Grophys. J . Int. ( 1996) 125,473-490
Deformation fields around a fault embedded in a non-linear ductile
medium
Terence D. Barr' and Gregory A. Houseman2
' VIEPS Department of Earth Sciences, Monash University. Clayton. Victoria. 3 168, Australiu
Austrulirin Guodynamics Coctpmitiur Rrseurch Center and Drpurtment of' Muthemutics. Cluvton, Victoriu. 3 168. A u s t r u h
Accepted 1995 December 15. Received 1995 December 14; in original form 1994 December 12
SUMMARY
The geological record of deformation is often characterized by a combination of
discontinuous deformation, in which strain is concentrated in faults, and continuous
deformation, in which strain is distributed through the material. Where slip occurs on
a fault that terminates, the surrounding material is deformed. In the lower crust and
in cases where large strains occur over long geological time-scales, it is appropriate
to model the deformation using a viscous (probably non-linear viscous) rheology.
We describe a method for practical finite-element solution of this problem using a
dynamically self-consistent formulation for stress and displacement on a fault of
arbitrary geometry; the accuracy of the method is tested by comparison with an
analytical solution for the linear rheology. We describe here the instantaneous deformation fields around a mode I1 fault under both plane-strain and plane-stress
conditions, and a range of rheological exponents n (where strain rate is proportional
to deviatoric stress to the nth power). The distributions of stress and strain rate around
the fault tip are controlled primarily by the rheological exponent n. A localized zone
of high strain rate projects beyond the end of the fault if n is about 3 or greater, and
the degree of localization of deformation increases with the value of n. The zone of
high shear-strain rate can be defined in practical terms by considering (1) the region
in which the creep velocity differs by more than 20 per cent from the velocity on the
nearby external boundary and (2) the region in which the maximum shear-strain rate
is greater than about twice the externally imposed shear-strain rate. For n = 1, the
volumes so defined differ considerably, but for large values of n, the two definitions
both describe the same narrow zone of deformation beyond the end of the fault.
Evaluation of the Navier-Coulomb criterion for brittle failure of the medium surrounding the fault tip shows first that brittle failure is much more likely on the extensional
side of the fault than the compressional side. It also shows that the volume of material
subject to brittle failure decreases rapidly with increasing n because of the relatively
weaker stress singularity. We analyse previously published displacement versus distance
data for faults terminating in sedimentary rocks at 0.1 to 100m length-scales under
different tectonic conditions, in order to determine the rheological exponent n. These
analyses result in n values between approximately 0.85 and 5 for the different faults,
with error bounds on n typically rt 1. The variation in n values may result from
differences in pressure, temperature and fluid conditions at the time of faulting. More
importantly, the analysis demonstrates a new method for the determination of the
effective rheological exponent under in situ geological conditions.
Key words: deformation, ductile fault models, fault slip, rheology, sediments, stress
distribution.
0 1996 RAS
473
414
T . D. Burr und G. A . Housernun
INTRODUCTION
In this paper we describe the mechanical behaviour of a
discrete fault embedded in a ductile medium. In particular, we
examine the relationship between slip on the fault and associated ductile deformation. There have been many studies of the
mechanics of a fault in an elastic medium, which are pertinent
to faults close to the Earth's surface and are valid for small
strains a n d short time-scales (examples can be found in Rice
1968; Jaeger & Cook 1969; Rudnicki 1980 Segall & Pollard
1980; Lin & Parmentier 1988). We attempt here to extend the
understanding of the problem of deformation in a faulted
ductile medium and to examine the geological implications of
ductile deformation around the end of a fault.
One of the motivations of this work is to understand better
how faults are related to shear zones in the lower crust. In the
lower crust, deformation is assumed to be dominated by a
non-linear ductile rheology, as opposed to the brittle deformation mechanisms that dominate in the upper crust. Faults
that cut through the upper crust must penetrate this ductile
medium and presumably terminate in some sort of ductile
shear zone. Another motivation is to understand better the
distributed deformation associated with crustal-scale strikeslip faults, evident as zones of transtension or transpression
near bends or around the terminations of large fault systems
(Sylvester 1988; Odonne 1990).
In general, geological deformation can be represented as a
continuum deformation interrupted by faults. A deformation
field can therefore be described in terms of slip on the faults
and continuum deformation in the surrounding medium.
Continuum deformation in geological materials may comprise
both elastic and ductile behaviour; the former is most evident
on human time-scales as earthquakes and elastic wave propagation. The latter is most evident in the geological record
when we observe folds, ductile shear zones and continuous
strain. In this paper we assume that ductile deformation can
be represented by a non-linear viscous constitutive relation.
While a component of elastic strain must be present in the
continuum deformation field, we make the assumption that we
can separate the elastic strain from the viscous strain. This
assumption is rigorous in a Maxwell viscoelastic body under
a constant applied stress; in this case the elastic strain is
constant with time, while viscous strain accumulates with time.
This separation of strain fields is valid in the case of linear or
non-linear rheology.
The aim of this paper is to describe deformation around the
end of a fault embedded in a non-linear viscous medium. We
omit the elastic component of the strain field, which has been
described previously (e.g. Rudnicki 1980), and which is independent of the long-term viscous strain. It is important to
quantify the viscous strain, because, in many geological
examples, the viscous strain is of the order of 100 per cent or
greater, while elastic strain can never be greater than about 1
or 2 per cent. Also, after the tectonic stress is relaxed, only the
viscous strain is preserved in the geological record.
To describe the deformation field we consider the following
measures of deformation: velocity distribution, strain rate,
stress, and failure criteria. We first investigate the deformation
around a simple example in which a planar fault cuts into a
block of non-linear ductile material under simple shear. We
then investigate criteria for the localization of strain rate
caused by the interaction of the non-linear rheology and
displacement on the fault. Then we consider some physical
criteria for further brittle failure in the vicinity of the fault.
In the last section of the paper, we examine the observed
displacement on outcrop-scale faults to determine the apparent
rheology of these faulted sediments at the time of faulting.
The problem of a planar fault under simple shear is approximately 2-D in the plane containing the slip vector and the
normal to the fault surface. In three dimensions, the fault
surface is bounded by a line that contains the slipped region.
The 2-D approximation is valid where the curvature of this
line is negligible and slip is perpendicular to the bounding line
(usually referred to as Mode I1 slip). The bounding line is a
point in the 2-D case that we call the fault tip. For normal or
thrust faults, we assume the deformation occurs under planestrain conditions. We also consider the case of crustal-scale
strike-slip faults, where the plane-stress approximation is more
appropriate. In general, the plane-strain and plane-stress solutions will bound the deformation field associated with an
arbitrarily oriented Mode I1 fault.
We present solutions for a non-linear viscous rheology, with
a range of rheological exponents. For simplicity, we describe
here only the instantaneous solutions. It is first necessary to
understand The instantaneous solutions before continuing on
to the finite deformation problem.
M O D E L PROBLEM
To model the stress and strain rate about a fault tip, we
consider two simple cases of incompressible plane strain and
incompressible plane stress (thin-viscous-sheet approximation),
and neglect buoyancy forces. We assume a non-linear viscous
rheology to represent the ductile behaviour of crustal materials,
since many common minerals found in the lithosphere (e.g.
quartz, plagioclase and olivine) have been found experimentally
to have a rheological exponent n of 3 to 5 (Brace & Kohlstedt
1980; Shelton & Tullis 1981). Karato, Paterson & Fitzgerald
(1986), however, have documented olivine as having a rheological exponent of n = 1 at low stress difference or high
temperature. England & McKenzie (1982, 1983) and
Houseman & England (1986) have shown that a rheological
exponent n of at least 3 is required to represent lithospheric
deformation in an active collision zone.
We take a constitutive relation that relates the components
of deviatoric stress and strain rate as follows:
is the strain
where zij is the deviatoric component of stress,
rate,
is the second invariant of the strain rate tensor, n is
the exponent that determines the non-linearity of the rheology,
and B is an empirical constant. If n = 1, B = 2q0, where qo is
the viscosity of the medium. The i and j subscripts represent
the Y, or y directions. The strain rate is defined in terms of the
x- and y-direction velocity components:
[-
i i j= -1 aui
2
ax,
+
21.
In this paper, we use the convention for strain rate, stress and
pressure that extensional values are positive and compressional
values are negative.
0 1996 RAS. G J I 125, 473-490
Dcformution ,fields uround u ,fuult
plane strain
To calculate the incompressible plane-strain deformation in
the non-linear viscous material, we use the above constitutive
relations with the force balance equations
a
- q j
axj
8
+ 7oxi
p=o,
(3)
where summation is over the j index, p is the pressure, and
gravitational body forces have been neglected. We also assume
that the material is incompressible:
(4)
gJ.J . = o .
With the above equations and appropriate boundary conditions, we solve for the velocity u and pressure p at every
point within the medium. The solution plane may be arbitrarily
oriented, but for the plane-strain case we generally assume
that it represents a vertical section.
475
boundary conditions that are applied to the perimeter of the
box approximate simple shear, as shown in Fig. 1. A velocity
of (ux,u,,) = (u,/2,0) is applied t o the top boundary (y = R,/2)
and the right-hand boundary above the fault, and a velocity
of (uy, u y )= (- 11,/2,0) is applied to the bottom boundary
( y = - R,/2) and the right-hand boundary below the fault.
This gives a slip rate u, on the fault at the right-hand boundary
(x=R,). O n the left-hand boundary ( x = -R,) there is a
constant normal stress g b , and the y-direction velocity on this
boundary is set to be u,=O. The stress o b is the background
stress affecting the region and may be equated to the overburden stress in some problems. We assume crb=crrz in the
plane-stress case. The solutions here are illustrated with
ob = 0; if cr, # 0 then displacements, strain rates and deviatoric
stresses are unaltered, but pressures, total stress components,
and the criteria for brittle failure are changed by the addition
of the constant g b .
Numerical technique
Thin viscous sheet
For the thin-viscous-sheet (plane-stress) approximation, the
solution plane is assumed to be horizontal, and the stress
balance (eq. 3) simplifies, by substitution of p = ozz- T,, and
averaging through the layer thickness, to
where z represents the vertical direction and oz=and zij are
now the vertical averages of the stress and deviatoric stress
components. The vertical deviatoric stress T,; is constrained
by the incompressibility (together with eq. 1 ) to be
T z z = - (7xx
+zyy).
(6)
Strain in the z direction (vertical) is thus permitted, but in the
solutions shown below, we only examine the instantaneous
solutions and therefore assume that crrz is constant.
We use the finite-element scheme described in the Appendix
to solve eqs 3 and 4 for the plane-strain case and eqs 5 and 6
for the thin-viscous-sheet case with the above boundary conditions and representative values of n = 1, 3, and 10. Accuracy
of the calculations is tested by comparison with an analytical
solution for the linear rheology (Appendix), and the difference
between the numerical and analytical solutions is quantified
for different mesh resolutions in Table A l .
The calculations use a length scale of R,, a velocity scale of
u,, a strain rate scale of u,/R,, and (for a linear rheology) a
stress scale of ~ , u o / R o .In presenting the results below, we
arbitrarily choose to set R, = I km, u, = 10 mm yrC1, and
for n = 1, q, = lo2" Pa s. For n > 1 , we choose a rheological
~
constant B (eq. 1 ) such that the total shear stress, T = T , dx,
along the external boundaries, y = RJ2, is the same for
n = 1, 3, and 10.
N U M E R I C A L RESULTS
Fault boundary conditions
A fault is introduced into the homogeneous viscous medium
as an internal boundary across which slip is allowed. The fault
We have obtained numerical solutions for the cases where the
rheology of the material surrounding the fault is characterized
is defined by the following boundary conditions:
(1) the normal stress across the fault is continuous;
( 2 ) the velocity normal to the fault is continuous but
otherwise unconstrained;
(3) the shear stress on the fault is equal to some constant,
which is set to be zero.
The actual shear stress on active faults is poorly constrained,
but the purpose of this paper is to examine how a very weak
slip surface affects the stress- and strain-rate fields. We choose
a reference frame in which the fault tip is at (0,O)and the fault
lies along the positive x axis. In this discussion, we refer to the
unfaulted negative x axis as the projection of the fault.
External boundary conditions
We consider a rectangular region (which has dimensions 2R,
by R,) centred about the fault tip (0,O). The fault of length
R , extends to the right-hand boundary at (R,, 0). The external
0 1996 RAS, G J I 125, 473-490
r
normal stress
Ob
i
---u
= -u0/2
Figure 1. Geometry and boundary conditions for the numerical solutions. We use a coordinate system with the origin at the fault tip, and
the box has a dimension of 2 R , by R,, as shown by the reference
points. The fault extends to the centre of the block from the middle
of the right-hand side. A dextral shear stress is imposed by a constant
velocity of u=u,/2 on the upper and u = -u,/2 on the lower surface,
a s shown by the arrows. On the left-hand boundary, a constant normal
stress, ob is applied. The model may represent a thrust, normal or
strike-slip fault by re-orienting the vertical direction as required.
476
T. D. Burr und G. A. Houseman
by power-law exponents of n = 1, 3 and 10. The strain-rate
and stress distributions near the fault tip are insensitive to the
details of the boundary velocity distribution. At distances from
the fault tip that are less than half way to the external
boundary, the solution is dominated by a singularity in the
strain-rate field, whose magnitude is determined only by the
rheological exponent n and the strain-rate intensity factor
(u,/R,) (Barr & Houseman 1992). The radial behaviour of the
velocity, strain rate, and stress fields near the crack tip in the
non-linear numerical solutions is as predicted by the analysis
of Rice & Rosengren (1968) and Hutchinson (1968) for the
analogous case of a traction-free crack in a non-linear elastic
medium. An analytical solution for the singular component is
described in the Appendix for n = 1. For non-linear rheology,
the radial dependence of the stress, strain-rate and velocity
components is summarized by Barr & Houseman (1992).
Velocity
The two components of the velocity field for the deformation
around the fault are shown in Fig. 2 for the plane-strain case
and Fig. 3 for the thin-viscous-sheet case. For each solution,
the velocities are predominantly parallel to the fault, with
only a minor component normal to the fault ( l u x [>> Iu,~),
as
illustrated in Fig. 4(a).
T h e x component of velocity increases with distance r from
the end of the fault approximately as ( B a n & Houseman 1992)
T h e slip rate on the fault can be determined by comparing
the difference in the x component of the velocity across the
fault. The slip rate shows the same radial dependence as the
velocity (eq. 7), since the x component of the velocity is antisymmetric about the fault. The slip rate for each rheology
varies from 0 to 10 mm yr-' from the fault tip to the righthand boundary, but the deformation is more concentrated
towards the fault tip with increasing n. As an example, the slip
rate drops to 5 mm yr-' at a distance of approximately R,/4,
R,/16 and RJ2" for n = 1, 3 and 10, respectively. For large
n, the material on either side of the fault essentially moves as
a rigid block, with most deformation concentrated close to the
fault tip.
For n = 1, there is a uniform gradient in the x component
of the velocity on the left-hand boundary. With increasing n
the gradient in the x component of the velocity at the left
boundary is increasingly localized into a narrow band centred
on y = 0, as evidenced by the closely spaced contours there
(Figs 2a and 3a). The net x-direction compression in the lower
half of the solution and x-direction extension in the upper half
require a y-direction transfer of material. The y-direction
velocity is > O almost everywhere in the plane-strain case, but
surprisingly is negative near the end of the fault in the thinviscous-sheet case. For n = 1 there is significant y-direction
movement both across the fault (x > 0) and around the end of
the fault (x < 0 ) . As n increases however, the movement across
the fault decreases dramatically (Figs 2b and 3b).
For n = 1, the y component of velocity shows a local
maximum on the fault and a second local maximum on the
fault projection (x < 0). As n increases, the maximum on the
fault decreases in magnitude and moves towards the fault tip,
whereas the maximum on the projection of the fault increases
in magnitude. The solutions illustrate a fundamental difference
between the n = 1 case and the n 2 3 cases in both plane strain
(Fig. 2) and plane stress (Fig. 3): as the fault tip is approached
along the fault (x > 0), y-direction displacement is observed to
increase towards the fault tip for n 2 3, whereas for n = 1, it
initially increases, then decreases. With finite deformation,
therefore, the maximum y-direction displacement will occur
on the fault for the n = 1 case and will occur beyond the end
of the fault (x < 0) for the n 2 3 cases.
Strain rate
The strain-rate components are related to gradients in the
velocity field, as shown in eq. (2). The principal strain-rate
directions are shown in Fig. 4(b). For the n = 1, 3 and 10
rheologies in both the plane-strain and thin-viscous-sheet cases,
the principal strain-rate directions appear almost identical in
this type of plot. Beyond the end of the fault (x<O), the
principal extensional component of strain rate i , is oriented
approximately at 8 = 45" (counter-clockwise) to the orientation
of the fault. We use 8 to define a direction relative to the fault
tip, i.e. 0 = tan-'(y/x). As the fault is approached, the principal
strain-late axes rotate so that i, becomes parallel to the fault
on the tqper side ( y > 0) and perpendicular to the fault on the
lower side ( y < O ) . On the boundary x = R,, the strain-rate
field is negligible and isotropic, so the directions shown on
that boundary are meaningless. The magnitudes of the principal
components of strain rate are shown in Fig. 2(c) for the planestrain case in which C, = -& and in Figs 3(c), (d) and (e) for
the thin-viscous-sheet case.
The strain-rate field has a singularity at the fault tip due to
the discontinuity of traction at the fault tip. The strength of
the singularity depends on the rheology of the material surrounding the fault tip; each component of the strain rate has
a power-law dependence on the radial distance from the fault
tip that is a function of the rheological exponent, n, as follows
(Barr & Houseman, 1992):
-
i (uo/Ro)(r/R0)-"~("' ) .
+
(8)
In the plane-strain case (Fig. 2c), the principal components
of strain rate, dl and i,, are symmetric with respect to the fault
[i.e. E,(x, y ) = dl(x, - y ) ] . In the thin-viscous-sheet case the
vertical component of strain rate izzis anti-symmetric with
respect to the fault, and the following symmetry relation holds
between the two horizontal principal strain rates in the
thin-viscous-sheet case: &(x, y ) = -i,(x, - y ) (Figs 3c, d and
e). Note that i , + E, + d,, = 0 for an incompressible material.
For n = 1, at a given radius from the fault tip, the principal
components of strain rate, 6, and 6,, are maximum along the
fault (0 = 0") and along the projection of the fault (0 = 180").
As n increases, strain rates adjacent to the fault decrease in
magnitude and strain rates on the projection of the fault
increase in magnitude. As n increases, strain rate becomes
more localized in an angular sense about the projection of the
fault and is more concentrated radially near the fault tip.
Stress
The deviatoric stress components are related to the strain rate
components by the non-linear constitutive equation ( 1). The
principal stress directions are the same as the principal
strain-rate directions, which are shown in Fig. 4 for n = 1.
Fig. 5 shows the magnitudes of the principal deviatoric stress
0 1996 RAS, G J I 125, 413-490
1
n=l
n=3
n=10
Figure 2. Contour plots for plane-strain deformation of the components of velocity (a) u, and (b) uy and the principal strain-rate components (c) i,
= -i3 ( i z=
z0) for each of n = 1, n = 3, and
ri = 10. The contour interval is 1 mm yr-' for u,, 0.1 mm y r - ' for ul. and 0.5 x lo-" s-' for i.
Regions with negative values are stippled. Strain rates in the n = 10 case are not resolved from zero
in the top right and bottom right corners, and contouring of the flat field in these regions produces contouring artefacts in some diagrams.
(c) E
Plane strain
u
n
'-L
"E;;.
-
a
a
3
2E
x
478
T. D. Burr. and G. A . Houseman
Plane stress
,
n=3
n=l
i
n=10
I
I
Figure 3. Contour plots for plane-stress deformation of the two components of velocity (a) u, and (b) up and the principal strain-rate components
for ug and
(c) d , , (d) 8, and (e) d,, = -(dl + d3) for each of n = 1, n = 3, and n = 10. The contour interval is 1 mm yr--l for u,, 0.1 mm yr
0.5 x
sK1 for 8. Regions with negative values are stippled.
K
component T , and the associated pressure anomaly near the
fault for the plane-strain case with n = 1, 3 and 10. The
corresponding quantities are shown in Fig. 6 for the thinviscous-sheet case. Note that z,, = --p in the thin-viscous-sheet
case, and T~ = -T, in the plane-strain case.
The mean pressure in these figures is constrained to be zero
by setting the background stress g h = O (Fig. 1 ) . The
corresponding solutions for cases where gb is a non-zero
'
constant are obtained simply by adding that constant to the
pressure field shown here. To determine the anomalous pressure field around a fault terminating at some depth where the
overburden stress is oZz,we assume that gh= crzz and add this
constant to the pressure solutions shown in Figs 5 and 6. This
is a good approximation if the solution region represents a
horizontal section, but it neglects vertical stress gradients due
to gravity if the section is vertical, and in that case is only
0 1996 RAS, G J I 125,473-490
Defirmution fields around u fuult
479
>< x x x x x x x x x x x x x >(
>: x x x x x x x x - % - % xx+ : <
::x x x x x x%++++++ x
>( x x x x x x %%+%+++:(
>( x x x x X % ( X % # % x X-t-):
::x x x x x x x x x x x x % ::
Figure 4. Arrow plots showing (a) the creep velocity and ( b ) the orientation of principal strain-rate axes for the plane-strain solution with n = 1.
In (a) the maximum velocity is 5 mm yr-’ and length of the arrow is proportional to velocity. In (b) all arrows are scaled to the same length in
order to show directions clearly; refer to Fig. 2(c) for amplitude of principal strain rates.
valid if the depth of the fault tip >>R,. We refer to the pressure
anomaly illustrated in Figs 5 and 6 as the tectonic component
of pressure, pt, due to the fault-tip singularity. The total
pressure at a point (x, y ) is given by p = pt(x, y ) ob.
Each of the stress components, including pressure, has a
singularity at the fault-tip. The stress and deviatoric stress
components have a radial dependence near the crack tip that
is dependent on the rheological exponent n as follows (Barr &
Houseman 1992):
+
7
- B(~o/Ro)l’n(r/Ro)-l’(n+l).
19)
The pressure anomaly has an r - l l * cos(0/2) dependence for
n = 1 (Appendix). As n increases, the pressure anomaly becomes
more elongated in the y direction, and its magnitude decreases
more slowly with r . The applied velocity boundary conditions
cause the lower half of the faulted block to be under compression and the upper half to be under extension. At a given
radial coordinate, the magnitude of the pressure anomaly in
the thin-viscous-sheet case is about half that in the planestrain case (Appendix).
In a mechanical sense, a singularity in stress is not possible
because at some very small scale, chemical bonds have only a
finite strength. This dilemma has been addressed in many
studies of cracks in elastic media (e.g. Rice 1968; Scholz 1990)
and we do not attempt to discuss it in detail here. There are
three ways to deal with the singularity in stress: (1) change the
geometry of the fault tip; (2) change the rheology of the
material surrounding the fault tip; or (3) increase the shear
stress on the fault near the tip, thus weakening the singularity.
If the fault tip has a finite curvature at some small scale (i.e.
it is not infinitely thin), the stress will not be singular. If the
rheology of the material becomes extremely non-linear (i.e. the
power law exponent n ---* a,or, in effect, it becomes plastic)
within some small radius about the fault tip, the stress will not
become singular.
THE RELATIONSHIP BETWEEN FAULTING
A N D DUCTILE DEFORMATION
Most observed shear zones in the geological context are zones
where the accumulated shear strain is relatively high. We
examine the distribution of strain rate around a fault tip and
assume that the area where strain rate is concentrated is
indicative of the formation of shear zones. In doing so, we
recognize that areas where the strain rate is high do not
necessarily correspond to areas where the accumulated strain
0 1996 RAS, G J l 125,473-490
will be high after a finite amount of deformation, as material
may move through the zone of high strain rate.
There are three aspects that should be examined in this
context in considering the relationship between a fault and an
associated ductile shear zone: (1) the width of the zone where
most of the ductile deformation is occurring in the vicinity of
the fault tip; (2) the region where shear strain rate is high; and
(3) the region where subsequent brittle failure is expected, and
the mode of failure. These three aspects are related, but d o not
necessarily coincide with each other.
For the purpose of this discussion, we define a displacement
shear zone off the end of the fault to be the region where the
x-direction velocity is between f4 mm yr-’. This region
accommodates 80 per cent of the relative displacement rate
imposed on the external boundaries, and is symmetric about
the fault. This shear zone is simply the region (Figs 7a and 8a)
where the velocities deviate significantly from those imposed
on the external boundary ( k 5 mm yr-I). For the linear case
where there is little deviation from a uniform simple-shear field
near the left boundary, this definition of a displacement shear
zone has little physical relevance, but for n 2 3 it becomes a
more useful definition.
For n = 1, the displacement shear zone is a ‘u’-shaped region
stretching from approximately x = (3/5)R, on the fault to the
left-hand boundary (Figs 7a and 8a). As n increases, the area
of the ductile shear zone decreases significantly and the shearstrain rate is more concentrated in front of the fault tip along
the projection of the fault (Figs 7a and 8a). For n = 3 and
n = 10, this ductile shear zone extends from the left boundary
(x = - R 0 ) to x = (2/5)R, or x = ( l/lO)R,, respectively. For
large n, almost all the ductile deformation is centred around
the projection of the fault.
The focusing of the velocity contours near the fault tip
indicates a concentration of the rate of shear deformation. We
show, in Figs 7(b) and 8(b), contours of maximum shear strain
rate for the same experiments (n = 1, 3 and 10). This measure
of the rate of shear deformation is independent of the orientation of coordinate axes. The maximum shear strain rate is
,1 =(1/2)1ci, -i31
where
,
ci, and ci3 are the greatest and least
principal strain rates; the orientation of the principal axes (for
n = 1, plane strain) is illustrated in Fig. 4. The planes that
experience maximum shear are at 45“ to the principal axes.
For the linear rheology, n = 1, the maximum shear-strain rate
shows a bimodal distribution with maxima centred on the
fault and on the projection of the fault, both of similar
magnitude (the analytical solution to the n = 1 problem has a
480
T . D.Barr und G. A . Houseman
Plane strain
n=l
n=3
n=10
(b) P
Figure 5. Contour plots of (a) principal deviatoric stress component ( T =
~ -z3) and (b) pzessure for plane-strain solutions for each of n = 1, n
and n = 10. Contour interval is 10 MPa for all plots, and stippled regions show negative d u e s .
Plane stress
n=l
n=3
= 3,
n=10
Figure 6. Contour plots of principal horizontal deviatoric stress components (a) z1 and (b) z3 and (c) pressure for plane-stress solutions with n =
1, n = 3, a n d n = 10. Contour interval is 10 MPa for all plots, and stippled regions show negative values.
0 1996 RAS, G J I 125, 473-490
Dqformation fields around a fault
Plane strain
n=3
n=l
48 1
n=10
I
I
.. . .._
. ... ..
.......
........
............ .....
3
.;;;:.. '
Figure 7. Two different measures of shear localization about the fault tip for the plane-strain solutions with n = 1, n = 3, and n = 10. (a) shows
contours of the u, component of velocity (interval 1 mm yr-I), with the shaded area showing where the magnitude of the velocity is less than
4 mrn yr-', and (b) shows contours of the maximum shear-strain rate (interval
SKI),
with the shaded region indicating where the shear-strain
approximately twice the shear-strain rate imposed by the boundary conditions.
rate is greater than 3 x
S
C
'
Plane stress
I
,
n=l
n=3
I
Figure 8. Two different measures of shear localization about the fault tip for the plane-stress solutions with n
presentation is the same as for Fig. 7.
0 1996 RAS, G J f 125,473-490
n=10
=
1, n = 3, and n = 10. The graphical
482
T. D. Burr und G. A . Housemun
cos 20 dependence, see Appendix). As n increases, the maximum
shear-strain rate increases along the projection of the fault and
decreases along the fault itself. To compare between the
different rheologies in Figs 7(b) and 8(b), we have shaded
the region where the maximum shear-strain rate exceeds
3 x lo-" s-', which is approximately twice the shear-strain
s-') that is nominally imposed by the exterrate (1.58 x
nal boundary conditions. This region where the maximum
strain rate is high both increases in area and becomes more
elongated about the projection of the fault as the material
around the fault becomes more non-linear.
Figs 7 and 8 thus illustrate two alternative, equally valid
and useful definitions of a ductile shear zone based on mechanical deformation-rate criteria. The two definitions are
roughly consistent for large n, and the differences for n = I
illustrate the limited applicability of the shear-zone concept in
the case of homogeneous linear rheology.
As the compression increases in magnitude ( p becomes more
negative), a greater shear stress is required for brittle failure
to occur. Conversely, given the predicted distribution of t,,
we may ask what pressure is required to prevent brittle failure
at any particular location in the ductile medium. The pressure
field includes the tectonic component due to faulting (illustrated in Figs 5 and 6), plus a constant component of background stress ( p = pt + ob). Because the relationship between
tectonic pressure pt and shear stress z, is defined by the faulttip solution, the only degree of freedom that determines
whether failure occurs at any location (.x, y) is the applied
background stress ob. We therefore define the variable o,(x, y )
as the value of background stress fib required to prevent brittle
failure under plane strain at location (x, y), such that
BRITTLE F A I L U R E A R O U N D T H E F A U L T
TIP
Contour plots of offor the three rheological models are shown
in Fig. 9(a) using a typical coefficient of friction, p =0.85
(Byerlee 1978). Note that the scale in Fig.9 is enlarged 10
times relative to the preceding figures to show detail around
the fault 'tip. The shaded region in Fig. 9(a) shows the area
that is subject to brittle failure for a given background stress
g b = - 160 M P a (corresponding roughly to a depth of 5 km).
If the magnitude of the background stress is decreased, the
region subject to brittle failure expands to fill the contour
labelled by the appropriate value of of. If the depth were
doubled, the region of brittle failure would be much smaller
in volume (enclosed by the -320 M P a contour).
At 5 km depth, the region where brittle failure is predicted
in plane strain is roughly 100 m wide for n = 1, and for II = 3
and 10, this region decreases to a width of 4 0 m and 10m,
respectively. This shaded region of predicted brittle failure is
asymmetric relative to the fault, because the upper side of the
fault is under relative tension compared to the lower side of
the fault. Eq. ( 1 1 ) can be used to determine the direction of
probable failure relative to the principal compressive stress
directions (shown in Fig. 4 for n = 1). As n increases, the region
of predicted brittle failure decreases in size very rapidly and
shifts more towards the upper side of the fault. Note that we
have not included the effect of pore-fluid pressure in the above
analysis. If pore-fluid pressure is a fixed ratio ,
Iof the background stress ob (which can represent the overburden), as is
often assumed, ofis increased by the ratio 1/( 1 - A).
The gravitational pressure gradient has been assumed as
negligible in these calculations. If the critical background stress
were due to overburden, it should be corrected for the pressure
gradient due to gravity, but this effect is negligible for the
length-scale shown in Fig. 9. For the 100 m wide cell, there
would be a corresponding change in overburden pressure of
approximately 3 MPa across the cell.
Another aspect to consider in the relationship between faulting
and distributed ductile deformation is whether the ductile
medium surrounding the end of the fault can support the
very high shear stresses, and where brittle failure would occur.
Because the shear stress on the fault is zero, stresses are
supported by the surrounding viscous medium. If these stresses
locally exceed some failure criterion, brittle failure may occur.
If the three principal stress axes, ul,o2 and o3 are known, the
Coulomb failure criterion can be applied:
Itl=
(10)
-Po"?
where p is the coefficient of friction, z is the shear stress and
the normal stress acting on an arbitrarily oriented plane.
We neglect cohesion as it is inconsequential, except in the
upper few kilometres of the crust. With brittle Coulomb failure,
the most probable planes for failure are oriented at an angle
+$ t o the direction of maximum compression t3,where
0, is
tan 2 4 = l/p.
(11)
The normal stress on and the shear stress z on these planes
can be written as
1
on= -(o,
2
+ 03)+ t, cos 24
and
z = -tmsin2$,
where ~ , , , = ( 1 / 2 ) ( (-03)
~ ~ is the maximum shear stress at a
given location. Resolving the normal and tangential stress
components o n these failure planes, eq. (10) may then be
rewritten as
Brittle failure in the thin-viscous-sheetsolution
Brittle failure in the plane-strain solution
For the plane-strain case [where p
t2 = 01, eq. (14) simplifies to
= (1/2)(a,
+ 03) and
The thin-viscous-sheet approximation permits z-direction (out
of plane) strain, so eq. (16) is modified. For this section we
assume that the solution plane is horizontal and the z direction
is vertical. If t, and z3 are the maximum and minimum
deviatoric stresses, then the background stress required to
(1 Ox mag)
(1 Ox mag)
(1Ox mag)
(1 Ox mag)
n=3
I
\
I
_
(lox ma;
(1Ox mag)
~
n=10
_
Figure 9. Contours enclosing arcas subject to brittle failure for a given background stress uf (contour interval 20 MPa). The shaded region in row ( a ) shows where brittle f;dure is likely to occur
under plane strain when the overburden stress is - 160 MPa (corresponding to a depth of about 5 km). If the overburden stress were doubled, for cxample, the region of likely brittle failure would
become much smaller (within the -320 MPa contour). The shaded region in row (b) shows where brittle failure is likely to occur under plane stress when the overburden stress is - 120 MPa. P'or
the plane-stress case, row (bl. the region is divided into three areas showing where brittle fsilure i h predominantly by normal faulting 1x1,thrust fai;!ting (TI o r strike-slip faulting ( S S I Note that
the frames in ro\vs (a) and ( h ) show a 10 times inagnificd view centred on the fault tip (i.e. the b o x 5 are 200 m aide). Row ( c ) shows a different distribution of t h e faulting types associated with the
calculated strain-rate fields for the plane-stress solutions These are the fault types predicted in a thin brittle layer overlying a thin viscous sheet (c.g. crust over mantle). 1-he shaded regions indicate
where thrust and strike-slip faulting occur and the non-shaded regions are where normal and strike-slip faulting occur (the solution plane is horizontal). The internal boundary separates those
regions where strike-slip faulting dominates over thrust (ST) or normal (SN) from those where either thrust or normal farilting dominates (TS or NS).
Plane stress
(b) O f
Plane stress
(a) Of
n=l
484
T . D. Burr und G. A . Housemun
prevent brittle failure at a given location is
If T~ and T~ are horizontal, of is the minimum background
stress that will prevent strike-slip faulting; if the maximum ( T ~
or the minimum (zl) compressive stress is in the vertical
direction, of is the background stress that will prevent normal
or thrust faulting, respectively.
The shaded area in Fig. 9( b) shows the area subject to brittle
failure for ob= - 120 MPa in the thin-viscous-sheet solutions,
and the contours of of show how the region of brittle failure
expands or contracts if the background stress changes. At each
point, we calculate separately the three background stress
values required to prevent strike-slip faulting, normal faulting
and thrust faulting, respectively. At every point in this figure,
the three values of ofare calculated, and the largest magnitude
(most negative) of the three is contoured. The regions in this
figure are denoted as strike-slip (SS), normal ( N ) , or thrust
(T), depending on which type of faulting requires the largest
magnitude of uf.If the background stress is of greater magnitude than the contoured value of ofat a given location, brittle
failure does not occur at that point; a background stress of
smaller magnitude will allow brittle failure, possibly by more
than one type of fault mechanism (e.g. strike-slip and normal
or strike-slip and thrust faulting).
Where brittle failure occurs under plane-stress conditions,
we expect normal and thrust faulting on either side of the fault
and strike-slip faulting beyond the end of the fault. Thrust
faulting occurs o n the side under relative compression and
normal faulting occurs on the side under relative extension.
With increasing n the thrust and normal failure regions increase
in area a t the expense of the strike-slip failure region, as shown
in Fig. 9(b).
The predicted brittle failure regions shown in Figs 9(a)
and ( b ) are based on a Coulomb failure criterion. It is, however,
also interesting to consider a kinematic criterion for surface
faulting, as opposed to the above dynamic failure criterion:
suppose the ductile thin viscous sheet is overlain by a thin
brittle layer ( e g crust over mantle) that is forced to accommodate the deformation field of the underlying ductile layer by
brittle failure. The deformation field at any point in this brittle
layer may be accommodated by a combination of thrust,
normal and/or strike-slip-type failure. Houseman & England
(1986) showed that the partitioning of strain in such a model
is simply dictated by the relative magnitudes of the two
principal horizontal strain rates, and in general two types of
faulting occur at any particular location.
Fig. 9(c) shows the distribution of faulting types that would
be associated with the calculated strain-rate fields under the
kinematic faulting criterion. The contours near the right end
of the block are not meaningful because the strain rates in
those regions are negligible. The region y > 0 is under relative
tension and thus deforms by a combination of normal and
strike-slip faulting. Within the closed contour in this region,
the magnitude of normal faulting exceeds the magnitude of
strike-slip faulting, and outside the contour, the opposite is
true. The lower half of the contoured region ( y < 0) is under
compression, and reverse faulting is substituted for normal
faulting in the above description. Strike-slip faulting dominates
in those regions where the vertical strain rate has an absolute
magnitude that is less than either of the horizontal principal
strain rates. Fig. 9(c) shows that in the area within which
normal or thrust faulting dominates, strike-slip faulting increases
as n is increased from 1 to 10 in the thin-viscous-sheet models.
)
GEOLOGICAL EXAMPLES
There have been several field studies which have attempted to
quantify the ductile deformation in the vicinity of a fault tip.
Williams & Chapman (1983) have measured variable slip near
the tips of thrust faults in Devon, England and Pembrokeshire,
Wales. Coward & Potts (1983) have measured strain ratios
and orientations in front of thrust faults in the Moine thrust
zone, Scotland. Muraoka & Kamata (1983) have measured
variable displacement on minor normal faults in Quaternary
sediments in Kyushu, Japan. Odonne (1990) has made estimates of the total strain around the Meyrueis wrench fault,
southern France, and compared his field observations with
experimental wax models. Hyett ( 1990) has measured both
variable slip and principal strain directions near the tip of the
Tutt Head thrust zone in Swansea, Wales. Peacock (1991) has
looked at variations in displacement along strike-slip faults at
Gypsy Poifit, Scotland. Scholz et ul. (1993) have studied how
maximum fa,& displacement scales with length in normal
faults near Bishop, California.
In the field studies of slip versus distance from the fault tip
(Williams & Chapman 1983; Muraoka & Kamata 1983; Hyett
1990; Peacock 1991), the slip on the faults is observed to
decrease rapidly to zero as the fault tip is approached. Where
the slip is large, deformation adjacent to the faults is accommodated by asymmetric folding. The extensional direction is
approximately parallel to the fault orientation on one side of
the fault and perpendicular to the fault on the opposite side.
In the viscous rheological model presented here, the slip rate
decreases as r l i ( " + l (eq.
)
71, where r is the distance to the fault
tip; therefore, for small displacements in the absence of fault
propagation, the slip should show the same radial behaviour
as the slip rate.
We fit displacement data (Table 1 ) published by Williams &
Chapman (1983; Fig. 7 ) , Peacock (1991; Fig. 8) and Muraoka
& Kamata (1983; Table 1) to this power-law equation to see
whether the model is consistent with this data, and if so, to
obtain an estimate of the value of n that best describes the
macroscopic rheology of these field examples. We assume the
displacement is proportional to the slip rate, and then fit
the displacement-versus-distance (Ad versus r ) data to the
equation
Ad = d,(r -
(18)
where ro, the unknown coordinate of the fault tip, rn, a powerlaw coefficient, and do, a proportionality constant, are to be
determined by least-squares fit to the data set. If the data are
consistent with the non-linear viscous rheology, rn = l/(n + 1).
In the data presented in Table 1, the distance r is measured
from an arbitrary reference point; thus, the distance from the
fault tip is r - r o . Values for rn, ro, and do that give the best
least-squares fit of eq. (18) to each data set are shown in
Table 1 and the data and best-fit curves are shown in Fig. 10,
together with estimates of the error in rheological exponent n.
For the minimization we used a coarse grid search of the three
variables, refined by a Levenburg-Marquardt algorithm (Press
et ul. 1992). We estimate errors on the three parameters by
0 1996 RAS, G J I 125, 473-490
Deformation fields around a fault
485
Table 1. Measurement of slip versus distance for various faults and best-fit model parameters.
~-
Peamck
Williams & Chapman
@)Fault G25
(a) Exe
Muraoka & Kamata
(c)
Fault G28
(4Fault
10
It 12
(el F
11
g) Fault 13
~
9
r (m)
5.36
8.92
9.45
13.11
14.34
15.08
0.80
6.93
7.35
9.35
10.08
10.29
ro = 3.2
do = 2.7
m = 0.54 -0.07+0.02
n = 0.86f0.06
1
1.39
1.88
2.51
3.16
3.45
3.69
4.37
4.78
5.04
5.61
7.55
8.14
8.37
8.65
9.43
10.84
0.75
0.80
0.75
0.72
0.72
0.68
i:
m = 0.31 -0.09+0.07
n = 2.2-0.6+1.7
9.55
10.08
12.06
13.24
1
0.09
0.20
0.50
0.55
0.58
0.70
0.77
0.79
0.84
0.99
0.95
0.96
0.96
0.93
0.90
0.97
0.95
ro = 0.75
do = 0.59
m = 0.22 -0.04+0.07
n = 3.5fl.l
r
~
2.5
7.3
14.0
r o = 3.7
do = 7.0
m = 0.17f0.05
n = 5.0-1.5t2.5
using the criterion that an acceptable fit to the data has a
residual (sum of the squares of the difference between model
slip and observed slip) that is at most 30 per cent greater than
the absolute minimum residual.
The best-fit model parameters for each of the faults studied
are given in Table 1, together with error estimates for the value
of n. Williams & Chapman (1983) report displacements on the
order of 0-10 m on a thrust fault in thinly bedded limestones
and slates in Devon, England. For their data, we obtained a
value of n =0.86f0.06, shown in Fig. 10a. Peacock (1991)
reports displacements on the order of 0-1 m on strike-slip
faults cutting sequences of interbedded shale and sandstone
units at Gypsy Point, Scotland. We analysed two of the faults
data set and obtained values of n = 3.5 & 1.1 and
from
n = 2.2(-0.6 + 1.4) (Figs 10b and c).
Muraoka & Kamata (1983) present a series of profiles
showing millimetre-scale slip on minor normal faults cutting
Quaternary sediments in Kyushu, Japan. We chose a subset
of these profiles that are consistent with a power-law relation,
and obtained values of the rheological exponent varying
between n = 1.5 and 5.4, with errors given in Table 1 and Figs
10(d)-(g). Not all of their examples show the smoothly varying
curve expected of a power-law relation. In some cases the
slip-versus-distance curve is stepped, suggesting that fault
propagation may have occurred during slip accumulation.
We analysed a subset of their data where there are no significant steps in the slip-versus-distance curve. It may also be less
appropriate to apply the model to these data because the
displacements in the Kyushu examples represent strains
of order 1 per cent, so it is possible that a significant component of the deformation is elastic rather than viscous, and
measurement errors may also be more significant.
The above analyses show that it is possible to estimate
rheological exponents for faulted rocks at the centimetre to
100m scale. These data show a broad range of n-value
estimates (between 0.86 and 5.4),which are generally within
the range of exponents measured in rock deformation experi-
%r
0 1996 RAS, GJI 125, 473-490
10.0
Ad (mm)
(cm)
0.9
2.7
4.4
5.0
6.5
10.6
18.3
23.3
24.2
24.9
29 I
35.3
35.7
36.6
42.2
2.0
2.5
4. I
4.7
::: 1
4.0
4.0
6.0
13.6
19.9
20.8
23.6
6.0
7.0
7.0
26.5
32.9
34.9
39.7
42.4
47.6
54.1
65.2
7.0
7.5
1.5
2.0
2.0
7.0
7.0
11.0
12.0
12.0
13.0
13.0
13.0
14.0
14.0
15.0
8.0
ro = 0 29
do = 1 7
m =0.394).12,+0.17
n = 1.5-0.7+1.2
1~
ro = 2.2
do=2.9
m =0.36+0.07
n = 1.8f0.5
1
ro =
19.6
d o = 8.2
m =0.164.05+0.08
n = 5.4t2.3
ments on quartz aggregates, as tabulated by Koch et al.
(1989; Table 1). However, our analysis also shows that the
error on the effective n value may be relatively large for any
particular fault. The range of n values we observe from the
published data may simply be representative of the range of
pressure, temperature, and fluid conditions at the time of
deformation.
Two factors were ignored in obtaining the above estimates
of n. We assumed (1) that the faults did not propagate (the fault
tip was fixed in the medium) while displacement occurred, and
(2) that the faults were very weak (with a shear stress z 0). If
either of these assumptions is not a good approximation, the
actual value of n will differ from that obtained by fitting the
data to this simplified model.
In addition to the variable slip in the vicinity of a fault tip,
there is also good qualitative agreement between field observations and the deformation predicted by this model. Coward
& Potts (1983) and Hyett (1990) observe that in the vicinity
of a fault tip, the greatest and least principal strain directions
flip across the fault, indicating fault-parallel compression on
one side of the fault and fault-parallel extension on the other
side; the same pattern in the principal strain-rate directions is
shown in our numerical solutions (Fig. 4), and also in the
analogue experiments of Odonne (1990). Elliott (1976),
Williams & Chapman (1983) and Scholz et al. (1993) observe
that there is a zone of intense ductile strain (known as a
‘ductile bead’) in front of a propagating fault tip. Blenkinsop
& Drury (1988) also interpreted a ‘process zone’ ahead of the
fault plane, preceding fault propagation of the Bayas Fault in
NW Spain. The process zone or ductile bead is represented in
our simulations by the zone of high strain rate in front of and
adjacent to the fault tip.
CONCLUSIONS
In studying the relationship between faulting and localized
ductile shearing, the numerical results presented here show
486
T. D.Burr and G. A . Houseman
2
6
4
8
10
12
14
16
distance (m)
distance (m)
distance (m)
12.
8
n
10.
v
i
g
j
8
6
oB 4
e
8
0'
2
0
0,.
2
4
6
10
8
12
10
14
20
distance (cm)
40
30
50
distance (cm)
20
38
I 5
Y
g
-3
10.
0
5.
00
0
5
10
15
distance (cm)
10
20
30
40
50
60
70
distance (cm)
Figure 10. Displacement versus offset from the fault tip for previously published studies of (a) small-scale thrust faults by Williams & Chapman
(1983), (b) and (c) strike-slip faults by Peacock (1991), and (d) to ( 8 ) small-scale normal faults by Muraoka & Kamata (1983). Best-fit curves
(thick line) with error estimates (thin lines) of the form described by eq. (18) are obtained by least-squares minimization; the best-fit parameters
are shown in Table 1 in similarly lettered columns. The best-fit values of n with error bounds that correspond to the plotted curves are labelled
on each plot.
that it is essential to consider the non-linearity of the rheology.
Compared to the n = 1 case, the ductile deformation is more
localized for n greater than 1, and the deformation occurs
more along the extension of the fault rather than adjacent to
it. Also, as n is increased, the volume of rock subject to large
shear-strain rate is increased, but the likelihood of brittle
failure in the immediate vicinity of the fault tip is decreased.
The existence of localized shear zones in a ductile medium
0 1996 RAS, GJI 125, 473-490
Deformation jields around a fault
has often been explained by positive feedback mechanisms that
cause strain softening, such as grain-size reduction or shear
heating (Brun & Cobbold 1980; Poirier 1980; White et ul.
1980; Kirby 1985; Kirby & Kronenberg 1987; Rutter & Brodie
1992). Our calculations show that even in a homogeneous
medium, zones of localized shear-strain rate will form about
the tip of a fault, and as the degree of non-linearity, n, increases,
the degree of localization will also increase. Localization of
shear may thus, in some cases, be a geometrical consequence
of terminating fault displacement in a homogeneous medium;
if strain softening occurs as well, the localization will be further
enhanced.
Our analysis of effective n values for the different examples
shown above is perhaps most significant in that it demonstrates
a method of measuring effective in situ n values for deformation
of geological formations. This method will obviously require
further validation by observation and analysis of other field
examples, with the aim of ensuring the applicability of the
model to the observed displacement fields. While n values can
be measured in the laboratory on small rock samples at high
temperature and high strain rate, the rheological relations
that result must be extrapolated over five to eight orders of
magnitude in the time-scale, in order to be applied to the
geological environment. Our method, based on analysis of the
deformation field around a fault, offers an alternative, which
might yield new insights into the rheology of geological
materials at geological time-scales.
ACKNOWLEDGMENTS
We would like to thank Lynn Evans for help in preparing the
figures. During part of this study, Terence Barr was supported
by an ARC Research Fellowship.
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APPENDIX
In addition to the constraints imposed by the external
boundary-velocity distribution, we also impose the conditions
of zero shear stress, continuous normal stress, and continuous
normal velocity ug on the fault O = 0. For the integer rn components of the solution, the latter two constraints are automatically satisfied, but the zero-shear-stress condition imposes
a particular constraint on the form of possible solutions,
Analytical solutions for faulted ductile media in plane
stress and plane strain
The deformation around a terminating fault in a viscous
medium is best described using a radial coordinate system, in
which the origin is the end of the fault, and 0 = 0 is along the
fault. Assume that velocity is specified along the perimeter of
a circular region r 5 Ro. If the material is incompressible and
if plane strain is enforced, then the velocity field may be
expressed in terms of a scalar stream function $,
Stream function and pressure are coupled by the NavierStokes equations and, in the case of Newtonian viscosity,
separately satisfy the biharmonic equation and Laplace's
equation obtained by using the curl and divergence operators,
respectively,
V411/=0 and V 2 p = 0 .
('42)
For the half-integer rn components of the solution, two more
constraints are provided by continuity of normal velocity and
normal stress, respectively,
C,+2
mfmdm
(rn+2)'
= --
-b,.
The combination of conditions (A7a) and (A7b) generally
implies that c, = d , = 0 for half-integer m.
Analytical solutions can only be found for boundary-velocity
distributions-that are consistent with the form of the solution
(A5) and the constraints (A7). In principle, any such boundaryvelocity distribution can be Fourier analysed to obtain a
set of necessary and sufficient constraints on the unknown
integration constants. To test the finite-element program on
the plane-strain problem we solved analytically the problem
defined by the following boundary-velocity distribution:
The general solution for stream function consists then of a
sum of terms ll/,,,(r,O),
ll/,=sin(rn8)[a,rm
+ bmr'mf2)]+ cos(rnO)[c,rm + dmr(mt2)],
(A31
(A74
sin(@)- sin(3O)- 2 sin(20) + 2 sin(40)
+ cos (;) + 3 ,,,(
31.
and the general solution for pressure consists of a similar sum
of terms p,(r, O),
pm= 4v(m
+ l ) r m [ d , sin(rn0)- b, cos(mO)],
(A41
where q is Newtonian viscosity, m is the angular order number
of the solution and a,, b,, c,, and d , are integration constants,
which are determined using boundary conditions. Differentiating
(A3), we obtain (for rn # 0) the plane-strain velocity field
u, = cos(rnO)[rna,r(m-l)
+ rnb,r('"+')]
- sin(rnO)[mcmr("-')
ug= -sin(rn8)[rnamr("-')
- cos(rnO)[rnc,r("-l)
+ rnd,,,r(m+l)],
+ rnfmb,r(m*l)]
+ rnf,dmr(m+l)],
(A51
- 3 sin(
:)
- 3 sin
( y)]
This boundary-velocity distribution implies both continuous and
discontinuous components to the internal velocity distribution.
Solving for the integration constants, the solution is
Uil
u, = -{(r/Ro)2[sin(0)- sin(38)I
4
where
UO
u@= - { ( r / ~ , ) ~ [ 3 cos(e)-cos(3~)-j
4
For 0 5 8 I 2x, an arbitrary boundary-velocity distribution
may be represented using the discrete Fourier series defined
by the complete set of rn values, {q/2}, where q is a nonnegative integer. If rn is a whole integer, deformation is
obviously continuous at 0 = 0. The half-integer series represents
discontinuous or fault-type deformation. Valid solutions to
(A2) are also obtained for the complete set of rn values, { -q/2},
but the requirement that velocity is bounded at r = 0 implies
that the only negative rn term which may be non-zero is
rn = - 112.
- (r/Ro)3[4C O S ( ~ ~ )2 co~(4e)]
+ (r/Ro)'/'[3
v UO
p = -[-2(r/R0)
RO
sin(8/2) + 3 cos(38/2)]},
(A9b)
sin(@)+ 3(r/R0)2sin(20)
+ (r/Ro)-"Zcos(O/2)].
('494
This solution illustrates deformation of the fault plane
synchronous with slip on the fault.
Solutions to the plane-stress problem are similar. Stream
0 1996 RAS, GJI 125,473-490
Deformation $fields around a fault
function cannot be used in the plane-stress case, so the coupled
radial and tangential stress-balance equations are solved simultaneously. The velocity solution may also be expressed in the
form (AS) if the constant f , is redefined,
-(r/Ro) sin(6)
3
+ -(r/Ro)2
sin(26)
2
+ i(r/R0)-’/’
1
cos(6/2) .
1
3m+8
fm=(3m-2)
In plane stress the pressure field can be obtained from the
velocity solution by
489
(A13c)
For both test problems the relative slip velocity on the fault
atr=Roand6=Ois2Uo.
Finite-element solutions
The constraints obtained from zero tangential stress and
continuous normal velocity, uB, across the fault are given by
(A7a) and (A7b), while continuity of normal stress provides a
separate constraint in place of ( A ~ c ) ,
- 3mb,
(3m - 2)
a,,,+’ = ___
As above, analytical solutions to the deformation are
only possible for boundary-velocity distributions consistent
with the form of the solutions (AS) and (A10) and the faultcondition constraints (A7a), (A7b) and (A12). To test the finiteelement program on the plane-stress problem we used the
boundary-velocity distribution
U , = -[sin(@ - 3 sin(36) - 4 sin(26) + 6 sin(46)
16
UO
+ 7 cos(e/2) + 9 cos(30/2)1,
(A12a)
U
uB= O [ i i cos(e)- 3 c o ~ ( 3 e-) 14 C O S ( ~ O +
) 6 Cos(46)
16
- 13 sin(6/2) - 9 sin(36/2)],
for which the solution is
UO
u, = -{ (r/Ro)’[sin(6) - sin( 36)]
16
8 1996 RAS, GJI 125, 473-490
(A12b)
In our finite-element representation of eqs (3) or (5) we
use triangular elements with quadratic interpolation functions for velocity and linear interpolation functions for
pressure, as described by Yamada et al. (1975) and Huebner
(1975). This formulation differs from the well-known
penalty-function approach in that it is not necessary to use
a pseudo-compressibility to approximate the continuity
condition.
We represent the interior fault surface using two spatially
coincident external boundary segments. In order to solve for
the velocity and pressure everywhere, it is then necessary to
introduce another set of unknowns, the set of normal and
tangential tractions, {s} and {t}, acting on the fault surface.
An additional set of constraints comes from continuity of
normal stress and velocity and specification of zero shear stress
on the fault. The complete system of equations is solved using
a pre-conditioned conjugate-gradient (PCG) method (e.g.
Ortega 1988, p. 196). The formulation is logically equivalent
to the slippery node technique of Melosh & Williams (1989).
We tested the finite-element solutions for accuracy, by
calculating solutions for the two constant-viscosity problems
described above [eq. (A9) for plane strain and eq. (A13) for
the thin viscous sheet]. In each case, finite-element nodes are
concentrated in the vicinity of the fault tip and we used three
different finite-element mesh discretizations, corresponding to
successive increases in the density of node points by factors of
approximately four. The maximum error in the velocity field,
shown for each test in Table Al, is approximately proportional
to N-’/’, where N is the number of nodes, whereas computation
time on a Sparc processor appears to increase approximately
as N3/’. An important confirmation of the accuracy of the
numerical solutions for n = 3 and n = 10 was obtained by
accurately reproducing the power-law radial dependence
described by eqs (7), (8) and (9).
490
T. D. Burr and G. A . Houseman
Table A l . Resolution tests for the finite-element solutions of the constant velocity problem.
No. of nodes
Plane strain
662
2342
8774
Thin viscous sheet
662
2342
8774
Error in velocity
CG iterations
Cpu time (Sparc2)
1 .oo%
0.31%
0.16%
336
953
2379
30 s
230 s
2213 s
1.35%
0.68%
0.34%
254
597
1322
18 s
123 s
1033 s
Difference between analytical and numerical solutions is expressed as a percentage of the
maximum velocity value in the solution.
0 1996 RAS, GJI 125, 473-490