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HAL Id: hal-02130595

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Thermodynamics of fiber bundles

Steven R. Pride, Renaud Toussaint

To cite this version:

Steven R. Pride, Renaud Toussaint. Thermodynamics of fiber bundles. Physica A, 2002, 312 (1-2),

pp.159-171. �10.1016/S0378-4371(02)00816-6�. �hal-02130595�

(2)

arXiv:cond-mat/0209131v1 [cond-mat.stat-mech] 5 Sep 2002

Steven R. Pride

a,

, Renaud Toussaint

b

a

Géos ien es Rennes, Université de Rennes1, 35042 RennesCedex, Fran e

b

Dept. of Physi s, University of Oslo,P.O.Box 1048 Blindern, 0316Oslo, Norway

Abstra t

A re ent theory that determines the properties of disordered solids as the solid

a umulates damage isapplied to thespe ial ase ofber bundles withglobal load

sharing andis shownto be exa t inthis ase. Thetheorypostulates thatthe

prob-ability of observing a given emergent damage state is obtained by maximizing the

emergententropyasdenedbyShannonsubje ttoenergeti onstraints.Thistheory

yields the known exa t results for theber-bundle model with global load sharing

andholdsforanyquen hed-disorder distribution.Itfurtherdeneshowtheentropy

evolves asa fun tion of stress,and shows denitively how the on epts of

temper-ature and entropy emerge in a problem where all statisti s derive from the initial

quen hed disorder. Apreviously unnoti edphasetransitionisshownto existasthe

entropygoesthroughamaximum.In general,thisentropy-maximumtransition

o - urs at a dierent point in strain history than the stress-maximumtransition with

thepre iselo ationdependingentirely on thequen hed-disorder distribution.

Key words: FiberBundles, Entropy Maximization,Phase Transitions

PACS: 46.50.+a,46.65.+g, 62.20.Mk, 64.60.Fr

1 INTRODUCTION

Theber-bundlemodelwithgloballoadsharingisasimplemodelforfailurein

tensionintrodu edalmost80years ago[13℄and havingre eived onsiderable

attention and extensions over the past 15 years [419℄ . Although this model

may have little pertinen e toreal brous systems su h as a rope breaking in

tension,it is ofinterestbe auseit possesses exa t analyti al properties.

Correspondingauthor. Fax: +33 223 23 6090

Email addresses: Steve.Prideuniv-rennes1.fr (Steven R.Pride),

(3)

terminingthepropertiesofadisorderedsolidthatisa umulatingirreversible

damage due to ra king under stress. The ensembles in this theory are

re-atedby onsideringdierentrealizationsof thequen hed disorderinthe lo al

breaking strength of the material. Su h realizations are made either for the

system as awhole, orof more pertinen e to real systems, by dividing a given

system into smaller mesovolumes and letting ea h mesovolume orrespond

toadierentrealizationof the quen hed disorder. The ensembles soobtained

have nothing to dowith thermal u tuations (mole ular dynami s). Given a

system withsu hquen hed disorder, ourtheory determines theprobability of

emergent ra k states by maximizing Shannon's measure of disorder subje t

to onstraints omingfrom the energeti s of the fra ture pro ess.

Inthepresentpaper, weapply thistheorytothespe i problemofber

bun-dleswith globalloadsharingand demonstratethatfor anyquen hed-disorder

distribution, ityields the known exa t results. Furthermore, apreviously

un-noti edphase transitionis demonstrated toexistwhereentropy goesthrough

amaximum.Thisphasetransitionisdistin tfromthewell-knownstress

maxi-mumtransitionandwasnoti edinthepresenttheorybe auseoftheprominent

role played by entropy.

However, the importan e of our theory is not that it yields a new result in

this old model, but that it applies and yields analyti al results about phase

transitions for any quasi-stati damage model; albeit, for models involving

ra k intera tions, approximate treatment of the fun tional integrations may

berequiredtoobtainanalyti alresults(su hasrenormalizationormean-eld

approximations).Usingourtheory,were entlytreatedtheproblemofhowthe

me hani alpropertiesofro ks hangedueto ra ksarrivingandintera tingin

ompressiveshear[2022℄.Weanalyti allydemonstrated thatthelo alization

transitionobserved in experiments isa se ond-order phase transitionwhen it

o urs.

2 THERMODYNAMICS

Ourtheory wasoriginallydeveloped foradisorderedsolid undertheinuen e

of astress tensor.Forberbundles, amu h simplers alartheory appliesand

soin this se tion, the formalismis rederived in this simplied ontext.

A ber bundle is depi ted in Fig. 1. A olle tion of

N

bers are stret hed between two rigid supports. One support is held xed, while the other is free

to displa e. A load

F

N

is applied to the bundle through the free support so thatthe bers are inastateof tension.Inthis paper, theload

F

N

willalways benormalizedby

N

todenetheoverall tension

τ = F

N

/N

.Ea hnon-broken

(4)

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1111111111111111111111111111111

1111111111111111111111111111111

1111111111111111111111111111111

1111111111111111111111111111111

1111111111111111111111111111111

0000000000000000000

0000000000000000000

0000000000000000000

0000000000000000000

1111111111111111111

1111111111111111111

1111111111111111111

1111111111111111111

F

N

Fig. 1.A bundle of

N

bers stret hed between two rigid supports with a load

F

N

applied throughthefree support.

berinabundlehasthe samelength

L

.If,when

τ = 0

,thislengthis

L

o

,then the measure of strainis

ε = L/L

o

− 1

.Experimentsmay be performedonthe bundleeither by ontrolling

τ

or

ε

.

Ea hofthebershasthe sameYoung'smoduluswhi histaken tobeunityso

thatthe axialstrain

ε

of ea hberisidenti al tothetensioninthe ber.The

N

bers have strengths

ε

1

,

ε

2

,...

ε

N

whi h are independent random variables sampled from a distribution

p(ε)

, whose umulative distribution is dened

P (ε) =

R

ε

0

p(x)dx

. As the strain

ε

of the bundle is in reased, the individual bers willbreak on etheirtension(strain)getstotheir xedstrength

thresh-old. The assumption of global load sharing is that when one of the bers

breaks atxed load, allthe other bers willextend by the sameamountthus

in reasing the tension in ea h of the surviving bers so that the load as a

whole is always supported entirely by the surviving bers. All of this denes

the berbundle modelwith globalload sharing. Of interest are the

me hani- alpropertiesof su h bundles as averaged over allpossible realizationsof the

berstrengths.

We rst need to know the probability

p

j

of observing one of the realizations to be in a parti ular state

j

of damage when the ensemble as a whole is at an applied strain

ε

. In the ber bundle model, a damage state

j

is dened by whi h of the

N

bers are broken. One ould dene

j

using a lo al order parameter that is1 if a beris inta t and 0 if the ber isbroken.

Our theorypostulates that thefra tion

p

j

of allrealizationsobserved tobein state

j

is obtained by maximizingShannon's measure of disorder

S = −

X

j

p

j

ln p

j

(1)

subje tto onstraints.Su h onstraintsmustinvolvetheindependentvariables

of

S

. To identify the independent variables, we onsider how both

S

and the average energy inthe ensembleof bundles hange asthe strainis in reased.

When

ε

in reases to

ε + dε

, there is both a work arried out in reversibly stret hing the bers and an additional work arried out due to irreversible

(5)

realizations of the disorder) willbe ledout of their urrent damagestate and

into state

j

, while others that were in state

j

will transfer to still dierent states. Ifthere is adieren e inthe numberof members entering and leaving

state

j

, therewillbea hange

dp

j

inthe o upationprobabilityof state

j

and su h hangesare what ause Shannon'sdisorder measure

S

to hange.

Theaverageenergydensity(averageenergynormalizedby

N

)intheensemble isgivenby

U =

P

j

p

j

E

j

.Here,

E

j

istheenergydensityrequiredto reatestate

j

at imposed strain

ε

and as averaged over all members that have been led to state

j

. Depending on the breaking strengths of a given realization, the work performed in arriving at state

j

an be dierent. It is through

E

j

that alldependen e onthequen hed-disorderdistributionentersthe problem.The

hange that o urs when

ε

in reases to

ε + dε

is

dU =

X

j

E

j

dp

j

+

X

j

p

j

dE

j

.

(2)

The rst term is the energy expended in hanging the disorder over the

ol-le tion of realizations. It is thus proportionalto the disorder hange and an

be written

T dS =

X

j

E

j

dp

j

.

(3)

The se ond term iswritten

f dε =

X

j

p

j

dE

j

.

(4)

and represents both the reversible stret hing energy in those members that

did not experien e breaks during the deformation in rement, as well as the

irreversible energy hanges due to all the breaks that did not result in a net

hange inthe o upation numbers of ea h state.

This de omposition of the energy in rement an be thought of as follows.

When breaks o ur throughout the ensembleof realizations as

ε

in reases to

ε+dε

,thereisaowofmembersbetween thestates.Thisowinvolvesenergy hanges due to bers breaking in the interval

. It may be resolved into a uniformin ompressible part whoseasso iatedenergy isin ludedwithin

f dε

as well as a non-uniform  ompressible part asso iated with more members

arriving in a given state than are leaving that state. The energy asso iated

with the non-uniform ow between states is entirely ontained in

T dS

.

From these expressions it may be on luded that if

U

is to be treated as a fundamentalfun tion,then

U = U(S, ε)

,orequivalentlyif

S

istobetreatedas the fundamentalfun tion then

S = S(U, ε)

. In other words, the independent variables that must be involved in the onstraints on the maximizationof

S

(6)

are

U

and

ε

. The proportionality onstants

T

and

f

are dened

T =

∂U

∂S

!

ε

and

f =

∂U

∂ε

!

S

.

(5)

Thestatefun tion

f

issomethingdierentthantheoverall tension

τ

sin e we alsohavethat

τ dε = dU

.Thus,ingeneral,

(τ −f)dε = T dS

sothat

f 6= τ

due to bers breaking in a positive in rement

. If strain were to be de reased, bers do not break and so

dS = 0

and the state fun tion

f

would be dened using onlythe purely elasti part of the energy hanges

dE

j

. Changes in

f

in this ase are equivalent to hanges in

τ

. Last, sin e we have taken

S

to be extensive(proportionalto

N

)while

U

isadensityindependentof

N

,

T

omes out being proportional to

1/N

. This hoi e is made so that fa tors of

N

do not lutter the equationsthat follow.

The onstraintinvolving

ε

isthat ea hnon-brokenberthroughouttheentire ensemble has the same length whi h implies

ε

j

= ε

. The onstraint involving

U

is that

U =

P

j

p

j

E

j

. Carrying out the maximizationof

S

subje t tothese onstraintsusing Lagrangianmultipliersgivesthe probabilitydistributionas

p

j

(β, ε) =

e

−βE

j

(ε)

Z(β, ε)

(6)

where

β = 1/T

and where the partitionfun tion

Z

isdened

Z(β, ε) =

X

j

e

−βE

j

(ε)

.

(7)

Thefreeenergy

F = F (β, ε)

inthisensemble(forthisset of onstraints)isthe Legendre transformof

U

withrespe t to

S

; i.e.,

F = U − T S

. Dierentiation then gives

dF = β

−2

Sdβ + f dε.

(8) Uponintrodu ing

U =

P

j

p

j

E

j

and

S = −

P

j

p

j

ln p

j

intotheLegendre trans-form

F = U − S/β

andusingthat

ln p

j

= −βE

j

− ln Z

weobtainthestandard result

βF = − ln Z.

(9)

Thus,thestandard anoni alensembleemergesinthisproblemwherequen hed

disorder alone (not mole ular u tuations) is responsible for the existen e of

ensembles.

Animportantquestioninsu hanapproa hiswhetheranythingpre ise anbe

said about the temperature

T = 1/β

.Indeed,

β

an formally be found asthe solutionof adierentialequation. This dierentialequation is obtained from

the previously stated but unused fa t that

dU = τ dε

whi h an bewritten

τ =

dU

=

X

j

dp

j

E

j

+

X

j

p

j

dE

j

.

(10)

(7)

ε

3

ε

2

ε

1

ε

τ

Fig.2.Astrain- ontrolledexperiment.Thesolidlineistheloadpathfollowedduring

theexperiment.Thedashedlinesrepresentthepaththatwouldbefollowedifstrain

were to be de reased at some point during theexperiment. Whena ber breaksat

onstant

ε

,the load

τ

mustberedu ed asrepresentedbytheverti al drops. Now fromEq. (6), we obtain that

dp

j

=

"

dE

j

β − E

j

d ln Z

#

p

j

,

(11)

while fromEqs. (8)and (9)

d ln Z

= −F

− β

dF

= −U

− βf.

(12)

Sin e ea h member is at the same

ε

, ea h member has its own

τ

j

, and so

τ =

P

j

p

j

τ

j

. We then obtain the dierential equationfor

β

inthe form

a

+ bβ + c = 0

(13)

with oe ients given by

a =

X

j

p

j

E

j

(U −E

j

); b =

X

j

p

j

E

j

f −

dE

j

!

; c =

X

j

p

j

dE

j

− τ

j

!

.

(14)

Sin e

p

j

= p

j

(β, ε)

, this equation is non-linear and thus di ult to solve. In the present work, wedemonstrate that a proposed fun tion

β = β(ε)

exa tly satisesthisequationand thusisthetrueber-bundle

β

.Tomakeprogress, wenext obtain

E

j

and

τ

j

forthespe i problemofberbundleswith global-load sharing.

3 FIBER BUNDLE MODEL

Figure2detailsthehistoryofhowtheoveralltension

τ

mightevolvewhen on-trolledvariationsin

ε

areappliedtoabundlehavingberstrengths

ε

1

, ε

2

...

ε

N

. In this parti ular example,

n = 3

bers have broken when the strain isat

ε

.

(8)

Theloadonthe bundleisequallysharedbythe

N − n

survivingbers sothat

τ

n

= ε



1 −

N

n



(15)

whi h is a relation independent of the history; i.e. it depends only on the

a tualstate of the bundle through the number ofbroken bers

n

, and not on the breaking thresholds

ε

1

, ε

2

...

ε

n

.

Thetotal workdensity

E

p

j

forstates

j

onsistingof

n

broken bers isthe area under the harging urve in a parti ular realization (the area under the solid

linein Fig.2)

E

j

p

=

Z

ε

0

τ (x) dx =

n

X

m=0

Z

ε

m+1

ε

m

τ

m

(x) dx =

n

X

m=0



1 −

m

N



ε

2

m+1

2

ε

2

m

2

!

where from Eq. (15) we have used

τ

m

(x) = x(1 − m/N)

and where by on-vention

ε

n+1

= ε

is the nal applied strain. A dire t re ursion gives exa tly

E

j

p

=



1 −

N

n



ε

2

2

+

n

X

m=1

ε

2

m

2N

.

(16)

Therst termhere istheelasti energythat an bereversiblyre overed upon

de reasing the strainwhile the se ondterm representsthe energy irreversibly

onsumed in the breaking pro ess. Both ontributions an be dire tly

vizual-ized inFig. 2.

Equation (16) is next averaged over the quen hed disorder to obtain the

av-erage energy

E

j

needed to reate state

j

. Ea h breaking threshold

ε

m

is an independantvariable,randomlydistributeda ording to

p(ε

m

)

underthe on-ditionthat

0 ≤ ε

m

≤ ε

. They are thereforedistributed a ording tothe prob-ability density

p(ε

m

)/P (ε)

where the normalization fa tor a ounts for the fa t that the upper limit

ε

is independent of the threshold values

ε

m

. Thus, averagingover the quen hed disorder gives

h(ε) =

*

ε

2

m

2

+

q.d.

=

1

P (ε)

Z

ε

0

x

2

2

p(x)dx

(17)

where

h(ε)

designates the average energy that islost when ea h berbreaks.

The average work density (Hamiltonian) required to reate the state

j

, aver-aged overall realizations of the quen hed disorderis then ingeneral

E

j

=



1 −

n

N

j



ε

2

2

+

n

j

N

h(ε) =

ε

2

2

n

j

N

"

ε

2

2

− h(ε)

#

.

(18)

For example, under the spe ial assumption that the breaking strengths are

randomly sampled from a uniform distribution on

0 ≤ ε

m

≤ 1

, we have that

p(ε) = 1

,

P (ε) = ε

,

h(ε) = ε

2

/6

,and

E

j

= ε

2

/2 − (n

(9)

We now apply the above theory and determine the thermodynami fun tions

asafun tionofappliedstrain

ε

.Thefollowinganalysisisvalidforanyproperly normalized quen hed-disorder distribution

p(ε)

.

The umulativedistribution

P (ε) =

R

ε

0

p(x) dx

is the probabilitythatany one ber has broken when the strain is at

ε

. Thus, the fra tion of all possible realizations having

n

j

broken bers and

N − n

j

unbroken bers is exa tly

p

exa t

j

= (1 − P )

N −n

j

P

n

j

.

(19) This distribution may be written

p

exa t

j

= p

0

exp



−n

j

ln



1 − P

P



(20) where

p

0

= (1−P )

N

istheprobabilityoftheunbroken state

j = 0

.Uponusing the Hamiltonianof Eq. (18), the postulate of entropy maximizationpredi ts

this same distribution tobe given by

p

j

= p

0

exp

"

n

j

β

N

ε

2

2

− h

!#

(21) where

p

0

= exp(−βε

2

/2)/Z

is the probability of the unbroken state.

These two distributions are both Boltzmannians in the number

n

j

of broken bers and are identi al if the temperature

T = 1/β

is given by

β(ε) =

−N

ε

2

/2 − h(ε)

ln

1 − P (ε)

P (ε)

!

.

(22)

Ifit anbeshownthatthis

β

satisesthedierentialequationofEq.(13),then our theory is exa t when appliedto berbundles with globalload sharing.

From the denition

P (ε)h(ε) =

R

ε

0

p(x)x

2

/2 dx

, one has

h(ε) < ε

2

/2

for any

distribution

p(ε)

.Thus,

β(ε)

isanegative in reasingfun tionup tothe strain point

ε = ε

β

whereitsmoothlygoestozero.Theine tionpoint

ε

β

isobtained fromthe onditionthat

P (ε

β

) = 1/2

anddenes apreviouslyunnoti edphase transitionthat willbeshowntobe distin tfromthe transitionatpeak stress.

When

ε > ε

β

,

β

be omes a positive in reasing fun tionof

ε

.

Therearetwokeyaveragepropertiesuponwhi hallthethermodynami

fun -tions depend; namely, the average fra tion of broken bers in ea h bundle

hn

j

/Ni

and the average of this fra tion squared

h(n

j

/N)

2

i

. Using the exa t

(10)



n

j

N



=

X

j

p

j

n

j

N

=

N

X

n=0

c

N

n

n

N

P

n

(1 − P )

N −n

(23) where

c

N

n

= N!/[n!(N − n)!]

denes the numberof ways of sele ting

n

obje ts froma group of

N

distinguishable items.The binomialtheorem states that

(x + y)

N

=

N

X

n=0

c

N

n

x

n

y

N −n

.

(24)

Upon dierentiating this equation with respe t to

x

and then multiplyingby

x/N

,givesthat when

x = P

and

y = 1 − P



n

j

N



= P

(25)

whi h is a known result onsistent with the meaning of

P

. Dierentiating a se ond time yields

*



n

j

N



2

+

= P

2

+

P (1 − P )

N

.

(26)

Usingthese tworesults, the otheraverages dening the thermodynami

vari-ables are easily reado.

The average stress

τ (ε)

is thus obtained tobe

τ =

X

j

p

j

τ

j

= (1 − P )ε

(27)

whi h is initiallyanin reasing fun tion of

ε

havingthe slope

= 1 − P − εp.

(28)

This slope goes tozero and the stress has a maximum at any strain point

ε

τ

that is asolution of

1 − P (ε

τ

) − ε

τ

p(ε

τ

) = 0

;i.e., atthe point(s) where

ε

τ

p(ε

τ

) = 1 −

Z

ε

τ

0

p(x) dx

(29)

admits a solution. This is a known exa t result [4,10℄. In general, we an

expe t that

ε

τ

6= ε

β

.The ondition requiredforequality ofthesestrain points is that simultaneously

εp(ε) = 1/2

and

R

ε

0

p(x) dx = 1/2

whi h for monotoni distributionfun tions

p(ε)

anonlyo urwiththeuniformdistribution

p(ε) =

1

in whi h ase

ε

τ

= ε

β

= 1/2

and the two transitions oin ide. If

p(ε)

is a monotoni in reasing fun tion of

ε

, then

ε

τ

< ε

β

while if it is a de reasing fun tionof

ε

,then

ε

τ

> ε

β

. Fornon-monotoni distributions, there an bean arbitrarynumberof stress maximasandEq. (29) an haveeithernosolutions

(11)

strain points

ε

β

and

ε

τ

is dis ussed inthe following se tion. The average energy inthe ensembleis

U =

X

j

p

j

E

j

= (1 − P )

ε

2

2

+ hP.

(30)

Againre allingthe denition of

h

fromEq. (17) gives

dU

= (1 − P )ε − p

ε

2

2

+

d(hP )

= τ

(31)

whi hisalsothe equationthat givesthedierentialequationfortemperature.

Thisissu ientfordemonstratingthatthe

β(ε)

ofEq.(22)satisesits dier-entialequation. Nonetheless, asa onsisten y test, the oe ients

a

,

b

, and

c

dened inEq. (14) willbederived and the dierentialequation willexpli itly

be shown to besatised.

To obtainthe state fun tion

f

, we needrst

dE

j

= ε −

n

j

N

"

ε −

P

p

ε

2

2

− h

!#

.

(32)

From Eq. (4)and the lemmaof Eq. (25) we then have

f =

X

j

p

j

dE

j

= (1 − P )ε + p

ε

2

2

− h

!

.

(33)

The variation of the entropy isobtained fromthe energy balan e as

dS

= β

dU

− f

!

= Np ln



1 − P

P



.

(34)

Togetherwiththe initial ondition

S(0) = 0

,thisisreadily integrated togive

S = −N [P ln P + (1 − P ) ln(1 − P )] .

(35) Thisexpression isthe lassi alShannonresultforasetof

N

randomvariables in two possible states having probabilities

P

and

1 − P

, whi h is pre isely the ase of the ber bundle with global sharing. This is another onsisten y

he k. This entropy

S

in reases fromzero [total ertainty that every member isinta t℄andgoessmoothlythroughamaximum[totalun ertaintyastowhat

stateamembermaybein℄atthesamestrainpoint

P (ε

β

) = 1/2

where

β

goes tozero.Afterthe smoothmaximum,

S

de reasestorea hzeroif

P (ε)

rea hes

1

in whi h ase there is total ertainty that ea h member is entirely broken. WenotethatEq.(35) analsobedire tlyobtainedfromtheShannonformula

(12)

Finally,the freeenergy

F

is again obtained fromitsLegendre transform de-nition

F = U − S/β

to be

F =

ε

2

2

ε

2

2

− h

!

ln(1 − P )

ln(1 − P ) − ln P

.

(36) At the point

ε

β

dened by

P (ε

β

) = 1/2

, the free energy diverges due to the fa t that

β(ε

β

) = 0

while

S(ε

β

)

remains nite. So long as

ε

β

6= ε

τ

, the free energy does not diverge when (if)

ε = ε

τ

. In passing,we alsonote that

Z =

h

(1 − P )

h

P

−ε

2

/2

i

N/(ε

2

/2−h)

(37)

is the exa t expression of the partitionfun tion.

Withthe aboveresultsestablished,wenowobtainthe oe ients

a, b, c

ofthe dierential equationfor the temperatureas

a = −

1 − P

N

P h

2

ε

2

2

!

2

; b =

1 − P

N

P h

2

ε

2

2

!

P ε − p

ε

2

2

+ ph

!

;

c = −p

h −

ε

2

2

!

.

(38)

Usingthese together withEq. (22) for

β

and its derivative

= −

Np

P (1 − P )(h − ε

2

/2)

+ N ln



1 − P

P



"

ε + (h − ε

2

/2)p/P

(h − ε

2

/2)

2

#

(39)

shows that the dierentialequation

a dβ/dε + bβ + c = 0

isexa tly satised.

5 Phase Transitions

5.1 The entropy maximum transition

The strain point

ε

β

dened by

P (ε

β

) = 1/2

is where simultaneously

β = 0

, the entropy is a maximum, and the free energy diverges. It is distin t from

the stress-maximum transition(s) at

ε

τ

. The interpretation of

ε = ε

β

as a phasetransitionis natural,sin e the most probable ongurationofa bundle

suddenly jumps from being entirely inta t to entirely broken. What are the

measurable manifestationsof this transitionat

ε

β

?

Sin etheentropyisamaximumatthispoint,theu tuationsbetween

(13)

brokenbersinstate

j

tobe

ρ

j

= n

j

/N

,anddenetheaverageofthisfra tion to be

ρ = hρ

j

i = P

where

P = P (ε)

is again the probability of a ber being broken. The quantity of interest here is the root-mean-square u tuation

∆ρ

inthe fra tionof broken bers given by

∆ρ =

r

D

∆ρ

2

j

E

=

r

D

ρ

2

j

E

− ρ

2

=

q

P (1 − P )

N

(40)

whi hindeedgoesthroughamaximumat

P (ε

β

) = 1/2

asexpe ted.This max-imum is something that an be dire tly measured in numeri al experiments

onber bundles but has never beforebeen ommented on. The reason ithas

been dis overed inthe presenttheory isbe ause entropy and temperatureare

expli itlypresent. We en ourage someone to numeri ally measure

∆ρ

and to verify that it is amaximum atthe transitionpoint

ε

β

. Re all that for mono-toni quen hed-disorder distributions, if

p(ε)

is a de reasing fun tion (more weak bers than strong bers), then

ε

β

< ε

τ

. So the numeri al observation inthis ase should be that

∆ρ

goesthrough a maximum prior topeak stress. Equivalent omments hold when

p(ε)

is anin reasing fun tion and

ε

β

> ε

τ

.

The other u tuations that are potentiallyof interest in ludethe

root-mean-square stress u tuation

∆τ =

r

D

∆τ

2

j

E

=

r

D

τ

2

j

E

− τ

2

= ε∆ρ

(41)

and the root-mean-squareenergy u tuation

∆U =

r

D

∆E

2

j

E

=

r

D

E

2

j

E

− U

2

=

ε

2

2

− h

!

∆ρ,

(42)

but sin e these are simply proportional to

∆ρ

it seems that the interesting signature of this phase transitionis the maximum in

∆ρ

.

Theorderofthistransitionisnot lassi allydenedintheEhrenfests heme;

however, it seems inappropriateto allit a ontinuous transitionbe ausethe

freeenergyissingularat

ε

β

asshownabove.Nonetheless,inthelimitas

ε → ε

β

the singular part of

F

diverges a ording tothe s aling law

F

s

= −

ln 2

8

2

β

− h(ε

β

)/2]

p(ε

β

)

(ε − ε

β

)

−1

(43)

whi h has a (trivial) universal exponent. Sin e

F

and itsderivativesare di- ult to numeri ally measure, the prin iplemanifestation of this phase

(14)

Thephasetransitionatpeakstress

ε = ε

τ

istheonethatresear hersuptonow havefo usedon.Sin e

S

anditsderivativesremain ontinuousandnitethere, an attempt to lassify this transition as being rst-order or se ond-order is

meaningless.Thelabelof ontinuous transitionseemsthe mostappropriate.

Thereis universality atthis transitiondueto thefa t thatfor any monotoni

analyti quen hed-disorderdistribution,

τ (ε)

isasmooth analyti fun tion so thatupondevelopingthisfun tionintheneighborhoodofitsmaximumoneis

guaranteed

|τ − τ(ε

τ

)| ∼ |ε − ε

τ

|

2

withthe exponent of ourse being

indepen-dentof the distribution

p(ε)

orother modelparameters. Su hdevelopment of an analyti fun tion about a riti al point is the way any mean-eld theory

a quires a universal s aling law. However, Kloster, Hansen and Hemmer [10℄

dodemonstratethatanon-analyti quen hed-disorderdistribution anleadto

highlynon-analyti stress-strainbehaviorthatthusfallsoutsidethequadrati

universality lass. Also, in other damage models in whi h elasti intera tion

between ra ks (damage points) is important, the simple observation of a

modelexhibitinganaveragedstress maximumdoesnot byitselfguarantee an

exa tlyquadrati stress-strain relationin the neighborhoodof the maximum.

Su h a relation may be non-analyti due to a diverging orrelation length in

the orrelation between ra ks.

The quadrati stress maximum means that

dε/dτ

diverges as

|τ − τ(ε

τ

)|

−1/2

[11,19℄ . A ordingly, the average rate at whi h the fra tion of broken bers

in reaseswithstress

dρ/dτ = p dε/dτ

alsodivergesas

|τ −τ(ε

τ

)|

−1/2

[4,15℄.

Us-ingadditional onsiderationsnot developedinthispaper,one an futhershow

that the average size of avalan hes alsodiverges as

|τ − τ(ε

τ

)|

−1/2

[10,15,17℄,

and that the distribution

D(∆)

of the size

of the avalan hes s ales as

D(∆) ∼ ∆

−5/2

[6,7,10,17℄ at the stress maximum. These are the prin ipal

observable hara teristi sof the stress-maximum transition.

6 Summary and Con lusions

Two prin ipal results have been obtained in this paper. First, it was

demon-strated that for any quen hed-disorder distribution used in the ber-bundle

modelwithgloballoadsharing,the probabilityofthe emergentdamagestates

may be exa tly al ulated by maximizing Shannon's entropy subje t to

on-straints.Itisthrough the onstraintsthatthenatureofthequen hed-disorder

distribution enters the ma ros opi thermodynami s.

(15)

Thisphasetransitionisdistin tfromthestress-maximumtransition;although,

for the uniform quen hed-disorder distribution, the two transitions oin ide.

The prin iple manifestation of this transition is that the root-mean-square

u tuationinthe numberof broken bers willgothrough amaximum,whi h

is aquantity that an bedire tly measured in numeri alexperiments.

To on lude, we postulatethat the probabilityof emergent states an always

be al ulatedthroughentropy maximizationforany damagemodelofinterest

in ludingmodels inwhi h there is elasti intera tion between the lo al

dam-age (order) parameters. Damage models with order-parameter intera tions

are not normally onsidered amenable toanalyti al treament;however, using

ourapproa haratherstandard anoni al-ensemblepartitionfun tionemerges

and the various fun tional integration pro edures available for studying the

partitionfun tion inthe neighborhood of riti alpointsmay be employed.

Su hgeneralityistheprin ipalutilityofourapproa h.A on ernistheability

toprodu eanexpression forthe modeltemperature. Thetemperature an be

found in prin iple as the solution of a well-posed initial-value problem.

Un-fortunately, the dierential problemis highly non-linear and thus di ult to

solve. Be ause stress maxima and entropy maxima donot normally oin ide,

astress-maximum transitionmaybe studiedby simplyassumingthe

temper-atureto bewell behaved inthe neighborhoodof thestress maxima. Butif an

expli it expression for the temperatureis desired,the following approa h an

be employed.

Inmodelsinvolving ra kintera tions,therearealwaysasubsetofdilutestates

inwhi hthe intera tions are negligible.The exa t probabilities ofsu h states

an usually be determined from the quen hed-disorder distribution alone. A

omparison between the probabilities al ulated from entropy maximization

and su h exa t probabilities then determines the model temperature to be

usedintheBoltzmannian.Withthetemperaturesodened,andwithaproper

model Hamiltonianinhand, the partition fun tion an beanalyzed.

Referen es

[1℄ F.T. Peir e, J. Text. Ind.17, 355 (1926).

[2℄ H.E. Daniels, Pro .R. So . A 183, 404 (1945).

[3℄ B.D. Coleman, J. Appl. Phys. 29, 968 (1958).

[4℄ J.V. Andersen, D. Sornette and K.W. Leung, Phys. Rev. Lett. 78, 2140

(1997).

[5℄ H.E. Daniels, Adv. Appl. Prob.21, 315 (1989).

[6℄ A. Hansen and P.C. Hemmer,Phys. Lett. A 184, 394 (1994).

(16)

(2002).

[9℄ R.C. Hidalgo, F. Kun, and H.J. Herrmann, Phys. Rev. E 64, 066122

(2001).

[10℄ M.Kloster,A.Hansen,andP.C.Hemmer,Phys. Rev.E56,2615(1997).

[11℄ F. Kun, S.Zapperi,and H.J. Herrmann, Eur. Phys. J.B 17, 269 (2000).

[12℄ D.Kraj inovi ,andM.A.G.Silva,Int.J.SolidsStru tures18,551(1982).

[13℄ I.L. Menezes-Sobrinho, A.T. Bernardes, and J.G. Moreira, Phys. Rev. E

63, 025104(R)(2001).

[14℄ L.Moral, Y.Morenoand A.F. Pa he o, Phys.Rev.E 63,066106(2001).

[15℄ Y. Moreno, J.B. Gomez and A.F. Pa he o, Phys. Rev. Lett. 85, 2865

(2000).

[16℄ Y. Moreno, J.B. Gomezand A.F. Pa he o, Physi a A 274, 400 (1999).

[17℄ S.Roux,A.Delapla eandG.Pijaudier-Cabot,Physi aA270,35(1999).

[18℄ S.Roux, Phys. Rev. E 62,6164 (2000).

[19℄ D. Sornette, J. Phys. A 22, L243 (1989).

[20℄ R. Toussaint and S.R. Pride, Phys. Rev. E (2002),(in press).

[21℄ R. Toussaint and S.R. Pride, Phys. Rev. E (2002),(in press).

[22℄ R. Toussaint and S.R. Pride, Phys. Rev. E (2002),(in press).

Figure

Fig. 1. A bundle of N bers strethed between two rigid supports with a load F N
Fig. 2. A strain-ontrolled experiment. The solid line is the load path followed during

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