PAGEOPH, Vol. 124, No. 3 ( 1 9 8 6 )
0033-4553/86/0531-3651.50+0.20/0
9 1986 Birkh/iuser Vedag, Basel
A Mechanical Model for Deformation and Earthquakes on Strike-Slip
Faults
B. ROWSHANDEL 1 a n d S. NEMAT-NASSER 2
Abstract--A two-dimensional model for stress accumulation and earthquake instability associated
with strike-slip faults is considered. The model consists of an elastic lithosphere overlying a viscous
asthenosphere, and a fault of finite width with an upper brittle zone having an elastoplastic response and
a lower ductile zone having an elastoviscoplastic response. For the brittle, or seismic, zone the behavior
of the fault material is assumed to be governed by a relation which involves strain hardening followed by
a softening regime, with strength increasing with depth. For the fault material in the ductile, or aseismic,
section, the viscous effect is included through use of a nonlinear creep law, and the strength is assumed to
decrease with depth. Hence, because of the lesser strength and the viscous effect, continuous flow occurs
at great depths, causing stress accumulation at the upper portion of the fault and leading to failure at the
bottom of the brittle zone. The failure is initially due to localized strain softening but, with further flow,
the material above the softened zone reaches its maximum strength and begins to soften. This process
accelerates and may result in an unstable upward rupture propagation.
'Relations are developed for the history of deformation within the lithosphere, specifically for the
velocity of particles within the fault and at the ground surface, The boundary-element method is used for
a quantitative study, and numerical results are obtained and compared with the recorded surface deformation of the San Andreas fault. The effects of geometry and material properties on instability, on the
history of the surface deformation, and on the earthquake recurrence time are studied. The results are
presented in terms of variations of ground-surface shear strain and shear strain rate, and velocity of
points within the fault at various times during the earthquake cycle.
It is found that the location of rupture initiation, the possibility of a sudden rupture as opposed to
stable creep, and also the ground deformation pattern and its history, all critically depend on the mechanical response of the material within the fault zone, especially that of the brittle section. Shorter earthquake recurrence times are obtained for shallower brittle zones and for a stiffer lithosphere. Lower
viscosities of the aseismic zone and the absence of asthenospheric coupling tend to suppress instability
and promote stable creep. The model results thus suggest that the overall viscosity of the ductile creeping
zone must exceed a minimum value for a sudden upward propagating rupture to take place within the
seismic section.
Key words: Fault mechanics; earthquakes; crustal deformation; San Andreas fault.
1 Department of Civil Engineering, Northwestern University, Evanston, Illinois 60201. Present
address: EWA, Inc., 133 First Ave. N., Minneapolis, Minnesota 55401.
2 Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La
Jolla, California 92093.
532
B. Rowshandeland S. Nemat-Nasser
PAGEOPH,
Introduction
Strike-slip earthquakes have been modeled by many investigators using various
possible mechanisms. Most of these are based on Reid's hypothesis of elastic rebound
and the theory of plate tectonics; see RUNDLE(1983) for a review.
Early models were based on the concept of dislocation within a linearly elastic
framework (e.g.: S~KETEE, 1958; CmNNERY, 1961, 1970; WEERTMAN,1965; RYBICKI,
1971) and also a linearly viscoelastic one (NUR and MAVKO, 1974). Since a fault may
be viewed as an inclusion embedded in a deforming elastic medium, ESHELBV'Ssolution (1957) of the inclusion problem has been used with some generalization to
analyze strike-slip faulting (e.g.: RUDNICIO, 1977; RICE and RCDNICKI, 1979; RUDNICIO, 1980a).
In another class of models the concept of fracture mechanics views faulting as a
fracture process in a linearly elastic medium (BURRIDGEand HALLIDAY,1971; LI and
RICE, 1983; see, RUDNICKI(1980b) for a review).
A postulated frictional resistance for fault materials has also been considered in
modeling strike-slip earthquakes (STUART, 1979, 1981; STUARTand MAVKO, 1979).
Many factors are now known to influence the frictional resistance of rocks, including normal stress, pore pressure, stress triaxiality, distortion-induced dilatancy,
temperature, time of stationary contact, sliding velocity, rock type, and recent history
of frictional resistance. Considerable progress has been made within the past
several years, in developing constitutive laws for the frictional behavior of rocks,
which incorporate such factors (e.g., NEMAT-NASSER, 1980; DIETERICH, 1981; RICE,
1983; RUINA,1983; TULLIS and WEEKS, 1986) and also in analyzing the stability of
systems with elastic coupling between rock surfaces obeying such laws and some
form of remote loading (e.g., RICE, 1983; RICE and RUINA, 1983; GU et al., 1984).
Closely related to the model analyzed in the present paper is the one proposed
by TURCOTTE and SPENCE (1974), which involves stress accumulation at an upper
locked section of a strike-slip fault prior to an earthquake, as a result of the relative
lithospheric motion. In this model free sliding takes place along the fault at great
depths. At failure, partial stress relief occurs at the upper locked section of the fault,
and stress relief at the lower section is regarded as responsible for aftershocks. SAVAGE
(1975) has suggested a modification of this model, in which the entire cycle of preseismic stress accumulation, seismic slip, and post-seismic adjustment is assumed to
be due to the transfer of the load along the depth of the fault. The concept of stress
distribution before, and stress redistribution after an earthquake along the depth of
the fault also underlies a model proposed by BUDIANSKYand AMAZIGO(1976), which
involves an accommodating aseismic linear creep, ascribed uniformly to the lithosphere. In this model the earthquake is represented by a prescribed stress drop
across the fault, whenever the stress in the fault reaches a critical value. A model for
earthquake repetition was later proposed by BURRIDGE (1977), which could be
regarded as a modification of the Budiansky and Amazigo model. In the Burridge
Vol. 124, 1986
A Mechanical Model for Deformation and Earthquakes on Strike-Slip Faults
533
model the stress drop is no longer prescribed. NUR (1981) has used a moving dislocation model to extend the Savage model to include aseismic as well as seismic
instability.
The majority of the studies mentioned above have focused on mathematical
descriptions of the faulting process. Examination of the mechanical properties of the
material within the fault zone and their effects on the earthquake phenomenon, and
the use of existing geodetic and seismological data have not been central to these
studies. For a more thorough consideration of the mechanics of faulting it is essential
to use the existing experimental results on rocks and minerals, as well as the existing
field data, together with modern theories of elastoplasticity and elastoviscoplasticity.
This is the aim of the present paper, where the fault is represented by a zone of finite
width with nonlinear mechanical behavior, situated between two linearly elastic
lithospheric plates. At shallow depths, where the rate effects are less prominent, the
fault material is assumed to be elastoplastic. The corresponding constitutive relation
is derived, with the use of nonassociative plasticity, as discussed by NEMAT-NASSER
and SHOKOOH(1980); see also ROWSHANDELand NEMAT-NASSER(1986). An attempt
is made to incorporate in this model some of the essential factors that affect shallow
crustal deformation within the fault zone. Rate effects are thought to become dominant at greater depths, and this is incorporated through a nonlinear creep law. In
addition, coupling between the lithosphere and the asthenosphere is included in the
model. The asthenospheric effect is represented by a basal traction acting at the
bottom of the lithospheric plate. We note that coupling between the lithosphere and
the asthenosphere has been studied by SAVAGEand PRESCOTT(1978) and TURCOTTE
et al. (1979). We also note that the nature of the basal shear stress acting on the
lithosphere is not completely clarified (see, for example: MELOSH, 1977; CHAPPLE
and TULLIS, 1977; DAVIES, 1978; RICHARDSONet al., 1979).
Laboratory data on the frictional characteristics of rocks imply the presence of
stresses of the order of kilobars at the faults. The absence, however, of a localized
heat anomaly at the San Andreas fault suggests a low stress there. Various factors,
such as the presence of water in the zone, have been considered, in order to explain
this phenomenon (LACHENBRUCH,1979). A fault model with a finite width, together
with the weakening effects of temperature and the effect of the strain rate, seems to
produce results that may help in resolving this paradox.
Formulation
The proposed model consists of a fault zone of finite width W, situated between two lithospheric plates of thickness h (Fig. la, lb). The fault extends
down to the bottom of the lithosphere, and its width is assumed to change with
depth (e.g., an increase with depth at large depths, see Fig. lb). The lithospheric
plates are linearly elastic. The fault material is elastoplastic at shallow depths (i.e.,
534
B. Rowshandel and S. Nemat-Nasser
PAGEOPH,
down to depth hu) and elastoviscoplastic at greater depths. It is convenient to use a
rectangular Cartesian c06rdinate system, with the xl axis as depth, the x2 axis normal
to the plane of the fault, and the x3 axis along the fault trace. For antiplane (strikeslip) deformation in the x 3 direction, the nonzero components of the stress and
velocity fields are z13 = rl(Xl, x2, t), 2723 = Z 2 ( X l , X2, t), and V 3 = V(x1, X2, t). The
following boundary conditions are employed:
L
P X2
ELASTIC
LITHOSPHERE
ASYHENOSPHERE
(a)
l
~-
FAULT WIDTH
W0
BRITTLE
(b)
DUCTILI--
3V
--=0
~XI
9
~X2=f
V2V=0
~V
I~X= g
I"
Wh
,t
I
v
Lith.
Asth.
X2
v0,2
X1
Figure 1
(a) Geometric representation of the model. (b) Variation of fault width with depth; brittle zone
(0 < x~ < h,) is characterized by uniform width Wo; width of ductile zone (h, < xx < h) increases exponentially with depth according to equations (20), (34), and (35). (c) Graphical representation of
boundary-value problem.
Vol. 124, 1986
A MechanicalModel for Deformationand Earthquakes on Strike-Slip Faults
535
1. The upper surface of the plate is traction-free; see Figure lc,
+1 = 0
at xl = 0
(la)
where the superposed dot denotes time derivative;
2. The lower plate boundary is subjected to shear stresses due to the asthenosphere,
+1 = #g(x2, t)
at x 1 = h
(lb)
where p is the shear modulus of the lithospheric plate, used as the normalizing factor
here, and the function g is t0 be defined;
3. The shear stress rate at the fault is governed by the fault constitutive relation,
+2 = "~f(Xl, t) =
I~f(xl, t)
at x 2 = 0
(lc)
where +f represents the stress rate carried by the fault and, hence, the function f
characterizes the constitutive behavior of the fault;
4. Points at large distances on opposite sides of the fault (x z = L >:>hu) are
moving away from each other with a constant relative velocity Vo,
VO
V(x 1, L, t) = ~-
(ld)
Among the many existing models of strike-slip faulting, the one defined above
seems to be most similar to that proposed by LACI-I~NBRUCnand SASS (1973), which
was later called the 'shear zone model' by PRESCOTTand NtJR (1981) who used it to
study some of the features of the San Andreas fault.
The deformation within the elastic lithosphere follows Hooke's law,
3V
+1 = ~ X 1,
OV
+2 ---- /1~X2
(2)
and for a long fault, where the variation of the field variables with distance along the
strike may be neglected, the equilibrium requires
c3+~
0+2
~X~- + (~X2 -~" 0
(3)
From (2) and (3) we obtain
32 V
•2 V
~x12 + Ox2~ = 0
(4)
Equation (4) and the boundary conditions (1) define a mixed boundary-value problem for the velocity field of the elastic lithosphere; See Figure lc. With the associated
Green's function G(x~, xz; ~, ~) the solution is
V(xDx2, t ) = ( ( G 3V
V ~G']
Js\ On-- OnJ dS
(5a)
536
B. Rowshandel and S. Nemat-Nasser
PAGEOPH,
where n measures length along the outward normal to the boundary S. With the
boundary conditions this becomes
V(Xl,
X2, t ) ~--
f o ( G(xx,
v~162
X2; 4, 0) f(~, t) + ~-
~[
= L )dr
-- ILG(xl, x2; h, () g((, t) d( (5b)
Jo
Here the functions f and g are the forcing terms introduced in equations (lc) and
(lb). With the eigenfunction expansion method, Green's function for the rectangular
region is
G(x1, x2; 4, ()
-
- - - +L
h
y>
2
m=:
mnL
.mzcy< . . m n ( y > -- L)
-- sech~-- cosn~
m~
smn
h
m~zx1
cos~
mg~
cos-~
(6)
where y > = max(x 2, () and y < = min(x 2, 0. Substitution of (6) into (5b) gives the
velocity field within the elastic lithosphere in terms of the resistive force f at the fault
and the driving (dragging) force g at the asthenospheric boundary:
V(Xx, x2, t) = TVO+
2
f~(X2-L
~ -
+
.rnrcL . mTz(x2-L )
~--1 mTz s e c n - ~ slnn
~
mrcx:
mrc~'~.~
cos--~cos--ff-jj[Gt)d~
(f(y>-L
-- ~ . - - h +
~__1(_ 1)mm~ s e c hmnL
. . mn(y>~ - L) c o s.tony<
~ - slnn
n ~ - - - - c o srnnxl'~
- ~ - - ) g(~, t) d~ (7)
Given the functions f and g, the above equation can be solved, by a numerical
method, to find the history of deformation within the elastic plate. In particular, we
are interested in the history of the fault deformation, obtained from equation (7),
with x z = 0,
/)f(Xl, t ) :
/)o
~-
+
h
f ;(~_~
+
+ re=f:
- - tanh--~--- c o s - ~ - - cos~---)Jt~, t) d~
m=l mrc
2
mTrL mrtx:
m~z(L-- O)
sech T
cos~ff-- sinh
~
g([, t) d[ (8)
mn
( -- 1)m
Vol. 124, 1986
A Mechanical Model for Deformation and Earthquakes on Strike-Slip Faults
Ke
f.
537
Kp(r)
,.~
f
a) Brittle zone
Ke
*l(r)
K~r)
b) Ductile zone
Figure 2
Model representation of fault materials.
and the ground surface deformation with x 1
vg(x2, t) = T +
h
2
rnTrL . . m l z ( x
,, =1 mn s e c n ~ - - slnn
/_.,~" (-- l),, 22_
m= 1
= O,
2
-
-
L)
m~r
~
cos -jj , ,) de
sechmlrL
--- sinhmn(y >
mn
-
L)
[L(y> _ L
- Jo \
h
+
.mzry < "~
cosn--Tj g(ff, t) dg (9)
h
Next we define functions f and g. Based on the continuity of stress across the
fault, f is given by
f (xl, t) = :cf(Xx, t )/#
(10)
At shallow depths the material of the fault zone is assumed to be elastoplastic. The
equivalent elastoplastic modulus therefore (Fig. 2a) is
K-
KeKp
(11)
Ko+Kp
where Ke (assumed to be constant) is the elastic modulus and Kp is the plastic
modulus. The plastic modulus Kp is a decreasing function of deformation, being
relatively large (positive) at first, and vanishing when the stress reaches the rock
shear strength. After that, Kp may be negative owing to strain softening. An expression for Kp may be developed on the basis of the elastoplastic flow of fault-type
materials (NEMAT-NASSER, 1980; NEMAT-NASSER and SHOKOOH, 1980; ROWSHANDEL
and NEMAT-NASSER, 1986). A brief account is given in the Appendix, and a set of
stress-strain curves of this kind, used in this study, are shown in Figure 3a by the
solid curves.
At the lower portion of the fault, i.e. xl > hu, as a result of elevated temperatures,
the viscous effects become important. A model for the elastoviscoplastic behavior of
the fault at great depths is sketched in Figure 2b. It includes a viscous constituent
538
B. Rowshandel and S. Nemat-Nasser
PAGEOPH,
FA~T
STRENGTH, kb
0.7
I0
0.7
T
0.6
1
O,S
vl
0.4
~
1
0
iiiiii::::
0.3
---7---" .......
0.2
. . . . . . . . . . . . . . . . 84
0.1
..............................
.........
-- . . . . . . . . .
1
30
-; . . . . . . . . .
f.- -40
2
3
50
SHEARSTRAIN,%
Figure 3
(a) Stress strain curves for the fault material at various depths: Solid curves, behavior within brittle zone
(rate-independent, elastoplastic); dashed curves, behavior within ductile zone for strain rate of 10-~2/sec
(rate-dependent, elastoviscoplastie). (b) Variation of fault shear strength with depth.
which represents the thermally activated creep at elevated temperatures and hence
m a y be described as follows:
~)v = AoT" exp { - E / R T }
(12)
where ~v is the creep-induced strain rate, 9 is the shear stress, E is the activation
energy, R is the gas constant, T is the absolute temperature, and n is a material
constant; see, for example, HEARD (1976), CARVER (1976), CARTER et al. (1981), and
KIRBY (1983).
Consider now a shear stress increment + at time t, a total strain rate ~ within the
fault, an elastic strain rate ~e, a plastic strain rate ~v, and a viscous strain rate ~v;
then we obtain for the brittle zone, where 0 < xl < h,,
":(xx, t) = K ( x x, t) ~(x a, t)
(13)
~(xl, t) = ~"(xl, t) + ~P(x x, t)
(14)
Vol. 124, 1986
A Mechanical Model for Deformation and Earthquakes on Strike-Slip Faults
539
and for the ductile zone, where h > x, > h,,
+(xi, t) = K(x,, t)[9(xa, t) - 9V(xl, t)] = ~](Xl, t)gV(xl, t)
(15)
9(x,, t) = 9~
(16)
t) + 9P(xl, t) + 9~(xl, t)
where r/ is an 'effective' (stress- and temperature-dependent) viscosity. From the
creep law we have
~;~(xl, t) = nA o v"-l(Xl, t) +(x l, t) exp { - E / R T ( x l ) }
(17)
t/(xl, t) = exp {E/RT(xi) } = ~/o(Xi)Z,_. (xl, t)
(18)
and hence
nAoz"-i(xl, t)
Let the two sides of the fault of width W move at the relative velocity vf(x~, t).
The total shear strain rate across the fault then is vf/W, and the function f becomes,
f ( x l , t) =
vf(xi, t )K(xi, t)
(19)
W(x,)
for an assumed uniform shear strain within the fault at a fixed depth. From (16) we
obtain
vf(xl, t)
t(xi, t)
W(Xl)
K(x1, t)
z"(xl, t)
+ -
-
(20)
nrlo(Xl )
Since both -~ and r are functions of the unknown variable vf, a numerical time
integration scheme is required. Assume that vr, z, and + are known at and prior to
time t - At. Denote z(xl, t - At) by Zo and +(xl, t - At) by "~o and consider the
following approximation of z(x~, t):
~(x~, t) = Zo(X~) + (1 - fl)t0At + fl~(x~, t)At,
0 ~< fl ~< 1
(21)
The choice of fl between 0 and 1 depends on the degree of nonlinearity of the
problem (e.g., through the parameter n), and also the change of the variables with
time. Here fl = 0.5 is chosen for the first iteration at every time step. Depending on
the size of At (e.g., 10 years, 5 years, and 1 year, used in this study) and the change of
stress at the fault with time, a few iterations may become necessary for every time
step. Using (21) in (20), we obtain
vf(xi, t)
+(xi, t)
W(x~)
K(x~,t)
-
-
T~(xi)
z~-i(xl)
+ - + - [(1 -- fl)+oAt + fl+(x~, t)At]
nno(Xl)
~o(Xl)
(22)
The expression for f ( x t , t ) - - f o r both the brittle zone, equation (19), and the ductile
zone, which may be obtained from (20)----van be written in the form
f ( x l, t) = P(xi, t) v~(xa, t) - Q(x~, t)
(23a)
540
B. Rowshandeland S. Nemat-Nasser
PAGEOPH,
where
K(xl, t)
~W(xl)'
O<xi
P(Xl, t) = { _ _
1
(23b)
qo(xl)K(xl, t)
#W(Xl)/']o(xl)
O,
<h.
-]-
g(x1, t ) z B - l ( x x ) f l A t
'
hu<Xl <h
<h.
r~(xl) + n+o(xt)zB-l(xi)(1 - fl) At
qo(Xl) + K(xl, t)rB-l(xl)flAt
'
O<Xl
Q(xi, t) = f K(Xa,
- - t)
n#
(23c)
h,<xl
<h
Equations (23) define the frictional resistance of the fault as a function of depth and
time.
To find an expression for the driving (dragging) tractions characterized by the
function g, we assume an asthenosphere of viscosity qa and thickness d. The corresponding basal traction za is then taken to be proportional to the velocity difference Av
between the bottom of the lithosphere and the asthenosphere and inversely proportional to the thickness d of the asthenosphere (see Figure 4a); that is,
a
za(x2, t) = ~Av(x2,
t)
a
F
A
U
L
(24)
X2
h
LITHOSPHERE
ASTIIENOSPHERE [-~-
T
*I
AV
.il
xi
pXi+l
Xl
Figure 4
(a) Lithosphere-asthenospherecoupling. (b) Part of the boundary-elementmesh for the lithospheric
plate.
This corresponds to a simple shear model for the flow within the asthenosphere
(MEeosrI, 1977). According to equation (24), the traction changes with time and
location of the point at the lithosphere-asthenosphere boundary, attaining constant
Vol. 124, 1986
A MechanicalModel for Deformationand Earthquakes on Strike-Slip Faults
541
values when the lithospheric flow reaches a steady state.
From equation (24) and the continuity of the stress increment across the lithosphere-asthenosphere boundary, the following expression is obtained for the
function g:
,
g(x2, t) = ~?]a
5 7 5A v~x2,
t)
(25)
where 6/fit represents the variation with time.
We now substitute equations (23a) and (25) for f and g in (8), rearrange terms,
and obtain the following equation for the fault velocity:
Uf(Xl, t) "q- vv\h- + ~--1 mrc t a n l a T
cos~h--- c o s T )
P(~, t)vf(~, t) d~
Vo f o ( L
~, 2
m~zL
mnxl
mlt~'~
= -m=l --mn t a n h ~ - c o s ~ - - - c o s ~ - - ) Q(~, t) d~
2 +
h + -f L ( L -- ~
~
2
m~zL
m~zxl . , m r f f L - ~)~
+ Jo--\ h
+ 2, ( - 1 ) ' - - s e c n - ~ - cos T
slnn
-h
] g((, t) d ( ( 2 6 )
m=t
mr~
All quantities on the right-hand side of this equation are assumed known. The first
term after the equality sign is the driving force associated with the far-field velocity
of the lithosphere, the second term is due to viscous relaxation in the fault, and the
last term is the asthenospheric force.
Numerical anaIys&
We solve equation (26) for the velocity at the fault by the boundary-element
method with linear elements, starting with zero initial conditions, and employing a
succession of time steps. We choose a number of nodal points on the boundary of
the lithospheric plate (see Fig. 4b) and in this manner reduce integral equations (7),
(8), (9), and (26) to a system of algebraic equations for the nodal velocities. Note
that in equations (7) and (8) functions f and g depend on the boundary velocities
and that, consequently, these equations are coupled.
Let the velocity at the typical node x i be denoted by vl and assume that there are
N nodes on the fault at x z -- 0. At an instant t the velocity distribution over the
incremental depth xi+ t - xi is expressed by
xi+,
vf(~, t) -
-
h~
~
~ -
xi
vi + --vi+thl
(27)
where hi = x~+ 1 - xi is the depth increment associated with the t~h element. Substituting (27) in (26) and approximating the integrals by summations over boundary
542
B. Rowshandeland S. Nemat-Nasser
PAGEOPH,
elements, we obtain the following system of equations for the nodal velocities,
Avf = F
(28a)
where vf is the nodal velocity vector at the fault and the matrix A is given by
AIj = 61j
Lhl
h2
~
+ PI-~ + Plff[ ~'-'-1~
2
mzrL cos~-~
rmrxj / 1
tanh--ff-
c o smrch
~ ) l "~
(28b)
ANj = 5Nj +
m z f f h ; hN))
t a n hmrcL
~ - - c o smrcxj/
T ~ ( - 1)" - cos
-
L h s + P Nh~2 ~
PN~-.=1 ~ 2
(28c)
Ao = 5ij +
~(Pihi
- P i_lh,_l)~L + h(P,_l h2pi_l
-1- --hi-1
~o
~1 ~
2
~
2
P,) ~--1 ~
mrcL
muxj . muxi
t a n h - f f - c o s ~ - - sm
m~L
mrcxj f
mrcxi
tanh--~ cos~(cos~
mnxi_l ~
cos~}
h2pi
2
mTrL m~zxj[" m~zxi
+ ~
~ (m~ tanh~~176
T
,.=1
- c o smT~xi+l
~ - - - ) "~ (28d)
Here 5~j is the Kronecker delta and P~ represents the average value of P, equation
(23b), over the element i. In equation (28a), F is the forcing vector defined by
=
2
m ~ "~
m~xi
hrn~L
.=1 rnn tan T
cos T
c o s ~ ) Q(~, t) d~
f L f L -- ~
, muL . ,mzt(L -- ~)
muxi~
+ J0 \ h + ,,=
~1 ( - 1),. ~mus e c n ~ - slnn
~
c o s T - ) g(r t) de
(28e)
Successive solutions of the system of algebraic equations (28a) for time steps yield
the history of the deformation of the fault. Note that the elements of both A and F
are time-dependent and hence must be evaluated at each time step. Using equation
(9), we also obtain the history of ground deformation.
Instability
As the deformation proceeds in a simulation, the stress in the ductile section of
the fault reaches the shear strength of the fault, and thereafter the materials in this
zone continue to undergo stable flow with increasing velocity. The velocity in this
zone, however, is bounded because of the presence of the viscous effect. At the same
time, the stress within the elastoplastic section of the fault continues to increase with
accumulation of elastoplastic strain, until at a certain point within this zone the
Vol. 124, 1986
A MechanicalModel for Deformationand Earthquakes on Strike-Slip Faults
543
stress reaches the fault strength. With further deformation, as the material softens,
the stress at this point begins to decrease. To maintain equilibrium, however, the
load then shifts to other points within the brittle zone, which in turn, reach their
maximum strength, and thereafter also soften, thus leading to an eventual degradation of the overall strength of the upper elastoplastic brittle zone.
Whether or not this process results in a 'global' instability (i.e., an earthquake)
depends on the distribution of stress along the depth of the fault and the overall
stiffness of the system (e.g., shear moduli and the geometry of the fault and the
lithosphere). Generally, the phenomenon of earthquake instability is represented by
the condition that the unloading stiffness of the inelastic fault exceeds that of the
elastic surrounding (RUDNICKI, 1980a); that is,
K
, -#/0~
(29)
where # is the shear modulus of the elastic lithosphere, K is the slope of the elastoplastic resistance curve (in the post-peak regime), and c~is a factor whose magnitude
depends on the geometry of the inelastic fault zone.
In the present model the increase in fault velocity, prior to instability, is
accompanied by strain softening within the brittle zone. Therefore, a stress drop
accompanies the instability, and from the condition (vf/v o
, oe) for instability,
one should be able to obtain a relation similar to equation (29).
Owing to the inhomogeneity of the material properties and the deformation
within the fault zone, for the general loading condition, a simple relation, similar to
(29), describing the instability condition in terms of the stiffness of the system cannot
be obtained. If certain simplifying assumptions are made, however, such a relation
can be derived by means of equation (26). Let us assume that prior to instability the
following are obtained.
A. The stress level within the ductile zone has reached the maximum shear
strength of the fault, and steady-state flow is occurring there:
K ( x l, t) = O,
h, < x 1 < h
(30a)
This corresponds to zero incremental loading and zero incremental resistance in the
ductile zone (see equation 23); that is,
P ( x l , t) = O,
h, < xl < h
(30b)
Q ( x l , t) = O,
h, < xl < h
(30c)
B. Instability occurs over the entire brittle section of the fault of uniform width
Wo. Moreover, the fault material in this zone has an average elastoplastic tangent
modulus/~; that is,
K ( x l , t) = K,
0 < x I < h.
(30d)
544
B. Rowshandeland S. Nemat-Nasser
PAGEOPH,
C. Material at the bottom of the lithospheric plate is undergoing a steady-state
motion, and therefore the incremental load due to the asthenosphere is zero:
g(x2, t) = 0,
0 <
X2
< L
(30e)
Substituting equations (30a) to (30e) in equation (26), we obtain
vf
--
vo
1
~
1
21 + [(L/Wo) (h./h) + Ao(h/Wo)] (R/x)
(31)
With the above equation the condition for instability becomes
g:
, -
~
(L/Wo) (hu/h) + Ao(h/Wo)
(32)
When (32) and (29) are compared, the shape factor ~ is
= Wo\h + Ao
(33)
which describes the effect of the system geometry on the instability. In equations
(31) to (33) A o is a parameter whose magnitude depends on the ratios L/h and hu/h
and the depth of the point under consideration. For L/h ~ 0.5 and hu/h~ 0.1-0.5,
for example, A o varies between 0.05 and 0.2. On the basis of the inclusion theory,
and by generalizing Eshelby's results, RtYONICIO (1980a) reports expressions for
for various simple inclusion geometries.
Data
For a quantitative study a constitutive equation for the elastoplastic behavior of
crustal rocks (equation 11) is developed on the basis of laboratory data for rock
deformation under high pressure and moderate temperatures (RowSHANDELand
NEMAT-NASSER, 1986). A summary is given in the Appendix. It must be mentioned,
however, that the laboratory data do not include all factors affecting the deformation
within the crust. For example, the laboratory loading rates are orders of magnitude
larger than the tectonic ones. Moreover, the material within the fault zone at shallow
depths usually consists of a mixture of crushed rock, breccia, and weak clayey
minerals called gouge, which are the result of previous faulting activity there (e.g.,
SmsoN, 1977). Therefore, the shear strength of the rock obtained by experiments on
intact specimens overestimates the actual in-situ values. In the present study we are
not specifically concerned with the detailed derivation of such constitutive equations,
even though the constitutive response of the fault material during an earthquake
cycle plays a major role in crustal deformation and instability. Instead, guided by
general relations obtained by ROWSnANDEL and NEMAT-NASSER (1986) for the
Vol. 124, 1986
A Mechanical Model for Deformation and Earthquakes on Strike-SlipFaults
545
elastoplastic behavior of rock-type materials, and taking into account the factors
contributing to the weakness of the fault rocks (as compared with the intact rocks),
such as size (scale) and the crushed nature of the material, we have prepared a set of
stress-strain curves (presented in Fig. 3a) to model the elastoplastic behavior of the
fault material (solid curves for elastoplastic behavior of the seismic zone xl < hu).
Furthermore, for the rate-dependent behavior within the ductile zone, the nonlinear
creep law, equation (12), is employed. The parameters in this equation can be fixed
by using the numerical values reported by various investigators (e.g.: TULLIS, 1979;
SHELTON and TtJLLIS, 1981; CARTER et al., 1981; KIRBY, 1983). From the results of
CARTER et al., (1981) for granitic rocks,
~v = 2 x 105"c3 e x p { - ( 1 . 3 x 104)/T}
T(x~) = To + ( x l - hu) A T
(34a)
(34b)
where the shear strain rate ~v is in rad/yr, -c is in kbar, the temperature T is in ~
and A T is the temperature gradient within the ductile zone, with T O representing the
temperature at the top of this zone.
The fault's maximum shear resistance for the ductile zone, therefore, depends on
the strain rate and temperature, and hence it is expected to decrease with depth (see
Fig. 3b). In fact, when equations (34a) and (34b) are combined with an expression
representing the variation of the fault width with depth within the ductile zone, a
relation describing the change of fault shear strength with depth can be obtained.
Adequate seismological or geophysical evidence, however, specifying the geometry
of transform faults at plate boundaries does not exist. Therefore, for the sake of
illustration, the following relation proposed by TURCOTTE et al. (1980) is taken to
represent the variation of fault strength with depth within the ductile zone:
z = "co exp{1 -- (xl/hu)b }
(35)
Here "co is the shear resistance at the base of the brittle zone. The values of "co and
the exponent b are reported by TURCOTTE et al. (1980) to be in the ranges 0.5-1.3
and 1.10-t.38, respectively. Here, we choose the values "co = 0.7 kbar and b = 1.1
for illustration. Notice that, for the case of steady-state flow within the ductile zone,
equations (34) and (35) together correspond to a fault zone which exponentially
widens with depth.
The dashed curves in Figure 3a show the stress-strain curves at various depths
within the aseismic section of the fault (xl > 10 km), where To = 400 ~ and
AT = 10 ~
and a strain rate of 10-12/sec (~0.3 x t0-6/yr) are used. The same
procedure as for the brittle zone is used to model the elastoplastic behavior in the
aseismic section. Since the purpose of the present work is an overall study of the
mechanics of earthquakes, the information contained in this figure should prove
adequate for obtaining order-of-magnitude estimates and for examining the effects
of various parameters.
546
B. Rowshandeland S. Nemat-Nasser
PAGEOPH,
The thickness of the lithosphere, h, is taken to be 50 km. For nonanomalous
continental regions the lithospheric thickness is reported to be on the order of 105115 km (WALCOTT, 1970). For regions of high tectonic activity, h is expected to be
smaller. For example, WALCOTT(1970) reports a value slightly greater than 20 km
for the Basin and Range province. For the San Andreas fault region, BRUNE (1974)
and THATCHER (1975a) have suggested a lithosphere thicker than 20 km, on the
basis of the narrow zone of strain release observed in the 1906 San Francisco
earthquake.
A value of 5 cm/yr is taken for the tectonic velocity vo. This value is comparable
to the long-term average tectonic velocity across the San Andreas fault. Conventional
ground-based geodetic measurements at different locations along the San Andreas
fault indicate a relative right-lateral movement rate of 30-50 mm/yr (e.g.: THATCHER,
1979a, 1979b; PRESCOTTet al., 1979). A value of 33 mm/yr is reported as the lower
bound for the relative plate motion rate for the central creeping part of the San
Andreas fault (THATCHER,1979a). SAVAGEand BURFORD(1973) estimated the relative
velocity across the central part to be 32 __+5 mm/yr for the period 1907-1971. All
these values are, however, somewhat smaller than the 56 ___ 3 mm/yr obtained by
MINSTER and JORDAN (1978). From plate tectonic reconstructions they report this
value for the relative velocity between the Pacific and the North American plates
across the San Andreas fault during the past three to four million years.
The value of L, the location of the uniform velocity boundary, to be used in the
proposed model, seems to depend on such factors as the lithospheric thickness,
especially the depth of the locked section of the fault, the existence of other faults in
the vicinity of the one under consideration, and the nature of the basal shear stress.
Surface deformation usually extends to distances several times the depth of the
locked section of the fault (CHINNERY, 1970; SAVAGEand BURFORD, 1973). An appropriate choice for L can be made by using the data available for the surface shear
strain rates reported for the San Andreas fault region (e.g.: SAVAGEand BURFORD,
1973; THATCHER, 1975a, 1975b, 1979a, 1979b, 1979c, 1981; SAVAGE, 1983). Data for
the shear strain rate at the northern and southern locked segments of the San Andreas fault have been collected by THATCHER (1983). The values range from
0.35 • 10 -6 to 2.5 • 10 -6 rad/yr. It must be noticed that in these two sections of
the fault the measurements are sensitive to the time of data collection during the
earthquake recurrence period and also to the location along the trace of the fault.
By contrast, in the central aseismic creeping section the deformation is relatively
uniform in both time and location along the fault and, owing to the rigid block
motion, most, if not all, of the deformation is concentrated within a few kilometers
of the San Andreas fault (SAVAGEand BURFORD, 1973; THATCHER, 1979a).
Based on these observations, if we choose 10- 6 rad/yr as the average engineering
shear strain rate and 5 cm/yr as the relative (tectonic) velocity across the locked
fault, then the width of the corresponding uniform shear deformation zone would be
50 km, and hence the estimate for L would be 25 kin.
Vol. 124, 1986
A MechanicalModelfor Deformationand Earthquakeson Strike-Slip Faults
547
The shear modulus of the elastic lithospheric plate, #, is taken to be of the order
of 500 kbar. For studies of the effect of the surrounding rigidity on the earthquake
phenomenon, different values are used for #, and the results are compared.
The reported viscosity of the asthenosphere mostly falls in the range 5 x 1019102i poise (e.g.: MCCONNELL, 1968; WALCOTT, 1970; ELSASSER, 1971; NUR and
MAVKO, 1974; CATHLES,1975; RICE, 1980; THATCHERet al., 1980), with lower values
corresponding to the tectonically active, and higher values to the stable, shield
areas. CATHLES(1975), for example, points out the existence of a zone 75 km thick,
of low viscosity channel (4 x 102~ poise), beneath the lithosphere. Here values of
this order are assumed for t/a, the 'effective asthenospheric Newtonian viscosity.'
From the relation reported by MELOSH (1977) for the basal shear (assuming a
Newtonian rheology and a simple shear flow within the asthenosphere), the range
of asthenospheric viscosity 10i9-1021 poise, combined with an asthenospheric thickness of 100 km and a relative velocity of 5 cm/yr across the top and bottom of
the asthenosphere, the corresponding maximum basal shear stress will be in the
range 0.16-16 bar.
High temperatures at both intermediate and great depths lower the flow stress,
making the material more ductile (GRIGGSet al., 1960; HEARD, 1972, 1976; PATERSON,
1970; BRACE and BYERLEE, 1970), and they also tend to increase the width of the
shear zone (YUEN et al., 1978; FLEITOUTand FROIDEVAUX, 1980). A wider shear zone
at greater depths corresponds to lower strain rates and leads to weak material, small
stress accumulation, and stable viscoplastic deformation. The viscosity in this region
is expected to be orders of magnitude smaller than the lithospheric one, because of
shear heating (HEARD, 1976; CARTER, 1976), that is, a high heat anomaly in a relatively narrow zone of intense shear deformation (YUEN et al., 1978; FLEITOUTand
FROIDEVAUX, 1980). YUEN et al. (1978), on the basis of a theoretical model have
obtained a viscosity of the order of 102i poise for wet olivine within a shear zone at
temperatures of 400-600 ~ and a relative velocity of 10 cm/yr across the zone.
Therefore, if the rate-dependent behavior of the material within the lower part of the
fault is modeled by a linearly viscous relation, then viscosities of the order of asthenospheric values are inferred. In this study, however, a stress-dependent viscosity is
used, based on the nonlinear creep law, equation (34a). The corresponding effective
viscosity then is in the range of 1019-1023 poise for a temperature range of 400800 ~ and shear stress given by equation (35).
An effective linear viscosity of 5 x 1020 poise for the ductile zone, with an elastic
shear modulus/z of 500 kbar for the lithosphere, would result in a relaxation time
t///2 of about 30 yr. Considering the post-seismic ground deformation reported for
the San Andreas fault region, this estimate seems reasonable, if some viscoelastic
deformation takes place in the lower part of the fault. THATCHER(1975a) reports an
aseismic strain release of about 1.2 • 10-6/yr near the fault for about 30 yr after the
1906 San Francisco earthquake. For a relatively thick (e.g., 50 Am) lithosphere with
a relatively shallow (e.g., 10 kin) brittle zone, such a large post-seismic deformation
548
B. Rowshandel and S. Nemat-Nasser
PAGEOPH,
rate close to the fault is more likely to be due to viscoelastic deformation in the
lower aseismic section of the fault than to the asthenosphere.
As far as the geometry of the fault is concerned, the upper locked section is
assumed to extend down to a depth of 10 to 15 km. This is similar to the structure of
the San Andreas fault. A 10-15 km depth of rupture is reported by SIEH (1978a) for
the great 1857 earthquake on the southern section of the San Andreas fault. He
suggests, however, that 15 km may be a better estimate of the depth of coseismic
faulting in 1857 than 10 km. A similar situation exists for the northern section of the
San Andreas fault. THATCHER(1975a) rules out a rupture depth greater than 10 km
for the 1906 San Francisco earthquake by examination of geodetically measured
surface deformation. Also, in support of ~ 15 km for the thickness of the seismogenic
section are the recorded earthquake history of the San Andreas fault and, to some
extent, laboratory evidence from rock deformation. Experimental results suggest that
rocks in general should flow at the temperatures and pressures prevailing at depths
of 10-15 km (e.g., BRACE and BYERLEE,1970).
For an investigation of the effect of fault width in the brittle zone W 0, on the
process of surface deformation and instability, values on the order of meters to tens of
meters are used for this parameter. In the case of interplate earthquakes the fault zone
is likely to extend down to the bottom of the lithosphere. A characteristic of the fault
zone at shallow depths is that both the geometry and the mechanical properties of the
material are functions of the amount of displacement across the fault and the time the
fault remains stationary (DIETERICH, 1980). Relative displacement across active, interplate faults has been, in most cases, going on for hundreds of thousands or even
millions of years, according to the theory of plate tectonics. This has led us to the
selection of a fault zone of finite width, consisting of crushed rocks, granular materials,
and clayey gouges at shallow crustal depths. The behavior of materials of this kind is
governed by highly nonlinear, constitutive relations.
The aseismic deformation in the lower section of the fault is assumed to take
place over a wider region. This would then correspond to the shear-zone model
suggested for the San Andreas fault, rather than to the deep-fault model, which
assumes that the fault plane extends down to the bottom of the lithosphere (PRESCOTT and NUR, 1981). The width of the shear zone depends on the existing temperature, the relative velocity across the zone, the rheology of the material, and the
history of the relative motion, according to YUEN et aL (1978), who, by starting with
a thermomechanical structure for the slip zone, and letting the deformation evolve in
time, estimates the width of the shear zone for the San Andreas fault to be a few
kilometers. TURCOTTE et aL (1980), too, on the basis of the thermal analysis of a
strike-slip fault, have developed a relation in which the width of the shear zone
increases exponentially with temperature (depth) and inversely with stress. Therefore,
in order to study the effect of the shear-zone width on the variation of surface
deformation with time and with distance from the fault, we have assumed that the
fault width is of the order of meters to tens of meters at the top of the aseismic zone
VoI. 124, 1986
A Mechanical Model for Deformation and Earthquakes on Strike-Slip Faults
549
(e.g., W h ~ Wo) and of the order of hundreds of meters to a kilometer at the bottom
of this zone. Moreover, the fault is assumed to have a downward-widening profile
within the ductile zone, where the width at any depth is estimated by means of
equations (20), (34), and (35).
Results
To solve the set of equations (28a) numerically, for the fault velocity vf, we start
with zero nodal velocities and gradually increase the far-field velocity V(xl, L, t)
from zero until the full velocity Vo/2, is attained. The gradual increase is necessary
for numerical stability. The results for one set of data are shown in Figure 5. In this
figure velocities for various depths are given as functions of time after the onset of
loading. As may be seen from this figure, velocities at greater depths increase more
rapidly. The velocity at the deepest point within the fault, that is, at the lithosphereasthenosphere boundary, eventually approaches the far-field velocity. In the aseismic
zone (depth 10-50 kin) the velocities are appreciable from the onset of loading and
increase slowly until the velocity profile within this zone reaches a nearly steady
state. The upper section of the fault (depth < 10 km) remains practically locked; that
is, the velocities in this zone are at least one order of magnitude smaller than those in
the underlying aseismic section. Owing to the falling resistance of the material (e.g.,
material softening) in the locked section, however, velocities in this section increase in
later stages of the deformation history. This could lead to instability within the locked
section. Whether this rapid deformation rate leads to instability (earthquake) or
continues in a stable manner (creep) depends on the overall stiffness of the system (sizes
and mechanical properties, see equation 32) and the distribution of stress within the
fault. An examination of the numerical results for the present model indicates that instability, when it occurs, will be accompanied by a stress drop in the brittle zone, a
ground-surface displacement close to the fault, and a drop in the ground-surface shear
strain (i.e., a negative ground-surface shear strain rate), all within one time step.
The velocity curves for the upper fault zone suggest that inelastic deformation in
this zone first begins at greater depths and then proceeds at shallower ones. In other
words, the inelastic deformation commences at the bottom of the locked section,
accelerates, and possibly turns into an unstable rupture which propagates upwards
toward the surface. The concept of the upward progression ofaseismic slip of the rupture
zone is addressed by several investigators (NUR, 1981; LI and RICE, 1983). Because of
the low strength of near-surface fault rocks (see Figure 3), some early (stable) strain
softening in the brittle zone at shallow depths is also observed.
Figure 5, which is based on the numerical values used here, shows that the time
from total rest to earthquake instability is about 770 yr. On the other hand, the
interarrival (recurrence) time of major interplate earthquakes (represented by T r
in Fig. 5) may be defined as the time required for the stress within the fault to
550
B. Rowshandel and S. Nemat-Nasser
PAGEOPH,
1.0
Dept h=50 kra
0.8
40
0 2
20
0
0
100
to
1
200
300
400
500
Tr
600
700
800
900
t , years
Figure 5
Time-velocity curves within fault at various times after onset of loading. Data: Wo = 1 0 - 2 km (uniform
with depth), Wh = 1 km, ha = 10 km, # = 500 kbar, tf = 1 0 1 9 poise, TO= 300 ~ AT = 10 Cr
Tr =
recurrence
time.
reach the pre-earthquake level, from a post-earthquake level. With this definition of
the earthquake recurrence time, T r becomes approximately 250 yr (see Fig. 5).
Earthquakes are believed to relieve only a small fraction of the stress on the
fault--a few bars to tens of bars stress drop, as compared with hundreds of bars to
even kilobars of total stress level. Even this stress drop occurs within a relatively
shallow seismogenic zone (e.g., a locked section 10-15 km deep for the San Andreas
fault). This is only a small fraction of the lithospheric thickness. The size of the
stress drop relative to the total stress on the fault suggests that the reference (initial)
time for an earthquake cycle (to) should be the time at which the stress within the
fault has reached its post-earthquake level. Here we choose the reference time to be
the time at which the stress in the ductile zone has reached its steady-state value,
according to the stress-strain curves of Figure 3a. With this earthquake-cycle reference time (t = to) we obtain a recurrence time on the order of 150-300 yr (see Fig.
5). This value is comparable to the numbers estimated for the recurrence times of
major earthquakes along the San Andreas fault. For example, SIEH (1978a) obtains
a recurrence time of 160-260 yr for the great 1857 earthquake on the southern San
Andreas fault. Large prehistoric earthquakes along the San Andreas fault have been
estimated to have recurrence times in the same range (SIEI-I, 1978b). For the northern
locked section of the San Andreas fault, THATCHER (1981), using crustal strain,
estimates a recurrence interval of 225 yr for the 1906 San Francisco earthquake.
Figure 6 shows the velocity curves of Figure 5 plotted as functions of depth for
Vol. 124, 1986
A Mechanical Model for Deformation and Earthquakes on Strike-Slip Faults
551
different times after the beginning of the cycle. Again, for the aseismic zone
(depth > 10 kin) based on the present model, a stable and slow increase in velocity
is observed. For the seismic zone, however, a rapid deformation begins prior to
instability at the base of this zone and propagates upward toward the surface, relatively slowly at first and then very rapidly.
1.0
~ 7 0 0
2OO
0,8
100
=
0.6
Vf
0.4
0.2
0
10
2O
3O
40
50
DEPTH, km
Figure 6
Velocity curves as function of depth at various times t. Data are same as for Figure 5.
In Figure 7 the stress distribution within the fault zone is shown at different
times t. The solid curves indicate the stress distribution within the fault at different
times after the start of loading (t = 0). The dashed lines indicate the state of stress
within the brittle section of the fault immediately after occurrence of the earthquake.
The corresponding stress drop is the difference between the stress level before and
that after the earthquake (the two curves labeled 770- and 770 § respectively).
Notice that because of the quasi-static nature of the model, the post-earthquake
stress and the seismic stress drop are determined by the strain softening constitutive
equation supplied for the brittle zone. (Because of the dynamic effect of an earthquake, however, the magnitudes of the postseismic stress and the coseismic stress drop
can be drastically different from those shown in this figure).
Neither the state of stress within natural fault zones nor the earthquake stress
drops are well established. Stresses on the order of several hundred bars, however,
have been reported in laboratory testing of rocks under high temperatures and
pressures. Most investigators consider the stress drop to range from a few bars to
approximately a hundred (HANKS, 1977; RICHARDSON and SOLOMON, 1977).
Figure 8 shows the ground-surface deformation at various instants after the
552
B. Rowshandel and S. Nemat-Nasser
PAGEOPH,
FAULTSTRESS.kb
0
0
0.I
0.2
0.3
0.4
0.5
0.6
0.7
10
20
DEPTH,
km
30
40
so
Figure 7
Stress distribution within the fault at various times after onset of loading. Curves: 770-, is stress distribution just before instability; 770 +, is stress distribution after instability. Data are same as for Figure 5.
E
5
r~
34~
y
r
s
z
o
00
0
0
0
5
10
15
2O
25
DISTANCEFROMFAULT,km
Figure 8
Ground-surface deformation at various times during earthquake cycle. Data are same as for Figure 5.
Vol. 124, 1986
A Mechanical Model for Deformation and Earthquakes on Strike-Slip Faults
Horizontal
553
Displacements
(in Meters)
2.0'
o
WEST
9
9 I".5
EAST
0
"
~.O
0
9
9
O
0.5
0
o
15
0
o
10
g
O
g
10
15km .....
§ Distance from Fault §
(a)
Horizontal Diplacements
(in Meters)
4.0
Point Arena
o Fort Ross
. Tomales Bay
. Colma
W~
IE
WEST
;%.
3.0
EAST
Oo
:go
O
QO
11~ ~
g~
9
1.0
x
9
§
9
Distance from Fault +
(b)
Figure 9
Observed (coseismie) falloff of horizontal displacements with distance from fault: (a) Tango earthquake;
(b) San Francisco earthquake. (After CI-nNNERY,1961).
554
B. Rowshandel and S. Nemat-Nasser
PAGEOPH,
earthquake-cycle reference time to. The curve for 220 yr shows the pre-earthquake
surface deformation. The maximum relative displacement within the region prior to
instability (the maximum preseismic surface displacement) is approximately 5 m,
which is on the order of magnitude of the maximum seismic surface displacement
recorded for large earthquakes. Seismic displacement across the San Andreas fault for
the 1857 earthquake is reported to have been within the range 3-9.5 m (SIEH, 1978a).
The nearly straight shape of the surface displacement curves in Figure 8 for large
distances from the fault, (beyond 10 km), which represents a uniform deformation
over a wide area, is the reflection of our choice of the deep fault with low strength at
depth. Figure 9 shows the observed displacement that accompanied two different
earthquakes, as reported by CHINNERY (1961). These field observations suggest a
region of relatively intense deformation in the vicinity of the fault. For a constant
shear modulus for the lithospheric plate, large drops in shear strain with distance
from the fault, as implied by Figure 9, would correspond to a large increase in shear
stress toward the fault. This, then, is equivalent to characterizing the fault as a zone
of high stress concentration, rather than a zone of intense deformation and weak
material. It must, however, be noticed that, for an active fault, because of previous
ground deformation and displacements along the fault, and also because of the possible existence of secondary faults in the vicinity, the stiffness and strength of crustal
rocks are expected to decrease toward the main fault.
Figures 10 and 11 show the model predictions of surface (engineering) shear
strain and shear strain rate, respectively. The curves are labeled with time after the
0-7 I
0,6i
o0.5
coo
z
~qo--t0
o~
0.4
400
~= o.31 f
~tO0
t =20Qyrs
0.1
o
i
~
,'o
,'5
2'o
i
25
DISTANCE FROM FAULT, kin
Figure 10
Variation of surface (engineering) shear strain with distance from fault at various times t. Data are same
as for Figure 5.
Vol. 124, 1986
A Mechanical Model for Deformation and Earthquakes on Strike-SlipFaults
555
~.1"5k
:Z
<
~o-sr
0
0
5
10
15
20
25
DISTANCE FROM FAULT, km
Figure 11
Variation of surface (engineering)shear strain rate with distance from fault at various times t. Data are
same as for Figure 5.
zero initial condition, t. From both Figures 10 and 11 it can be seen that at large
distances from the fault, comparable to the depth of the brittle zone and deeper
(e.g., beyond 10 km), there exists a relatively uniform shear strain whose magnitude
increases smoothly with time. The uniform shear strain and shear strain rate, at
distances beyond 10 km from the fault, mainly reflect the steady, stable deformation
of the material within the fault in the aseismic zone.
For points close to the fault (0-10 km), on the other hand, Figures 10 and 11
show noticeable spatial and temporal variations of surface deformation. For earlier
stages of loading the level of shear strain, from Figure 10, seems to be relatively
uniform with distance from the fault. With further loading, however, owing to a
greater rate of deformation close to the fault and at shallow depths (Fig. 11), the
shear strain becomes greater in this region. A temporal fluctuation in the rate of
surface shear strain is observed in Figure 11; it results from the deformation process
within the brittle zone. For example, the onset of softening within the locked section
of the fault is accompanied by a change in the strain rate distribution (that is, as the
lower part of the brittle zone softens, the peak value of the shear strain rate shifts
toward the fault); see Figure 11. TRATCI4ER (1975a) reports a possible accelerated
shear strain accumulation on the northern San Andreas fault over approximately 50
yr preceding the 1906 San Francisco earthquake. Temporal fluctuations of as much
as a factor of 2 are observed in geodetic measurements of interseismic shear strain
rates. For example, THATCHER (1979a, 1979b) identifies the epidosic strain rate
changes with slip on unspecified buried faults.
Despite the temporal fluctuations close to the fault, the strain rate curves at later
556
B. Rowshandel and S. Nemat-Nasser
PAGEOPH,
stages of loading exhibit profiles which are maximum at the fault and fall relatively
rapidly with distance from it. Negative shear strain rates are also obtained, with the
present model, at times close to earthquakes. This happens when inelastic deformation dominates in the brittle zone, resulting in some stress relief due to material
softening at shallow depths.
For comparison with observations, we have presented the tensor shear strain
rate i/2 as a function of distance from fault in Figure 12. This figure is taken from
THATCHER (1979C). It shows the variation of the measured shear strain rate with
distance from the San Andreas fault. (Notice that, for comparison, the readings in
Figure 12 must be multiplied by 2.) The magnitude of the tensorial shear strain rate
at the fault just before an earthquake, as predicted by the model, reaches a value of
0.7 x 10-6/yr. This value appears to be too large, compared to the field data presented in Figure 12, but tensorial shear strain rates in the range of 0.4 x 10-6/yr to
0.4(
c;
.s o.2q
t
*
.0
-80
'
=
-40
=
!.
0
PERPENDICULAR DISTANCE
,
=
40
TO .
Zt~ hm
,
FROM F A U L T , KM
Figure 12
Maximum (tensor) shear strain rate plotted against perpendiculardistance from San Andreas fault: solid
circles, from triangulation measurements; open circles, from geodimeter observations. One-standard-deviation error bars from least-squares fits of data shown for reference.After THATCr~R,1979C.
Vol. 124, 1986
A Mechanical Model for Deformation and Earthquakes on Strike-Slip Faults
557
0.9 x 10-6/yr are reported for a period of 50 yr before the 1906 San Francisco
earthquake (THATCHER,1975a).
Next, using the model, we investigated the effect that changing various parameters
would have on earthquake instability and accompanying phenomena. The parameters considered are the crustal shear modulus #, the linear viscosity t/a of the asthenosphere, and the geometry of the fault (the depth of the seismic zone, hu, the
width of the seismic zone, Wo, and the width at the b o t t o m of the aseismic zone,
Wh). F r o m the corresponding velocity, surface shear strain, and surface shear strain
rate curves, the following conclusions may be drawn.
A. The depth hu of the locked section of the fault plays a major role in the
process of stress accumulation and instability. A deeper seismogenie zone results in a
longer recurrence time Tr and a more widespread surface deformation. Consequently,
the energy release during an earthquake becomes greater. The results shown in the
previous figures are for a brittle zone 10 k m deep. The earthquake recurrence time
Tr, obtained for a brittle zone 10 km deep, ranges from 150 to 300 yrs, and the
relative preseismic surface displacements (of points across the fault at large distance)
are on the order of 5 m. Using 8 km for the depth of the brittle zone results in a
recurrence time of 100 to 150 yr (Fig. 13) and a relative surface displacement on the
order of 3 m across the fault. The surface deformation for this case is concentrated in
a narrower zone around the fault. This may be seen in Figures 14 and 15. The
former figure, for a shallower brittle zone, shows that the surface strain is appreciably
DepthfSOk
o.~ r
~
f/,~/-;.,~.....~
o.or,f , , /
~-o
" I//I/
o.,t,! / /
I/
./
I/
/.^/
/
/'3 /
,,-'/
./
I'--"_..-'~'"
L,
O0
,,,
I
100
__--
___huo,O~o
/
/
/
..._t-~
I
200
/ /
/
/
_.f- ....
,,-!~
I-
T,
~,
_ / ~~___----~-k--k-~
"
......
I
).
_J
300
400
500
600
I
700
I
800
i
900
t, years
Figure 13
Time velocity curves for points at various depths within fault at various times t; effect of brittle zone
depth h.. Data are same as for Figure 5.
558
B. Rowshande] and S. Nemat-Nasser
•.---••
0.7
0.8
..
PAGEOPH,
~hu
....
= 8 kin
h.= 10 km
. . . .
? 0.5
................
i 0.4
_
0.3
,.,.,
,f
0.2 ',..
~
___--2-~_. to
300
t--2_OOYr._s.....
.............
0.1
Io
1's
2'0
2's
DISTANCE FROMFAULT, km
Figure 14
Variation of surface (engineering) shear strain with distance from the fault at various times, t; effect of
brittle zone depth. Data are same as for Figure 5.
~ 1.5
~ \
o
___hu=lokm
///
;~ 1.0
~,~.~
r /"
- .
~ - - - ~ _
.
.
.
.
.
.
"<~-.._"x..
/
.
.
.
-
.
.
.
......
.
.
.
.
.
r.,.z 0.5
w
0
i
I
i
DISTANCE FROM FAULT, km
Figure 15
Variation of surface (engineering) shear strain rate with distance from fault at various times t; effect of
brittle zone depth h.. D a t a are same as for Figure 5.
larger in the vicinity of the fault and smaller at distances on the order of hu from the
fault.
B. A larger width 141o for the fault in the brittle zone, results in larger (stable)
creep along the fault, whereas a small fault width results in a m o r e abrupt instability.
Vol. 124, 1986
A Mechanical Model for Deformation and Earthquakes on Strike-Slip Faults
559
The instability for a wide seismogenic zone can be suppressed once the stress in that
zone reaches the fault strength, and, instead, an accelerated creep episode may take
place. On the other hand, a shorter recurrence time Tr and a more pronounced
instability result for a narrower upper fault zone. As the fault width Wo for the
brittle zone decreases, surface deformation becomes more concentrated close to the
fault. Also, for a narrower brittle zone the model predicts the occurrence of inelastic
strain softening deformation, first at the base of the seismic zone; whereas for larger
Wo, strain softening is more likely to start from the upper (ground) surface.
C. The velocity in a wider aseismic zone (large Wu) attains higher values and
becomes more uniform with depth. The surface deformation at greater distances
from the fault also becomes more uniform with distance. The size of Wh, as mentioned
earlier, depends on the temperature and stress within the fault and the tectonic
velocity across the fault. Therefore, for the effect of fault width at depth the values of
Wh are chosen according to equations (34) and (35).
D. For a larger shear modulus # and the same rate of deformation (i.e., Vo) the
rate of strain energy accumulation becomes higher. Therefore, the recurrence time
becomes shorter. Also, for a larger # the velocities in the aseismic zone become
larger (rigid block motion at depth). Smaller pre-earthquake surface deformation
and shear strain and less fluctuation in the surface shear strain rate are observed for
large #.
E. Results for various asthenospheric viscosities r/a (within the range 1019-1021
poise) are obtained. In a comparison of the results it is concluded that a larger
asthenospheric viscosity introduces larger velocities within the ductile zone of the
fault in the early stages of loading. A larger asthenospheric viscosity also causes a
faster rate of surface deformation, especially in the vicinity of the fault. Of course,
owing to the form of equation (24), representing the mechanical effect of the asthenosphere, a driving force is supplied to the bottom of the lithosphere for as long as
the lithospheric velocity is less than the mantle velocity. In the early stages of loading
and just before instability, the asthenosphere has its greatest effect; during the
intermediate times within the earthquake recurrence period, on the other hand, its
effect is low.
F. To study the effect of fault viscosity, we change the temperature level within
the ductile zone while keeping the shear resistance in that zone constant (notice that
this requires the fault width in the ductile zone to be decreased, or the velocity
across the fault to be increased). A larger temperature corresponds to a smaller
effective viscosity for constant stress; see equation (18). It is observed, from the
model, that a large temperature within the ductile zone tends to suppress the instability and promote stable creep within the brittle zone. If Wh is so chosen that the
fault strength follows from equation (35), then for a temperature gradient of
10 ~
T o must be approximately within the range 250-350 ~ to have a
pronounced instability. In fact, with r/a.~ 10 z~ poise, T o > 350~
and
AT = 10 ~
we obtain stable rupture in the brittle zone, with strain softening
560
B. Rowshandeland S. Nemat-Nasser
PAGEOPH,
initially occurring at the top of the brittle zone (the ground surface) and propagating
downward. With TO < 250 ~ and AT = 10 ~
on the other hand, we still observe a stable rupture in the brittle zone, but strain softening starts from the bottom
of the brittle zone and propagates upward. For TO in the range 250-350 ~ and
AT = 10 ~
unstable rupture occurs within the brittle zone. In general, the
model predicts unstable rupture over a larger area of the fault for lower temperatures
(i.e., larger effective fault viscosity), whereas higher temperatures (lower fault viscosity) result in smaller areas of unstable rupture (larger area of pre-instability strain
softening).
For temperatures in the range discussed above, and from equations (18), (34),
and (35), effective linear fault viscosities of the order of 102~
poise and relaxation times (r//#) in the range 6-60 yr are obtained. In other words, with the
model presented here, for instability, the relaxation time should be a fraction of the
earthquake recurrence time. On the basis of different models, BUDIANSKYand AMazic,o (1976) and BONAVEDE et aL (1982) have obtained similar values for the
minimum linear viscosity for the San Andreas fault region.
Discussion and conclusions
To understand fully the mechanics of faulting and earthquakes one should have
a good knowledge of the constitutive behavior of the fault material. To characterize
the mechanical interaction of the fault with the loading system one also needs to
know the geometry of the system--the width of the fault zone and its variation with
depth, and the depth above which the fault may be considered seismogenic and
below which the viscous properties of rock become important during major
earthquake recurrence periods.
Ongoing experimental and theoretical research, aimed at developing representative constitutive equations for fault materials, can help considerably toward a
better understanding and a more quantitative description of the mechanical behavior
of such materials.
A large body of data is also being accumulated by continuous worldwide geodetic
surveys around major faults, such as the San Andreas fault. As the precision of the
measuring devices improves, better geodetic data will be compiled.
More advanced geophysical survey methods are also contributing to a better
characterization of fault geometry and structure at depths.
Issues concerning lithosphere-asthenosphere and crust-mantle coupling can be
partly resolved through various geophysical and seismological studies.
The data gathered through these various activities should be used for investigating the phenomenon of faulting and earthquakes. For systematically integrating and using a growing body of such data, and in resolving further the many
unanswered questions regarding faulting and earthquakes, conceptual models that
Vol. 124, 1986
A MechanicalModel for Deformationand Earthquakes on Strike-Slip Faults
561
incorporate various laws of mechanics become extremely helpful.
The model that was presented in this paper is believed to be useful for such a
purpose. In addition, this model can be modified, as more field and laboratory data
become available and as the significance of various factors in faulting and earthquakes becomes better understood. Various combinations of data have been used
for the numerical analysis in the present study, of which only a few sets of figures
have been included in this paper. From the results obtained in this study, several
conclusions may be drawn concerning the relative significance of different parameters in deformation and in earthquakes on strike-slip faults.
The constitutive behavior of the fault material, especially the brittle zone, was
observed to have a strong influence on the nature of the deformation at the fault and
within the crust, and the time of instability. Assuming a strain softening behavior for
the fault material in the brittle zone, then whether precursory ground-surface deformation close to the fault (e.g., noticeable change in surface shear strain rate prior
to an earthquake) is observed, or whether it is not observed, depends on the variations of the fault strength and the width of the brittle zone with depth, and also on
the strength and the 'effective viscosity' of the material in the ductile zone.
The effect of the stiffness of the system (e.g., the shear modulus of the lithosphere,
the tangent modulus of the fault rocks, and the geometry of the fault-lithosphere
system) on the ground-surface deformation history and on the instability of the
fault is also found to be considerable.
A finite width for the fault (narrow brittle zone and downward-widening ductile
zone), as was adopted here, is based more on reasoning and inferences from related
tectonophysical studies, especially for the ductile zone (e.g.: TURCOTTE et al., 1980;
YUEN et al., 1978), than on direct physical evidence. The effects, according to the
present model, turn out to be noticeable enough for the fault width to be considered
in future modelings,
As far as the effect of the asthenosphere is concerned, its inclusion in this study
has been based on the currently inadequate understanding of the coupling between
the lithosphere and the asthenosphere and the nature of the forces that drive the
plates. A greater undestanding of the role of the mantle in crustal deformation and
plate motion is necessary before any firm conclusion concerning the retardation or
acceleration effects of the asthenosphere on deformation and instability of a fault
can be reached.
Although qualitatively similar behavior was observed for different cycles (except
for the beginning of the first one), results presented in this paper are based mainly
on the first cycle. Extending the model to earthquake cycles beyond the first one
would necessitate certain assumptions, especially concerning the residual stresses in
the brittle zone after an earthquake. Even though the constitutive equation used here
(see the Appendix and Fig. 3a) provides these residual stresses, the results obtained
are based on quasi-static loading conditions. Post-earthquake fault stress (in the
brittle zone) is expected to be lower due to the earthquake's dynamic effect. Beyond
562
B. Rowshandel and S. Nemat-Nasser
PAGEOPH,
the first cycle, one expects the strength of the fault to increase with time, as a healing
effect. The model, nevertheless, can be used to give results for further cycles if a
certain fault stress-strain behavior is assumed for the brittle zone, after the first
instability (i.e., beyond the post-peak softening shown by stress-strain curves of
Figure 3a). Since our results for the first cycle, however, were found to be sensitive
to the constitutive behavior of the material in the brittle zone, and since the constitutive equation used in this study does not incorporate the post-earthquake healing
effect, extension to higher cycles was not considered. For any objective conclusions
to be reached with regard to earthquake cycles beyond the first one, it is crucial to
use a constitutive equation which also takes into account the role of the period of
stationary contact on the fault (elastoplastic) strength.
Acknowledgements
This work has been supported by the U.S. Air Force Office of Scientific Research
under Grant No. AFOSR-84-0004 to Northwestern University. We are grateful to J.
W. Rudnicki and G. A. Kriegsmann for helpful discussions.
Appendix
The derivation of the plastic modulus Kp for simple shearing under .confining
pressure is summarized here, including (parametrically) the effect of temperature; for
more details see NEMAT-NASSER (1980), NEMAT-NASSER and SHOKOOH (1980), and
ROWSHANDEL and NEMAT-NASSER (1986).
Only the rate-independent plastic part of the deformation rate is considered. The
plastic shear strain rate and the dilatational strain rate are denoted by p and (9,
respectively, the shear stress and pressure being z and p. Both ? and | are loadpath-dependent. Let f = 0 define the yield surface and g be the plastic potential;
then
F(p, 7, |
T)
g - r + G(p, 7, |
T)
f =- z -
(A.1)
where T is the temperature. We then have
= ~ Og/& = 2, (9 = 2 Og/dp = 2 OG/Op
(a.2)
The parameter 2 = ~ is given by the consistency condition that, for continued plastic
flow, we must have f = 0. For deformation occurring at constant pressure and
temperature (i.e., at a given depth in the fault), we obtain
Vol. 124, 1986
A Mechanical Model for Deformation and Earthquakes on Strike-Slip Faults
OF 0G
0fly{/~)
;=f=
/TOTp+
563
(A.3)
To obtain the dilatancy parameter OG/Op= 0/9 = dO/d7 in terms of the pressure-sensitivity parameter OF/Op,we equate the rate of stress-work (per unit volume),
# = 39 + pO
(A.4)
with the rate of plastic dissipation,
+ 0F
where zr is the cohesive stress; note that
(A.5)
(OF/Op)pis the frictional force. This
OG OF + zc -- z
ap
Op
yields
(A.6)
p
Then from (A.3) it follows that
dz
KP=dy
-
OF07+ OF(OFc3| + 3~-3)
P
(A.7)
Note that in the simple shearing considered here, OF/Opis the coefficient of overall
friction; it is always positive. Equation (A.6) gives the variation of the volumetric plastic
strain rate with increasing shear stress z. When z is small, OG/Opis positive (volumetric compaction). After z attains a suitably large value, OG/Opbecomes negative
(volumetric expansion).
The parameter OF~07characterizes material-hardening due to shearing. It has
been shown by NEMAT-NASSER and SHOKOOH (1980) that, for earth materials, one
may assume
OF
O7
- e exp{-pT},
p > 0
(A.8)
where c~ and p (as well as OF/Op,OF/O| and 3c) are material parameters. In the
present case these parameters are regarded as depending on pressure and temperature; see ROWSHANDELand NEMAT-NASSER (1986) for a detailed account.
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(Received 16th September 1985, revised 17th June 1986, accepted 18th June 1986)