1
It is assumed that there is a preexisting composite, of
which the elastic constants are known to be Ck. We consider
the case that a matrix part of the composite is replaced with
a number of inclusions under the displacement condition.
Here, the volume fraction of newly introduced inclusions
is indicated by vk+1. If vk+1 is so small that the variation
in strain in preexisting matrix and inclusions are negligibly
small, the strain energy Wsk+1 after this replacement may be
written as
k
ꢂ
+
vj τ Ij .
(B.1)
j=1
The strain e and the stress τ with superscript M or I mean
the strain and the stress averaged over matrix or an inclusion.
The subscript j indicates the quantities of the jth inclusion.
vj is the volume fraction of the jth inclusion. The total
volume fraction of inclusions v is thus written by
2Wsk+1 = (ea · Ck+1ea)
k
ꢂ
v =
vj .
≈ (ea · Ckea) + vk+1(ea · CM ekp+1).
(B.5)
j=1
Here, ekP+1 is the polarization strain. The elastic constants
The following expressions are derived from (B.1);
Ck+1 after the replacement may be thus determined by eval-
uating ekp+1
.
k
ꢂ
ea = eu +
vj ePj ,
In principle, ekp+1 cannot be determined without solving
the deformation of matrix and inclusions in newly produced
composite under the displacement condition. However, it is
not the present task to solve this problem. Eshelby (1957)
presented the expression for the strain in a spheroidal inclu-
sion embedded in infinite isotropic matrix strained by ea. He
yielded further the polarization strain or the stress free strain
ep in the following form,
j=1
where ePj = −(SIj − SM )τ Ij and eu = SM τa
(B.2)
k
ꢂ
τa = τu +
vj τ Pj ,
j=1
where τ Pj = (CIj − CM )eIj and τu = CM ea
ep = −Tea.
(B.6)
and
ePj = SM τ Pj and τ Pj = CM ePj .
The expressions for T are given in his paper. We employ
his solution as that for the finite size of composite in the
expectation that the precision of the solution gradually drops
with an increase in volume fraction of inclusions.
We assume here that the spheroidal inclusions have the
same elastic property and the same shape and are oriented
to the same direction. It may be rational to consider that the
matrix to be newly replaced with the inclusions is strained by
euk for the polarization strain ekp+1, as seen from (B.2). Using
the expression for eu in (B.2) and Ck, ekp+1 may be expressed
by
Here, ePj and τ Pj are the polarization strain and the polar-
ization stress, respectively (Hill, 1963). eu and τu are the
uniform strain and the uniform stress in the matrix that is
presumed to fill up the composite. When the displacements
are given at the outer boundary of the composite, the average
strain field in a composite should satisfy the upper equation
of (B.2) and when the traction is applied to the outer bound-
ary, the average stress field in a composite should satisfy the
lower equation of (B.2).
The strain energy density W is generally defined by
ekp+1 = −TSM Ckea.
Making use of (B.7), we can derive the following expres-
sions,
(B.7)
ꢃ
1
W =
(τ(x) · e(x))dv,
(B.3)
2
V
x being the position in the composite. Denoting the strain
energy density for the displacement condition by Ws and that
for the stress condition by Wc. For the composite whose
effective elastic constants are Ck and Sk, the strain energy
densities may be expressed by
(ea · Ck+1ea) = (ea · Ckea) − vk+1(ea · CM TSM Ckea)
= (ea · CM (I − vk+1T)SM Ckea),
(B.8)
on the condition that vk+1 is sufficiently small. Equating the
coefficients of eiaeaj on the both sides of the equation, the
2Wsk = (ea · Ckea)
effective elastic constants Ck+1 are determined as follows;
k
ꢂ
vj (ea · CM epj )
1
= (ea · CM ea) +
{Ck+1
}
i j
=
[{CM (I − vk+1T)SM Ck}i j
+ {CM (I − vk+1T)SM Ck}ji ].
2
j
(B.4)
2Wck = (τa · Skτa)
= (τa · SM τa) +
(B.9)
k
ꢂ
vj (τa · SM τ pj ).
From the stress condition, we obtain
j
1
{Sk+1
}
i j
=
[{(I + vk+1T)SM CM Sk}i j
+ {(I + vk+1T)SM CM Sk}ji ].
2
The equations (B.4) hold for macroscopically uniform com-
posites, when there are no body forces and no internal
(B.10)

K. YAMAMOTO et al.: THE ROLE OF DAMAGED ZONE ON TNE STRENGTH OF FAULTS
1193
By successive use of (B.9) or (B.10) and
where Gikj ’s are
Gk11 = 2P11
k
ꢂ
Gk12 = (P12 + P21 + P13 + P31)/2
v =
vj + vk+1
,
Gk13 = (P12 + P13 + P21 + P31)/2
Gk22 = 2{3(P11 + P33) + P44 + P13 + P31}/8
Gk23 = 2{P22 + 3(P23 + P32) + P33 − P44}/8
Gk33 = 2P22
j=1
(B.15)
for j from 1 to k + 1, we can determine the effective elastic
constants of composites with the inclusions oriented to the
same direction. It has been confirmed that the same effective
elastic constants are determined either from (B.9) or (B.10).
Application to Arbitrarily Oriented Inclusion
G44 = (P11 + P33 + P44 − P23 − P32)/2
G55 = G66 = (P55 + P66)/2.
The tensor T is represented in the coordinate system that
is taken parallel to the principal directions of a spheroid that
is defined as the shape of each inclusion. Here, we denote
(strain, stress) in the coordinate of an inclusion by (eꢀ, τꢀ),
and (strain, stress) in the coordinate taken to the composite
by (e, τ). For the inclusion of which orientation is specified
to the coordinate taken to the composite by (ϑ, ϕ), (eꢀ, τꢀ)
may be related to (e, τ) by
In the case of isotropic orientation, the changes are written
as follows;
A ≡ [2(P13 + P31) + 3P33 + 2P44]/15
B ≡ [8(P13 + P31) + 2P33 − 2P44]/15
C ≡ 2(2P33 + 3P44)/15
dC1k1 = 8P11/15 + A
dC1k2 = [{2P11 + 5(P12 + P21)}/15 + B]/2
dC1k3 = dC1k2
eꢀ = L(ϑ, ϕ)e and τꢀ = L(ϑ, ϕ)τ,
(B.11)
dC2k2 = 0.45P11 + [(P12 + P21) + P66]/24 + A
(B.16)
dC2k3 = {0.3P11 + 0.25(P12 + P21)
− P66/12 + B}/2
where L(ϑ, ϕ) is the matrix for the transformation of tensor.
For an inclusion of which orientation is specified by (ϑ, ϕ),
the expression (B.7) may be rewritten as
dC4k4 = [{9P11 − 4(P13 + P31)}/15
+ {P66 − (P12 + P21)}/6 + C]/2
dC5k5 = [{4(P11 − P13 − P31) + 5P66}/15 + C]/2
dC6k6 = dC5k5.
ꢀp
e
= −TSM L(ϑ, ϕ)Ckea.
(B.12)
k+1
Further, we define
We consider the case that a number of inclusions are re-
placed with matrix at the (k + 1)th step of iteration. If
the total volume vk+1 of the newly introduced inclusions is
very small and the distribution of the orientation is given by
ꢁ(ϑ, ϕ), Eq. (B.8) may be expressed by
α ≡ (dCik1 + dCik2 + dCik3)/(CiM1 + CiM2 + CiM3 )
(B.17)
for i = 1, 2, or 3
β ≡ dCkj j /CMj j for j = 4, 5, or 6,
the incompressibility K and the rigidity μ of composite hav-
ing inclusions of the volume fraction v can be obtained from
the following expressions,
(ea · Ck+1ea) = (ea · Ckea)
− vk+1 ꢁ(ϑ, ϕ)(L(ϑ, ϕ)ea · PL(ϑ, ϕ)Ckea)dϑdϕ
P ≡ CM TSM .
ꢃ
κ = κM exp(−αv)
(B.18)
(B.13)
μ = μM exp(−βv).
Here, K M and μM are the incompressibility and the rigidity
of matrix.
Equating the coefficients of eiaeaj on the both sides of the
equation, we can get the approximate values of the effec-
tive elastic constants of composite having arbitrarily oriented
inclusions in isotropic matrix. The second term on the right
hand side of (B.13) yields the increment of the effective elas-
tic constants {dCikj }. In order to yield explicit formulae, we
take x3ꢀ -axis to the normal direction of circular cross-section
of inclusion, where xiꢀ (i = 1, 2, 3) denotes the coordinate
taken to each inclusion.
When inclusions are oriented in order that their circular
cross-sections are parallel to the x1-axis and their normal di-
rections distribute symmetrically around the axis, the incre-
ments of the effective elastic constants may be written by
References
Awata, Y., K. Mizuno, Y. Sugiyama, R. Imura, K. Shimokawa, K. Okumura,
and E. Tsukuda, Surface fault ruptures on the northwest coast of Awaji
Island associated with the Hyogo-ken Nanbu Earthquake of 1995, Japan,
Zisin 2, 49, 113–124, 1996 (in Japanese with English abstract).
Barton, C. A., M. D. Zoback, and D. Moos, Fluid flow along potentially
active fault in crystalline rock, Geology, 23, 683–686, 1995.
Brune, J. N., T. L. Henyey, and R. F. Roy, Heat flow, stress and rate of slip
along the San Andreas fault, California, J. Geophys. Res., 74, 3821–3827,
1969.
Byerlee, J., Friction of rocks, Pure and Appl. Geophys., 116, 615–626, 1978.
Chester, F. M., J. P. Evans, and R. L. Biegel, Internal structure and weaken-
ing mechanisms of the San Andreas fault, J. Geophys. Res., 98, 771–786,
1993.
Coyle, B. J. and M. D. Zoback, In situ permeability and fluid pressure
measurements at ∼2 km depth in the Cajon Pass research well, Geophys.
Res. Lett., 15, 1029–1032, 1988.
1
dCikj = − vk+1Gikj ,
(B.14)
2

1194
K. YAMAMOTO et al.: THE ROLE OF DAMAGED ZONE ON TNE STRENGTH OF FAULTS
Eshelby, J. D., The determination of the elastic field of an ellipsoidal inclu-
Mechanics of Materials, 4, 1–16, 1985.
sions, and related problems, Proc. Roy. Soc. Ser. A, 241, 376–396, 1957. Oppenheimer, D. H., P. Reasenberg, and R. W. Simpson, Fault plane solu-
Feng, R. and T. V. McEvilly, Interpretation of seismic reflection profiling
data for the structure of the San Andreas fault zone, Bull. Seism. Soc.
Am., 73, 1701–1720, 1983.
tions for the 1984 Morgan Hill, California, earthquake sequence: Evi-
dence for the state of stress on the Calaveras Fault, J. Geophys. Res., 93,
9007–9027, 1988.
Freudenthal, A. M., Statistical approach to brittle fracture, in Fracture, An Paterson, M. S., Experimental Rock Deformation-Brittle Field, 254 pp.,
Advanced Treatise, II, edited by H. Liebowitz, pp. 591–619, Academic
Press, New York San Francisco London, 1968.
Geographical Survey Institute, The horizontal strain in Japan, II. 1994–
1883, Technological data of Geographical Survey Institute, F-1 No. 10.
Geographical Survey Institute, Tsukuba, 1997.
Handin, J., R. V. Hhager, Jr., M. Friedman, and J. N, Feather, Experimental
deformation of sedimentary rocks under confining pressure: Pore pres-
sure tests, Bull. Am. Soc. Petrol. Geol., 47, 717–755, 1963.
Hill, R., New derivation of some elastic extremum principles, in Progress
in Applied Mechanics, The Prager Anniversary Volume, pp. 99–106,
MacMillan, New York, 1963.
Huenges, E., J. Erzinger, J. Ku¨ck, B. Engeser, and W. Kessels, The perme-
able crust: Geohydraulic properties down to 9101 m depth, J. Geophys.
Res., 102, 18,255–18,265, 1997.
Springer-Verlag, Berlin Heidelberg New York, 1978.
Sato, N., Estimation of stresses in the vicinity of the Nojima fault, Awaji
Is., Hyogo Pref. Japan, from core samples, Master thesis, Tohoku Univ.,
87 pp., 1999 (in Japanese).
Sato, N., Y. Yabe, K. Yamamoto, and T. Hirasawa, Stresses at sites close
to the Nojima earthquake fault estimated from core samples: III, Prog.
Abst. Seism. Soc. Japan, 1999 Fall Meeting, C14, 1999 (in Japanese).
Sibson, R. H., F. Robert, and K. H. Poulsen, High-angle reverse faults,
fluid-pressure cycling, and mesothermal gold-quartz deposits, Geology,
16, 551–555, 1988.
Tanaka, H., N. Tomida, N. Sekiya, Y. Tsukiyama, K. Fujimoto, T. Ohtani,
and H. Ito, Distribution, deformation and alteration of fault rocks along
the GSJ core penetrating the Nojima fault, Awaji Island, Southwest Japan,
in Proc. Int. W/S on the Nojima fault core and borehole data analysis,
edited by H. Ito, K. Fujimoto, H. Tanaka, and D. Lockner, GSJ Interrim
Rep., No. EQ/00/1, USGS Open-file Report 00-129, 81–101, 1999.
Tsukahara, H., R. Ikeda, and K. Yamamoto, In situ stress measurements in
a borehole close to the Nojima Fault, The Island Arc, 10, 261–265, 2001.
Iio, Y., Frictional coefficient on faults in a seismogenic region inferred from
earthquake mechanism solutions, J. Geophys. Res., 102, 5403–5412,
1997.
Ikeda, R., Y. Iio, and K. Omura, In-situ stress measurements in NIED
boreholes in and around the fault zone near the 1995 Hyogoken-Nanbu Watt, J. P., G. F. Davices, and R. J. O’Connell, The elastic properties of
earthquake, Japan, The Island Arc, 10, 252–260, 2001. composite materials, Rev. Geophys. Space Phys. Res., 14, 541–563, 1976.
Ito, H., Y. Kuwahara, T. Miyazaki, O. Nishizawa, T. Kiguchi, K. Fujimoto, Yamamoto, K., Theoretical determination of effective elastic constants of
T. Ohtani, H. Tanaka, T. Higuchi, S. Agar, A. Brie, and H. Yamamoto,
Structure and physical properties of the Nojima fault, by the active fault
drilling, Butsuri-Tansa, 49, 522–535, 1996 (in Japanese).
Ito, H., Y. Kuwahara, and O. Nishizawa, Stress measurements by the hy-
draulic fracturing in the 1995 Hyogoken-nanbu earthquake source region,
composite and its application to seismology, Doctoral thesis, Tohoku
Univ., 199 pp., 1980.
Yamamoto, K., Strength distribution of microfracture elements in granites
under compression test, in Proc. 3rd SEGJ/SEG Symp., pp. 327–334, Soc.
Exploration Geophys. Jpn., Tokyo, 1995.
in Rock Stress, edited by K. Sugawara and Y. Obara, pp. 351–354, A. A. Yamamoto, K., Estimation of fracture stress for intact rocks and possibility
Balkema., Rotterdam, 1997.
of long-term earthquake prediction, Zisin 2, 50, Sup. 169–180, 1998 (in
Japanese with English abstract).
Yamamoto, K. and Y. Yabe, Stresses at sites close to the Nojima Fault
measured from core samples, The Island Arc, 10, 266–281, 2001.
Jones, L. M., Focal mechanisms and the state of stress on the San Andreas
Fault in southern California, J. Geophys. Res., 93, 8869–8891, 1988.
Kuwahara, Y. and H. Ito, Deep structure of the Nojima fault by trapped
wave analysis, Proc. Int. W/S on the Nojima fault core and borehole data Yamamoto, K., M. Kosuga, and T. Hirasawa, A theoretical method for
analysis, Nov. 22–23, Tsukuba, Japan, (GSJ Interim Rep. No. EQ/00/1:
USGS Open-file Rep. 00-129), 283–289, 1999.
determination of effective elastic constants of isotropic composites, Sci.
Rep. Tohoku Univ., Ser. 5, Geophysics, 28, 47–67, 1981.
Li, Y.-G., J. E. Vidale, K. Aki, F. Xu, and T. Burdette, Evidence of shallow Yamamoto, K., Y. Kuwahara, N. Kato, and T. Hirasawa, Deformation rate
fault zone strengthening after the 1992 M7.4 Landers, California, Earth-
quake, Science, 279, 217–219, 1998.
Li, Y.-G., K. Aki, J. E. Vidale, and M. G. Alvarez, A delineation of the
analysis: A new method for in situ stress estimation from inelastic defor-
mation of rock samples under uni-axial compressions, Tohoku Geophys.
Journ. (Sci. Rep. Tohoku Univ., Ser. 5), 33, 127–147, 1990.
Nojima fault ruptured in the M7.2 Kobe, Japan, earthquake of 1995 using Yamamoto, K., N. Sato, and Y. Yabe, Strength of fault as inferred from the
fault zone trapped waves, J. Geophys. Res., 103, 7247–7263, 1998.
Li, Y.-G., J. E. Vidale, K. Aki, and F. Xu, Depth-dependent structure of
the Landers fault zone from trapped waves generated by aftershocks, J.
Geophys. Res., 105, 6237–6254, 2000.
Matsushima, S., Variation of the elastic wave velocities of rocks in the pro-
cess of deformation and fracture under high pressure, Disas. Prevention
Res. Inst. Kyoto Univ., Bull., 32, 2–8, 1960.
McGarr, A., M. D. Zoback, and T. C. Hanks, Implications of an elastic
analysis of in situ stress measurements near the San Andreas Fault, J.
Geophys. Res., 87, 7797–7806, 1982.
Nishigami, K., Investigation of deep structure of active faults using scattered
waves and trapped waves, Seismogenic Process Monitoring, edited by
H. Ogasawara, T. Yanagidani, and M. Ando, pp. 245–256, Balkema,
Rotterdam, 2000.
stresses measured in the vicinity of the Nojima fault (Extended abstract),
Tohoku Geophys. Journ. (Sci. Rep. Tohoku Univ., Ser. 5), 36, 272–290,
2001.
Zoback, M. D. and J. H. Healy, In situ stress measurements to 3.5 km
depth in the Cajon Pass Scientific Research Borehole: Implications for
the mechanics of crustal faulting, J. Geophys. Res., 97, 5039–5057, 1992.
Zoback, M. D. and J. Townend, Implication of hydrostatic pore pressures
and high crustal strength for the deformation of intraplate lithosphere,
Tectonophys., 336, 19–30, 2001.
Zoback, M. D. et al., New evidence on the state of stress of the San Andreas
fault system, Science, 238, 1105–1111, 1987.
K. Yamamoto (e-mail: yama@aob.geophys.tohoku.ac.jp), N. Sato, and
Y. Yabe
Norris, A. N., A differential scheme for the effective moduli of composites,