Int J Fract
DOI 10.1007/s10704-016-0175-x

ORIGINAL PAPER

A fiber-bundle model for the continuum deformationof brittle material
K. Z. Nanjo

Received: 15 April 2016 / Accepted: 7 December 2016
© Springer Science+Business Media Dordrecht 2016

Abstract The deformation of brittle material is primarily accompanied by micro-cracking and faulting.
However, it has often been found that continuum fluid
models, usually based on a non-Newtonian viscosity,
are applicable. To explain this rheology, we use a fiberbundle model, which is a model of damage mechanics.
In our analyses, yield stress was introduced. Above thisstress, we hypothesize that the fibers begin to fail and afailed fiber is replaced by a new fiber. This replacement
is analogous to a micro-crack or an earthquake and its
iteration is analogous to stick–slip motion. Below theyield stress, we assume that no fiber failure occurs, andthe material behaves elastically. We show that deformation above yield stress under a constant strain rate
for a sufficient amount of time can be modeled as an
equation similar to that used for non-Newtonian viscous flow. We expand our rheological model to treat
viscoelasticity and consider a stress relaxation problem. The solution can be used to understand aftershock
temporal decay following an earthquake. Our results
provide justification for the use of a non-Newtonianviscous flow to model the continuum deformation ofbrittle materials.
K. Z. Nanjo (B)Global Center for Asian and Regional Research, Universityof Shizuoka, 3-6-1, Takajo, Aoi, Shizuoka 420-0839, Japane-mail: nanjo@u-shizuoka-ken.ac.jp
K. Z. Nanjo
Institute of Advanced Sciences, Yokohama National University,
79-5, Tokiwadai, Hodogaya, Yokohama 240-8501, Japan

Keywords Fracture · Brittle deformation · Rheology ·Fiber-bundle · Yield stress · Viscoelasticity

1 Introduction
Material fracture is a complicated phenomenon. Even
if the material appears to be homogenous, there will
be a distribution of dislocations, flaws, and other heterogeneities present. As the applied stress is increased,
uncorrelated micro-cracks occur randomly on the heterogeneities. As the density of micro-cracks increases,
the stress fields of the micro-cracks interact and themicro-cracks become correlated. The micro-cracks
eventually coalesce to form a through-going fracture.
Even in an idealized case where propagation of a single fracture goes through a homogenous solid, thisis poorly understood by dynamic fracture mechanics
because of the singularities at the crack tip (Freund1990). However, this irreversible process can be treatedas a part of damage mechanics. Generally, the irreversible deformation of a solid is referred as “damage”(Kachanov 1986; Krajcinovic 1996). Thus, all deformation associated with decohesion between inclusions,accumulation of dislocations leading to the nucleationof micro-cracks, debonding of fibers and matrix in composite materials and other events can be covered by thisterm.
Brittle and ductile deformation plays important rolesin the irreversible deformation of a solid. In the brittle
process, significant deformation is localized at planar

123

K. Z. Nanjo

surfaces, on which a through-going fracture is formed,
as described above. Other examples of brittle deformation are associated with displacements on faults inthe Earth’s crust (e.g. King 1983; King et al. 1994;
Thatcher 1995; Jackson 2002) and stick–slip motion
between elastic bodies (e.g. Yamaguchi and Ohmata
2009; Morishita et al. 2010) and other materials.
An alternative approach to irreversible deformationis the ductile deformation, which can be describedby utilizing creep (Newtonian and non-Newtonian)and plastic rheology. An empirical power-law equation
between strain rate ε̇ and stress σ̄ , often calle Dorn’s
equation, has been widely used (e.g. Dorn 1954; Nicolas and Poirier 1976; Karato and Wu 1993)

Q + P Va,ε̇ = Aσ̄ n exp −RT(1)where A and n are 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. It isvalid for both diffusion creep (n = 1) and dislocation
creep (n = 3−5) which are thermally activated.
Both brittle and ductile processes can be quantified
using the concept of dislocations. Dislocation theories
have been developed by many scientists to explain not
only the mechanical properties but also the optical andelectromagnetic properties of crystals. Taylor (1934),
Orowan (1934) and Polanyi (1934) applied dislocation studies to the plastic deformation of simple crystals whose dislocation is much lower than the theoretical values calculated from atomic theory assuming aperfect-lattice state. Mura (1969) developed a methodof continuously distributed dislocations to consider the
relationship between macroscopic plasticity and dislocation theory. However, these studies have not considered how micro-cracks contribute to the deformationof solids. Deformation associated with micro-cracks,
stick–slips, and faults can be treated as dislocations,which are shear deformations across planar surfaces.
Linking between large-scale or long-term ductile
deformation to small-scale or short-term brittle process
is of great interest in engineering, physics, material science, and geophysics (e.g. Sornette and Virieux 1992;
Ma and Kuang 1995; Kovács et al. 2013; Alava et al.
2006; Lyakhovsky et al. 1997; Kun et al. 2006; Hansen
et al. 2015). For example, Sornette and Virieux (1992)
theoretically derived a link between short-timescale
deformation due to slips on faults to long-timescale

123

tectonics. Computer simulation of composite materials (Kovács et al. 2013) showed how brittle failure at the microscopic level leads to a ductile macroscopic response. An avenue for irreversible behavior
associated with brittle and ductile processes is damage mechanics. The concept of damage mechanics has
been utilized in resolving engineering problems (Krajcinovic 1996; Skrzypek and Ganczarski 1999; Voyiadjis and Kattan 1999). Two models that have beenutilized to do this are the fiber-bundle model (FBM)
and the continuum damage model (CDM).
CDM is widely used in civil and mechanical engineering (e.g. Kachanov 1986) and also applied to
Earth’s tectonic processes (e.g. Lyakhovsky et al. 1997;
Ben-Zion and Lyakhovsky 2002). A damage variable
α is introduced to quantify deviation from linear elasticity and the distribution of micro-cracks in the material being considered. By definition, α assumes values
between 0 and 1 and failure occurs when α = 1. Damage evolution is a transient process so that we have α(t)until failure occurs (α = 1).
Another approach to the irreversible deformation ofmaterials is provided by FBM, a discrete model of damage mechanics. FBM was applied to fatigue in structural materials and earthquakes in geophysical settings(e.g. Hemmer and Hansen 1992; Moreno et al. 2001;
Hansen et al. 2015). The behavior of irreversible deformation is quantified by the original number of fibers inthe bundle n 0 and the number of remaining fibers n f .
Damage evolution is a transient process of (n 0 −n f )/n 0
from 0 to 1 (as an equivalent, the decrease of n f fromn 0 to 0). CDM was shown to be equivalent to FBM forassessing the occurrence of a failure in a simple geometry (Krajcinovic 1996; Turcotte et al. 2003; Turcotteand Glasscoe 2004).
Our aim is to explain the continuous deformation ofheterogeneous solid material with dislocations. Continuously deformed material is hypothesized to include
heterogeneity that influences motion during dislocations. We assume that displacements on dislocations
dominate. We show that when FBM is applied to thebrittle deformation of a solid, a non-Newtonian powerlaw viscous rheology is obtained. In our analyses, yieldstress is introduced as follows: below this stress thesolid behaves elastically and can act as a stress guide,
and above this stress the continuum deformations canbe modeled as a power-law viscous fluid. Yield stressneeds to be defined in a transition from brittle or elastic
behavior to ductile or plastic behavior. This expandsA fiber-bundle model for the continuum deformation of brittle material

upon previous studies (Turcotte and Glasscoe 2004;
Nanjo and Turcotte 2005) that did not consider yieldstress.
As an application example, we use a viscoelastic
version of our model under a constant strain applied toa sample. We show that because of damage, the stresson the sample relaxes and this relaxation can reproduce
the power-law temporal decay. Our result shows good
agreement with Omori–Utsu’s law for aftershock decay
following an earthquake (Omori 1894; Utsu 1961).
Our rheological model, which we will show in thispaper, is generic to understand the continuum deformation of brittle materials in not only geophysics but also
engineering and material science. We believe that thesolution to a relatively simple problem in this paper isillustrative.

2 FBM
2.1 Fiber failure criterion
While FBM strives to establish a link between microscopic deformation features and macroscopic observations, the model is phenomenological on many levels(e.g. in the distribution rules). Thus, the first requirement is to specify the failure criterion for the fibers.
When stress is applied to a fiber bundle, the fibers
begin to fail. Local failure events are usually dynamic in
reality so that inertial effects should be taken into consideration during episodic fiber failures. Therefore, weused the dynamic time-dependent failure model where
the distribution of failure time of fibers is specified interms of the stress on the fibers (Coleman 1958). The
alternative model is static in that the distribution of
the failure strengths of the fiber is specified (Daniels
1945).
It is also necessary to specify how the stress ona failed fiber is redistributed to the remaining fibers(Smith and Phoenix 1981; Kun et al. 2006). Continuum
mechanics established rules for interaction between
forces and deformations at distances (e.g. long range
elastic interaction, force field due to a dislocation,
dipoles, quadruples, and so on). In order to treat thislong range interaction, we used the uniform loadsharing hypothesis in which the stress from a failed
fiber is redistributed equally to the remaining fibers(e.g. Hemmer and Hansen 1992; Turcotte et al. 2003).
The alternative is the local load-sharing hypothesis,

where stress from a failed fiber is redistributed to neighboring fibers (usually nearest neighbors) (Newman andPhoenix 2001). We understand that the local load sharing and the uniform load sharing are two extreme formsof the load sharing rule. In continuum mechanics ofelastic materials, the stress distribution around cracks
follows a power law relation between stress increase
and distance from the crack tip (e.g. Lawn and Wilshaw
1975). Motivated by this result of fracture mechanics(Kun et al. 2006), an important future work is to extend
FBM by introducing a load-sharing rule of the powerlaw form, which we do not address in this paper.
In order to apply FBM to a continuously deforming solid material, a failed fiber is replaced by a newfiber. The fiber replacement hypothesis has been previously used by Zapperi et al. (1997), Kun et al. (2000),Moreno et al. (2001), and Halász and Kun (2009). We
hypothesize that the replacement of a broken fiber by
a new fiber is analogous to an earthquake rupture, amicro-crack or the migration of a dislocation. Replacing fibers allows us to model the repetitive occurrenceof earthquakes on a fault, migration of dislocations ina deformed solid, or the stick–slip motion between two
mediums.
To model damage evolution from an undamaged toa damaged state, we introduce a yield stress σy andcorresponding yield strain εy . If the stress is less than
the yield stress σ ≤ σy there is no damage and linear
elasticity is applicable. If the stress is greater than theyield stress σ > σy , damage occurs and fiber failure
occurs to model this irreversible behavior. We consider
that a brittle solid obeys linear elasticity for stresses in
the range 0 ≤ σ ≤ σy . We also assume that Hooke’s law
is applicable so that the dependence of stress σ on strain
ε is given by σ = E 0 ε, where E 0 is Young’s modulus, aconstant. From this equation, the corresponding yieldstrain is given by εy = σ y /E 0 .
The standard approach to the dynamic timedependent failure of a fiber bundle is to specify an
expression for the rate of failure of fibers (Coleman
1956, 1958; Newman and Phoenix 2001; Turcotte et al.
2003). The form of this breakdown rule is given by
dn f (t)= −ν(σ )n f (t),dt(2)

where n f (t) is the number of original fibers that remain
unbroken at time t and ν(σ) is known as the hazard rate,
which is a function of the fiber stress σ(t).

123

K. Z. Nanjo

It is also necessary to prescribe the dependence of thehazard rate ν on the fiber stress σ. Turcotte and Glasscoe (2004) and Nanjo et al. (2005) used the equation
ν(σ) = νf [σ(t)/E 0 ]β , where νf is the reference hazard
rate corresponding to a stress equal to Young’s modulus of each fiber E 0 and β is constant. For composite
materials, it was empirically found that the values of βfall in the range of 2–5 (Newman and Phoenix 2001).
We modify their results to include yield stress σy :
σ − σy βν(σ ) = ν fif σ > σy .
(3)E0
If 0 < σ ≤ σy , ν(σ) = 0 and no fiber failure occurs.
Equations 2 and 3 show variability in the time delay,
even if the stress on the bundle is carried equally by allfibers. A fundamental question is if the cause of thetime delay is associated with the damage. This cause
is in a close association with the thermal fluctuations
in phase changes. The temporal delay of the damage isthat it takes time to nucleate micro-cracks. An in-depth
discussion is given in Zapperi et al. (1997), Moreno
et al. (2001), Shcherbakov and Turcotte (2003), andKovács et al. (2008).
In order to illustrate the failure of a simple fiberbundle under uniform loading that introduces yieldstress σ y and yield strain ε y = σ y /E 0 , we consider
two examples without fiber replacement. The first is
the instantaneous application of a uniform strain ε0 toa fiber-bundle. We show that catastrophic failure in a
finite time period does not occur. The second is theinstantaneous application of a constant tensional force
F0 to a fiber-bundle. The solution obtained explains
for the occurrence of a catastrophic failure of the fiberbundle.
We first consider the case in which a uniform strain
ε0 is applied at time t = 0 and maintained upon thefiber-bundle for t > 0. In this case, the stress oneach fiber has a constant value σ0 = E 0 ε0 . From
Eq. 3 with σ = σ0 (= E 0 ε0 ), the hazard rate isβν = ν f ε0 − ε y which is independent of time t anddependent of the excess strain ε0 −εy . Equation 2 can beβ 
integrated to give n f (t) = n 0 exp −ν f ε0 − ε y t ,where the initial condition n f (0) = n 0 has been used.
The total excess force F(t) carried by the fiber bundle at time t is given by F(t) = n f (t)a σ0 − σ y ,where a is the area of a fiber and σ0 − σ y is
the excess stress. The total excess
is given forceβ by F(t) = n 0 a E 0 ε0 − ε y exp −ν f t ε0 − ε y .

123

Because there is no fiber replacement, the total excess
force F(t) decreases as fibers fail. Catastrophic failure
in a finite period of time does not occur.
Next, we consider the case in which a constant tensional force is applied to the fiber-bundle at time t = 0.
Similar to the first case, we take no fiber replacement
into account. The initial stress on each fiber at t = 0
is given by σ0 = F0 /n 0 a. The applied tensional force
remains constant, so that when a fiber fails the force
carried by that fiber is redistributed to other fibers.
Thus, the stress on surviving fibers increases with timet. This is uniform load sharing and is a mean-fieldapproximation. One implication of this assumption isthat all the remaining fibers have the same stress σ(t).
Equating the total excess force at time t to the initial
total excess force at t = 0, the stress on the surviving
fibers is related
to the number of sound fibers n f (t) by
σ(t) = nn 0 σ0 − σ y + σ y . Substituting this equationfand Eq. 3 into Eq. 2 and integrating with the initial con1

dition n f (0) = n 0 , we obtain n f (t) = n 0 (1 − t/tc ) β ,where tc is the time to failure of a fiber-bundle given
−1 ε − ε −β . Catastrophic failure
by tc = ν −10yf βoccurs at t = tc . The stress in each of the remaining fibers at time t is obtained by substituting n f (t) =1
n 0 (1 − t/tc ) β into σ(t) = nn 0f σ0 − σ y + σ y with the−1result σ(t) = σ0 − σ y (1 − t/tc ) β + σ y . Each fiber
satisfies linear elasticity until it fails, ε(t) = σ(t)/E 0 .
−1Substitution of σ(t) = σ0 − σ y (1 − t/tc ) β + σ yinto ε(t) = σ(t)/E 0 gives the strain ε(t) of each remain
−1
ing fiber, ε(t) = 0 − ε y (1 − t/tc ) β + ε y where
ε0 = σ0 /E 0 and ε y = σ y /E 0 . The stress and strainof each remaining fiber approach infinity as the time
approaches the time to failure at t = tc , which is finite.

2.2 Non-Newtonian viscous rheology model
We consider the uniform extension of a rod that is madeup of n 0 fibers, in which the rod is being extended at
a constant strain rate ε̇. The statistical distribution offiber lifetimes was determined. We assume that all n 0
fibers have stress equal to σy at t = 0. The stress in eachfiber at subsequent times is given by σ(t) = σy + E 0 t ε̇.
Substitution of this equation and Eq. 3 into Eq. 2 andintegration with the initial condition n f (0) = n 0 given f (t) = n 0 exp −ν f ε̇β t β+1 /(β + 1) . We rewrite thisform with the relation

A fiber-bundle model for the continuum deformation of brittle material

1

β=2
β=3
β=5
β = 100n (τ)/n(b)nf(τ)/n0 = 1

0.8

f

0.6
0.4

β=2
β=3
β=5
β = 10

3
2

f(t )

(a)

1
0.2
0
0

1

2

0

3

0

1

2

t

(c) 1.3(d)0.2

νf = 1log10 t

τ

0
1.25

-0.2
-0.4

2

-0.6

4

6

8

10

-1

0

0.5

Fig. 1 a Dependence of the normalized number of unbroken
fibers n f (τ )/n 0 on the non-dimensional time τ for several values of β. Also included for comparison is n f (τ )/n 0 = 1 forthe case in which no fiber failure occurs. b Dependence of thenon-dimensional probability density function f (τ ) on the non-

τ β+1
,
β +1

(4)

where τ is the non-dimensional time given by τ =β

ν fβ+1 ε̇ β+1 t. n f (τ ) is shown in Fig. 1a as a functionof τ while assuming several values of β. High values ofβ show that n f (τ ) quickly decreases with τ, showing
quick damage evolution. If σ ≤ σy , n f (τ ) = n 0 , thenthere is no damage evolution. The non-dimensional
probability distribution function f (τ ) for the distribution of fiber lifetime is given byf (τ ) = −τ β+1
1 dn f (τ )= τ β exp −.
n 0 dτ
β +1(5)This is a Weibull probability density function. A
well-known case equation to show the mean nondimensional fiber lifetime τ̄ is given by
τ̄ =

∞
0

f (τ )τ dτ = (β + 1)

1
β+1





β +2
,
β +1

1

1.5

2

log10 εb1

τ ~ 1/εβ=2
β=5β = 10β = 1000

-0.8

1.2

n f (τ ) = n 0 exp −

3

t

(6)

where  (z) is the gamma function of z (Abramowitz
and Stegun 1965). We now use the form of f (τ ) anddimensional time τ for several values of β (modified from Nanjo
and Turcotte 2005). c Dependence of the mean non-dimensional
fiber lifetime τ̄ on β (modified from Nanjo and Turcotte 2005). d
Dependence of the dimensional fiber lifetime t¯ on the strain rateε̇ for several values of β taking νf = 1

the values of τ̄ for various values of β. The relation
between f (τ ) and τ (Eq. 5) is given in Fig. 1b for several values of β. This figure illustrates that the spread of
the function f (τ ) increases as the value of β decreases.
Figure 1c represents Eq. 6 relating τ̄ to β. The lifetimeτ̄ decreases as β increases.
The dimensional mean fiber lifetime t¯ is obtained by1βsubstation of Eq. 6 into τ = ν fβ+1 ε̇ β+1 t to give

t¯ =

β +1
νf

 1

β+1





β
β + 2 − β+1
ε̇
.
β +1

(7)

t¯ is shown in Fig. 1d as a function of ε̇, assuming νf = 1
and several values of β. For large β values, t¯ is proportional to the inverse of ε̇.
In order to get the mean stress on the fiber bundle, we consider the mean stress on a single fiber thathas been replaced many times (Fig. 2). For this single
fiber, we assume that our replacement hypothesis forthe fiber bundle described above is applicable. When
this fiber fails, it is instantaneously replaced by a newfiber with stress equal to the yield stress σy . Time intervals between failure events satisfy the probability dis-123

K. Z. Nanjo

2

σ/σy

1.5
1

σ/σ = 1

0.5

y

σ/σy
0

0

1

2

3

t

Fig. 2 Schematic time-stress diagram associated with fiber failure events. The normalized stress σ/σ y is given as a functionof time t for a fiber. The fiber is extended at a constant rate ε̇.
This rate linearly increases stress with time t. A failed fiber isreplaced by a new one with stress σy (dashed line). The replacement of new fibers and their rupture have been randomized forsufficient times. The mean stress on the considered fiber after
many replacements is given by σ̄ (dash-dotted line)

tribution of the lifetime given by Eq. 5. The strength ofa newly created fiber is proportional to the period fromthe time at which it is replaced to the time at which it
failed. The fiber strength before and after replacement
is not related. The fiber stress is σ = σy at t = 0, andthe mean stress during the period from 0 to t on a fiberσ̃ is given by σ̃ = E 0 t ε̇/2 + σ y .
The mean stress on the fiber after many replacementsσ̄ is given by∞tσ̃ − σ y f (t)dt + σ y
σ̄ =
t¯
0
E 0 ε̇ ∞ 2=
t f (t)dt + σ y .
(8)
2t¯ 0
1

β

Using τ = ν fβ+1 ε̇ β+1 t, we derive   1
 β+1 2 σ̄ − σ yε̇ β+1 1 ∞ β+2
τ
dτ.
=
τ
exp −
E0
νf
τ̄ 0
β +1

 1




β +1 β+1
β +3
β +2
=
ε̇

/
.
νf
β +1
β +1
(9)
This is rewritten as
1 σ̄ − σ y n
,
ε̇ =
τc
E0

(10)

where n = β + 1 and τc is a characteristic time given


β+1
β+3
β+2
1
/


.
by τc = β+1
νf
2
β+1
β+1

123

Each fiber in the bundle behaves like the single fiber
we have considered. We further assume that the temporal distribution of the mean stress for a large numberof fibers is equal to that for the single fiber considered. Thus Eq. 10 is regarded as the expression relating
the mean stress σ̄ on our fiber bundle to the applied
strain rate ε̇. This is a non-Newtonian viscous rheology. This form of damage rheology is equivalent to theform of rheology (Eq. 1) that is derived from considerations of dislocation densities and atomic diffusivities. An advantage of the former rheology in Eq. (10)over the latter one (Eq. 1) is to provide justification for
the use of non-Newtonian viscous flow for the continuum deformation of brittle materials. The rheology in
Eq. (10) has applicability to cases where materials are
too cold to justify the use of the rheology in Eq. 1 associated with dislocation creep and diffusion creep, bothof which are thermally activated.
The characteristic time τc is now related to the hazard rate νf . Physically, this is the reference of thedelay time associated with damage evolution. However, Nanjo and Turcotte (2005) empirically fitted it toobserved rheology in the Earth’s crust.

3 Application to viscoelasticity
It is useful for a number of problems to combine a fluid
rheology on a long timescale with elastic behavior on ashort timescale. For this purpose, viscoelastic rheology
is usually used. The models, which include the Maxwell
model, the Kelvin–Voigt model and their generalizedversions, are used to predict a material’s response under
different loading conditions. As an application example of our rheological model in Eq. 10, we consider
the Maxwell viscoelasticity. We further consider a casewhere a constant strain is applied to the viscoelastic
medium. The response of the medium is applicable toan understanding of the time-dependent decay of aftershocks. Detailed discussion of the application to viscoelasticity is given in Nanjo et al. (2005).
The Maxwell model for viscoelasticity considers amaterial in which the total strain rate ε̇ve is hypothesized
to be the sum of an elastic strain rate ε̇el and viscousstrain rate ε̇v given as ε̇ve = ε̇el + ε̇v . The elastic strainof the material is εel = σ̄ /E 0 and the time derivativeis ε̇el = E10 ddtσ̄ . If the stress on the medium is lessthan the yield stress σ̄ < σ y there is no viscoelastic
deformation and the material behaves with elasticity. If

A fiber-bundle model for the continuum deformation of brittle material

B

1

C
D

σ/σy

σ0/σy

We consider the viscoelastic medium to which a constant strain has been applied. A strain ε0 > ε y is applied
instantaneously at t = 0 and is held constant. The
behavior of the material is elastic during very rapid
application of the strain. Thus, the initial stress σ0 is
σ0 = E 0 ε0 . If ε0 ≤ ε y , no damage occurs and theinitial stress remains unchanged σ̄ = σ0 for t > 0. If
ε0 > ε y , the material is strained elastically along thepath AB in Fig. 3. The total strain is then maintained
constant ε0 so that damage evolves and repetitive failures occur. Due to the damage and failures, the stresson the sample relaxes from the initial stress σ0 to theyield stress σ y . This relaxation takes places along thepath BCD illustrated in Fig. 3. This solution will givethe time dependence of stress σ(t) during stress relaxation.
For t > 0, ε̇ve = 0 and Eq. 11 reduces to

A
0

0

F E

ε /ε

1

0 y

ε /ε

ve y

Fig. 3 Schematic stress–strain diagram for Maxwell viscoelastic
material (modified from Nanjo et al. 2005). Stress σ0 > σ y andstrain ε0 > ε y are instantaneously applied to a material at time
t = 0. The material elastically behaves and follows the linear
path AB. Subsequently the strain εve = ε0 is maintained constantand the material damage and failure associated with the nonNewtonian flow relax the applied stress σ0 to the yield stress σ yalong the path BCD. In order to determine the energy associatedwith the relaxation process, we instantaneously remove the stressσ̄ and strain ε0 at point C. The subsequent linear elastic behaviorof the material follows the path CF. The total lost energy e AF T
in the limit t → ∞ corresponds to the area ABCDEF. The lostenergy in aftershocks e AF corresponds to the area ABCF0=σ̄ − σ yE0σ̄ − σ yE0σ̄ − σ y=
σ0 − σ y
1 + (n − 1)

n
.

(12)

1

σ0 −σ y n−1
E0

t
τc

1/(n−1) .
(13)

n
.

(11)

In the limit t → is σ̄ (∞)
 = σy . The nor ∞, theresult
malized stress σ̄ − σ y / σ0 − σ y is given as a function of non-dimensional time t/τc in Fig.
 into 4b taking
account the normalized excess stress σ0 − σ y /E 0 =1.0 and several values of n. Increasing the values of nThis is the rheological law relating strain rate, stress,
and the rate of change of stress for our Maxwell viscoelastic material.

(a)

(b)

0

-0.5

0

y

(σ - σ )/(σ - σ )-1yFig. 4 Dependence of thenormalized stress σ̄ − σ y / σ0 − σ y on thenormalized time t/τc a forseveral values of n withnormalized initial stress
σ0 − σ y /E 0 = 1, and bfor several values of
σ0 − σ y /E 0 with n = 3

Integration with the initial condition σ̄ = σ0 at t = 0
gives

the stress on the medium is greater than the yield stressσ̄ > σ y viscous strain will occur and the viscous strainσ̄ −σ y nrate is given by Eq. 10 as ε̇v = τ1c. We thenE0
have
1
1 d σ̄
+ε̇ve =
E 0 dtτc

1 d σ̄
1+
E 0 dtτcn=3(σ0 - σy)/E0 = 1(σ - σ )/E = 0.50y0(σ0 - σy)/E0 = 1
-1.5

n=2
n=3
n=5
n = 10(σ0 - σy)/E0 = 1.5(σ - σ )/E = 2
0

-2
-2

-1

y

0

0

t/τc

1

2

-2

-1

0

1

2

t/τc

123

K. Z. Nanjo

greatlyslows
  stress relaxation. The normalized stressσ̄ − σ y / σ0 − σ y is given as a function of t/τc inFig. 4a taking
 the exponent n = 3 and several valuesof σ0 − σ y /E 0 . High initial stresses relax quickly
followed by a power-law relaxation.
This relaxation process is applicable to an understanding of the time-dependent decay of aftershocks
that follow the main shock. A universal scaling law
applicable to the temporal decay of aftershock activity
is known as the Omori–Utsu law (Omori 1894; Utsu
1961). The most widely used form is given as
K
dN
=
.
dt
(c + t) p(14)

where N is the number of aftershocks with magnitude
greater than a specified value, c and K are constants,
and the power-law exponent p has a value somewhat
greater than unity.
Following Shcherbakov and Turcotte (2003) and
Shcherbakov et al. (2005), our working hypothesis isthat stress transfer during a main shock increases thestress σ̄ and strain εve above the yield values σ y and ε yin some regions adjacent to the fault on which the main
shock occurred. The increases to 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 also neglect any increasein stress due to large aftershocks because only 3% of thetotal energy is associated with the aftershock sequence
while 97% is associated with the main shock (Nanjo
and Nagahama 2000; Shcherbakov et al. 2005). We
hypothesize that the applied strain ε0 remains constantand that aftershocks relax the stress σ̄ to its yield value
σ y . The occurrence of aftershocks is attributed to thisrelaxation process as given in Eq. 13. The time delay ofaftershocks relative to the main shock is directly analogous to the time delay of the damage. This is because
it takes time to nucleate micro-cracks, i.e. aftershocks.
In order to quantify the rate of aftershock occurrence, we determined the rate of energy release in therelaxation process. The stored elastic energy release(per unit mass) e0 in a material after an instantaneous strain ε0 has been applied along path AB ise0 = E 0 ε02 /2. Since the strain is constant during stressrelaxation, no work is done on the sample. If the applied
strain (stress) is instantaneously removed at point C, we
hypothesize that the sample will follow the elastic path123

CF that is parallel to path AB. The elastic energy e1recovered during stress relaxation on this path is givenby e1 = σ̄ ε0 /2. We assume that the difference between
the energy added e0 and the energy recovered e1 islost in aftershocks. This energy e AF , which is given bye AF = e0 − e1 , corresponds to the area ABCF. The
total energy of aftershocks e AF T obtained in the limitt → ∞ corresponds to the area ABCDEF with theresult e AF T = E 0 ε0 ε0 − ε y /2. Using this equationand integrating the time derivative of e AF = e0 − e1 ,we obtain the rate of energy release
 1
1
n−1
n−1 c
1 deAF( p − 1) c p−1
=
.
(15)
=
n
eAFT dt(c + t) p(c + t) n−1p
and c =
where we take n = p−1

τcn−1 . See(n−1)(ε0 −ε y )“Appendix 1” for detailed derivation of Eq. 15.
Following Newman et al. (1995), Shcherbakov andTurcotte (2003), Turcotte and Glasscoe (2004), andShcherbakov et al. (2005), we hypothesize that therate of energy release is equal to the rate of occurrence of earthquakes deAF /dt = dN/dt. If we assume
K = eAFT ( p − 1) c p−1 , Eq. 15 is identical to Eq. 14.
There are several attempts to explain aftershock
decay patterns within damage models (e.g. Ben-Zionand Lyakhovsky 2006) and within the context of rateand-state dependent frictional rheology (e.g. Dieterich
1994; Kaneko and Lapusta 2008). What new featurethat our model introduces into this problem is to associate aftershock relaxation, i.e. months to years with thelong-term crustal deformation, i.e. millions of years.
The behavior of the deforming the Earth’s crust can bemodeled as the non-Newtonian viscous flow in Eq. 10.
Once a large earthquake (a main shock) occurs in thecrust, stress suddenly increases in some regions adjacent to the fault on which the main shock occurred.
Stress relaxation is accompanied by the aftershock
sequence. Using a viscoelastic version of our model,
we got the Omori–Utsu law temporal aftershock decay
in Eq. 15. The “healing” (modeled by fiber replacementin our study), needed for the continuum deformation of
the crust, is introduced into the problem of aftershock
decay.

4 Discussion
The basis of our mechanism is the application of arenewable FBM to brittle deformations. FBM has beenA fiber-bundle model for the continuum deformation of brittle material

applied successfully to the failure of composite materials. This model, as defined in Eqs. 2 and 3, is inherently
dependent on time through the hazard rate. This timedependence is associated with the nucleation and coalescence of micro-cracks. In order to represent the continuum deformation of a solid, it is necessary to introduce “healing”. Following Zapperi et al. (1997), Kun
et al. (2000); Moreno et al. (2001); Kun et al. (2006),
and Halász and Kun (2009), we do this by replacing
a failed fiber by a new fiber with stress equal to yieldstress. The result is a non-Newtonian viscous rheology
as previously found by Turcotte and Glasscoe (2004)
and Nanjo and Turcotte (2005). However, fiber replacement is not damage mechanics but depends on how to
set an ad-hoc boundary condition to constrain the solution of the model.
Continuum damage in the context of fiber bundle with multiple failure events allowed have beenaddressed including stick slip generation by Kun et al.
(2006) (see also Halász and Kun 2009). The feature
introduced by their research is the introduction of thesudden stiffness degradation. These authors assumed
that at the failure point of a fiber, the stiffness of thefiber gets reduced. The obtained relation between the
average load on a fiber and the strain shows plasticity
response during the period of multiple failure events
allowed. Despite mechanistic details, the microscopic
processes and the macroscopic constitutive behavior look very similar between our research and theirresearch. Our assumption that single fibers have linear
elastic behavior up to fiber failure, which is similar to
Kun et al. (2006). We derived and used the Weibull
probability density function for fiber lifetimes in Eq. 5,
which is equivalent to the equation assumed for thedisordered breaking thresholds in Kun et al. (2006).
We calculated the mean stress on the fiber after many
replacements σ̄ in Eq. 8, which is similar to the average load on a fiber in Kun et al. (2006). They show,by analytical calculation, that plastic response of theirfiber bundle model on the macro-scale emerges, whichis the same as Eq. 10 with assuming yield stress σ y = 0
in the limit of the power-law exponent n to be infinity.
The similarity between Kun et al. (2006) and our study
suggests that incorporating physical evidence of healing, such as the sudden stiffness degradation, into ourmodel is an important theoretical development that is
worthy of further exploration.
Following Shcherbakov and Turcotte (2003),
Shcherbakov et al. (2005), Nanjo et al. (2005), andManaker et al. (2006), we introduced the concept ofyield stress into FBM. As a consequence, the powerlaw rheology (Eq. 10) relating between strain rate ε̇ andexcess stress σ̄ − σ y is obtained. This theoretical development expands upon our previous works (Turcotte andGlasscoe 2004; Nanjo and Turcotte 2005) that did notconsider yield stress. This is different from normal constitutive creep laws such as in Eq. 1 without yield stress.
However, an empirical equation similar to Eq. 10 has
been used in engineering to predict more accurately
the rheology of deformation with stress above yieldstress. This is called the yield-power law (YPL) model(Hamphill et al. 1993; Houwen and Geehan 1986; Skelland 1967; Reed and Pilehvari 1993; Zamora and Lord
1974) that has been applied to predict the rheologicalbehavior of ductile materials such as mud. Comparison
with the YPL models shows that the power-law rheology (Eq. 10) relating between strain rate ε̇ and excess
stress σ̄ − σ y is appropriate to precisely predict boththe continuum deformation of ductile materials and thecontinuum deformation of brittle materials.
We now compare the results derived above with alaboratory experiment and geophysical observations.
Yamaguchi and others (Yamaguchi and Ohmata 2009;
Morishita et al. 2010) conducted an experiment of continuum shear deformation of a material that consists of
two elastic bodies. Two elastic bodies, when slid against
each other, exhibited stick–slip motion repeating lockand sliding. A constant pull velocity Vpull was applied to
one body while the other remained fixed. Tensile force(frictional force) F over many occurrences of stick–slips was monitored for several constant pull velocities
from Vpull = 1 to Vpull = 1000 μm/s. No clear stick–slip behavior was observed below Vpull = 200 μm/s.
Above this value, stick–slip behavior was observed: F
increased with time t (lock) followed by a sudden force
drop (sliding). The average force over many stick–slip
motions F̄ was correlated to the velocity Vpull , given
by V pull ∝ F̄ 1/0.15 . This result is in good agreementwith Eq. 10 taking n = 6.7 if σ̄ /σ y  1.
Houseman and England (1986) and England andMolnar (1997) considered an indenter model for continental deformation, which significantly takes place
on faults (e.g. King 1983; King et al. 1994; Thatcher
1995; Jackson 2002). England and others applied thefinite element method to a thin non-Newtonian viscous
sheet to obtain solutions. Using the power-law rheology
given in an equation similar to Eq. 10, they obtained
results for n = 3 and 10. These authors compared123

K. Z. Nanjo

their results with observations in the Indian–Asian collision zone and found broad agreement provided thatthe power-law exponent is large (n > 3). Thus, the useof a non-Newtonian viscous flow rheology given as in
Eq. 10 is common to modelling the continuum deformation of brittle materials in a wide range of scales:from a laboratory experiment with stick–slip motion
to orogenies such as the Indian–Asian collision withfaulting.
The normal constitutive law in Eq. 1 is also derived
from considerations of atomic diffusivities and dislocation densities. The exponent is n = 1 for diffusion creep
and n = 3−5 for dislocation creep. The large value ofthe exponent for the laboratory experiment (n = 6.7)
and broad agreement of the large values of the exponent
for the continental deformation (n > 3) are certainly
not surprising for aftershocks. Nanjo et al. (2007) studied the decay of aftershock activity for four Japanese
earthquakes: 1995 in Kobe (magnitude M = 7.3),
2000 in Tottori (M = 7.3), 2004 in Niigata (M =6.8), 2005 in Fukuoka (M = 7.0). These authors
determined the rates of occurrence of aftershocks innumbers per day as a function of time for 1000 days
(378 days for Niigata and 225 days for Fukuoka) after
a main shock. They fitted an equation similar to
Eq. 14 to the data and found p = 1.19∼1.23 for
Kobe, p = 1.21∼1.24 for Tottori, p = 1.32∼1.34 for
Niigata, and p = 1.28∼1.29 for Fukuoka. Shcherbakov
et al. (2005) studied aftershock sequences following
four
California
earthquakes
and
found
p = 1.22 ± 0.03 for 1992 in Landers (M = 7.3),
p = 1.18 ± 0.02 for 1994 in Northridge (M = 6.7),
p = 1.21 ± 0.05 for 1999 in Hector Mine (M = 7.1),
and p = 1.12 ± 0.02 for 2003 in San Simeon (M =6.5). From n = p/( p − 1), we found that the values of the exponent n fell in the range n = 4∼11.
Reasenberg and Jones (1989) also carried out a detailed
study of aftershocks for major earthquakes in California and found the mean value p = 1.07 ± 0.03.
From n = p/( p − 1), we found n = 15. Moreno
et al. (2001) used a large value (β = 30) to performtheir simulations of a fiber-bundle model for complex
aftershock sequences. From n = β +1, we observedn = 31. Although applicable values of n are not wellconstrained, one important aspect of our study is thatthe power-law exponent n is likely large for the continuum deformation of brittle solids.
In the limit of the power-law exponent n to infinity, there is no dependence of stress on strain rate, and123

the rheology is perfectly plastic. The large power-law
exponents we found show the behavior of the deforming materials approaches that of a perfectly plastic
material. From β = n − 1 and Eq. 5 (Fig. 1a) showingthat the spread of fiber-lifetime distribution decreases
with increasing β, the physics behind this perfect plasticity dictates that recurrence of failures approaches
perfectly periodic. This is applicable to a frictional rheology without taking frictional hysteresis into consideration, such as the Anderson theory of faulting (Anderson 1951; Scholz 2002). The continuum deformationof brittle solids modeled by a power-law rheology in
Eq. 10 with large n values shows intermediate between
perfectly plastic rheology based on periodic recurrence
failures and power-law rheology with normal range of
exponents (typically, n = 2−5: Newman and Phoenix
2001) based on randomized recurrence failures.
Some forms of “damage” that we did not consider
in this paper are clearly thermally activated. The deformation of solids by diffusion and dislocation creep isan 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 a thermally activated rupture process to find that seismic decay rates
after main shocks following the Omori–Utsu law in
Eq. 14. The continuum deformation of the continental crust has already been considered as a thermally
activated process by Nanjo and Turcotte (2005) who
utilized a fiber-bundle model. These authors assumed
that fiber failure is a thermally activated earthquake andmodified the hazard rate in Eq. 3 to a form that links
the dependence of the hazard rate to the absolute temperature. They obtained a power-law relation betweenσ̄ and ε̇ and the rheology was exponentially dependent
on the inverse absolute temperature as given by Eq. 1.
Their analyses, based on laboratory experiments (e.g.
Nakatani 2001), argued in favor of thermally activateddamage in order to find the strength envelope of thecontinental lithosphere.
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.
The analysis given in this paper is for a uniaxial problem. This is the reason why the solution to this relatively

A fiber-bundle model for the continuum deformation of brittle material

simple problem is illustrative and the methodology canbe readily adapted to understand the irreversible deformation of brittle materials not only in geophysics but
also in engineering and material science. However, it isclearly desirable to extend the analysis to a fully threedimensional formulation to treat relatively realistic but
complicated problems. One way to do this is to reduce
the shear modulus by damage but maintain the bulk
modulus as an invariant. This approach has been discussed by Lyakhovsky et al. (1997) and Manaker et al.
(2006).

5 Conclusion

Using the stored elastic energy e0 = E 0 ε02 /2 andthe recovered elastic energy e1 = σ̄ ε0 /2, we obtainE ε2

the energy lost in aftershocks eAF = 02 0 − σ̄2ε0 . Substitution of Eq. 13 into this equation givesE 0 ε0 
ε0 − ε y
eAF =
2⎡
⎤
1
⎢
× ⎣

n−1
1+(n − 1) ε0 −ε yAcknowledgements The author thanks the Editor K. RaviChandar and two anonymous reviewers for constructive comments. A part of this study was conducted under the auspicesof the MEXT Program for Promoting the Reform of National
Universities.

Appendix 1
Derivation of the equation of the rate of energy releasein Eq. 15.

⎥
1/(n−1) ⎦ .
(16)

Using
the

 total energy of aftershocks e AF T = E 0 ε0ε0 − ε y /2, we rewrite Eq. 16 aseAFThis paper uses a particular version of the fiber bundle model (FBM), often used in damage mechanics,
to explain the success of non-Newtonian viscous flow
models for the continuum deformation of brittle materials. The particular version of the model uses a timedependent variation on a model introduced by Coleman (1956, 1958). In particular, it introduces a yieldstress, which not only provides a minimum stress level
for the breakdown process to occur (a power-law inactual stress minus yield stress), but also becomes thestress carried by the fiber (rather than zero) immediately
after the fiber fails. In fact, a second assumption is thatthe broken fiber is “replaced” with a new fiber that isalready at the yield stress and which then has the same
failure rate in terms of increasing stress. After considering two simple examples, we show that when the
FBM is applied to brittle deformation of a solid, a nonNewtonian, power law, viscous rheology is obtained.
The model is also used to explain stress relaxation inviscoelasticity through a stress relaxation problem. In
this context the focus is on making connection to the
Omori–Utsu law for the rate of aftershocks following
an earthquake.

t
τc

⎡

1
⎢
= eAFT ⎣ n−1
1+(n−1) ε0 −ε y

⎤
t
τc

⎥
1/(n−1) ⎦ .
(17)

Taking the time derivative of Eq. 17, we obtain the rateof energy releasen−1
1
1 deAFτc ε0 − ε y=n .
n−1 t  n−1
eAFT dt
1 + (n − 1) ε0 − ε y
τc
(18)
τc
Using c =n−1 , we rewrite Eq. 18 as(n−1)(ε0 −ε y )

 1
1
n−1
n−1 c
1 deAF=.
neAFT dt(c + t) n−1(19)p, Eq. 19 is rewritten as
If we take n = p−1

1 deAF( p − 1) c p−1=.
eAFT dt(c + t) p(20)

This is the equation of the rate of energy release inaftershocks in Eq. 15.

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