JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110, B07403, doi:10.1029/2004JB003438, 2005
A model of damage mechanics for the deformation
of the continental crust
K. Z. Nanjo
Center for Computational Science and Engineering, Department of Physics, University of California, Davis, California, USA
Institute of Statistical Mathematics, Tokyo, Japan
D. L. Turcotte
Department of Geology, University of California, Davis, California, USA
R. Shcherbakov
Center for Computational Science and Engineering, Department of Physics, University of California, Davis, California, USA
Received 16 September 2004; revised 24 March 2005; accepted 4 April 2005; published 14 July 2005.
[1] We derive a continuum rheology for the deformation of the continental crust using
continuum damage mechanics. It is hypothesized that when a constant strain rate e_ is
applied to a solid material, the stress s and damage increase until failure occurs, which is
analogous to an earthquake. We further assume that this process is repeated, in analogy
to the reoccurrence of earthquakes on a fault. Our model assumes that the continental
crust behaves elastically below a yield stress sy. Above this stress the continuum
deformations can be modeled as a non-Newtonian viscous flow with e_ / (s sy)n, where
n is constant. We derive the modified Omori’s law for aftershock decay using a
viscoelastic version of our model and get good agreement with observations taking n = 6.
Using parameter values appropriate for aftershocks, we obtain a continuum crustal
rheology that can explain major orogenies such as the Indian-Asian collision.
Citation: Nanjo, K. Z., D. L. Turcotte, and R. Shcherbakov (2005), A model of damage mechanics for the deformation of the
continental crust, J. Geophys. Res., 110, B07403, doi:10.1029/2004JB003438.
Q þ PVa
e_ ¼ Asn exp ;
RT
1. Introduction
[2] Brittle and ductile deformation plays major roles in the
deformation of the continental crust. Block models and
continuum models are associated with brittle and ductile
processes, respectively, and have been proposed as a means
of quantifying the deformation of the continental crust.
Significant continental deformation takes place on major
faults. Models in which deformation occurs on a specified
array of faults are the block models. King et al. [1994a],
Thatcher [1995], and Jackson [2002] have discussed the
merits of a block (microplate) description of active continental tectonics. However, the frequency-size distribution of
faults is fractal so that we can conclude that faults are present
on a wide range of scales. Tectonic models for fractal
distributions of faults have been proposed by King [1983,
1986], Turcotte [1986], King et al. [1988], and Hirata [1989].
[3] An alternative approach to the deformation of the
continental crust is to use continuum models. In these
models ductile deformation is assumed and creep (Newtonian and non-Newtonian) and plastic rheologies are utilized. To describe the ductile behavior of rocks at high
temperature, an empirical power law equation between
strain rate e_ and stress s has been proposed as
Copyright 2005 by the American Geophysical Union.
0148-0227/05/2004JB003438$09.00
ð1Þ
where A and n are the constants, Q is the activation energy,
Va is the activation volume, R is the gas constant, T is the
absolute temperature and P is the pressure [e.g., Ashby and
Verall, 1977; Kirby and Kronenberg, 1987; Karato and Wu,
1993]. This empirical relation is called Dorn’s equation for
the steady state flow of a solid [e.g., Dorn 1954; Nicolas
and Poirier, 1976] and it is valid for both diffusion creep
(n = 1) and dislocation creep (n = 3 –5). An example of a
continuum model for the deformation of the continental
crust is the viscous sheet model where equations similar to
equation (1) were used [e.g., Bird and Piper, 1980;
England and McKenzie, 1982; Houseman and England,
1986, 1993; England and Houseman, 1986]. However,
these studies did not consider the role of brittle processes
in the deformation of the continental crust.
[4] Both ductile and brittle processes can also be quantified using the concept of dislocations. Dislocation theories
have been developed to explain not only the mechanical
properties of crystals but also their optical and electromagnetic properties. Taylor [1934], Orowan [1934], and Polanyi
[1934] applied the concept of dislocations to the plastic
deformation of simple crystals in order to explain why the
observed yield stresses of crystals are much lower than the
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NANJO ET AL.: DAMAGE MECHANICS AND CRUSTAL DEFORMATION
theoretical values calculated from atomic theory assuming a
perfect lattice state. Mura [1969] developed a method of
continuously distributed dislocations to consider the relationship between macroscopic plasticity and dislocation
theory. However, these studies have not considered how
microcracks contribute to the deformation of solids.
[5] We hypothesize that deformation in the continental crust
is dominated by displacements on faults at all scales. Our
objective is to derive a continuum rheology that is applicable to
this deformation. An avenue for irreversible behavior associated with these brittle and ductile processes is damage
mechanics. The concept of damage mechanics has been
utilized widely in engineering problems [Krajcinovic, 1996;
Skrzypek and Ganczarski, 1999; Voyiadjis and Kattan, 1999].
[6] Damage refers to irreversible deformation of solids.
Some examples include plasticity, brittle microcracking,
thermally activated creep and so on [Krajcinovic, 1996].
In order to quantify the brittle deformation of solids
associated with microcracking, an empirical continuum
model of damage mechanics was introduced and is widely
used in civil and mechanical engineering [Kachanov, 1986].
This type of model is often called continuum damage model
and has also been applied to tectonic processes [Lyakhovsky
et al., 1997; Ben-Zion and Lyakhovsky, 2002; Shcherbakov
et al., 2005]. Another approach to the anelastic deformation
of composite materials is provided by the fiber-bundle
model, a discrete model of damage mechanics. The fiberbundle model was applied to fatigue in structural materials
and earthquakes in geophysical settings [e.g., Hemmer and
Hansen, 1992; Moreno et al., 2001]. The continuum
damage model was shown to be equivalent to the fiberbundle model for the occurrence of a failure in a simple
geometry [Krajcinovic, 1996; Turcotte et al., 2003]. The
concept of the yield stress was used for damage mechanics
by Shcherbakov and Turcotte [2003, 2004]. These authors
considered that damage occurs in the form of microcracking at stresses greater than the yield stress.
[7] In this paper we show that when the continuum damage
model is applied to the brittle deformation of the continental
crust, a non-Newtonian power law viscous rheology is
obtained. In our analyses a yield stress is introduced as follows:
below this stress the continental crust behaves elastically and
can act as a stress guide, above the yield stress the continuum
deformations can be modeled as a power law viscous fluid.
This expands upon the previous work of Turcotte and
Glasscoe [2004] that did not consider a yield stress.
[8] Another consequence of the viscoelastic behavior of
the crust is the occurrence of aftershock sequences. A main
shock suddenly increases the stress in some regions of the
crust adjacent to the fault on which the main shock
occurred. Stress relaxation is accompanied by the aftershock
sequences. The modified Omori’s law [Utsu, 1961] gives a
temporal decay in the rate of aftershock occurrence [e.g.,
Scholz, 2002]. We show that the rate of energy release in our
model corresponds to the rate of aftershock decay described
by the modified Omori’s law.
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stress in a solid material on time at a constant strain rate
e_ . The stress and damage increase until failure occurs,
which is analogous to an earthquake. The process is then
repeated: this allows us to model the reoccurrence of
earthquakes on a fault. To model damage evolution from
undamaged to damaged state, we introduce a yield stress
sy and corresponding yield strain ey. If the stress is less
than the yield stress s sy, there is no damage and linear
elasticity is applicable. If the stress is greater than the
yield stress s > sy, damage occurs and a state variable a
can be introduced to model this irreversible behavior.
[10] We first consider that a brittle solid obeys linear
elasticity for stresses in the range 0 s sy. We also
assume that Hooke’s law is applicable so that the dependence of stress s on strain e is given by
s ¼ E0 e;
ð2Þ
where E0 is Young’s modulus, a constant. From equation (2)
the corresponding yield strain ey is given by
ey ¼
sy
:
E0
ð3Þ
[11] We further assume that damage occurs in the form of
microcracks at stresses greater than the yield stress s > sy.
As a consequence the behavior of the solid will deviate from
that predicted by linear elasticity. Following Shcherbakov et
al. [2005], we introduce a damage variable a according to
e ey ¼
s sy
:
E0 ð1 aÞ
ð4Þ
The damage variable a quantifies deviations from linear
elasticity and the distribution of microcracks in the material
considered. By definition the damage variable a takes
values between 0 and 1 (0 a 1) and failure occurs
when a = 1. Damage evolution is a transient process so
that we have a(t) and the damage variable increases as
long as s > sy or until failure occurs (a = 1). The
following two examples are representative for damage
evolution. If a constant stress s0 > sy is applied
instantaneously, microcraking will occur and a increases
until failure occurs at a = 1. If constant strain e0 > ey is
applied the stress will relax until s = sy with again a = 1.
[12] On the basis of thermodynamic considerations
[Kachanov, 1986; Krajcinovic, 1996; Lyakhovsky et al.,
1997], the time evolution of the damage variable a was
related to stress s and strain e. Shcherbakov and Turcotte
[2003, 2004] and Shcherbakov et al. [2005] modified their
results to include a yield stress sy and corresponding yield
strain ey. In this formulation the time evolution of damage is
given by
2
da
e
1 ;
¼ AðsÞ
dt
ey
ð5Þ
where
AðsÞ ¼ 0 if
2. Continuum Damage Model
[9] Utilizing a continuum damage model, we will show
that the result obtained is identical to a non-Newtonian
fluid rheology. We assume a linear dependence of the
2 of 10
AðsÞ ¼
0 s sy ;
r
1 s
1
td s y
if
s > sy ;
ð6Þ
ð7Þ
NANJO ET AL.: DAMAGE MECHANICS AND CRUSTAL DEFORMATION
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with td the characteristic timescale for damage and r a
power law exponent to be determined from experiments.
[13] For strains e greater than ey and corresponding
stresses s greater than sy, we substitute equations (4) and
(7) into equation (5) and obtain
rþ2
da 1 s
1
1
:
¼
dt
td sy
ð1 aÞ2
ð8Þ
We assume that the applied stress increases linearly with
time. With the initial condition of stress s = sy at t = 0, we
assume that the stress at subsequent times is given by
s ¼ sy þ E0 e_ t:
ð9Þ
Substituting equation (9) into equation (8) using equation
(3), we obtain
da 1 e_ t rþ2
1
:
¼
dt
td ey
ð1 aÞ2
ð10Þ
Integrating with the initial condition a = 0 when t = 0, we
find
(
a¼1
)13
3 e_ rþ2 t rþ3
1
:
rþ3
td ey
ð11Þ
Failure occurs at time tf when a = 1, thus we have
tf ¼
1
rþ2
ðr þ 3Þtd rþ3 ey rþ3
:
3
e_
ð12Þ
The failure time tf increases with the inverse of the strain
rate e_ for large values of the power law exponent r. From
equation (9) the mean stress during the run-up to failure is
given by
1
s ¼ sy þ E0 e_ tf :
2
ð13Þ
Furthermore, substitution of equations (3) and (12) into
equation (13) gives
rþ3
3
s
e_
¼
1
:
2
sy
ey td ðr þ 3Þ
ð14Þ
It is convenient to introduce a new exponent n = r + 3 and a
new characteristic time
tc ¼
ntd sy n
:
3ey 2E0
ð15Þ
In terms of this characteristic time tc, we can rewrite
equation (14) as
e_ ¼
1 s sy n
:
E0
tc
ð16Þ
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So far we have considered the failure of a single element. In
order to model the continuum deformation of the continental
crust we assume that a failed element is immediately
replaced by a new undamaged element with the stress in the
element equal to the yield stress. The repetitive failure of the
element is hypothesized to model the reoccurrence of
earthquakes on a fault. The failure stress is equivalent to
the static frictional stress on faults and the yield stress is
equivalent to the stress which is assumed to be the stress on
the fault after an earthquake. We assume that the continental
crust is made up of many independent units, and each of
these units experiences the periodic failures considered
above. The failures on a unit are the periodic earthquakes
that occur on a fault or fault segment.
[14] The application of equation (16) to the continental
crust results in a non-Newtonian viscous rheology in terms
of the relationship between strain rate e_ and the excess stress
over the yield stress s sy. Many authors have applied a
non-Newtonian viscosity to the deformation of the continental crust [e.g., Bird and Piper, 1980; England and
McKenzie, 1982; Houseman and England, 1986, 1993;
England and Houseman, 1986]. However, many plate
interiors do not appear to exhibit any internal deformation.
Thus the inclusion of a yield stress in any crustal rheology
appears to be a requirement. No continuum deformation
occurs if the stress is less than the yield stress.
[15] Turcotte and Glasscoe [2004] modeled the repetitive
failure of faults using a fiber-bundle model. They also
derived a non-Newtonian viscous rheology but their analysis did not include a yield stress. The similarity of results
using fiber-bundle and continuum damage models is not
surprising since they can be shown to be equivalent under
some conditions [Turcotte et al., 2003].
[16] It is of interest to consider the microphysics of the
rheological model presented above. We relate this physics to
time delays associated with brittle failure as the time
evolution of damage plays an important role in our rheological model. We explore these time delays from the
viewpoint of metastable phase changes. In the derivation
of a nonlinear viscous rheology for the continental crust
given above, the characteristic timescale for damage td plays
a crucial role. What is the physical basis for introducing this
timescale? Although the damage mechanics model has been
shown empirically to quantify the time delays associated
with brittle failure, no rigorous derivation of the model to
explain the time delays has been provided. The time delays
associated with the nucleation and growth of a single crack
are often attributed to stress corrosion [Freund, 1990]. This
explanation has also applied to the time delays associated
with aftershocks [Das and Scholz, 1981].
[17] More recently, a number of statistical physicists have
related time delays associated with brittle fracture to the
time delays associated with metastable phase changes.
Buchel and Sethna [1997], Zapperi et al. [1997], and Kun
and Herrmann [1999] have drawn an analogy between
brittle fracture and a first-order phase transition. Sornette
and Andersen [1998] and Gluzman and Sornette [2001]
argue that brittle rupture is analogous to a critical point (a
second-order phase change). Selinger et al. [1991], Rundle
et al. [1999, 2000], and Zapperi et al. [1999] have drawn an
analogy between brittle failure and spinodal nucleation. The
basic idea is that there is an analogy between the nucleation
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NANJO ET AL.: DAMAGE MECHANICS AND CRUSTAL DEFORMATION
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of bubbles in a superheated liquid and the nucleation of
microcracks in a brittle solid.
[18] The analogy between phase transitions and fracture
also has a thermodynamic basis. Thermal fluctuations are
crucial in phase transitions of solids and liquids. A fundamental question is whether temperature plays a significant
role in the damage of brittle materials. Guarino et al. [1998]
varied the temperature in their experiments on the fracture
of chipboard and found no effect. However, a systematic
temperature dependence has been found for the lifetime
statistics of Kevlar fibers [Wu et al., 1988].
[19] Time delays associated with bubble nucleation in a
superheated liquid are explained in terms of thermal fluctuations. The fluctuations must become large enough to
overcome the stability associated with surface tension in a
bubble. The fundamental question in damage mechanics is
the cause of the delay in the occurrence of damage. This
problem has been considered in some detail by Ciliberto et
al. [2001] and Scorretti et al. [2001]. These authors attributed damage to the ‘‘thermal’’ activation of microcracks. An
effective ‘‘temperature’’ can be defined in terms of the
spatial disorder (heterogeneity) of the solid. The spatial
variability of stress in the solid is caused by the microcracking itself, not by thermal fluctuations. This microcracking occurs on a wide range of scales.
[20] A related time delay is associated with friction.
Laboratory studies of friction between rock surfaces are
empirically correlated with rate-and-state friction laws
[Dieterich, 1978; Ruina, 1983]. When the slip velocity is
changed there is a time delay associated with the establishment of a new steady state coefficient of friction. The state
variable in the friction law has also been used to quantify
the delay in rupture on a frictional surface. Dieterich [1994]
has derived Omori’s law for the time delay of earthquake
aftershocks from the equations of rate-and-state friction.
There is clearly a close association between this work and
the damage mechanics approach.
3. Viscoelasticity
[21] In the previous section we derived a continuum fluid
rheology for the continental crust which is non-Newtonian
power law with a yield stress. It is useful for a number of
problems to combine a fluid rheology on long timescales
with an elastic behavior on short timescales. For this
purpose a viscoelastic rheology is usually used [Turcotte
and Schubert, 2002]. The Maxwell model for viscoelasticity
considers a material in which the total strain rate e_ ve is
hypothesized to be the sum of an elastic strain rate e_ el and a
viscous strain rate e_ v
e_ ve ¼ e_ el þ e_ v ¼
s_
s
þ ;
E0 m
ð17Þ
where m is the viscosity of the material. If a constant strain
e0 is applied instantaneously the resulting stress will relax in
a viscoelastic relaxation time tve = m/E0. For the Earth’s
mantle we take E0 = 7 1010 Pa and m = 1021 Pa s so that
tve = 450 years. The simple Maxwell model combines the
short timescale elastic behavior associated with seismic
waves with geological timescale fluid behavior associated
with mantle creep and convection.
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[22] In this paper we hypothesize that the Maxwell
viscoelastic model is also applicable to the deformation of
the continental crust. This will explain the seismic waves on
short timescales and the continuous deformation observed
on geological timescales. The elastic strain of the material
as given by equation (2) is
eel ¼
s
:
E0
ð18Þ
If the stress on the medium is less than the yield stress s <
sy there is no viscous deformation and the material behaves
elastically. If the stress on the medium is greater than the
yield stress s > sy viscous strain will occur and the viscous
strain rate is given by equation (16) as
e_ v ¼
sy n
1 s
:
E0
tc
ð19Þ
The total strain rate e_ ve in equation (17) is the sum of the
strain rate e_ v from equation (19) and the time derivative e_ el
of equation (18), thus we have
e_ ve ¼
1 ds 1 s sy n
:
þ
E0
E0 dt tc
ð20Þ
This is the rheological law relating strain rate, stress, and
rate of change of stress for our Maxwell viscoelastic
material. Following Turcotte and Schubert [2002], we
assume that the deformation of the continental crust can be
treated as the behavior of the Maxwell viscoelastic material.
4. Constant Applied Strain
[23] As a first example, we consider the response of the
viscoelastic medium to a sudden application of constant
strain e0 at time t = 0, the strain is maintained constant for
t > 0. As discussed above, the behavior of the material is
elastic during the very rapid application of strain. Therefore the initial stress s0 is
s0 ¼ E0 e0 :
ð21Þ
If e0 ey no damage occurs and the initial stress remains
unchanged s = s0 for t > 0. If e0 > ey, the material is strained
elastically along the path ABD in Figure 1. The strain is
then maintained constant e0 so that damage evolves and
repetitive material failures occur. Because of the damage
and failures, the stress on the sample relaxes from the initial
stress s0 to the yield stress sy. This relaxation takes place
along the path DFG illustrated in Figure 1. The solution will
give the time dependence of stress s(t) during the stress
relaxation.
[24] For t > 0 e_ ve = 0 and equation (20) reduces to
0¼
1 ds 1 s sy n
:
þ
E0
E0 dt tc
ð22Þ
Integrating with the initial condition s = s0 at t = 0, we find
4 of 10
s
sy
1
¼
1 :
)n1
s0 sy (
s0 sy n1 t
1 þ ðn 1Þ
E0
tc
ð23Þ
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NANJO ET AL.: DAMAGE MECHANICS AND CRUSTAL DEFORMATION
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Figure 1. Schematic stress-strain diagram for the Maxwell viscoelastic material as it is applied to the
crustal deformation. Stress s0 > sy and strain e0 > ey are applied instantaneously to a material at t = 0. The
material behaves elastically and follows the linear path ABD. (1) Subsequently, strain e0 is maintained
constant and the material damage and failure (aftershocks) associated with the non-Newtonian flow relax
the applied stress s0 to the yield stress sy along the path DG. In order to determine the energy associated
with the relaxation process, we instantaneously remove the stress s and strain e0 at the point F. The
subsequent linear elastic behavior of the material follows the path FH. (2) Subsequently, the stress s0 is
maintained at a constant value and the strain increase along the path DE.
In the limit t ! 1, s ! sy and no further damage can
occur, as expected. Before discussing the application of this
solution to earthquake aftershocks, we discuss the temporal
decay of aftershock activity.
[25] The modified Omori’s law [Utsu, 1961] describes the
temporal decay of aftershock activity and is given in the
form [Scholz, 2002]
dN
K
;
¼
dt
ðc þ t Þp
ð24Þ
where dN/dt is the rate of occurrence of aftershocks with
magnitudes greater than m, t is the time that has elapsed
since the main shock and K, p and c are parameters. This
law is a manifestation of temporal correlations in aftershock
sequences which can be viewed as complex relaxation
processes occurring after main shocks. Omori [1894]
introduced equation (24) with p = 1, usually a good
approximation. For t c the dependence of dN/dt on t in
equation (24) gives a power law behavior for aftershock
decay. This type of temporal fractal property is called a long
time tail [Takayasu, 1990].
[26] Shcherbakov et al. [2005] derived an alternative form
for the temporal decay of aftershock activity. They called
this a generalized Omori’s law. They obtained it from a
combination of the Gutenberg-Richter relation [Gutenberg
and Richter, 1954], Båth’s law [Båth, 1965], and the
modified Omori’s law given in equation (24) with the result
1 dN p 1
1
;
¼
Nt dt
c ð1 þ t=cÞp
ð25Þ
where Nt is the total number of aftershocks with magnitudes
greater than m. In order for this relation to be valid it is
necessary that p > 1.
[27] We now model the time-dependent decay of
aftershocks using our viscoelastic analysis. Following
Shcherbakov and Turcotte [2003, 2004] and Shcherbakov
et al. [2005], our working hypothesis is that stress transfer
during a main shock increases the stress s and strain eve
above the yield values sy and ey in some regions adjacent
to the fault on which the main shock occurred. The
increases of stress and strain are essentially instantaneous
and follow linear elasticity. We believe that it is a good
approximation to neglect any increase in regional stress
due to tectonics during the aftershock sequence. We
hypothesize that the applied strain e0 remains constant
and that aftershocks relax the stress s to its yield value sy.
The occurrence of aftershocks is attributed to this relaxation process as given in equation (23).
[28] In order to quantify the rate of aftershock occurrence
we determine the rate of energy release in the relaxation
process. The stored elastic energy density (per unit mass) e0
in a material after an instantaneous strain e0 has been
applied along the path ABD in Figure 1 is
1
e0 ¼ E0 e20 :
2
ð26Þ
Since the strain is constant during the stress relaxation, no
work is done on the sample. If the applied strain (stress) is
instantaneously removed at point F we hypothesize that the
sample will follow the elastic path FH which is parallel to
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Figure 2. Values of the characteristic time c for aftershock decay obtained using equation (25) to
correlate with the lower cutoff magnitude m for the Landers, Northridge, Hector Mine, and San Simeon
earthquakes as given by Shcherbakov et al. [2005].
the path ABD. The elastic energy e1 recovered during the
stress relaxation on this path is given by
1
e1 ¼ se0 :
2
Taking the time derivative of equation (29) and using
equation (30) we obtain the rate of energy release
n1
1
e0 ey
1 deAF
tc
¼
n :
n1
eAFT dt
n1 t
1 þ ðn 1Þ e0 ey
tc
ð27Þ
Substitution of equation (23) into equation (27) with
equations (3) and (21) gives
1
E0 e0 e0 ey
1
2
e1 ¼ E0 e0 ey þ 1 :
n1
2
n1 t
1 þ ðn 1Þ e0 ey
tc
ð28Þ
This relation is clearly very similar to the generalized
Omori’s law given in equation (25).
[29] Since Gutenberg-Richter frequency statistics is applicable to aftershocks it is appropriate to assume
1 dN
1 deAF
:
¼
Nt dt
eAFT dt
We assume that the difference between the energy added
e0 and the energy recovered e1 is lost in aftershocks. This
energy eAF corresponds to the area ABDFH in Figure 1
and is given by subtracting the energy recovered e1 in
equation (28) from the energy added e0 in equation (27)
as
eAF
3
6
7
1
6
7
61 1 7:
4
n1 5
n1 t
1 þ ðn 1Þ e0 ey
tc
ð32Þ
With this condition equations (25) and (31) are identical if
we take
p
;
p1
ð33Þ
tc
ð p 1Þtc
n1 ¼ 1 :
ðn 1Þ e0 ey
e0 ey p1
ð34Þ
n¼
c¼
1
¼ e0 e1 ¼ E0 e0 e0 ey
2
2
ð31Þ
Substituting equations (33) and (34) into equation (31) with
equation (32) gives
ð29Þ
1 dN
1 deAF p 1
1
:
¼
¼
Nt dt
eAFT dt
c ð1 þ t=cÞn
ð35Þ
The total energy of aftershocks eAFT is obtained in the
limit t ! 1 with the result
[30] We will now consider the applicable values of c and
p for aftershock sequences and derive the applicable values
of n, tc and e0 ey with
1
eAFT ¼ E0 e0 e0 ey :
2
Ds s0 sy ¼ E0 e0 ey :
ð30Þ
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ð36Þ
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Figure 3. Dependence of the excess stress Ds = s0 sy and excess strain e0 ey on the characteristic
decay time c from equations (34) and (36) taking tc = 1020 s, E0 = 5 1010 Pa, and p = 1.2 (n = 6). Also
shown are the characteristic decay times for the four earthquakes as given in Figure 2.
[31] Shcherbakov et al. [2005] have studied aftershock
sequences following four California earthquakes: Landers
(magnitude MM = 7.3), Northridge (MM = 6.7), Hector Mine
(MM = 7.1) and San Simeon (MM = 6.5). These authors
determined the rates of occurrence of aftershocks with
magnitudes greater than m in number per day as a function
of time for 1460 days (100 days for San Simeon earthquake)
after a main shock. They fit equation (25) to the data and the
c values were obtained for lower cutoff magnitudes m = 1.5,
2.0, 2.5, 3.0, 3.5, 4.0. They showed that the c values are not
constant but scale with the lower cutoff magnitude m. Their
values for c for the four aftershock sequences are given in
Figure 2. These authors found p = 1.22 for Landers, p =
1.18 for Northridge, p = 1.21 for Hector Mine, and p = 1.08
for San Simeon.
[32] For our further analyses we take the typical value p =
1.2 and from equation (33) we find n = 6. Following Turcotte
and Schubert [2002], we take E0 to be 5 1010 Pa for crustal
deformation. We further assume that tc = 1020 s and will
discuss this choice in some detail in section 5. The
corresponding values of Ds = s0 sy and e0 ey are given
in Figure 3 as a function of the characteristic time c from
equations (34) and (36). Also included in Figure 3 are the
values of c given in Figure 2 for the four earthquakes.
[33] We create Figure 4 based on the values of m given in
Figure 2 and the values of Ds = s0 sy and e0 ey given
Figure 4. The results from Figures 2 and 3 are combined so that required excess stress Ds = s0 sy
and excess strain e0 ey are given as a function of the lower-magnitude aftershock cutoff m.
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Figure 5. Dependence of s sy on e_ from equation (37). Also included are results for a linear viscous
rheology from equation (39).
in Figure 3. The values of the excess stresses Ds = s0 sy
and strains e0 ey are given as a function of the lowermagnitude cutoff for aftershocks m. The values of the
excess stress appear to be reasonable. We see that the excess
stress increases with the magnitude of the aftershocks. This
can be attributed to the association between higher stress
levels and larger aftershocks. It is reasonable that the large
aftershocks occur in regions where the stress transfer from
the main shock is large. We also see for the four cases
considered that the stresses required to generate aftershocks
are less for larger earthquake than for smaller earthquakes.
One explanation for these results is that the high amplitude
surface waves generated by large earthquakes weaken the
faults in which aftershocks occur.
5. Constant Applied Stress
[34] We next consider the behavior of our viscoelastic
medium if a constant stress s0 > sy is applied at t = 0 and
maintained constant for t > 0. During the very rapid
application of stress the material behaves elastically along
the path ABD in Figure 1. Therefore we have the initial
strain e0 at t = 0 as given by equation (21). Subsequently,
there is no change in the stress, ds/dt = 0 along the path DE
illustrated in Figure 1. Equation (20) reduces to
e_ ve ¼
1 s0 sy n
:
E0
tc
ð37Þ
We get a power law relationship between the strain rate
e_ ve and the excess of stress s0 sy. Our Maxwell model
represents a non-Newtonian viscous fluid rheology with a
stress threshold sy. Equation (37) can be integrated with
equations (3) and (21) and the initial condition e = e0 at
t = 0 to give
eve ¼
n
t e0 ey þ e0 :
tc
ð38Þ
We obtain a linear dependence of the total strain eve on
time t.
[35] We next apply the derived result in equation (37) to
the continuum deformation of the continental crust. We take
E0 = 5 1010 Pa and n = 6 as determined in the previous
section. The rheology is determined when the characteristic
time tc is specified.
[36] For the continuum deformation of the continental
crust we expect the stress s sy to be in the range of 103
to 101 MPa and the strain rate e_ ve to be in the range of 1018
to 1010 s1. We take the characteristic time tc = 1028,
1024, 1020, 1016, and 1012 s. The resulting dependence
of stress s sy on strain rate e_ ve is given in Figure 5. It is
seen that the stress varies little with increasing strain rate. If
there were no dependence, constant stress independent of
strain rate, the rheology would be perfectly plastic.
[37] Also included for comparison is the dependence of
stress on strain rate for a linear Newtonian viscous rheology,
s sy ¼ h_eve ;
ð39Þ
where h is the viscosity. The range of equivalent crustal
viscosities is from h = 1019 to h = 1023 Pa s. There is a strong
dependence of s sy on e_ . With the nonlinear viscous
rheology with large power law exponent the behavior of the
deforming upper continental crust approaches that of a
perfect plastic material. A similar discussion was given by
Turcotte and Glasscoe [2004], but this did not include the
stress threshold.
[38] Although the continuum deformation of the continental crust can be associated with a wide range of stresses
and strain rates, a representative value would be s sy =
0.1 MPa and e_ ve = 1014 s1; these values correspond to the
solid circle in Figure 5. The characteristic time that gives
these values is tc = 1020 s. This value was the value used
for the deformation of the continental crust in the previous
section. Thus the same characteristic time tc = 1020 s gives
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reasonable values for the aftershock relaxation, i.e., weeks
to months and also reasonable values for the long-term
crustal deformation, i.e., millions of years.
6. Discussion
[39] At low confining pressure rock behaves as a brittle
material; that is, it fractures when a large stress is applied.
However, when the confining pressure approaches a rock’s
brittle strength, the yield stress can be defined in a transition
from brittle or elastic behavior to ductile or plastic behavior.
We introduced the concept of yield stress into our continuum-damage model. Our model with yield stress is applicable to the deformation of the ‘‘damaged’’ continental crust
that is loaded at high-confining pressures. This stress was
not included into previous analyses [Turcotte and Glasscoe,
2004].
[40] The large value of the power law exponent (n = 6) is
certainly not surprising for orogenic zones. Houseman and
England [1986] considered an indenter model for continental deformation and applied the finite element method to
thin non-Newtonian viscous sheet to obtain solutions. Using
the power law rheology given in equation similar to
equation (37), they obtained results for n = 2 and 9. England
and Houseman [1986] compared these results with observations in the Indian-Asian collision zone and found broad
agreement provided that the power law exponent is large
(n > 2). Our continuum rheology can explain the brittle
deformation in orogenies such as this collision.
[41] The values of the excess stress are consistent with
other results. To understand whether Landers earthquake
changed the proximity to failure on San Andreas Fault
system, King et al. [1994b] explored how changes in
Coulomb conditions associated with one or more earthquakes may trigger subsequent events. These authors found
the distribution of aftershocks for Landers earthquake, as
well as for several other events in its vicinity, can be
explained by the Coulomb stress as follows: aftershocks
are abundant where the Coulomb stress on optimally
oriented faults rose by more than 0.15 MPa, and aftershocks
are sparse where the Coulomb stress dropped by a similar
amount. If the stress in our model is comparable to the
Coulomb stress on optimally oriented faults, this result of
raised stress is in agreement with our excess stress given in
Figures 3 and 4.
[42] Shcherbakov et al. [2005] studied the stress relaxation process in a continuum damage model and found that
the rate of energy release in the model is identical to the rate
of aftershocks described by the generalized Omori formula
in equation (25). We used the time evolution of damage in
equations (5), (6), and (7) that is the same as that given by
Shcherbakov et al. [2005] and found that our result is
identical to their result. One aspect of damage mechanics
that was considered in our paper but not by Shcherbakov
et al. [2005] is ‘‘healing.’’ If a material heals, then the
damage and the damage variable decrease. When studying
material failure it is not necessary to consider healing.
However, any steady state deformation of a brittle material
requires both the generation and healing of the damage.
The continental crust is by definition a damaged, brittle
material. Earthquakes associated with displacements on
faults are analogous to the acoustic emissions from micro-
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cracking during the failure of brittle solid. However,
earthquakes are repetitive so that quasi steady state deformations of the continental crust can occur. This requires a
balance between damage, the creation of new faults and
increased displacements on existing faults, and the ‘‘healing’’ of faults. In this paper we considered the repetitive
occurrence of earthquakes. This was modeled by the
increase of the damage variable, from zero (undamage)
to one (failure), and the repetition of the process. This
repetition includes material healing. We derived an applicable rheology in equation (16) for the continuum deformation of the continental crust. Therefore, in order for a
quasi steady state behavior, our model requires that active
faults must become inactive, i.e., they must ‘‘heal’’.
[43] Some forms of ‘‘damage’’ that we did not considered
in this paper are clearly thermally activated. The deformation of solids by diffusion and dislocation creep is an
example. The ability of vacancies and dislocations to move
through a crystal is governed by an exponential dependence
on absolute temperature. Another example is given by
Nakatani [2001], who documented a systematic temperature
dependence of rate and state friction. Sornette and Ouillon
[2005] used thermally activated rupture process to find that
seismic decay rates after main shocks follow the modified
Omori law in equation (24). The continuum deformation of
the continental crust has already been considered as a
thermally activated process by Nanjo and Turcotte [2005]
utilized the fiber-bundle model. These authors assumed that
fiber failure is a thermally activated earthquake. They
obtained a power law relation between stress and strain rate
and the rheology is exponentially dependent on the inverse
absolute temperature as given by equation (1). Their analyses based on laboratory experiments [e.g., Nakatani, 2001]
argued in favor of thermally activated damage in order to
find the strength envelope of the continental lithosphere.
[44] However, it is a matter of controversy whether
temperature plays a significant role in the damage of brittle
failure of materials. Guarino et al. [1998] varied the
temperature in their experiments on the fracture of chipboard and found no effect.
[45] In this paper, we showed that the continuum damage
model gives non-Newtonian viscous fluid behavior above
the yield stress. Below the yield stress elastic deformation
was assumed. Applying this model to viscoelastic problems,
we considered crustal deformation. In our model the deformation of the continental crust above the yield stress is due to
the repetitive occurrence of faulting on a wide range of
scales. The model reproduces the temporal power law decay
of aftershock occurrence associated with stress relaxation
after a main shock. With parameter values appropriate for our
aftershock observation, the model provides a continuum
rheology applicable to the brittle deformation in orogenic
zones.
[46] Acknowledgments. The authors would like to thank Richard
Arculus, the Associate Editor, and two anonymous reviewers for helpful
comments. This work is supported by the National Science Foundation
(USA) under grant NSF ATM-03-27571 (DLT, RS), NASA/JPL grant
1247848 (RS), and JSPS Research Fellowships for Young Scientists (KZN).
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K. Z. Nanjo and R. Shcherbakov, Center for Computational Science and
Engineering, Department of Physics, University of California, One Shields
Avenue, Davis, CA 95616, USA. ([email protected]; roshch@cse.
ucdavis.edu)
D. L. Turcotte, Department of Geology, University of California, One
Shields Avenue, Davis, CA 95616, USA. ([email protected])
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