Effect of discontinuity in threshold distribution on the critical behaviour of a random fiber bundle

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by  Uma Divakaran

Effect of discontinuity in threshold distribution on the critical behaviour of a random
fiber bundle
Uma Divakaran∗ and Amit Dutta†
Department of Physics, Indian Institute of Technology Kanpur - 208016, India
7 (Dated: February 6, 2008)
0
The critical behaviour of a Random Fiber Bundle Model with mixed uniform distribution of
0
threshold strengths and global load sharing rule is studied with a special emphasis on the nature
2
of distribution of avalanches for different parameters of the distribution. The discontinuity in the

n threshold strength distribution of fibers non-trivially modifies the critical stress as well as puts a
a restriction on the allowed values of parameters for which the recursive dynamics approach holds
J
good. The discontinuity leads to a non-universal behaviour in the avalanche size distribution for

5 smaller values of avalanche size. We observe that apart from the mean field behaviour for larger
2 avalanches,anewbehaviourforsmalleravalanchesizeisobservedasacriticalthresholddistribution
isapproached. Thephenomenologicalunderstandingoftheaboveresultisprovidedusingtheexact
]
analytical result for the avalanche size distribution. Most interestingly, the prominence of non-
h
universalbehaviour in avalanchesize distribution dependson thesystem parameters.
c
e
m PACSnumbers: 64.60.Ht,81.05.Ni,46.50.+a,62.20.Mk
-
t
a I. INTRODUCTION Hansen[6] studied the avalanche size distribution D(∆)
t
s ofanavalancheofsize∆inaRFBMwiththeglobalload
t. Breakdownphenomena in nature has capturedthe at- sharing(GLS)scheme,inwhichtheadditionalstressdue
a to a broken fiber is distributed equally to the remaining
m tentionofscientistsforyears[1]. Astudyofthisphenom-
intactfibers. Theyestablishedauniversalpower-lawdis-
ena plays a major role for the prediction of failure and
- tributioninthe large∆limit givenasD(∆)∝∆−ξ with
d design of materials and structures. One of the paradig-
ξ =5/2.
n matic model mimicking the fracture processesis random
o fiber bundle model (RFBM) which is simple yet subtle
c
enoughtocapturetheessentialphysicsofthebreakdown
[
phenomena.

2 1
Random fiber bundle models [2, 3, 4, 5] have been
v σ σ
3 studied extensively in recent years. Typically a RFBM 1−( 2− 1 )
consists of N parallel fibers with randomly distributed
2
2 thresholdstrength(σth) takenfrom a givendistribution. )
8 If the stress generateddue to anexternalforce is greater h
t
0 thanσ ofafiber,itbreaks. Thedynamicsofthemodel σ
th (
6 isinitiatedbyapplyingasmallexternalforcejustenough ρ
0
to break the weakest fiber present in the bundle. The 0 σ σ
/ 1
t load carried by this broken fiber is shared amongst the 1 2
a
remainingintactfibersfollowingaloadsharingrulecaus- σ
m
th
ing further failures. When no further failure takes place,
-
d the external force is once again increased quasistatically FIG. 1: Mixed Uniform Distribution
n to break the weakest intact element present in the bun-
o dleandtheprocesscontinuestillthebundlebreaksdown
RFBM with threshold strengths which are continuous
c completely atanexternalstresscalledthe criticalstress.
: and are uniformly distributed between 0 to σ and also
v Even though the threshold distribution of real materials 1
betweenσ to 1[3, 4],havebeenstudiedseparately. But
i maynotbe knownexactly,intheoreticalmodels the dis- 2
X what happens when two such bundles are merged is not
tributions are usually approximated by either a uniform
r distribution or a Weibull distribution [3]. known, especially when σ1 <σ2 such that there exists a
a
discontinuity in the threshold strength distribution. In
The avalanche size is defined as the number of bro-
thispaper,weinvestigatetheroleofsuchadiscontinuity
ken fibers between two successive loadings. The distri-
onthe criticalbehaviourofaRFBM.Thedistributionof
bution of avalanche size turns out to be a key factor in
threshold strength of fibers used in the present work is
charaterising any breakdown phenomena. Hemmer and
given as (See Fig. 1)
1
ρ(σ ) = 0<σ ≤σ
th 1−(σ −σ ) th 1
∗Electronicaddress: [email protected] 2 1
†Electronicaddress: [email protected] = 0 σ1 <σth <σ2
2
1
= σ ≤σ ≤1. (1) allowedvalues ofthe parameterσ1 as shownbelow. Any
1−(σ2−σ1) 2 th value of σ1 > f leads to a value of σ2 smaller than σ1
The discontinuity as defined above introduces dilution which is not an acceptable distribution (see Eq. 1).
in the model in the sense that two types of fibers sepa- We now define Ut as the fraction of unbroken fibers
rated by a gap in their threshold distributions coexist in after a time step t. Then the redistributed stress at the
the same bundle. It is shown below that this disconti- instant t is σt = F/Nt = σ/Ut where the applied force
nuity plays a crucial role in the dynamics of the model. F = Nσ and Nt = NUt. The recurrence relations be-
Here, a fraction f of fibers belong to the weaker thresh- tween Ut,Ut+1 and between σt,σt+1 for the GLS are ob-
old distribution with strengths uniformly lying between tained as [4]:
0 and σ1 (Class A) whereas the remaining fraction of σ
fibers have stronger threshold strengths between σ2 and Ut+1 = 1−P(σt)=1−P(U )
1 (Class B). Clearly, (σ2−σ1) is the measure of the dis- σ σ t
continuity which vanishes in the limit of purely uniform and σt+1 = U = (1−P(σ )) (3)
distribution[4]. t+1 t
In a recent paper, Pradhan, Hansen and Hemmer [7] where P(σ ) is the fraction of broken fibers with the re-
t
showed that for a bundle which is close to the complete distributed stress σ , and is given as
t
breakdown(i.e.,imminentfailure),acrossoverinξ from
a value 5/2to 3/2 is observedwhen the threshold distri- σt
P(σ )= ρ(σ )dσ .
bution approaches the critical distribution. Critical dis- t Z th th
0
tribution in their case is the distribution in which the
lowest threshold of the remaining intact fibers is equal Thisdynamicspropagatesuntilnofurtherbreakingtakes
to half that of the strongest. This crossover has also place. It should be emphasised that the initial load is so
been observed with other load sharing rules[8]. We are small that the redistributed stress is always less than σ2
however interested in looking at the distribution of to- and therefore fibers from class B cannot fail. Thus to
tal avalanche size and a crossover from a power-law be- initiate the breaking of class B fibers, the redistributed
haviour with ξ =5/2 to a non-mean field, non-universal stress at a later time t must exceed σ2. The fixed point
behaviourforsmaller∆isobservednearacriticaldistri- solution for U (= U∗) and σ(= σ∗) at which no further
bution. Thisinterestingbehaviourisduetothepresence failure takes place can be obtained using the standard
of class A fibers. In the present model, the critical dis- technique of solving the above recursive relations. By
tributioncorrespondsto σ2 =0.5. We emphasize thatin substituting P(σt) in Eq. (3) we get
Ref.[7],theproximityofx (thethresholdoftheweakest
0 σ
intact fiber) to the the critical threshold (=0.5) leads to Ut+1 = 1−P( )
U
the crossover behaviour whereas in the present case it is t
σ 1 σ
the proximity of σ2 to the criticalvalue (0.5) which is at = 1−[1−(σ1−σ ) + 1−(σ −σ )(U −σ2)].
the rootof the observedcrossoverbehaviour. In a sense, 2 1 2 1 t
σ is playing a role analogous to x in our model.
2 0 At the fixed point,
The paper is organized as follows: We have already
introduced the model above. The critical stress and ex- σ 1 σ
U∗ =1−[ 1 + ( −σ )]
ponents are obtained in Sect. II(A) with a special em- 1−(σ −σ ) 1−(σ −σ ) U∗ 2
2 1 2 1
phasis on the avalanche size distribution in Sect. II(B).
The concluding remarks are presented in Sect. III. giving the following stable fixed point solutions
1 σ
U∗ = [1+ 1− ] and
II. RESULTS AND DISCUSSIONS 2(1−(σ −σ )) r σ
2 1 c
1 1 σ
A. Critical stress and exponents σ∗ = − 1− with (4)
2 2r σ
c
1
To study the dynamics of failure of fibers, we use the σc = 4[1−(σ −σ )]. (5)
recursive dynamics approach [4]. If a fraction f of the 2 1
totalfibers belongto ClassAandthe remaining1−f to If the external applied stress is less than σ , the system
c
class B, then the uniformity of the distribution demands reaches a fixed point. For σ > σ , the bundle breaks
c
1 σ1 down completely as both the U∗ and σ∗ become imagi-
f = dσ; nary. Equation (4) suggests that the redistributed stress
1−(σ −σ )Z
2 1 0 σ∗ attains the maximumvalue (=0.5)atσ . The critical
f c
so that, σ1 = 1−f(1−σ2). (2) stress of the mixed model varies with the gap (σ2−σ1)
and reduces to the value σ = 1/4 for the uniform dis-
c
Theaboveequationprovidesarelationshipbetweenf, tribution as (σ −σ )→0. In deriving Eq. (4), σ is as-
2 1 t
σ andσ andatthe same time puts a restrictiononthe sumedtobegreaterthanσ whiletheappliedstresswhen
1 2 2
3
plotted againstthe redistributed stress shows a disconti- We study the avalanche size distribution numerically
nuity at σ . Beyond σ , the external force is increased by using the method of breaking of the weakest fiber[6].
1 1
to break the fiber with threshold strength σ , i.e., the Inthesimulation,thefibersarearrangedinanincreasing
2
gap in the threshold distribution is also reflected in the orderoftheir thresholdstrengths. An externalforcesuf-
constitutive behaviour of the model. ficient to break the weakestfiber is applied and the load
The discontinuity on the other hand imposes some re- duetothebreakingofthisfibergetsredistributedamong
strictions on the parameters for this calculation to be the remaining intact fibers following the GLS scheme.
valid. Since the maximum value of the redistributed The number of failed fibers for a fixed external load is
stress is equal to 0.5, σ must be less than 0.5 so that recorded till the dynamics reaches a fixed point. There-
2
some fibers from class B also fail at the critical point. after,theexternalloadisincreasedfurtherandtheabove
The condition σ < 0.5, eventually restricts the value process is repeated till the critical stress is reached.
2
of critical stress to be less than 0.5 for any chosen dis- Following interesting observations (Fig. 2) are clearly
tribution. However, σ = 0.5 is a limiting case when highlighted. (i) For the cases 1, 2 and 5 of the table,
2
the redistributed stress at the critical point marginally the avalanche size exponent is 5/2. (ii) For the cases
reaches class B fibers. In short, we must have 3,4 and 6, there is an apparent power law behaviour for
smaller ∆ with the exponent which is found to depend
σ1 <f,σ2 <0.5 . on the system parameters. In the examples chosen here
the exponent happens to be close to 3. For larger ∆,
OnecandefineanorderparameterOassociatedwiththe however, we retrieve the universal mean field behaviour
transition [4] as shown below: with ξ = 5/2. Also, (iii) an increase in the region with
ξ ≈ 3 is seen as σ → 0.5. These observations establish
2
the following: (i) there is a non-universal behaviour of
O =2[1−(σ −σ )]U∗−1=(σ −σ)1/2 =(σ −σ)β.
2 1 c c D(∆) in the small ∆ limit, (ii) there is a crossover to
the universal behaviour in large ∆ limit, and (iii) the
The order parameter goes to 0 as σ → σ following a
c crossoverbehaviour is prominent as σ →0.5.
power law (σ − σ)1/2 . Susceptibility can be defined 2
c The above mentioned results can be explained by ex-
as the increment in the number of broken fibers for an
tendingtheanalyticalresultfortheavalanchesizedistri-
infinitesimal increase of load. Therefore,
butionobtainedbyHemmerandHansen[6]tothemixed
dm model. Thegeneralexpressionforthe avalanchesizedis-
χ= where m=N[1−U∗(σ)]. tribution with GLS is given as
dσ
D(∆) ∆∆−1 xc xρ(x)
Hence, = dxρ(x)(1− )
N ∆! Z Q(x)
0
χ∝(σ −σ)−1/2 =(σ −σ)−γ.
c c ∆−1
xρ(x) xρ(x)
exp(−∆ ) (6)
The exponents β and γ stick to their mean field values (cid:20)Q(x)(cid:21) Q(x)
[4, 5] i.e.β = γ = 1/2 . We conclude that though the
where x is the redistributed stress and Q(x) is the frac-
discontinuity alters the critical stress, the critical expo-
tion of unbroken fibers at x. The upper limit of the
nents remainunaltered. It is to be noted that the model
integration (x ) is the redistributed stress at the criti-
reduces to the already obtained results in various lim- c
cal point. The right hand side of Eq. (6) is broken into
its. For example, if class A fibers are absent (f=0),
two parts, D (∆) and D (∆) for the mixed model with
σ = 1/4(1−σ ) and an elastic to plastic deformation 1 2
c 2 any allowed values of σ and σ (where ρ(x) = ρ =
is observed [4]. 1 2
1/(1−σ +σ )):
2 1
As long as the redistributed stress is restricted to the
class A fibers,
B. Avalanche size distribution
1−σ +σ −x
2 1
Q(x)= ,
Let us now focus on the avalanche size distribution 1−σ2+σ1
exponent ξ. Below is shown some of the allowed distri-
we have
butions which satisfy the above mentioned restrictions.
∆∆−1 1 σ1 1−σ +σ −2x
2 1
D (∆)= dx( )
Case f σ1 σ2 σc 1 ∆! 1−σ2+σ1 Z0 x
1 0.10 0.08 0.28 0.31 x x ∆
×exp(− ) .(7)
2 0.20 0.19 0.24 0.26 (cid:20)1−σ +σ −x 1−σ +σ −x (cid:21)
2 1 2 1
3 0.20 0.15 0.40 0.33 Whentheredistributedstressbelongstothesecondblock
4 0.30 0.25 0.42 0.30 (class B), we have
5 0.30 0.29 0.33 0.26 1−x
Q(x)= , and
6 0.40 0.35 0.47 0.29 1−σ +σ
2 1
4
0 0
case1 case4 (a) case1
(a) (b) −2 log10D2(∆)
−2 −5x −5x log10D1(∆)
2 2
−4
−4
−6
0
)
∆ N−6 (b) case4
(D ) −2 log10D2(∆)
log10 0 case6 Integration ∆(D0N−4 log10D1(∆)
(c) simulation og1
−2 (d) l −6
−5x
2 0
(c) case6
−3x
−4 −2 log10D2(∆)
log D(∆)
10 1
−4
−6
−6
0 0.5 1 1.5 2 0 0.5 1 1.5 2
0 0.5 1 1.5 2
log ∆ log10∆ log ∆
10 10
FIG. 2: Fig. (2a) corresponds to case1 of the table where FIG. 3: Figs. a, b and c present the numerical integration
no crossover is observed. (2b) shows a crossover from a non- resultsshowinganincreasein∆c(definedinthetext)asσ2 →
universal behaviour to the universal behaviour with ξ = 5/2 0.5. Thesymbolcorrespondstolog10D(∆)/N=log10(D1(∆)+
as∆isincreased(case4). Fig.(2c)showsanincreaseinregion D2(∆)) whereas log10D1(∆) and log10D2(∆) are drawn to
with non-universal behaviour (case6). A line with slope -5/2 compare their relative magnitudes.
is drawn in each figure for comparison. In Fig. (2c), a line
withslope-3isalsoshowntoindicatetheapparentpower-law
behaviour. Fig. (2d) compares the simulation and numerical
for three differentcases areshowninFig. 3. Incidentally
integration data for case4 which match identically with each
in the examples presented here, ∆ increases with σ .
other. c 2
If we look at a particular critical distribution (Fig. 4), a
crossoverfromanapparentpower-lawbehaviourofD(∆)
∆∆−1 1 0.5 1−2x with exponent ξ close to 3 for small ∆ to a universalbe-
D2(∆)= ∆! 1−σ +σ Z dx( x ) haviour with ξ =3/2 is observed.
2 1 σ2 We now discuss the numerical integration and simula-
∆
× x exp(− x ) . (8) tionresultsinthelightofEqs.(7)and(8). Withachange
(cid:20)1−x 1−x (cid:21) ofvariablex→x/(1−σ +σ −x),wecanrewriteEq.(7)
2 1
in the following form
AcloseinspectionshowsthatEq.(8)correspondsto a
situation where class A fibers are absent and the results ∆∆−1 xm 1−x
match exactly with Pradhan et.al [7] where a crossover D1(∆)= ∆! Z x(1+x)2e(logx−x)∆dx (9)
inthe avalanchesizeexponentfrom5/2to avalue 3/2is 0
observedasσ →0.5. Here,thepresenceofclassAfibers
2 where x = σ /(1 −σ ). The argument of the expo-
necessitates the simultaneous study of Eqs. (7) and (8) m 1 2
nential term has a maximum at x = 1 which is outside
forthetotalavalanchesizedistribution. Fordistributions
the range of integration and hence the saddle point in-
whichareclosertothecriticaldistribution,Eq.(8)yields
tegration method can not be applied [9]. Expanding the
exponent ξ = 3/2 for smaller ∆, but the contribution
term 1/(1+x)2 in a power series and then using the in-
from the Eq. (7) modifies this small ∆ behaviour.
complete gamma function[10], we arrive at the following
The results obtained by numerically integrating
result (see the Appendix)
Eqs. (7) and (8) match identically with the numerical
simulation results [Fig. (2d)]. We observe a sharp fall of
e(1−xm)∆ ∞
Dof1s(m∆a)l[lEerq.s7iz]ewshaircehcpoonintrtisbtuotethdemfaacitntlyhabtythteheavbarlaeankchinegs D1(∆)= ∆3/2 x∆mXq=0(−1)q(q+1)xqm×
ofclassAfibers(Fig.3). The pointofintersectionofthe
∞ (x ∆)k
twointegralsyields∆cwhichisthevalueoftheavalanche ( m
sizewherethe crossovertakesplace. The variationof∆c Xk=0(∆+q)(∆+q+1)...(∆+q+k)
5
0
exponents are unchanged, there is a non-trivial change
−1 in the burst avalanche distribution behaviour where dis-
continuity leads to a non-universal, non-mean field be-
−2
haviour for small ∆. We would like to emphasise that
−3 non-universalitybecomesprominentonlywhenσ2 →0.5.
Forlarge∆limit,thebehaviourishoweveruniversaland
−4
∆) mean field. If f =0 or σ2 <<0.5, the non-universal be-
N
(D −5 haviour completely disappears. The non-universality in
0 D(∆) is also seen for other distributions[9]. The beauty
og1−6 of our model is that the non-universal behaviour is tun-
l
−7 ablewith the systemparametersandthere is a crossover
fromnon-universalityto universalityinthe limit oflarge
−8
∆. One should also note that the imminent failure of
−9 thebundle(i.e.,finalstagesofthebreakdownprocess)is
0 0.5 1 1.5 2 2.5
the same as in Ref.[7] because the effect of class A fibers
log ∆ essentially vanishes in that limit.
10
Acknowledgments
FIG. 4: Avalanche size distribution for a particular critical
threshold distribution with f = 0.4. Thick line corresponds The authors thank S. Banerjee, P. Bhattacharyya,B. K.
to a slope of -3 and dotted line to -3/2. The asymptotic Chakrabarti, Y. Moreno S. Pradhan and S. K. Rai for
behaviourclearly is D(∆)∼∆−3/2. useful help and suggestions.
Appendix
− ∞ xm(xm∆)k ). (10) In this appendix, we shall indicate how to arrive at
(∆+q+1)(∆+q+2)...(∆+q+1+k) Eq. (10) of the text starting from Eq. (9). Equation (9)
Xk=0 can be written as
The leading behaviour of the infinite series (10) is e∆ xm 1−x
∆−5/2e(1−xm)∆x∆m which justifies the non-universality D1(∆)= ∆3/2 Z (1+x)2 x∆−1 e−∆xdx
0
observed in numerical simulations in the small ∆ limit.
The question therefore remains why does the non- xm 1−x
universalbehaviourbecomeprominentasσ →0.5. The Let f(∆) = x∆−1 e−∆xdx
2 Z (1+x)2
behaviour of D (∆) [Eq. (8)] for smaller ∆(<< ∆ ) is 0
at the root of 2this. When σ2 → 0.5, D2(∆) goecs as = xm(1−x) x∆−1 e−∆x( ∞ (−1)q(q+1)xq)dx
∆−3/2 andhence the contributionfromD (∆)wins over Z
1 0 Xq=0
to produce a prominent non-universal behaviour. The
small ∆ behaviour of D(∆) is therefore non-mean field = ∞ (−1)q(q+1) xm(1−x)x∆+q−1 e−∆xdx.
and non-universal when σ2 →0.5. On the other hand if Xq=0 Z0
σ <<0.5, x is relatively smaller and the contribution
2 m
from class A fibers decays very fast as ∆ increases and The integral can be further written as
one observesa mean-field universalbehaviour almostfor xm xm
the entire range of ∆. We also observe a non-powerlaw x∆+q−1 e−∆xdx− x∆+q e−∆xdx
Z Z
behaviour for small ∆ when σ is very small and σ is 0 0
1 2 1 xm∆
close to criticality as is expected from the analytical re- = y∆+q−1 e−ydy
sult. However for a given σ (∼ 0.5), larger x leads to ∆∆+q Z
2 m 0
a non-universalbehaviour up to larger∆. The crossover 1 xm∆
value∆ isthusroughlygivenbythevalueof∆forwhich − y∆+q e−ydy.
c ∆∆+q+1 Z
D (∆)crossesoverfroma∆−3/2 behaviourto∆−5/2 be- 0
2
haviour.
Using the power series expansion of Incomplete Gamma
Function [10]
III. CONCLUSIONS 1 ∞ (x ∆)∆+q+k
= [e−xm∆ m ]
∆∆+q (∆+q)(∆+q+1)...(∆+q+k)
In conclusion, we have studied a mixed fiber bun- Xk=0
1
dle with a discontinuous but uniform threshold distri- − ×
∆∆+q+1
bution and GLS. Discontinuity leads to a functional de-
pendence of the critical stress on the system parame- [ e−xm∆ ∞ (xm∆)∆+q+1+k ]
taellroswσe1d,vσa2luaensdoffthaensdepaalrsoamimetpeross.esAlrtehsotruigchtiotnhsecornititchael kX=0(∆+q+1)(∆+q+2)...(∆+q+1+k)
6
Rearranging terms, we get ∞ x (x ∆)k
− m m ).
(∆+q+1)(∆+q+2)...(∆+q+1+k)
∞ Xk=0
f(∆)=e−xm∆x∆ (−1)q(q+1)xq ×
m m
Xq=0
∞ (x ∆)k
m
( i.e., Eq. (10).
(∆+q)(∆+q+1)...(∆+q+k)
Xk=0
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