Fracture process of a fiber bundle with strong disorder

Abstract

We investigate the eect of the amount of disorder on the fracture process of heterogeneous materials in the framework of a fiber bundle model. The limit of high disorder is realized by introducing a power law distribution of fiber strength over an infinite range. We show that on decreasing the amount of disorder by controlling the exponent of the power law the system undergoes a transition from the quasi-brittle phase where fracture proceeds in bursts to the phase of perfectly brittle failure where the first fiber breaking triggers a catastrophic collapse. For equal load sharing in the quasi-brittle phase the fat tailed disorder distribution gives rise to a homogeneous fracture process where the sequence of breaking bursts does not show any acceleration as the load increases quasi-statically. The size of bursts is power law distributed with an exponent smaller than the usual mean field exponent of fiber bundles. We demonstrate by means of analytical and numerical calculations that the quasi-brittle to brittle transition is analogous to continuous phase transitions and determine the corresponding critical exponents. When the load sharing is localized to nearest neighbor intact fibers the overall characteristics of the failure process prove to be the same, however, with dierent critical exponents. We show that in the limit of the highest disorder considered the spatial structure of damage is identical with site percolation—however, approaching the critical point of perfect brittleness spatial correlations play an increasing role, which results in a dierent cluster structure of failed elements.

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Fracture process of a fiber bundle with strong disorder
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J. Stat. Mech. (2016) 073211
Fracture process of a fiber bundle with
strong disorder
Zsuzsa Danku and Ferenc Kun
Department of Theoretical Physics, University of Debrecen, PO Box 5, H-4010
Debrecen, Hungary
E-mail: ferenc.kun@science.unideb.hu
Received 12 May 2016, revised 6 June 2016
Accepted for publication 11 June 2016
Published 28 July 2016
Online at stacks.iop.org/JSTAT/2016/073211
doi:10.1088/1742-5468/2016/07/073211
Abstract. We investigate the eect of the amount of disorder on the fracture
process of heterogeneous materials in the framework of a fiber bundle model.
The limit of high disorder is realized by introducing a power law distribution of
fiber strength over an infinite range. We show that on decreasing the amount
of disorder by controlling the exponent of the power law the system undergoes
a transition from the quasi-brittle phase where fracture proceeds in bursts to
the phase of perfectly brittle failure where the first fiber breaking triggers a
catastrophic collapse. For equal load sharing in the quasi-brittle phase the
fat tailed disorder distribution gives rise to a homogeneous fracture process
where the sequence of breaking bursts does not show any acceleration as the
load increases quasi-statically. The size of bursts is power law distributed
with an exponent smaller than the usual mean field exponent of fiber bundles.
We demonstrate by means of analytical and numerical calculations that the
quasi-brittle to brittle transition is analogous to continuous phase transitions
and determine the corresponding critical exponents. When the load sharing is
localized to nearest neighbor intact fibers the overall characteristics of the failure
process prove to be the same, however, with dierent critical exponents. We
show that in the limit of the highest disorder considered the spatial structure
of damage is identical with site percolation—however, approaching the critical
point of perfect brittleness spatial correlations play an increasing role, which
results in a dierent cluster structure of failed elements.
Keywords: avalanches, fracture, classical phase transitions
Z Danku and F Kun
Fracture process of a fiber bundle with strong disorder
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PAPER: Classical statistical mechanics, equilibrium and non-equilibrium
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Fracture process of a fiber bundle with strong disorder
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1. Introduction
Natural materials and most of the artificially made ones have an inherent disorder
which appears at dierent length scales in the form of dislocations, flaws, microcracks,
grain boundaries, or internal frictional interfaces [1]. When subject to mechanical load,
this quenched structural disorder plays a decisive role in the emerging fracture process:
disorder gives rise to strength reduction by introducing week locations where cracking
can be initiated. For a fixed sample size the tensile strength is a stochastic variable
described by a probability distribution [1–3]. Increasing the extension of samples a size
eect emerges, i.e. the ultimate strength of disordered materials is a decreasing function
of their size, which has to be taken into account in engineering design [3, 4]. The main
benefit of disorder is that it stabilizes the fracture process, making it possible to arrest
propagating cracks. As a consequence the fracture process of disordered materials is
composed of a large number of crack nucleation—propagation—arrest steps which gen-
erate a sequence of precursory cracking avalanches. This crackling noise is of ultimate
importance to work out forecasting technologies for natural catastrophes such as land-
slides and earthquakes, and for the catastrophic failure of engineering constructions
[5, 6]. The degree of disorder present in materials aects fracture processes both on the
macro- and micro-scales; however, detailed understanding of the relevant mechanisms
is still lacking.
It is rather dicult to perform laboratory experiments precisely tuning the amount
of a material’s disorder. The length scale of disorder was controlled by heat treatment
in phase-separated glasses [7] which proved to have an eect on the roughness of the
generated crack surface. Sub-critical crack propagation was investigated in a sheet of
paper under a constant external load where the paper was softened by introducing
holes in dierent geometries. It was found that the increasing disorder slows down the
Contents
1. Introduction 2
2. Fiber bundle model with strong disorder 3
3. Fracture strength 5
4. Crackling noise 5
5. Critical exponents 8
6. Localized load sharing 10
7. Discussion 14
Acknowledgments 16
References 16

Fracture process of a fiber bundle with strong disorder
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propagation of the crack [8]. Recently, it has been shown for the compressive failure
of porous materials that the amount of structural disorder is crucial for forecasting the
global failure, i.e. the higher the disorder is the more intensive precursory activity is
obtained, which improves the precision of forecasting [9].
In theoretical studies of the eect of quenched disorder on fracture, discrete models
of materials are indispensable. In the framework of discrete models either random-
ness is introduced for the strength of cohesive elements or random dilution is applied
on a regular lattice, where the amount of disorder is controlled by the width of the
strength distribution [10–14] and by the degree of dilution [2, 15] respectively. In these
investigations the fiber bundle model (FBM) is a very useful tool since it captures the
relevant aspects of fracture processes but it is still simple enough to obtain analytical
solutions and to design ecient simulation techniques [16, 17]. In the present paper
we use fiber bundle modeling to investigate the limiting case of high disorder for the
fracture process of heterogeneous materials. A power law distribution of fiber strength
is considered over an infinite range where the exponent of the distribution controls
the amount of disorder. We demonstrate that on varying the amount of disorder the
system undergoes a transition from a quasi-brittle phase with precursory bursting
activity to a perfectly brittle phase where the first fiber breaking triggers catastrophic
failure. In the quasi-brittle phase the high disorder has interesting consequences both
on the macro- and micro-scales: the ultimate strength of the bundle increases with the
system size, which is controlled by extreme order statistics of fibers’ strength. Under
quasi-statically increasing load the cracking bursts of fibers form a stationary sequence
without any acceleration and signature of the imminent global failure of the system.
Burst sizes are power law distributed, however, with a significantly lower exponent
than with moderate disorder. Simulations revealed that even if strong stress concentra-
tion is introduced in the system by locally redistributing the load after breaking events,
the overall behavior of the fracture process remains the same and the spatial structure
of damage closely resembles percolation.
2. Fiber bundle model with strong disorder
In the model we consider N parallel fibers with linearly elastic behavior described by
the same Young’s modulus E = 1. Disorder is introduced in the system such that the
failure threshold of fibers
σth
is a random variable, which takes values in the interval
⩽σ σ
<+
∞
th
min
th
according to the probability density function (PDF)
()σµσ=µ−−
p.
th th
1
(1)
The lower bound
σth
mi
n
has a finite value
σ
=
1
th
min but the strength values
σth
cover an
infinite range. A very important feature of
()σpth
is that the amount of disorder can
be controlled by varying the exponent μ in such a way that increasing μ results in
a lower disorder. In our study we focus on the parameter range
⩽µ<0 1
, where the
disorder is so high that even the first moment of the distribution (1) does not exist;
however, normalizability is ensured (see figure 1(a) for illustration). As a first step,
we assume that under an increasing external load σ when a fiber breaks its load is

Fracture process of a fiber bundle with strong disorder
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equally redistributed over all intact fibers. In this equal load sharing (ELS) limit the
constitutive equation
()σ ε
of the model can be obtained from the generic expression
() [()]σεε ε=−EPE1
where P denotes the cumulative distribution function (CDF) of
failure thresholds, and the term
()ε−PE1
is the fraction of fibers that are intact at the
deformation ε. For our model the constitutive equation
()σ ε
can be cast in the form
()
 ⩽

⎧
⎨
⎩
σε
εε
ε
εε
ε
=>
µ−
E
E
fo
r,
fo
r,
0
10
(2)
where the threshold strain
ε0
has the value
/ε σ==
E
1
0th
min. Figure 1(b) shows that
below
ε0
linearly elastic behavior is obtained since no fibers can break. As breaking
sets on for
ε ε>0
, non-linearity of
()σ ε
emerges, which gets stronger with increasing μ.
It can be seen that on varying μ as a control parameter the constitutive behavior of
the system has two qualitatively dierent regimes: for
µ<1
the non-linear increase of
()σ ε
implies a quasi-brittle response of the system where under stress controlled loading
the fracture of the bundle would proceed by stable damaging as fibers gradually break.
However, increasing μ above 1 the constitutive curve becomes decreasing beyond the
threshold strain
ε0
, having a sharp maximum at
ε0
. This functional form implies that
all fibers break immediately in a catastrophic avalanche as the external load surpasses
the value
εE0
. Since linear response is followed by sudden collapse the behavior of the
Figure 1. (a) Probability density of failure thresholds for several values of the
control parameter μ. (b) Constitutive curve
()σ ε
of the system for several dierent
values of the exponent μ. In the parameter range
1µ<
the constitutive curve
monotonically increases, while for
1µ>
it has a sharp maximum at
1
th
min
σ
=. (c)
Average critical strain
c
ε
of the bundle where global failure occurs as a function of
the system size N for several values of the exponent μ. The straight lines represent
power law fits. (d) Exponent of the size dependence of the critical strain
c
ε
. The
straight line represents the power law for exponent −1.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
p (th)
012345
th
0.25
0.5
0.75
1.0
1.4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0123
45
0.25
0.5
0.75
1.0
1.4
104
1014
1024
1034
<
c
>
104105106107
N
0.2
0.3
0.4
0.6
0.81.0
1
10
10-1
1
a) b)
c)
d)

Fracture process of a fiber bundle with strong disorder
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system is perfectly brittle in the parameter range
µ>1
. The transition from the quasi-
brittle to the brittle phase occurs at the critical point
µ=1
c
where the stress σ becomes
constant
σ ε=E0
, independent of the strain ε (see figure 1(b)).
3. Fracture strength
An interesting feature of the constitutive equation (2) is that in the quasi-brittle phase
it does not have a local maximum so that it is monotonically increasing until the last
fiber breaks. This has the consequence that even under stress controlled loading no
catastrophic avalanche of breaking emerges so that in a finite bundle of N fibers the
fracture strength
σ
c
and the corresponding fracture strain
εc
are determined by the
breaking threshold of the strongest fiber. Hence, the macroscopic strength of the system
is controlled by the extreme order statistics of fibers’ strength [18, 19]. The average
failure strain
εc
of the bundle can simply be obtained as the average value of the larg-
est threshold strain
εth
max
of fibers, where the relation
σ ε=E
th th
holds for the stress and
strain thresholds of single fibers. According to the generic result of extreme order statis-
tics [18, 19] the average
εN
th
max
of the largest of N independent identically distributed
random variables can be determined as
⎛
⎝
⎜⎞
⎠
⎟
εε==−
+
−
P
N
11
1,
N
cth
max1
(3)
where P−1 denotes the inverse of the cumulative distribution function P. Substituting
the distribution function P of our model the above equation yields
/
/
⎛
⎝
⎜⎞
⎠
⎟
ε=
+
≈
µ
µ
−
N
N
1
1.
c
1
1
(4)
The result shows that the strength of the bundle increases as a power law of the system
size N, which is in a strong contrast with the usual decreasing strength of the disorder
dominated size eect of heterogeneous materials. In the simulations
εc
was determined
by directly averaging the strain at which the last avalanche occurred. Figure 1(c) dem-
onstrates that the numerical results are consistent with the theoretical expectations
and can be very well fitted by the power law
ε∼α
N
c
, where the exponent α has an
excellent agreement with the analytical prediction
/αµ=1
(see figure 1(d)).
4. Crackling noise
The quasi-static loading of the sample is carried out by increasing the external load to
provoke the breaking of a single fiber. The failure event is followed by the redistribu-
tion of load where each intact fiber receives the same load increment under ELS condi-
tions. The elevated load may give rise to additional breakings, and in turn the repeated
steps of load redistribution and breaking can generate a failure avalanche. The size
∆
of
the avalanche can be characterized by the number of fibers breaking in the correlated

Fracture process of a fiber bundle with strong disorder
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trail. The microscopic origin of the perfectly brittle behavior for
µ>1
is that the first
breaking event, i.e. the breaking of the weakest fiber immediately triggers an avalanche
that cannot stop leading to catastrophic collapse. However, in the quasi-brittle phase
the fracture of the bundle proceeds in a sequence of bursts with size
∆
spanning a
broad range. It can be observed in figure 2(a) for a single sample that the sequence of
breaking bursts has an astonishing stationarity, i.e. in spite of the increasing load the
moving average of event sizes and their fluctuations remain practically constant. Note
that the absence of a catastrophic avalanche means that even the last avalanche obeys
the same statistics as all others. This behavior is in strong contrast to what is usually
observed when the disorder is moderate: as the load increases the size of bursts spans
a broader and broader range when approaching global failure so that the average event
size rapidly increases towards failure [16, 20, 21].
The statistics of crackling avalanches is characterized by the distribution of their
size
()∆p
, which is presented in figure 2(b) for several values of the exponent μ. Power
law behavior is evidenced, which is followed by an exponential cuto. It can be observed
in the figure that as μ approaches 1 the cuto burst size increases and finally diverges
so that for
µ=1
the complete distribution can be described by a single power law. The
most remarkable feature of the results is that the exponent ξ of the power law regime
is
/ξ=32
—significantly lower than the usual mean field exponent
/ξ=52
of fiber bun-
dles found for a broad class of disorder distributions. The lower value of the exponent
implies a higher fraction of larger avalanches during the breaking process of the highly
disordered system.
In order to understand the absence of acceleration in the avalanche activity and
the emergence of the low exponent of the size distribution it is instructive to calcu-
late the average number a of fiber breakings triggered immediately by the failure of a
single fiber at the strain ε [20, 21]. Since the load
σ ε=E
dropped by the broken fiber
is equally shared by all the intact fibers of number
[()]σ−NP1
the stress increment
σ∆
they experience is
/[ ()]σσ σ∆=−NP1
. Eventually, a follows by multiplying
σ∆
with
the probability density
()εp E
of failure thresholds and with the total number of fibers N
()
()
()
ε
εε
ε
µ=
−
=
aEpE
PE1.
(5)
The right-hand side of the equation was obtained by substituting the PDF p and the
CDF P of failure thresholds of our model. It follows that in our FBM the probability
of triggering avalanches does not depend on where the system is during the loading
process, and hence, from the viewpoint of avalanches all points of the constitutive
curve are equivalent to each other. This mechanism explains the absence of increasing
bursting activity in figure 2 with increasing load. Note that a catastrophic avalanche
occurs when
()σ>a1
[20], which can only be obtained in our case for
µ>1
. Hence, for
any
µ<1
the system approaches failure in a stable way, all fibers breaking in finite
avalanches.
The complete size distribution
()∆p
can be obtained analytically by substituting
()εa
into the generic form [20–22]
p
N
pxax ax
x
e
!1e
d
xax
1
0
1
c
()
()()[()] ()
∫
∆
=
∆
∆
−
∆− −∆
∆− ∆
(6)

Fracture process of a fiber bundle with strong disorder
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where for the upper limit of integration xc we have to insert the strength of the bundle.
Inserting our PDF and the expression (5) of the average number of triggered failures a,
and utilizing the approximation
!e2π∆∆∆
∆−∆

the analytic result can be cast into
the final form
() //
∆
∆
−−∆∆

p
Ne.
32 c
(7)
A power law of exponent
/ξ=32
is obtained followed by an exponential cuto, in agree-
ment with the numerical results. Here,
∆c
denotes the characteristic burst size which
controls the cuto of the distribution. The value of
∆c
depends solely on the control
parameter
µµ
∆
=
−−
1
1ln
.
c
(8)
To demonstrate the consistency of the results in figure 2(c) we present the scaling plot
of the avalanche size distributions where the two axes of figure 2(b) are rescaled with
powers of
∆c
, using
κ=1
and
/κξ =32
for the exponents on the horizontal and vertical
axes respectively. The high quality data collapse underlines that the analytical solution
provides a comprehensive description of the avalanche activity of the model.
Figure 2. (a) Sequence of bursts emerging during the quasi-static loading of a
bundle of N = 105 fibers with
0.9µ=
. The burst size
∆
is plotted as a function
of the order number i of the crackling event. The bold yellow line represents the
moving average of the event size considering 100 consecutive bursts. No acceleration
towards global failure can be pointed out. (b) Size distribution of avalanches
p()∆
for several values of the exponent μ. Power law functional form is obtained,
followed by an exponential cuto. The straight line represents a power law with
exponent 3/2. (c) Data collapse analysis of the avalanche size distributions. The
data presented in (b) is replotted such that the two axes are rescaled with powers
of the characteristic burst size
c()µ∆
.
1
10
102
10
3
01000 2000 30004000500060007000800090001
0000
i
10-13
10-11
10-9
10-7
10-5
10-3
10-1
p( )
10 10210310410510 6107
0.1
0.3
0.5
0.7
0.8
0.9
0.94
0.96
0.98
0.99
0.995
0.999
1.0
10-8
10-4
1
104
108
p ( ) c( )
10-5 10-4 10-3 10-2 10 -1 110
/
c
( )
a)
b) c)

Fracture process of a fiber bundle with strong disorder
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It has been shown in equal load sharing FBMs that the probability distribution of
the size of bursts has a power law functional form where the exponent exhibits a high
degree of universality with respect to the form and amount of disorder: for disorder dis-
tributions, where the constitutive curve
()σ ε
of the system has a quadratic maximum,
the burst size exponent is 5/2 [16]. Reducing the disorder by making the distribution
of fibers’ strength narrower, a crossover is obtained to a lower exponent 3/2 [13, 20,
23, 24]. The same happens when constraining the avalanche statistics to windows
shrinking towards global failure—hence, the crossover has been suggested as an early
signature of the imminent failure event [24]. An important consequence of our results
is that at any point of the loading process avalanches of the same size range can pop
up, and hence, the distribution does not evolve, both the exponent and the cuto size
remain the same wherever we measure them during the loading process.
5. Critical exponents
As the control parameter μ approaches the critical value
µc
from below the system
undergoes a phase transition from quasi-brittle to perfectly brittle response. We used
analytical calculations and finite size scaling of the simulated data to determine the
critical exponents of the transition. Based on the Taylor expansion
() /+≈−xxxln1 2
2
in (8) it can be shown that as μ approaches 1 the cuto burst size has a power law
divergence as a function of the distance from the critical value
µ=1
c
()
/
µµ∆∼− σ−
∆
,
c
c
1
(9)
where the value of the cuto exponent of avalanche sizes is
/σ=
∆12
.
Of course, in a finite system of N fibers deviations occur from the analytical solution
of the infinite system size where even the critical point
µc
has size dependence. In order
to obtain a detailed characterization of how the system approaches the critical point of
perfectly brittle behavior with increasing μ, we determined the average burst size
∆
as a function of μ for several system sizes N. For
∆
first the average burst size of sin-
gle samples is calculated as the second moment
=∑∆Mii
22
of burst sizes divided by the
first one
=∑∆Mii1
, and then this quantity is averaged over a large number of samples.
Note that the largest burst is always omitted in the summation. Figure 3(a) presents
that for low μ values the average burst size falls close to one, indicating that nearly all
fibers break one-by-one. When approaching the brittle phase larger and larger bursts
can emerge and
∆
develops a sharp peak in the vicinity of
µ=1
c
. The finite values
∆>0
observed for
µ>1
are obtained due to the finite size eect on the critical point.
Assuming that the quasi-brittle to brittle transition is analogous to continuous phase
transitions power law divergence of
∆
can be expected
()µµ∆∼ −γ−,
c
(10)
which defines the γ exponent of the transition. Since the exponent ξ of the burst size
distribution is less than 2, both the first M1 and second M2 moments of burst sizes
diverge in an infinite system. This has the consequence that the average burst size is
proportional to the cuto size
∆∼∆c
, which implies the relation
/γσ=1
of the two

Fracture process of a fiber bundle with strong disorder
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critical exponents. In order to numerically verify the analytical predictions, we plotted
the average burst size
∆
as a function of the distance from the critical point
µ µ−
c
,
varying the value of
µc
until the best straight line is obtained on a double logarithmic
plot. The procedure is illustrated in figure 3(b) for the system size N = 107. A high qual-
ity power law of exponent 2 is obtained in the figure with the finite size critical point
() ()µ==
N10 1.0009 2
c7
in an excellent agreement with the analytical considerations.
The value of the finite size critical point
()µN
c
is presented in figure 3(c) for system
sizes N covering three orders of magnitude. It can be observed that
()µN
c
converges
towards 1 with increasing N and obeys the scaling law
() () /
µ µ=∞+ν−
NB
N
,
cc 1
(11)
characteristic for continuous phase transitions. Here
()µ∞=1
c
denotes the critical
point of the infinite system, and ν is the correlation length exponent of the transition.
It can be seen in figure 3(d) that the dierence
() ()µ µ−∞N
cc
decreases as a power law
of N with an exponent 1/2, which implies
ν=2
for our model.
When studying the phase transition nature of fracture phenomena the order para-
meter is typically defined in terms of the fraction of fibers which break up to the criti-
cal point of the loading process. However, for the quasi-brittle to brittle transition of
highly disordered systems this fraction is always 1 and 0 in the quasi-brittle and brittle
phases respectively, without any dependence on the distance from the critical point.
Figure 3. (a) Average size of bursts
∆
as a function of μ for system sizes covering
three orders of magnitude. (b) The average burst size
∆
of N = 107 is replotted as
a function of the distance from the critical point
c
µ
, where
1.0009
c
µ=
was used.
The straight line represents a power law of exponent 2. (c) The critical point
c
µ
of
finite size systems as a function of the number of fibers N. (d) The dierence of the
finite size critical point and that of the infinite system
N
cc
() ()µ µ−∞
as a function
of N. The straight line represents a power law of exponent 1/2.
10
102
103
104
105
< >
10-1 1
N
104
105
106
107
10
102
103
104
105
10
6
< >
10-3 10-2 10-1
c(N)-
1.0
1.005
1.01
1.015
1.02
1.025
c
(N)
104105106107
N
10-3
10-2
c(N)- c( )
10410510610
7
N
a) b)
c) d)

Fracture process of a fiber bundle with strong disorder
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J. Stat. Mech. (2016) 073211
To characterize in which phase the system is when changing the control parameter μ,
we define the order parameter of the transition
∆
n
as the average
∆
N
of the number
of bursts
∆
N
before global failure normalized by the total number of fibers N so that
/=
∆∆
nNN
. Since far below the critical point
µ µc
all avalanches are small (most
of them have size 1) the control parameter has the value
≈
∆
n1
, while it tends to zero
when approaching
µc
from below and it is zero in the brittle phase. For the continuous
quasi-brittle to brittle transition power law functional form
()µµ∼−
β
∆
n
c
(12)
is expected for
µ µ<c
, which defines the order parameter exponent β of the transition.
Figure 4 presents the scaling collapse of the order parameter obtained at dierent sys-
tem sizes assuming the scaling structure
() (( ()))
//
µµµ=Ψ−∞
βν ν
∆−
nN
NN,,
c1
(13)
where
()Ψx
denotes the scaling function. Best collapse is obtained with the critical
exponents
β=1
and
ν=2
.
6. Localized load sharing
In the case of equal load sharing studied so far, all fibers keep the same load. As the
external load increases fibers gradually break but in spite of this the acceleration
towards failure is completely missing. The qualitative explanation is that although the
load bearing cross section of the bundle decreases and the load per fiber increases, the
remaining fibers are always strong enough to ensure stability.
When the load sharing is localized (LLS) stress concentration develops around failed
regions which in turn induces spatial correlation in the breaking process [13, 21, 22,
25, 26]. It has been shown in FBMs that as a consequence, for moderate disorder the
system becomes more brittle and fails earlier at lower loads than in ELS [13, 26–28].
Figure 4. Scaling collapse of the order parameter of the transition. Rescaling the
two axes according to (12) a high quality data collapse is obtained. Inset: the order
parameter
n∆
as a function of the distance from the critical point
c
µ µ−
for the
system of N = 107 fibers. The straight line represents a power law of exponent 1.
10-3
10-2
10-1
1
10
102
103
<n >N
/
-3000 -2000 -1000 0
1000
( -
c
())N1/
104
5104
105
5105
106
5106
107
N
104
105
106
107
<n >
10-3 10-2 10-1 1
-c

Fracture process of a fiber bundle with strong disorder
11
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J. Stat. Mech. (2016) 073211
In order to see if this qualitative picture is valid when the disorder is high we carried
out LLS simulations on a square lattice of size L = 2001, equally redistributing the load
of broken fibers on their intact nearest neighbors in the lattice.
Figure 5 shows that for strong disorder the size dependence of the macroscopic
strength of the LLS bundle has the same functional form as in the ELS case, i.e.
εc
increases as a power law of N (figure 5(a)) and the μ dependence of
εc
is also consistent
with the analytic prediction of extreme order statistics (figures 5(b), (c)). The result
shows that on the macro-level the spatial correlation introduced by the localized load
sharing does not have any apparent consequence, even for the lowest disorder
→µ1
the
macroscopic strength is controlled by the strongest fibers.
Figure 6 demonstrates that the burst size distributions
()∆p
have the same trend
when the exponent μ of the disorder distribution approaches 1 as for the ELS counter-
part: power law distributions are obtained with a diverging cuto in the limit of
→µ1
.
It is interesting to note that the stress concentration around failed regions gives rise
to a higher exponent
ξ=±1.80.05
of
()∆p
which implies a somewhat lower frequency
of large size bursts compared to the ELS. The high quality data collapse of the dis-
tributions of dierent μ values in figure 6(a) was obtained with the cuto exponent
()σ=
∆0.33
.
In order to perform scaling analysis in terms of system size N simulations were
carried out on square lattices of size L = 101, 201, 501, 1001, 1501, 2001, 3151. The
correlation length critical exponent
()ν=2.55
was determined by analyzing the system
size dependence of the critical point
()µN
c
, which is highlighted in figure 6(b). The finite
size scaling of the order parameter was used to obtain the β exponent
()β=0.83
of the
quasi-brittle to brittle transition (see figure 6(c)). The average burst size
∆
was also
found to have the same diverging behavior as in ELS described by the critical exponent
()γ=1.92
(not presented in figure).
The localized redistribution of load has the consequence that fibers breaking in a
correlated avalanche form a connected cluster. In later stages of the failure process it
may occur that intact fibers get isolated so that when they break the range of load
redistribution is gradually extended until at least one intact fiber is found which then
Figure 5. (a) Critical deformation
c
ε
of LLS bundles as a function of the system
size N for several values of μ. Power law behavior is obtained. (b) The exponent
α of the size dependence as a function of the control parameter μ. (c) The
μ-dependence of α is described by a power law of exponent 1, similarly to the ELS
case.
1024
1044
1064
1084
<
c
>
105106
N
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
2
4
6
8
10
12
0.20.4 0.60.8 1.01
10
10-1 1

Fracture process of a fiber bundle with strong disorder
12
doi:10.1088/1742-5468/2016/07/073211
J. Stat. Mech. (2016) 073211
gets the load. Intact fibers along the perimeter of clusters are highly stressed since they
share the total load dropped by the fibers of the interior of the clusters. This damage
structure has the consequence that for moderate disorder the clusters are space filling
compact objects which remain small compared to the system size until failure. The
final catastrophic avalanche is typically initiated by the breaking of a perimeter fiber.
Figure 7 presents snapshots of the evolution of an LLS bundle of size L = 2001 for
µ=0.9
where avalanches are highlighted by dierent colors. It can be observed that
due to the high disorder the avalanches are not compact but they have a rather diuse
interior. At the beginning of the breaking process spreading avalanches do not aect
each other; however, as the number of broken fibers Nb increases avalanches merge and
form large broken clusters (cracks in the model). Large avalanches already occur at
early stages of the fracture process; due to their diuse structure in later stages small
avalanches may nucleate even in the internal holes of the extended ones. The degree of
damage in the bundle can be characterized by the fraction of broken fibers
/=nNN
bb
,
which increases from 0 to 1 as the loading proceeds. It is interesting to note that even
at high values of nb, where large clusters dominate the damage structure, the stability
of the LLS system is retained, which is in strong contrast to the highly brittle behav-
iour of LLS bundles with moderate disorder [13, 26, 29].
In order to quantify the evolution of the cluster structure of broken fibers during
the loading process, we determined the average value
S
of the size S of clusters as a
function of nb.
S
is defined as the ratio of the second and first moments
=∑
MS
ii
2
2
,
=∑
MS
ii1
of cluster sizes, where the largest cluster is omitted in the summation. The
value of
/MM
21
is averaged over 5000 simulations in bins of nb. The results are pre-
sented in figure 8 for four values of the μ exponent. It can be observed that for all μ
the
()Sn
b
curves have a sharp maximum, which indicates the emergence of a giant
cluster at a critical damage fraction
nb
c
. Since the failure process is dominated by the
disorder of fibers’ strength the cluster structure can be expected to be similar to the
site percolation problem on a square lattice where nb is analogous to the site occupation
probability [30]. This is confirmed by the fact that when μ decreases, i.e. the amount
of disorder increases, the critical point
nb
c
gradually shifts to the site percolation criti-
cal point
≈n0.5923
b
on a square lattice, while for higher values
→µ1
the giant cluster
Figure 6. (a) Scaling collapse of the burst size distributions obtained at dierent μ
values. (b) The critical point of finite systems
N
c()µ
was determined based on the
average bursts size. Using the scaling ansatz (11) the correlation length exponent
ν could be obtained. (c) The order parameter obeys the same scaling form (13),
which yields the β exponent.
10-5
1
105
1010
P( ) ( c- )
- /
10-5 10-3 10-1 10
/(
c
- )-1/
~-1.8
0.9999
0.999
0.995
0.992
0.99
0.98
0.95
0.92
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.110-3
10-2
c(N)- c( )
104105106107
N
10-3
10-1
10
<n >N /
-600 -400 -200 0200
400
( - c())N
1/
101
201
501
1001
1501
2001
3151
a) b) c)
L

Fracture process of a fiber bundle with strong disorder
13
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J. Stat. Mech. (2016) 073211
emerges earlier. The reason is that for low μ exponents all avalanches are small so that
in the limit of
→µ0
the disorder is so high that the fibers practically break one-by-one
and the entire breaking process can be considered as a sequence of random nucleations,
Figure 7. Snapshots of the evolving breaking process on a square lattice of
size L = 2001 with
0.9µ=
using periodic boundary conditions. Avalanches are
highlighted by randomly assigned colors. At early stages of the fracture process
((a), (b)) bursts can evolve independently of each other; however, later on the
merging of bursts dominates.
a)
c) d)
b)
Figure 8. Average size of clusters
S
as a function of the fraction of broken fibers
nb. As μ decreases the position of the maximum
nb
c
tends to critical occupation
probability pc of site percolation on a square lattice indicated by the vertical
straight line.
10-6
10-5
10-4
10-3
10-2
<S>/N
0.00.2 0.40.6 0.
81.0
n
b
0.95
0.9
0.5
0.1

Fracture process of a fiber bundle with strong disorder
14
doi:10.1088/1742-5468/2016/07/073211
J. Stat. Mech. (2016) 073211
as it is in percolation. At higher μ the lower disorder gives more room for the stress
concentration, which in turn gives rise to extended avalanches and a stronger spacial
correlation of local breakings. Since large avalanches can already be triggered at the
beginning of the fracture process (see also figure 7), large clusters can appear even at
low damage fractions, which makes the
()Sn
b
strongly asymmetric in figure 8 for high
μ values.
A more detailed picture is provided by figure 9 which shows the size distribution
of clusters p(S ) at several values of nb both below and above the corresponding criti-
cal point
nb
c
for two values of μ. Power law distributions are obtained, followed by an
exponential cuto
() (/)∼−
τ−
pSS SSexp,
c
(14)
where the cuto cluster size Sc is controlled by the value of nb. It can be observed that
Sc tends to diverge as the critical damage fraction
nb
c
is approached from both sides.
Careful scaling analysis revealed that below and above the critical point
nb
c
the exponent
τ of the power law regime has dierent values. In figures 9((b), (c)) and ((e), ( f )) we
rescaled the avalanche size distributions with powers of the distance from the critical
point
|−|nn
bb
c
, tuning the scaling exponents along the horizontal and vertical axis until
best collapse is achieved. The scaling functions can be well fitted with the exponents
τ=1.67
and
τ=2.1
(
µ=0.1
), and
τ=1.77
and
τ=2.0
(
µ=0.9
), respectively below
and above the corresponding
nb
c
(see figure 9). Clusters of broken fibers are generated by
avalanches. At the beginning of the fracture process the merging of the clusters of indi-
vidual avalanches is practically negligible—hence, below the critical damage fraction
nb
c
the value of the exponent τ of the cluster size distribution p(S) should be close to the
exponent ξ of the avalanche size distribution
()∆p
. Above
nb
c
the merging of avalanches
dominates, which gives rise to a steeper cluster size distribution with a τ greater than
ξ. The values of τ determined numerically slightly depend on the control parameter μ
falling in the range
τ=1.65
–1.95 below and
τ=1.95
–2.1 above
nb
c
, which is consistent
with the above arguments.
The good quality data collapse of p(S) also implies that the cuto cluster size Sc has
a power law dependence on the distance from the critical point
nb
c
/
∼| −|σ−
Sn
n
.
bb
c
c1
S
(15)
The value of the cuto exponent
σ
S
falls in the range 0.25–0.5, depending on the value
of the control parameter μ.
7. Discussion
The presence of disorder makes the fracture process jerky where damage accumulates in
intermittent avalanches that can be recorded in the form of crackling noise. Forecasting
technologies of global failure of engineering construction or natural catastrophes like
landslides, collapse of rockwalls, earthquakes, volcanic eruptions strongly rely on iden-
tifying signatures of the imminent failure based on the acceleration of crackling signals.
In the present paper we showed in a fiber bundle model that this very important eect

Fracture process of a fiber bundle with strong disorder
15
doi:10.1088/1742-5468/2016/07/073211
J. Stat. Mech. (2016) 073211
of disorder is limited, i.e. when the disorder gets high fracture becomes unpredictable
again. Heavy-tailed distributions of the failure thresholds of material elements give
rise to a homogeneous fracture process which does not exhibit any sign of acceleration.
Reducing the amount of disorder, the system undergoes a continuous phase transition
to perfectly brittle failure, without restoring the ability of forecasting. In the mean field
limit of the fiber bundle models (ELS) we determined analytically and numerically the
critical exponents of the transition. On the macro-level the fracture strength of the
bundle proved to increase with the system size—which is the direct consequence of
the heavy-tailed distribution of fibers’ strength defined over an infinite support. For
practical purposes the case of a large but finite upper cuto of local strength is also
of high importance. Controlling the cuto value a crossover is expected between the
decreasing size dependence typical for moderate disorder FBMs and the increasing one
revealed by the present study. The crossover is accompanied by the changing degree of
stationarity of the time series of breaking bursts, which addresses an interesting ques-
tion for failure forecast methods, as well.
In order to clarify how the inhomogeneous stress field developing around failed
regions changes the evolution of the fracture process, we also studied the limit of local-
ized load sharing where the load of a broken fiber is equally redistributed over its intact
nearest neighbors. Even in this case fat tailed distributions proved to ensure the domi-
nance of disorder over spatially correlated stress enhancements: the size distribution of
Figure 9. Size distribution of clusters p(S) at several damage fractions nb for
0.1µ=
and
0.9µ=
in the upper and lower rows respectively. In (a) and (d )
distributions are presented covering the entire range of the damage fraction nb.
Data collapse of the curves is presented separately below (b), (e), and above (c),
( f ) the corresponding critical point
nb
c
:
n0.598
b
c=
for
0.1µ=
and
n0.548
b
c=
for
0.9µ=
. The legend used in (a) and (d) is the same as in ((b), (c)) and ((e), ( f ))
respectively.
10-12
10-10
10-8
10-6
10-4
10-2
p(S)
10 102103104105106
S
10-9
10-7
10-5
10-3
10-1
10
103
p(S)(nb
c-nb)- / S
10-2 10-1 110102
S/(n
b
c-n
b
)-1/ S
0.01
0.1
0.2
0.3
0.4
0.5
0.52
~S-1.77
10-3
1
103
106
109
1012
p(S)(nb-nb
c)- / S
10-6 10-5 10-4 10-3 10-2 10-1 1
S/(n
b
-n
b
c)-1/ S
~S-2.0
0.54
0.56
0.58
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
10-13
10-11
10-9
10-7
10-5
10-3
10
-1
p(S)
10 102103104105106
S
10-7
10-4
10-1
102
105
p(S)(nb
c-nb)- / S
10-3 10-2 10-1 1
S/(nb
c-nb)-1/ S
0.01
0.1
0.2
0.3
0.4
0.5
0.52
0.54
0.56
0.59
~S-1.67
10-3
1
103
106
109
p(S)(nb-nb
c)- / S
10-5 10-4 10-3 10-2 10-1 1
S/(nb-nb
c)-1/ S
~S-2.1
0.6
0.65
0.7
0.75
0.8
0.85
0.90
0.95
)c)b)a
)f)e)d
nb/N
nb/N
nbnN/ b/N

Fracture process of a fiber bundle with strong disorder
16
doi:10.1088/1742-5468/2016/07/073211
J. Stat. Mech. (2016) 073211
avalanches has a power law functional form with an exponent close to the mean field
value. This is in strong contrast to what is usually found in LLS FBMs, i.e. a very rap-
idly decreasing distribution of avalanche sizes is usually obtained, which is described
either by a power law of exponent 9/2 or by an exponential. Additionally, the continu-
ous nature of the quasi-brittle to brittle transition remains in LLS, although the critical
exponents have somewhat dierent values in the two limiting cases of load sharing.
For LLS fiber bundles the spatial structure of damage strongly resembles the site per-
colation problem; deviations due to the presence of spatial correlations are obtained
in the vicinity of the quasi-brittle to brittle phase transition. Although, for the lowest
disorder
→µ1
the avalanche statistics and cluster structure of the LLS system shows
the increasing role of spatial correlations, the macroscopic strength of the bundle is still
consistent with the extreme order statistics obtained in ELS. The reason is that the
time series of avalanches still exhibits a high degree of stationarity with the absence
of a relevant acceleration and a catastrophic avalanche so that the strongest fibers can
control macroscopic failure.
Based on our analytical and numerical results we conjecture that fat tailed strength
distributions determine a unique universality class of the quasi-static fracture of fiber
bundles. Recently, it has been demonstrated that 3D printing technology can be used
to produce materials with finely tuned structural properties [31, 32]. In the near future
it may also become possible to realize experimentally the limit of high disorder studied
here.
Acknowledgments
We thank the projects TAMOP-4.2.2.A-11/1/KONV-2012-0036. This research was sup-
ported by the European Union and the State of Hungary, co-financed by the European
Social Fund in the framework of TMOP-4.2.4.A/2-11/1-2012-0001 National Excellence
Program.
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